Cognitive Foundations of Strategic Behavior:
ARCHME
From Games to Revolutions
MASSACHUSETTS INSTITUTE
OF TECHNOLOLGY
by
David Jimenez-Gomez
JUN 09 2015
Licenciado, Universidad de Murcia (2008)
M.A., Universitat Autonoma de Barcelona (2010)
LIBRARIES
Submitted to the Department of Economics
in partial fulfillment of the requirements for the degree of
PhD in Economics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ David Jimenez-Gomez, MMXV. All rights reserved.
The author hereby grants to MIT permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document
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Department of Economics
May 8, 2015
Certified by..
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Daron Acemoglu
Elizabeth and James Killian Professor of Economics
Thesis Supervisor
Certified
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Abhijit Banerjee
Ford Foundation International Professor of Economics
--- I---'Thesis
Supervisor
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ted...................
Accepted by. Sig nature
Ricardo Caballero
Ford International Professor of Economics
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2
Cognitive Foundations of Strategic Behavior:
From Games to Revolutions
by
David Jimenez-Gomez
Submitted to the Department of Economics
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
PhD in Economics
Abstract
Game theory is one of the main tools currently used in Economics to model strategic behavior. However, game theory has come under attack because of the strong
assumptions it makes on people's behavior. Because of that, alternative models of
bounded rationality, with more realistic assumptions, have been proposed.
In the first chapter, I develop a game theoretic model where players use two
different reasoning processes: cooperative and competitive. The model generalizes
Level-k and team reasoning, and provides a unified explanation for several important phenomena. In Rubinstein's Email game, players coordinate successfully
upon receiving enough messages. In 2 x 2 games of complete information, the solution concept lies between Pareto dominance and risk-dominance. In coordination
games, the model explains several experimental facts that cannot be accounted
for by global games, especially the fact people coordinate more with public rather
than private information. I show the importance of public events in revolutions: a
self-interested government prevents the generation of common knowledge among
the citizenry when times are bad.
In the second chapter, I develop a model of cognitive type spaces which incorporates Level-k and Cognitive Hierarchy (CH) models into games of incomplete
information. CH models make two assumptions: agents of higher level have richer
3
beliefs and can perform more computations. In my model, like in Level-k and CH
models, an agent's level determines how complex her beliefs are. However, given
their beliefs, agents are fully rational and behave according to Interim Correlated
Rationalizability. My main result is that, restricted to cognitive type spaces, the
product topology and the uniform strategic topology coincide, what implies that
two players with similar beliefs behave similarly. This means that, unlike for
general type spaces, predictions will be robust to small specification errors and
suggests that incorporating cognitively plausible assumptions into game theory
can increase robustness. As an application, I show that in the Email game, when
players receive few messages they never attack; however, when they receive enough
messages, they behave as if there was complete information, and both actions are
rationalizable.
In the third chapter, I develop a dynamic model of forward-looking agents in
the presence of social pressure. I show that social pressure is effective in generating public good provision: after an agent starts contributing to the public good,
other agents decide to contribute as well because of fear of being punished, what
generates contagion in the network. In contrast with the previous literature, contagion happens fast as part of the best response of fully rational individuals. The
network topology has implications for whether the contagion starts and the extent
to which it spreads. I find conditions under which an agent decides to be the first
to contribute in order to generate contagion in the network, as well as conditions
for contribution due to a self-fulfilling fear of social pressure.
Thesis Supervisor: Daron Acemoglu
Title: Elizabeth and James Killian Professor of Economics
Thesis Supervisor: Abhijit Banerjee
Title: Ford Foundation International Professor of Economics
4
Acknowledgments
I want to express my gratitude to my advisors, Daron Acemoglu, Abhijit Banerjee and Muhamet Yildiz, for their invaluable guidance throughout the PhD. They
provided me with advice at all levels of research, from the big ideas to the small
details. Their breath of knowledge and commitment to research was an example
for me to follow and derive inspiration from. As I explored the frontiers of the
knowledge in Economics, they gave me honest and encouraging feedback, and devoted many hours of their time to help me in my endeavor.
I have built on the work of countless academics, too many to be mentioned here;
they are cited where it corresponds in the respective chapters. Some of them kindly
devoted time to give me feedback on my work. For the first chapter, I thank Gabriel
Carroll, Alp Simsek, Tomasz Strzalecki, Nils Wernerfelt, and the participants of
seminars at MIT and Chicago. For the second chapter, I thank Nemanja Antic,
Dan Barron, Ben Brooks, Gabriel Carroll, Elliot Lipnowski, Georgy A. Lukyanov,
Stephen Morris, Jawwad Noor, Alp Simsek, Tomasz Strzalecki, Nils Wernerfelt,
and the participants of seminars at MIT, Universitat Autnoma de Barcelona
and EconCon. For the third chapter, I thank Glenn Ellison, Ben Golub, Giovanni
Reggiani, John Tsitsiklis, Xiao Yu Wang, and the participants of seminars at MIT.
Prior to my studies at MIT I had several teachers and mentors who inspired
me through their example, and supported my lifelong desire of learning. Among
5
them, I am especially indebted to Joss Orihuela Calatayud at Universidad de Murcia, and Salvador Barbera and Miguel Angel Ballestcr at Universitat Aut6noma
de Barcelona, for transmitting to me their passion for research and encouraging
me in pursuing such a fascinating path.
I am thankful to my friends for many moments of inspiration and joy. I am
especially indebted to Dana Chandler, Caitlin Lee Cohen, David Colino, Dong Jae
Eun, Sara Herndndez, Elena Manresa, Giovani Reggiani, Alejandro Rivera, Xiao
Yu Wang, Nils Wernerfelt and Leonardo Zepeda-Nninez in Cambridge, and Pedro
Garcia-Ares, Tugce Cuhadaroglu, Ezgi Kaya and Pau PujolAs Fons in Barcelona,
for their support and affection; and to my many friends, in Murcia and elsewhere,
who were far but always felt so close.
Finally, I am extremely grateful to my family. My grandmother Soledad taught
me how to read at an early age, thus setting me on a path of curiosity and discovery. My grandfather Prudencio showed me the value of effort through his living
example. My parents, Prudencio and Marisol, transmitted to me their love for
learning and teaching, and have always done everything in their hand to help me
advancing in my path, with unwavering love and encouragement. My sister Alicia
and I have rejoiced together in the good moments, and supported each other in
the less good ones. It is because of all of you that I am where I am today.
6
I gratefully acknowledge financial support from "La Caixa" Foundation, the
Bank of Spain, Rafael del Pino Foundation, the MIT Economics department and
the George and Obie Shultz Fund. The opinions in this thesis are exclusively my
own and do not represent those of the aforementioned institutions.
7
8
Contents
1
Cooperative and Competitive Reasoning: from Games to Revo11
lutions
1.1
1.1.1
1.2
1.3
1.4
........
Introduction .............
...
........
11
Literature review . . . . . . . . . . . . . . . . . . . . . . . . 16
The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
. . . . . . . . . . . . . . . . . . . . 20
1.2.1
Bounded representation
1.2.2
Cooperative reasoning . . . . . . . . . . . . . . . . . . . . . 25
1.2.3
Competitive reasoning: Level-k . . . . . . . . . . . . . . . . 30
1.2.4
Putting it all together: Solving the Email game . . . . . . . 31
1.2.5
Games of complete information . . . . . . . . . . . . . . . . 34
Global games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.3.1
Theoretical predictions . . . . . . . . . . . . . . . . . . . . . 38
1.3.2
Revolutions and Collective Action . . . . . . . . . . . . . . . 42
Generalized Model and Applications
1.4.1
. . . . . . . . . . . . . . . . . 43
Normally distributed signals . . . . . . . . . . . . . . . . . . 46
9
50
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
.
1.5
Signal jamming by the government
.
. . . . . . . . . . . . .
1.4.2
2 You Are Just Like Me: Bounded Reasoning and Recursive Beliefs 73
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.1
Introduction ...
2.2
The Model ....
80
2.3
Application: The Email Game
85
2.3.1
88
Analysis
Main Result ...
90
2.5
Conclusion . .
93
.
2.4
3 Social Pressure in Networks Induces Public Good Provision
3.1
Introduction . ..
3.2
The model
3.3
Contagion
............
111
.......
119
..............................
122
Simpl e-:Case: An Illustration . . . . . . . . . . . . . .
122
3.3.2
ad..qulri.......................
.. Case
. . .. . . . . . . . . . .. . . . . . . .
Gener al
125
.
.
3.3.1
132
Leadership
Spear headed equilibria . . . . . . . . . . . . . . . . .
.
135
3.4.2
Social pressure equilibria: Supermodularity and MPE
137
3..5
Bounded rationality and visibility . . . . . . . . . . . . . . .
142
3.6
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . .
145
.
.
.
3.4.1
.
.
3.4
al
.. . .
111
10
Chapter 1
Cooperative and Competitive
Reasoning: from Games to
Revolutions
1.1
Introduction
"Fornearly thirty years, the price of a loaf of bread in Egypt was held constant;
Anwar el-Sadat's attempt in 1977 to raise the price was met with major riots. Since
then, one government tactic has been to make the loaves smaller gradually, another
has been to replace a fraction of the wheat flour with cheaper corn flour." - Chwe
(2001)
Strategic behavior has baffled economists for decades. Consider the following
11
example from Crawford et at. (2008): two people must choose between X and Y;
if their choices coincide they both receive $5. Around 80% of people choose X in
this situation. What happens if we tell one person she will receive 10 extra cents
for coordinating on X, and the same to the other person about coordinating on Y?
Now around 80% of people choose the action for which they are not paid extra;
and this results in massive miscoordination, and therefore lower payoffs. If X was
salient, why do players stop coordinating on it once the small extra incentive is
added? Another example comes from Chwe's quote above: people could not coordinate their protests when the subsidized bread became smaller or of lower quality;
but when the price rose, protests erupted. Why these differences in coordination?
I propose a model which provides a unified explanation for these apparently
disconnected phenomena. The main idea is that people use two different kinds
of reasoning: cooperative and competitive. Seemingly small differences in the
structure of a game, or of a political scenario, lead to a switch from one reasoning
modality to the other, and can therefore generate a large change in behavior. The
model has three main components.
1. Players have a bounded understanding of the game, indexed by a player's
level k: the higher k is, the better the agent is able to understand the game.
Intuitively, a player is unable to consider the other player's beliefs of very high
order, and k roughly measures the order at which a player stops thinking.
2. Cooperative reasoning: players attempt to play a Pareto dominant equi12
librium a* whenever there is one. The likelihood of this happening depends
on a parameter 0, which measures the probability that others reason cooperatively, as well as the riskiness of a*.
3. Competitive reasoning when cooperative reasoning fails, players behave
as level-k thinkers: level-0 players randomize uniformly, and level-k players
best-respond to players of level k - 1.
The solution concept is dual reasoning: players first attempt to engage in
cooperative reasoning, and use competitive reasoning otherwise. The model is
psychological, in the sense that it attempts to capture, as much as possible, the
reasoning process that people follow in strategic interactions. Consider the wellknown Email game, where players exchange emails of confirmation back and forth
until a message is lost. Because no player is ever sure whether she was the last to
receive a message, there is no common knowledge in this game, Rubinstein (1989);
however, most people would agree that after receiving a large quantity of confirmation emails (millions for example), then both players have common knowledge,
for all practical purposes, that the message is known. This intuition is captured
by my model because, after having received enough messages, players represent
the game as if there was complete information, and are able to coordinate on the
Pareto dominant equilibrium (Theorem 1). This is consistent with experimental
evidence for this game, Camerer (2003). Note that we need all three components
of the model to obtain this result. Firstly, the boundedly rational representation is
necessary so that players who have received enough emails consider the message is
13
commonly known. Secondly, cooperative reasoning is needed to have players coordinate on Attacking when they receive enough messages - without it both actions
would become rationalizable. 1 Finally, we need competitive reasoning in order to
have a default behavior that happens when players cannot engage in cooperative
reasoning, and Level-k reasoning is a particularly appropriate because it has been
widely applied and it coincides with risk-dominance in several important scenarios.
Global games (which are games of incomplete information where payoffs are
observed with a small amount of noise) usually have a unique equilibrium, Morris
and Shin (2003).2
However, this relies on assuming that players are extremely
rational, and able to perform a long chain of reasoning. Heinemann et al. (2004)
showed that when tested experimentally, subjects follow some of the predictions
of global games but violate several others. In particular, players are able to coordinate, and to use public signals, better than global games predict; facts which are
captures by dual reasoning (Theorems 4 and 5). This is because dual reasoning
captures the cognitive limitations of players, as well as their ability to engage in
cooperative reasoning.
In the limit when the noise is zero, global games become games of complete
information. I show that in 2 x 2 complete information games with two pure Nash
'In Jimenez-Gomez (2013) both actions are rationalizable after players observe enough messages. In that paper this is something desirable, as we want to use the Interim Correlated
Rationalizability as a solution concept.
2
Under certain conditions, which are usually satisfied in applications.
14
equilibria, the Pareto dominant equilibrium or the risk-dominant will be chosen,
as a function of risk-dominance and parameter o1 (Proposition 2). This is remarkable because game theorists have long debated whether the Pareto dominant or
the risk dominant equilibrium should be chosen in such situations, Harsanyi and
Selten (1988). I provide a simple condition under which the Pareto dominant equilibrium will be selected over the risk-dominant: this imposes a clear structure to
the discussion on when we will expect players to coordinate on either equilibrium.
I model revolutions as coordination global games, where a mass of citizens
chooses whether to participate or not and payoffs increase in the level of coordination and the weakness of the fundamentals of the state. In that context I show that
the government will choose to make public announcements when the fundamentals
are strong and players assign low probability to others using cooperative reasoning
(Proposition 6). When the government can jam the signal to make individuals believe fundamentals are better than they truly are, the government jams the signal
for intermediate values of the fundamentals: when they are very good, there is no
risk of revolution; when they are very bad, revolution cannot be averted, Proposition 10. This is in contrast to the fully rational global games result, in which
the government is always able to prevent a revolution, as the signals become more
precise.
The model attempts to capture the psychological reasoning that happens in the
15
mind of the players, when facing a strategic situation. The formal way to do so is by
merging three assumptions which have been successful in their respective domains:
players have a bounded understanding of the game, they attempt to cooperate
when doing so is not too risky (team reasoning), and they otherwise attempt to
predict what the other player will choose, and best-respond to it (Level-k). There
are two justifications for doing so. The first is the evidence: Decety et at. (2004)
showed that different neural circuits are involved in cooperative vs. competitive
reasoning (cooperative reasoning engages orbitofrontal cortex differentially, and
competitive reasoning engages inferior parietal and medial prefrontal cortices);
Crawford et al. (2008) and Bardsley et al. (2010) showed that their experimental
data could be explained by a combination of team reasoning and Level-k. The
second, and perhaps more convincing reason, is that the model works: it offers a
series of novel predictions and is better able to explain evidence from experiments
(Heinemann et al., 2004; Cornand, 2006) than global games and Level-k models,
what opens the possibility of applying the model to several interesting applications;
I discuss this in more detail in the Conclusion.
1.1.1
Literature review
It has been long recognized by economists that game theory models assume too
much rationality on the part of players. While rationality has a formal gametheoretic definition (that of maximizing utility given beliefs), what economists
usually refer to is not a failure of this narrowly defined notion of rationality, but
16
rather the notion that people exhibit biases in their beliefs and limits in their
computational capacity.3 The paper is connected to the literature on bounded rationality on game theory, and especially to Level-k (Nagel (1995), Stahl and Wilson
(1995, 1994)) and Cognitive Hierarchy Camerer et al. (2002) models. Two recent
papers in this literature are especially relevant: Strzalecki (2014), who applies a
Cognitive Hierarchy model to the Email game, and Kneeland (2014), who applies
it to global games. In Kets (2014) agents can have finite and infinite hierarchies of
beliefs, and players can have common certainty of events, even if they have finite
hierarchies of beliefs. Her model is very general and encompasses standard game
theory; in contrast I offer a model which is very simple and attempts to improve
the accuracy of predictions, following the tradition of Level-k models.
In Economics, there is a literature, pioneered by Michael Bacharach and Robert
Sugden on team reasoning: the game-theoretic idea that players are able, under
certain circumstances, to coordinate on Pareto optimal outcomes, and therefore
"reason as a team"; Sugden (1993), Bacharach (1999), Bacharach and Stahl (2000),
Bacharach (2006). Outside of Economics, the literature on collective intentionality has been dominated by philosophers, Sellars (1968, 1980), Bratman (1999),
Gilbert (2009). This literature has studied fundamental issues such as how do
people represent a shared intention in their minds but, unlike the literature on
team reasoning, little attention is paid to whether the actions taken by the players
3
Yildiz (2007) is an excellent example of a model where players are fully rational in this
narrow sense, and yet they are wishful thinkers because they hold delusional beliefs.
17
are incentive compatible. Because of that, this literature is less relevant for the
model in this paper, although it provides an intellectual basis for future research
into the cognitive foundations of cooperative reasoning.
The false consensus effect refers to a cognitive bias by which a person overestimates the extent to which her beliefs are representative of those of others,
Ross et al. (1977). There is a long tradition in the psychology, and more recently
game theory literatures of eliciting beliefs of players in games, which shows that
the false consensus effects is also relevant to strategic interactions, Dawes et al.
(1977), Mess6 and Sivacek (1979), Rubinstein and Salant (2014). Rubinstein and
Salant (2014) is especially relevant, as they elicit both beliefs about other player's
actions as well as profiles in the Chicken game. They find that people have a
disproportionate tendency to assign their own beliefs to others. 4 In the model, the
assumption about how players represent information about the game implies that
individuals exhibit the false consensus effect.
This paper is also deeply connected to the literature on equilibrium selection,
4
Other papers related to the false consensus effect includes Kelley and Stahelski (1970), who
found that in a repeated prisoner's dilemma, players who tended to defect often believed that
others were like themselves, whereas those who were more cooperative where aware of the true
distribution of defectors. Kuhlrnan and Wimuberley (1976) classified players as "individualistic",
"competitive" and "cooperative" in a series of one-shot games and showed that each type assumed
most others to be of their own type. A modern study is Iriberri and Rey-Biel (2013), which
studies beliefs of players in a Dictator game, and find that selfish players exhibit more false
consensus than other categories of players. These observations fit nicely within my framework,
where players who engage in cooperative reasoning are one level of awareness above those who
engage in competitive reasoning.
18
and global games in particular. Carlsson and van Damme (1993) show that in
2 x 2 games, by adding infinitesimally small noise to the original games, players
conform to risk dominance. In "A General Theory of Equilibrium Selection in
Games", Harsanyi and Selten endeavor to find a selection criterion that is universal. They developed the risk dominance criterion, but concluded that whenever in
conflict, payoff dominance should have precedence has a selection criterion. My
model reconciles both criteria: risk-dominance affects the behavior in competitive
reasoning, and Pareto dominance in cooperative reasoning.
The paper is also connected to notions of correlated equilibrium, Aumann
(1987), Forges (1993), Dekel et al. (2007). While in correlated equilibrium there is
some external correlation device which enlarges the set of equilibria, in this paper
the correlation arises endogenously from the reasoning processes of the players:
when players engage in cooperative reasoning, they are de facto coordinating their
actions. This means that, in contrast to this literature, correlation reduces the set
of outcomes when using dual reasoning.
The remainder of the paper is organized as follows. Section 3.2 presents the
model, using the Email game as a running example. At the end of the section
the model is applied to games of complete information, to discuss the connection
between Pareto dominance and risk dominance. Section 1.3 discusses the empirical
evidence on global games, presents the main theoretical results (consistent with the
19
evidence), and applies the model to revolutions. Section 1.4 ofers a generalization
of model, which is applied to normally distributed signals and the case when the
government can jam the signals. Section 3.6 concludes. The Appendix contains
proofs of all the theoretical results.
1.2
The model
We start the description of the model with some preliminary notation and concepts. There are only two players, I =
{1, 2}.
Player i represents an arbitrary
player, and -i represents the Player who is not i. Let
E
be the set of states of
the world; each Player i has utility ui(O, ai, a-i). Given a set X, we represent by
A(X) the set of probability measures on X. Let T be the type space for Player
i, such that there is a belief function
fl
:T
-+
/3
Elti) is
A(E x Ti), so that #(O,
the probability that type ti assigns to the state being 0 E
E
and Player -i being
of type t-i E E C T-i. Finally, if Y is a random variable, we will denote by E[Y]
the expectation of Y.
1.2.1
Bounded representation
We turn now to describe how players use a boundedly rational representation of
the game. In particular, Player i with level k can only reason about the first
k-order beliefs: beyond that, Player i will "approximate" her higher order beliefs
20
using her lower order beliefs. Before expressing this idea formally, we need to
introduce some standard concepts from the literature on epistemic game theory.
For each E C E x T, x T2 and each type tj E T, we define Et, as5
Eti
=
{(0, t-i) : (9, t1 , t 2) E E}.
We define the belief of Player i as,
Bi(E) = {ti C
: ,3(Et~Itj) = 1}.
We can define mutual belief B 1 (E) as:'
B'(E) =
Ex
B1 (E) x B2(E),
and k-mutual belief as
k-i
Bk (E) =
f
Bm (E) n B1(Bk-1 (E)).
m=1
We define common certainty as
00
C(E) =
B(E).
k
k=1
That is, C(E) is the set of states of the world at which players believe E, believe
others believe E, believe others believe others believe E, etc. Note that common
5I follow Chen et al. (2014) in their exposition.
6
By convention, B0 (E) = E.
21
certainty requires that players be able to perform an arbitrarily long chain: "I
believe you believe I believe..." However, most people are unable to perform such
types of reasonings for more than 4 iterations, Kinderrnan et al. (1998). Because
of that, we will assume that when there is mutual k-belief of an event, a player of
level-k considers that as evidence that there is common certainty of the event.
Assumption 1. If ti of level k believes E is k-mutually believed, then she believes
E is common certainty:
,i (Bk(E)Iti) = 1 ==-> #i(C(E)Iti) = 1.
Let T be the type space of the game, which we will call the objective type
space. Assumption 1 implies that each type ti has a bounded representation of
the game, which we will denote by Tk(ti) (where k is the level of the type), and
call the k-representation. In this bounded representation, whenever there is kmutual belief of an event, ti believes the event is common certainty, even if in T the
event E might not be common certainty.8 In order to illustrate this, we analyze
the email game, popularized by Rubinstein (1989).
Example 1 (Email game, Part I). There are two generals, and each of them in
charge of a division. Their objective is to conquer an enemy city, and in order to
7
An exception is Kets (2014), where players can have common certainty of an event, even
when their beliefs are of finite order.
8 Using the terminology of Morris et al. (1995), Tk(ti) has depth of knowledge equal to k.
22
accomplish so, each of them must attack from a different location. Because of that,
they can only communicate through email. Each general has two actions: to attack
(A), or not to attack (N). The enemy can be of two types: strong or weak. Both
generals share a common prior: the enemy is either weak or strong with probability
1/2 each.
Because the generals cannot communicate directly, they have implemented a
communication system through email. General 1, who is closer to the enemy,
learns whether the enemy is strong or weak. If the enemy is weak, an email is
sent automatically to general 2, who sends an email back with the confirmation
that 2 received the email from 1, etc. Each time an email is sent, it is lost with
probabilitye. This process of emailing back and forth is repeated until an email is
lost, in which case no more emails are sent.
Let's denote by tT the type for Player i who sent m messages. Note that to
is the type who is certain that the enemy is strong and of Player 2 having sent 0
messages. For other types, their beliefs are as follows:
1. Player 2 who sends 0 messages (i.e. type t|) assigns probability 1/2 to the
enemy being strong, and e/2 to the enemy being weak and the message lost.
Therefore type tO assigns probability 1/(1 +,e)
to the enemy being strong and
type to, and e/(1 + c) to the enemy being weak and t' respectively.
2. In any other situation, Playeri who sent m messages (i.e. type tT,) places
23
probabilityE on the message being lost before Player -i received it, and (1- 6)E
on the message being lost after Player -i received. Therefore type t' assigns
probability -I- and 1- to t'"
- and
1
and t' respectively; and t"j assigns probability
to tm and tm+1 respectively.
Therefore, the objective type space T can be represented as follows:
01(tI)
)3(to) W
to
to
0
S
tl
0
0
1+
0
l
S
E
T+E
1
2
(t),)
W
l
W
t1
S
02 (t2)
...
S
0
2
0
2-E
2i
W
-..
2-E
1
02(t )
01 (tM)
S
0
W
1
t"'
2-c
0
W
S
1
0
2-c
t2
1
1-E
2-E
t+1
1
1-C
2-E
Let's consider now how different types would represent the game. For example,
consider Player 1 who sent 2 messages and has k = 1. Note that it is mutual
belief that the enemy is weak, because both players know that the other received at
least 1 message. Assumption 1 implies that in T1 (t2), t2 believes that it is common
certainty that the enemy is weak.
On the other hand, considerPlayer 1 who sent 1 message, and has k = 2. Now
24
there is no mutual belief that the enemy is weak, because Player 1 is not certain
whether her email was received by Player 2. Because of that, the condition in
Assumption 1 is not satisfied, and in T 2 (t') there is no common certainty that the
enemy is weak. This hints to a very important role for k: the lower the k, the
easier it is for a player to (maybe mistakenly) believe there is common certainty of
an event.
1.2.2
Cooperative reasoning
Now that we have defined what a bounded representation of the game means,
we need to properly address what is it that cooperative reasoning accomplishes.
In doing so, I build on a vast literature from Philosophy and Economics, which
have delved on the intricacies of cooperative reasoning and intention.9 A central
concept in all of the traditions is the idea of a collective intention (also known
as a "we-intention"): an intention that is shared by a group of people, and which
encompasses the individuals in that group. For example, if Alice and Bob want to
go for a walk together, this is a collective intention: it is not sufficient that each
of them attempts to walk next to other person, but that is is a goal shared by
both. I will avoid a philosophical analysis and will instead use a game theoretic
approach, following the Bacharach-Sugden tradition in Economics.
9
See Bacharach (2006) for a review of the idea in Economics, and Gold and Sugden (2007)
for a philosophical game-theoretic approach.
25
Collective intention
In game theoretic terms, we will consider that a collective intention involves agents
who attempt to play a Pareto dominant equilibrium.
is a Pareto dominant equilibrium at a given E
We say action profile a*
cE
x T, x T2 , if a* is a
Pareto dominant equilibrium of the game of complete information defined by 0,
for each 0 E proj9E. We next introduce a slight generalization of the notion of
q-dominance from Kajii and Morris (1997).
E
We say a* is q-dominant at E C
x T, x T2 if for any type tj in E, it is a best response to play a* whenever other
types in E play a*i with probability at least q.
Definition 1. An action profile a* is q-dominant at E C 8 x T x T2 if for all
tj E projrE and all i E I, it holds that:
/A-
XT-[ui(0,
Iti)
a*, a-j) - ui(0, a', a-i)] drq(a-i, tj)do(0, ti
for any a' =L a* and for any rj E A(Ai x T-i) such that ri(a*i, tj)
0,
q for all
t_i E projT_.E.
We are now ready to define the key concept of p-collective intention, which
is an action profile a* such that, for some types, it is common certainty that a* is
a Pareto optimal equilibrium which is p-dominant.'(
10The definition of collective intention is based on the similar concept by Shoham and Leyton-
Brown (2009).
26
Definition 2. Given action profile a* and given E C
Ex
T x T2 , we say that a*
is a p-collective intention at E if
1. E c C(E)
2. a* is a Pareto optimal equilibrium at E
3. a* is p-dominant at E.
The intuition behind a collective intention a* is that all players believe that
a* is a Pareto dominant equilibrium, all players believe that all players believe
this, etc. Therefore, there is common certainty that it is in the best interest of
everybody to coordinate on a*. However, playing a* might be risky, depending
on the game payoffs. Because a* is p-dominant, the higher p is, the more risky
is to play a*. When p = 0 playing a* is dominant for Player i, and when p = 1,
Player i would play a* only if she is certain that everybody else is playing a*_ with
probability 1.
To simplify the notation, given F C T, if there is S C
E
such that a* is a
p-collective intention at E = S x F, we will also say a* is a p-collective intention
at F.
Example 2 (Email game, part II). We continue our example of the Email game.
The payoffs are as follows."
"Payoffs as in Dekel et al. (2006) and Strzalecki (2014).
27
When the enemy is strong:
A
N
A
-2,-2
-2,0
N
0, -2
0,0
A
When the enemy is weak:
N
A
1,1
-2,0
N
0,-2
0,0
In Part I of the example, we saw that in T1 (t2) there is common certainty that
0 = W. Moreover, when 0 = W, a* = (A, A) is a Pareto optimal equilibrium, so
the first two conditions in Definition 2 are satisfied. Note that a* is 2/3-dominant
because it the other player chooses A with probability at least 2/3, then the payoff
from choosing A is at least 2/3 - (-2)/3 = 0. Therefore, a* is a 2/3-collective
intention in E.
We saw in PartI of the example, in T 2 (tI) there is no common certainty either
that 0 = W or that 0 = S. The only event which is common certainty at T 2(ti) is
the event E =
E x T 2(t'), but there is no action profile which is a Pareto dominant
equilibrium at E. Therefore, there is no collective intention for t' of level k = 2.
28
From intention to action
Let
'
E [0, 1] be a parameter that measures the probability that others will engage
in cooperative reasoning. In other words, if there is a collective intention a*, V) is
the probability each player assigns to other playing according to a*. The following
assumption determines when do players engage in cooperative reasoning.
Assumption 2 (Cooperative reasoning). For each Player i E I and each type
ti E T, if there is a p-collective intention a* at Tk(ti) with
> p, then ti plays ai.
Otherwise, ti engages in competitive reasoning.
The intuition behind Assumption 2 is as follows. Player ti conjectures that
t-i will engage in cooperative reasoning and therefore play a* i with probability
at least 0 and, because a* is p-dominant, a* is a best response to such conjecture
whenever 0 > p. When this condition is not met, players cannot use cooperative
reasoning, and will engage in competitive reasoning, which will be described in the
Section below.
Example 3 (Email game, Part II continued). Because (A, A) is a 2/3-collective
intention at T'(t'), Assumption 2 implies that type t2 plays A whenever 4 > 2/3.
Intuitively, t2 of level 1 has a low ability to represent the game, and upon sending 2
messages she believes that "enough messages have been sent" so that it has become
common certainty that 0 = W.
While this is not the case in the objective type
space T, it is the case in T1 (t2). Then, t2 will engage in cooperative reasoning and
29
play Attack when she assigns enough probability 4' on Player 2 also Attacking.
On the other hand, type t' of level 2 cannot engage in cooperative reasoning,
because, as we saw, in her representationT 2 (tl) there exists no collective intention.
1.2.3
Competitive reasoning: Level-k
In standard Level-k models of complete information, a Level-0 player chooses according to a given exogenous distribution, usually taken to be uniform, and a
player of level k > 1 best responds to the belief that she is playing against a player
of level k -1.
We will preserve that basic intuition, and will apply it to games of in-
complete information. Given a type ti, and a conjecture q (a-iItUi) E A(A-i x Ti),
we define the best response as
BRt, (,q) = arg max
ui(0, di, a-i)d (a-ilti)di(0, ti ti)
Given a type ti with level k, we will define the behavior of ti recursively. Let
Lk(tj) denote the behavior of type tj of level k. The level-0 behavior is defined,
as is usual in the literature, to be a uniform distribution over the set of actions.
L k(ti) =BRt, (L k-1)
30
%
Then, we can define the behavior of types of level k > 1 recursively.
)
When Lk-1(t-i) is a singleton, Equation 1.1 is well-defined. When Lkt-1
is not a singleton, Equation 1.1 should be interpreted as the set that results from
all possible conjectures for Lk-1 (t_. Lk(-) is the intuitive extension of Level-k to
games of incomplete information. Type ti believes that the other players are of
level k - 1; because it is a game of incomplete information, type ti also has beliefs
about the state of the world and types of the other players. The combination of
the beliefs about the level of others and the beliefs about the state of the world
jointly determine behavior. We make this explicit in the following assumption.
Assumption 3 (Competitive reasoning). Type ti of level k who engages in competitive reasoningplays according to Lk(t,).
Example 4 (Email game, Part III). Let's consider the behavior of a type of Levelk. First, level-O players randomize uniformly. Because of that, it is a best-response
for a player of level 1 to play N, independently on her beliefs about 9 (when 0 = S,
N is dominant, and when 0 = W, N is a best response to uniform randomization).
Therefore, all level 1 players choose N. This means that, for all k > 1, it is a best
response to choose N, and therefore all players with k > 1 choose N
1.2.4
Putting it all together: Solving the Email game
Assumptions 2 and 3 imply that a type ti E T with level-k engages in a dual
reasoning process: first she attempts to engage in cooperative reasoning, and if
she cannot, she engages in cooperative reasoning.
We can define this process
formally as the solution concept we will use in the remainder of the paper.
31
Definition 3. Given k and 4', we define the solution concept of dual reasoning
Di(ti) as
Di (ti) -{i
a*
if V' > p, where a* is a collective p-intention,
Lk(t,)
otherwise.
Going back to the Email game, let m be the number of messages received by
tv', and k her level. Rubinstein (1989) showed that the only rationalizable action,
irrespectively of how many messages a player receives, is to never attack. The
intuition for his result is as follows. Players who sent 0 messages never attack,
because they put enough probability on 0 = S (so that A is dominated by N).
Player 1 who sent 1 message puts high probability on Player 2 having sent 0 messages, and therefore best respond by playing N.
Player 2 who sent 1 message
puts high probability on Player 1 having sent only 1 message, and also best responds by playing N. Proceeding inductively, all players best respond by playing
N, irrespectively of how many messages they receive. We see here that there is a
contagion of behavior, by which the actions chosen by those who sent 0 messages
affect the actions of everybody else. Note, however, that this argument requires
that players be extremely rational, and able to follow such a long chain of reasoning.
On the other hand, when tI engages in dual reasoning, we obtain a very dif32
ferent result: tT attacks when m and 0 are large enough.12
Theorem 1. Player tT Attacks if and only if m >
k+i=1
and ) > 2/3.
The intuition for this result is as follows. Because tT uses a boundedly rational
k-representation, she believes that there is common certainty of 0 = W upon receiving enough messages. In that case, (A, A) is a 2/3-collective intention, and she
attacks whenever 0 > 2/3. When she receives too few messages, or ' is not high
enough, tn engages in competitive reasoning and, as we saw in Example 4, that
means she plays N. Dual reasoning formalizes our intuition that after observing
enough messages there is "common certainty" of 0 = W for all practical purposes,
and players Attack when they believe others engage in cooperative reasoning with
high enough probability.
Compare Theorem 1 with the following result by Strzalecki (2014). He assumes
that players of Level-0 always Attack (instead of randomizing), and shows that if
m >
k+i=,
Player i Attacks. Note that Theorem 1 and Strzalecki's result are
remarkably similar in the prediction of what will happen when enough messages
are received. Their explanation, however, relies on mechanisms which are almost
opposite. Strzalecki's result relies on a contagion process starting from Level-0
players, and showing that when enough messages have been received, a Player of
12
Unless otherwise stated, all the results in this paper are for k > 1. I follow the literature of
Level-k models, which assumes that level-O agents do not exist in the population, but only on
the minds of the players.
33
level k + 1 is certain that Players of level k will Attack. The proof of Theorem 1
relies on the fact that when m is large enough, players cannot include the type who
receives no messages in their representation, and therefore the contagion process
from Rubinstein (1989) is prevented to happen. The game is thus transformed
into a complete information game where players coordinate on the Pareto optimal
equilibrium.
1.2.5
'3
Games of complete information
Note that in a game of complete information, the first step of representing the
objective type space is redundant, because
e is a singleton.
However, the model
still has interesting predictions to make in this context. In order to obtain the
cleanest possible results, I focus on the simplest possible games: 2 x 2 symmetric
2-player games of complete information, which can be represented by 4 parameters
as follows:
L
L
R
R
mw xy
z
y, xz,
Table 1.1: Symmetric game
Let's consider the case where the games has two pure-strategy Nash equilibria:
13
Theorem 1 is also connected to a result in Jimnenez-Gomez (2013). In that model, agents
also represent the game as if there was complete information when they receive enough messages;
unlike here, agents cannot engage in competitive reasoning, but behave according to interim
correlated rationalizability, Dekel et at. (2007).
34
(L, L) and (R, R). Moreover, consider that (L, L) is the Pareto optimal equilibrium
(hence w > z). Note that (L, L) is p-dominant, for p defined as:
==
pw+(1-p)x=py+(1-p)z
p=
z- x
If p < 1/2, then (L, L) is risk-dominant, otherwise (R, R) is risk-dominant.
We have the following result.
Proposition 2. Player i chooses L if and only if 1/2 > p or 4 > p."
The intuition is as follows: because this is a game of complete information,
the first two conditions of Definition 2 of collective action (i.e. that it is common
certainty that (L, L) is Pareto dominant) are immediately satisfied. Therefore,
players choose action L, corresponding to the Pareto dominant equilibrium, when
this is not too risky, i.e. when
4
> p (by Assumption 2). Otherwise, Player i
engages in competitive thinking, which will lead to playing according the riskdominant equilibrium (and hence choosing L when p<1/2).
Proposition 2 has two conditions under which players choose L. The first condition of 1/2 > p corresponds to risk-dominance, Harsanyi and Selten (1988). The
second condition of
4'
> p happens when players engage in cooperative reason-
ing and choose the Pareto dominant equilibrium. In their classical discussion,
Harsanyi and Selten (1988) argued that Pareto dominance should be used as an
"At p = 1/2, dual reasoning predicts that either L or R are played, as individuals are indifferent. I will omit conditions for those knife-edge cases in the rest of the paper, as they would
needlessly complicate the statements of the theorems.
35
equilibrium refinement and, in the absence of a Pareto dominant equilibrium, risk
dominance should be used.- 5 Proposition 2 gives a very clear prediction for this
simple class of games: when 4 is high enough as compared to p, players coordinate on the Pareto dominant outcome; when 4' is not large enough, they choose
the risk dominant outcome. This captures Harsayi and Selten's intuition that in
games where the Pareto dominant equilibrium is more rewarding as compared to
the risk dominant equilibrium, players are more likely to coordinate on the Pareto
dominant equilibrium. Moreover, the model as applied to 2 x 2 symmetric games
is a generalization of Pareto dominance (when 4 = 1) and risk dominance (when
4' = 0). Following Matthew Rabin's argument for extending economic models in
a testable way, Rabin (2013b,a), the model allows us to perform a statistical test
the hypothesis HO : 0 = 0 (which corresponds to a Level-k model) or HO : 4'
1
(which corresponds to a team-reasoning model).
1.3
Global games
The idea behind global games is to turn a game of complete information into
a game of incomplete information, by adding a small amount of noise, Carlsson
"They wrote: "risk dominance is important only in those situations where the players would
be initially uncertain whether the other players would choose one equilibrium or the other. Yet,
if one equilibrium would give every player higher payoffs than the other would ( ...) every player
can be quite certain that the other players will opt for this equilibrium which will make risk
dominance considerations irrelevant" p. 358); "a theory that considers both payoff and risk,
dominance is more in agreement with the usual image of what constitutes rational behavior"
(pp. 89-90) - cited in Carlsson and van Dammne (1993).
36
and van Damme (1993). Heinemann et al. (2004) took this idea to the lab, and
run an experiment. The game is a coordination game with two actions: action
A is "safe" and gives a constant payoff T. Action B is "risky", yielding a payoff
of Y when the number of subjects who choose B is at least C1 - C2 - Y, (where
C1,
C2 are constants). Subjects had access to this formula and to a table in the
instructions. Let U[a, b] denote a uniform distribution of the interval [a, b]. The
state Y is randomly selected from U[10, 90]. In the Common Information (CI)
treatment, Y is common certainty. In the Private Information (PI) treatment,
each individual receives a signal Xi, i.i.d. U[Y - 10, Y + 10]. The following is a
summary of the evidence from Heinemann et al. (2004) :16
1. Undominated thresholds: strategies are consistent with undominated
thresholds
2. PI thresholds: with PI, mean thresholds are close to or below the global
games equilibrium
3. CI thresholds: with CI, there is more coordination on the Pareto dominant
equilibrium than what global games predict
4. Comparative statics: mean thresholds follow the predicted comparative
statics
5. Information effect: with PI, mean thresholds are higher than with CI
"The original fact 6 in Heinemann et al. (2004)has been omitted because it is about how
ordering affects play, and none of the theories is concerned with that.
37
6. Coordination: with PI, there is more variation in individual thresholds
than with CI
7. Predictability: variation in mean thresholds across sessions is similar for
CI and PI
The following table summarizes whether these experimental facts are predicted
by three different theories: global games, Level-k model as in Kneeland (2014),
and the dual reasoning prediction offered in this paper. As we can see, the theory
of global games predicts too conservative behavior with PI, and fails to predict
that there is more coordination with CI.
Global games
Level-k model Dual reasoning
Y
Y
Y
2. PI thresholds
Y/N
Y
Y
3. CI thresholds
N
N
Y
4. Comparative statics
Y
Y
Y
5. Information effect
N
'Y/N
Y
6. Coordination
N
Y
Y
7. Predictability
N
Y
Y
1. Undomin. thresholds
1.3.1
Theoretical predictions
Consider the following game, which has been used extensively in the global games
literature.
38
Attack
Not attack
Attack
7,
0, 0 - 1
Not attack
- 17,0
0,0
When 0 > 1, Attack is strictly dominant, whereas if 0 < 0, Not attack is
strictly dominant. When 0 E [0, 1] both actions are rationalizable. With complete
information, a direct consequence of Proposition 2 is the following corollary.
Corollary 3. Player i Attacks if and only if 0 > 1/2 or 0 > 1 -
).
Now suppose that each player observes a signal xi = 0 + ej, where ei ~ i.i.d
U[-u, o-].
The original game of complete information has two pure equilibria;
the powerful result from global games is that by adding a small amount of noise,
the equilibrium is unique and it coincides with risk dominance, Carlsson and van
Damme (1993). In particular, as the noise vanishes (i.e. a -+ 0), players attack if
and only if they receive a signal greater than 1/2, Morris and Shin (1998). However,
this result depends on players following an arbitrarily long chain of reasoning. The
argument is as follows: players with xi + o < 0 have Not attack as a dominant
action, and those with xi - o > 1 have Attack as dominant. Given this, players
receiving nearby signals will play the same action (i.e. those with xi + a close to
0 will not attack, those with xi - a close to 1 will attack). Proceeding in this way
a contagion happens, where the action of types closer and closer to intermediate
signals is determined by a chain of best-responses started by those with extreme
signals. For a -+ 0, however, this contagion takes an unbounded number of steps,
39
and our intuition should warn us that players might not be using such reasoning
process. This intuition is formally captured in the following result, in which we see
that, for contagion to happen, the cognitive level k needs to increase as o- becomes
small.
Theorem 4. Playeri with signal xi Attacks if and only if xi > 1/2 or xi -o-- k >
I - b.
Theorem 4 maintains xi > 1/2 as a sufficient condition to coordinate on Attack,
but there is another condition under which Attack is possible: if the noise is
sufficiently small (as compared to k), then players can approximate the type space
as having common certainty that E[O] > 0, in which case players can coordinate
on Attacking through cooperative reasoning. Therefore dual reasoning predicts
successful coordination on Attack in strictly more cases than global games do, as
the the experimental evidence suggests. For comparison, it is instructive to observe
what happens when the players observe a public signal y ~ U[O - a, 0 + -].
Theorem 5. With a public signal y, players Attack if and only if y > 1/2 or
y
1 -0.
Theorem 5 shows that, when players observe a public signal, the conditions are
analogous to those we found in Corollary 3 for the game of complete information.
It is instructive to compare this result to Izmalkov and Yildiz (2010). In their
model, players believe that others received a higher signal than themselves with
probability q: when q > 1/2, agents are overoptimistic. They find that Player i
40
attacks if and only if xi
1- q, a condition which is analogous to xi
1- 0. Dual
reasoning, in addition, is able to explain why players are better able to coordinate
with public than with private information.
Taken together, Theorems 4 and 5 prove that dual reasoning is consistent with
the evidence from Heinemann et al. (2004). Players use undominated thresholds
(Fact 1), since they play Attack when their signal is higher than a certain threshold in [0,11. The thresholds with private signals are lower than 1/2 and lower
than the threshold for a public signal, and this implies that there is more coordination with the public signal (Facts 2, 5). Thresholds with private signals are
more disperse than with a public signal, and the latter is between the threshold
for risk-dominance and Pareto dominance (Facts 3, 6). The comparative statics
hold: players attack less as 0 decreases (or if the cost increased from 1), Fact 4.
Finally, the predictability of a successful attack is the same in both conditions: it
depends on 0, 0 and k, Fact 7 (in particular, there is no multiplicity that could
generate excess variability in public signal case).
Note that the fact that the model is able to accommodate the experimenta
evidence so well is only possible because of the three components of the model:
bounded rationality, cooperative reasoning and competitive reasoning. Without
bounded rationality in the representation of the game, the conditions for cooperative reasoning would never be fulfilled, and the model would reduce to a Level-k
41
model. If we did not have cooperative reasoning, bounded rationality would imply
that there would be multiple equilibria, as players with low enough k would have
both actions rationalizable. Finally, competitive reasoning provides the "default"
behavior when cooperative reasoning fails, which is why we maintain the sufficient
condition for attacking when agents receive a signal greater than 1/2. It is all
three components together that make the model fit the empirical results.
1.3.2
Revolutions and Collective Action
To model revolutions, I will use the simplest possible extension to the coordination
model in Section 1.3. There is a continuum of mass 1 of citizens, and each of them
chooses whether to Attack or Not attack. If an individual chooses not to attack,
her payoff is constant and equal to 0. If the individual chooses to Attack, she
obtains 0 if the revolution succeeds and 0 - 1 if it does not succeed. When fraction
q of the citizens attack, the revolution succeeds with probability q. Therefore,
conditional on a fraction q of individuals attacking, the expected payoffs are given
by:
Attack
Not attack
Fraction q attack
0-(1-q)
0
Suppose that a self-interested government wants to avoid revolution. The government observes 0 perfectly, and can choose one of two options:
42
. to disclose 9 as a public signal with perfect accuracy
* not to disclose any public signal, in which case each citizen receives a private
signal xi
-
i.i.d U[9 - o-, 9 + o].
Proposition 6. The government discloses 9 if and only if 9 < 1/2 and '$ < 1 -0.
The intuition of this result is as follows. When 9 > 1/2, if players learn the true
value of 9, they always attack because it is risk-dominant (and hence selected by
competitive reasoning). Therefore, in that case, the government does not disclose
9. When 9 < 1/2, the government faces a tradeoff: disclosing 9 informs individuals
that times are bad for them, but also aids them in coordination. The smaller 0
is, the larger the first effect; when
4
< 1 - 9, the informational effect is more
important than the coordination effect.
1.4
Generalized Model and Applications
Up to now, we have required that players have common certainty of an event E in
order for a* to be a collective intention at E. That requirement, however, seems
intuitively too strict: what if agents are almost certain about E, but consider
that things might be otherwise with a small probability? In this section we will
extend the concept of collective intention for the case when E is common p-belief,
Monderer and Samet (1989). Recall from Section 1.2.1, that for each E C
T, x T2 and each type tj c T, we define Et as
43
E
x
Eti = {(0, t-i) : (0, t1 , t2 ) E E}.
We define the p-belief of Player i as,
B(E) = {tj E T : i (Eti Iti) > p},
and the mutual p-belief BP(E) as:
BP(E) =9 x B'(E) x B2(E).
We define B"'P(E) = BP(E), and the k-mutual p-belief for k > 1 by:
k-1
Bk'P(E)
-
n
BM"'P(E) n BP(Bk-l'P(E)),
m=1
and we define common p-belief as
CP(E)
B p(E).
=
k=1
In order to extend the model to the case of common p-belief, we need to modify
Assumption 1, so that it accommodates the concept of p-belief.
Assumption 4. If tj of level k p-believes E is k-mutually p-believed, then she
p-believes E is commonly p-believed. Formally:
!3(Bk'P(E)|ti) >
p
==>
44
13(CP(E)|ti) > p.
Note that Assumption 4 generalizes Assumption 1, because the latter is a
particular case when p = 1. We also need to re-define the concept of collective
intention, to take into account the fact that the event E might believed with
probability less than 1.
Definition 4. Given action profile a* and given E C E) x T x T2 , we say that a*
is a p-collective intention at E if p = q/s, with
1. E c Cs(E)
2. a* is a Pareto optimal equilibrium at E
3. a* is q-dominant at E.
Definition 4 generalizes Definition 2, and Assumptions 2 and 3 remain unchanged: players engage in cooperative reasoning whenever a* is a p-collective
intention, and 0 > p; otherwise they engage in competitive (Level-k) reasoning.
Note that when s = 1, this boils down to the condition we had in Section 3.2;
however when s < 1, the condition ?/ > q/s requires either that 0 is higher or q
lower. This is intuitive: when event E is commonly s-believed for s < 1, there
is probability 1 - s that the other player does not believe in E. Because of that,
a* has to be less risky (hence q be lower), or that the other player engages in
cooperative reasoning more often (hence ib be higher).
45
1.4.1
Normally distributed signals
In this section we consider global games where noise is normally distributed; unlike the model in Section 1.3, the generalized model can be applied to the case of
normal signals. The results will be very similar to the ones we found previously,
showcasing the adaptability and robustness of the model.
When signals are normal, and unlike in the case of uniform signals, the support
of the beliefs for each agent is unbounded: all agents put a positive (even if small)
probability on 0 having extreme values.
In particular, this precludes common
certainty of E[9] > 0. However, agents can still have common s-belief of E[O]
0,
for s < 1, and that is where the generalized model shows its value. We consider
the following four scenarios:
-
A(O , a 2
)
Private signal: each player i receives independent signal xi
)
Common signal: all players receive the same signal y ~ K(9, r2
Private and common signal (PCS): players receive xi and y
Two common signals (TCS): all players receive y' and y 2 , distributed
T2
)
independently A(9,
Proposition 7. The following are sufficient and necessary conditions for Player
i to Attack in the different scenarios
9 Private signal: xi > 1/2 or ' > minx*,
46
1*
Common signal: y > 1/2 or 4 > 1 - y
* Private and common signal: or2 Y +
(a .-2 + 7 2 ) or
> min
xb
. 2 common signals:
22y2
y
Va2+3(2+2
> 1/2 or i > 1 -
.
[*
.
(1.2)
X* /+
(1+)
-
9
'+Y2
The conditions for one or two common signals are quite intuitive: because there
is common certainty about the signals, the conditions are analogous to their counterpart in a game of complete information. When there is only a private signal,
Player with signal xi needs to consider what is the lowest signal x* such that others will Attack. There is a tradeoff: a lower x* means that there will be k-mutual
s-belief that E[9] > x* for higher s (and, by Assumption 4, common s-belief that
E[9] ;> x*); but on the other hand, Attack is q = (1 - x*)-dominant for a player
with signal x* and hence lower x* raises q. Type xi wants to minimize q/s, in
order to have V) > q/s, and therefore there is an optimal x*, which is given by
the condition in Proposition 7. With a private and a public signal, the intuition
is quite similar, except that players need to take into account both y and xi when
figuring out who's the player with the lowest x* who Attacks. Note that y and
xi are given weights l and
(Jar
in the denominator of Equation 1.2: the
higher k and the ratio u/'r, the more players put weight on the public signal y
rather than on their own private signal xi.
47
Cornand (2006) compared the outcomes of having a private and common signal,
versus two common signals, in an experimental setup quite similar to Heinemann
et al. (2004).'7 She found the following facts:
1. Players used undominated threshold strategies
2. With PCS, agents consider more the public signal
3. The probability of an attack is higher with TCS than with PCS
4. There is no significant difference in coordination between TCS and PCS' 8
Equipped with the conditions from Proposition 7, we turn to analyze the predictions of the model for the different scenarios, and compare them to the results
from Cornand (2006).
Theorem 8. Let #b> 1/2.
1. Players use undominated thresholds strategies in all scenarios
2. When players receive a public and a private signal with
02
= 2, they put
more weight on the public than on the private signal
3. Coordination is identical with two common signals than with one common
signal
17
1n her experiment, the noise is distributed uniformly.
1 Cornand (2006) includes a fifth fact: predictability of an attack seems to be higher in PCS
than in TCS. However, the difference is small, and there is no statistical test to determine whether
it is significant.
8
48
4. Players coordinate more with one or two common signals than with a private
and common signal
5. Players Attack more with two public signals than with one public signal (for
0 > 1 - 7)
6. On expectation, players Attack more with one public signal than with a public
and a private signal when
U.2 =r2
and 0 > 1 - 0)
Players put more weight on the public than on the private signal in PCS,
because the public signal is more important for generating common s-belief that
when r = -: both signals convey the same information, but y is observed by everybody else. With one or two common signals, coordination is identical, in the sense
that all agents' decision is based on whether y or "1"2 is greater or smaller than
1 - 0; conditional on that, all agents behave likewise. However, agents coordinate
more with common signals than in PCS, because adding a private signal means
that some agents might decide to Attack based on a high xi, whereas some agents
might decide to Not Attack based on a low xi. In terms of attacking, players attack more with two public signals than with one public signal when 9 > 1 - V, for
the simple reason that
1 -
4
,
2
is more precise than y, and therefore it will be above
more often. Finally, on expectation players attack more with one common
signal than in PCS: this is because half of individuals will receive (on expectation)
a signal xi lower than y, in which case, from Equation 1.2, we have that those
individuals are less likely to attack than if they had just received only y. Because
at least half of individuals attack less on PCS than with one signal, it must be
49
that individuals attack (weakly) less with PCS than with one signal.
Comparing the results with those of Cornand (2006), we observe that the first
three facts are corroborated by the model: undominated thresholds, agents consider more the public than the private signal, and probability of an attack is higher
with TCS than with PCS (this follows from 5 and 6 from Theorem 8). Fact 4 (coordination is not significantly different between TCS and PCS) is not predicted by
the model although, unlike standard global games, the model predicts that behavior will converge to the complete information case as noise vanishes: so for small
amounts of noise, behavior will become increasingly undistinguishable between
TCS and PCS. This is an area of future research, which leaves the possibility of
designing an experiment to test the difference between standard global games and
dual reasoning in different informational scenarios.
1.4.2
Signal jamming by the government
We return now to the scenario of Section 1.3.2, where a government is choosing
a property of the signal that will be observed by the citizens. I follow Edmond
(2014), where instead of choosing between a public or a private signal, the government now has to decide the precision of a private signal. There is a fixed cost
c of Attacking, and the individual obtains an extra payoff of 1 if the attack is
successful. The payoff for Not attacking is normalized to 0. Let q be the fraction
50
of individuals who Attack: an attack is successful if and only if q > 1 - 0."
The government observes 0, and can take a hidden action a < 0, that incurs a
convex cost C(a), such that C(0) = 0, C'(jaj) > 0 and C"(Ial) > 0. The government's payoff if the attack is not successful is 1 - 0 - q. If the attack is successful,
the government's payoff is normalized to 0. After the government chooses a, each
agent receives a signal xi = 0 + a + ei, where ej ~ i.i.d.
IA(0,
u 2 ).
In this setup, Edmond (2014) shows the following counter-intuitive result: as
o- -+ 0, then a(9)
-*
0 for all 9, and the individuals never Attack. That is, as
the signals xi become extremely precise, the government stops manipulating the
signals, and yet no individual ever Attacks. The intuition for this result is that
governments always manipulate the signal a little, so that given a player with signal xi, she believes that others believe 0 is lower than xi. But this is true as well
for those who received a signal lower than xi, who believe 9 is even lower, etc. In
a sense, the opposite of Izmalkov and Yildiz (2010) happens: every individual is
pessimistic about the beliefs of others, what makes coordination more difficult.
What happens when individuals are boundedly rational and are able engage in
cooperative reasoning? Edmond (2014) considers fully sophisticated agents, who
know that the government is manipulating the signal. We could also consider
19
Edmond uses the condition q > 0, so that higher 0 is worse for the individuals, but I am
using 1 - 0 to be consistent with the rest of the models in this paper.
51
naive agents, who are not aware that the government can manipulate the signal,
and therefore take it at face value. Note that sophisticated agents always Attack
strictly more than naive agents, because their beliefs over 0 first-order stochastically dominate those of sophisticated agents. If we find conditions for naive agents
to Attack, the sophisticated agents should Attack in at least the same conditions
(and maybe more). Because of that, and the simplicity of the analysis, I will only
consider the case for naive agents.
Proposition 9. Player with signal xi Attacks if and only if xi > 1/2 + u-V- 4 (c),
or
> min
1 -x* + o-<b(c)(13
+(13)
Note that the conditions for Attacking in Proposition 9 are analogous to the
conditions we found in Proposition 7 for the case of private signals, with the extra
term v/2-<4I((c) which measures how the cost of Attacking c affects the tradeoffs.
When the cost is c = 1/2, then <b' (c) = 0 and the conditions are identical; higher
costs make Attack less likely in this scenario, whereas lower costs make Attack
more likely than in the scenario in Proposition 7.
What happens when o
--
0? As long as xi > x*, the denominator of Equation
1.3 tends to 1. Therefore, the minimum on the right hand side of equation is
achieved when x*
-+
xi. In the limit, Player i Attacks if and only if 0 > 1/2 or
0 > 1 - xi. We see here that bounded rationality, either in the form of team rea52
soning or Level-k reasoning, predicts that individuals are able to coordinate their
attack agains the government in this model. Edmond notes that if citizens are
able to coordinate their actions, then all regimes would be overthrown; that would
correspond to the case of V) = 1, in which case indeed we have that all individuals
with xi > 0 Attack as - -+ 0. Our model however formalizes this intuition, and
shows that it is also valid for intermediate values of 0: as 0 decreases, individuals
need to obtain higher signals in order to Attack, but Attack still takes place as
--+ 0. It is only in the extreme 0
-+
0, that no individual ever Attacks.
Proposition 10. As --+ 0, there are t and 9 such that:
If 9 < 2,
If
< 9 < 9,
If 9 < 9,
a = 0 and players choose No attack,
a = i - 9 and players choose No attack,
a = 0 and players choose Attack,
where : is defined as the xi such that Equation 1.3 is an equality, and 9 is
defined so that 1 - 9 = C|
- 0|).
The intuition is as follows: it is the lowest signal that makes a player choose
to Attack, and 9 is the highest 9 such that the government finds it optimal to
jam the signal in order to prevent an attack. Therefore, when 9 < t, the government does not need to jam the signal for the attack to fail. When
<
9 < 9,
the government prevents an attack by jamming the signal. Finally, when 9 <9 , it
53
would be too costly for the government to prevent an attack, and therefore the government "gives up" and does not jam the signal, and a successful attack takes place.
When using models to derive policy implications, it is crucial that we are certain of the robustness of the results, and this is another example of the advantages
of modeling bounded rationality explicitly: by having players use dual reasoning
instead of an equilibrium concept, we recover the intuitive result that as the noise
of the signal vanishes, players are more and more certain of the true value of 0
and therefore, for 0 high enough, they will be able to coordinate on Attack. This
paper represents a step at introducing psychologically-grounded assumptions into
game theory, and Proposition 10 (and its contrast to the result in Edmond (2014))
show that this is indeed a promising direction.
1.5
Conclusion
Cooperation and coordination have been crucial parts of human life throughout
our evolutionary history. Indeed, there are strong evolutionary forces that select
those who are able to cooperate and coordinate their behavior. 2 O I developed a
model which explicitly incorporates these two reasoning modalities, and showed
that it explains several important pieces of evidence, such as behavior in Rubin2
'Crawford
and Haller (1990), Kandori et
al. (1993), Fudenberg and Levine (1993), Crawford
(1995), Young (1996).
54
stein's email game, in experimental global games, and why governments would
want to prevent the generation of public information.
Cooperative and non-cooperative game theory remain largely separated areas
of research. Their concepts and methods are also disconnected, although there are
a few exceptions that provide non-cooperative foundations for cooperative game
theory, Acemoglu et al. (2008). Our framework could act as a stepping stone between both literatures, given that players behave largely like their non-cooperative
counterparts, and yet are able to achieve higher degrees of coordination through
correlation in their actions (correlation which is taken for granted in cooperative
game theory). Moreover, by assuming that there are either two, or a mass of
players, I have avoided dealing with coalition-formation, a topic which is as fascinating as it is daunting, Acemoglu et al. (2009, 2008). A clear next step is the
generalization of the model, and in particular of the notion of collective intention,
to any number of players, possibly adapting some of the insights advanced by the
philosophical literature cited in the Introduction.
In this paper I have taken
4 as exogenous and constant for all players.
Bacharach
(2006) considers the possibility that an individual belongs to different groups, and
therefore
4'
would be endogenous and dependent on group identity. Jackson and
Xing (2014) shows that in a simple coordination game, players of different cultural backgrounds (citizens from USA vs. India) were able coordinate better with
55
others of their same group. The interaction between group identity and strategic
reasoning is an area of future research.
This paper also offers new insights into mechanism design. Some authors have
criticized certain mechanisms for being too convoluted and assuming too much
rationality; for example Aghion et al. (2012) show that the Moore and Repullo
(1988) mechanism is not robust to even small amounts of uncertainty. I suggest
that using the model in this paper in a mechanism design context can alleviate
many of these concerns: on the one hand agents use a subjective representation
of the objective type space, which rules out mechanism that require infinite steps
of reasoning; moreover the model implicitly distinguishes between when equilibrium (cooperative) reasoning vs. competitive reasoning will be employed. This
is fundamental because the evidence suggests that players cooperate more than
our models predict, and therefore the mechanism can avoid collusion by exploiting
competitive reasoning and stifling cooperative reasoning, for example in the context of auction design.
The results of this model might have important policy implications.
Global
games have been used to analyze how disclosure of information by central banks
can influence the behavior of agents; Hieinemann et al. (2004) observe that crises
happen more often with private information (which, as we observed in Section
1.3, is consistent with our framework). On the other hand, in a similar context,
56
Cornand (2006) claims that "it is important for central banks to disclose a single,
clear signal". Proposition 6 sheds new light on these conclusions, and suggest that
Cornand's suggestion might only be valid when the fundamentals are strong. The
model offers a new perspective on this important problem, and a cautionary warning that the context of the game is important as to whether players will engage in
cooperative or competitive reasoning.
To conclude, this paper offers a psychologically-informed model, which merges
the traditional rigor of game theory with the pragmatic approach of Level-k and
team reasoning models, and shows how incorporating more realistic cognitive assumptions can serve to increase the understanding of economic phenomena while
preserving the tractability of our models.
Appendix
PROOF OF THEOREM 1. Let tT be the type of Player i who sent m messages.
First, we will prove that Player i believes there is k-mutual belief that 0 = W if
and only if m > kll=1. We will do so by induction: for k = 0, Player 1 believes
that 0 = W whenever m > 1/2, and Player 2 whenever m > 0, which is true.
Now, suppose that the induction hypothesis is true up to k - 1, and we want
to prove it for k. Player 1 who sent m messages and is of level k believes that
Player 2 received at most m - 1 messages. By the induction hypothesis, there is
k - 1-mutual belief that 9 = W if and only if k2
57
< m - 1, that is, whenever
< m. Therefore, when this condition holds, Player 1 believes that Player 2
k-1
believes there is k - 1-mutual belief that 0 = W, and therefore Player 1 believes
there is k-mutual belief that 0 = W, as we wanted to prove. The proof for Player
2 is analogous.
Therefore, by Assumption 1, in Tk(tT) there is common certainty that 0= W
if and only if m > k
>2. (Attack, Attack) is a Pareto optimal equilibrium, and
it is 2/3-dominant: therefore it is a 2/3-collective intention. In that case, by Assumption 2, type tT plays A whenever V > 2/3.
When this condition is not met, tT engages in competitive reasoning. Players
of level-0 randomize, and therefore it is a best response for all types of level 1 to
play N. Consequently, all types of level k > 1 play N.
PROOF OF PROPOSITION 2. Because
E=
II
{0* is a singleton, the k-representation
is the same as the original type space for all k, i.e. because it is a game of complete
information, players of any level k believe that 0 = 0* is common certainty. Note
that (L, L) is the Pareto dominant equilibrium, which is p-dominant. Therefore
(L, L) is a p-collective intention and, by Assumption 2, Player i chooses L whenever
'7
> p. If this condition is not satisfied, Player i uses competitive (Level-k)
reasoning. Players of level 0 randomize uniformly; Players of level 1 best-respond
by choosing L if and only if 1/2 > p. Because of that, players of level k > 2 choose
58
L if and only if 1/2 > p, what concludes the proof.
El
PROOF OF THEOREM 4. First, if (and only if) xi - o -k > 0, in Tk(x,) there
is common certainty that E[9] > 0. This is because the agent who receives signal
x' = xi-a-k believes that E[0] > 0, the agent who receives signal x" = xi-or-(k-1)
believes it is 1-mutual belief that E[9] > 0, and by induction the agent who receives
signal xi believes that it is mutual k-belief that E[9] > 0. By Assumption 1, in
Tk(xi) there is common certainty that E[9] > 0. In that case (A, A) is Pareto
optimal and p-dominant for p given by
p(xi - o - k) + (1 - p)(xi - a-- k - 1) = 0 -> 1 - p = xi - o- - k.
By Assumption 2, Player i with signal xi will play A whenever 0 > p, i.e.
whenever xi -
-- k > 1 -
4.
If this condition is not satisfied, Player i with signal
xi will engage in competitive (Level-k) reasoning. Players of level 0 randomize;
players of level 1 for which xj > 1/2 play A, and those with xj < 1/2 play N.
Proceeding inductively, types of all levels k > 1 play A if xj > 1/2 and N if
xj < 1/2, what concludes the proof.
0
PROOF OF THEOREM 5. When y > 0, there is common certainty that (Attack,
Attack) is a Pareto dominant equilibrium, which is (1 - y)-dominant. By Assumption 2, each player will Attack whenever y > 1 -
4.
When this condition does
not hold, players will engage in competitive (Level-k) reasoning. Level-0 players
59
randomize, and level-i players best respond by playing Attack if and only y > 1/2.
Because of that, all players with k > 2 best-respond by choosing Attack if and
only if y
1/2, what concludes the proof.
El
PROOF OF PROPOSITION 6. When 0 > 1/2, Theorem 5 guarantees that if the
government discloses 0, all individuals will Attack. Therefore, the government
discloses a private signal, in which case, by Theorem 4 the attack succeeds with
probability D (O-min(1/2,1-O+-K)),
which is less than 1.
When 0 < 1/2, there are two cases:
* If / < 1 -0, then even when the government discloses the private signal, by
Theorem 5, no individual ever Attacks. Therefore, the government discloses
0
" If
'/>
1 -9, by Theorem 5 all individuals Attack upon observing 9. There-
fore, the government does not disclose a public signal and the individuals
observe a private signal, and by Theorem 4 the probability of a successful
attack is <b (-min(1/21-+o-K)),
which is less than 1.
D
PROOF OF PROPOSITION 7. Let's analyze first the cases with one or two common signals. With one common signal y, the expected value of 0 is precisely y,
60
and this is common certainty. Therefore, by Assumption 2, players will Attack
whenever 0 > 1- y. Otherwise, they engage in competitive reasoning, and Attack
whenever y > 1/2. The reasoning for the case with 2 common signals is analogous,
except that the expectation of 9 is given by
y1,2
Before we analyze the other two cases, it is useful to note two facts from Morris
and Shin (2003):
1. Player i with signal xi believes that 9 -
;a2+r2
when there is
AF(xi, 0.2
)
only a private signal 9
(~
L722+2
2. Player i with signal xi believes that the signal xj for player j is distributed
~
(2Y+72Xi
2,7 2
when there is only a private signal xj
+04;
+2
Private signal The case k
=
-
Af(xi,
2U 2
)
X
1 is trivial. For illustrative purposes I will show
first the case k = 2. Type xi believes the signal for other players x' is distributed
according to K(xi, 2U 2). Therefore, the probability xi assigns to the signal of the
other player being greater than a certain threshold x1 is21
x1
-
xi)
xi -
x1
For the same reason, a type with signal x1 assigns probability
11
0-
x1) =4
21
x1 - x*)
In the rest of the proof, we will use xk to represent signals where k is an index that is not
associated to the identity of the player; i.e. x1 might be a signal for Player 1 or Player 2.
61
to others receiving a signal greater than or equal to x*. Notice that the set
with the largest s such that it is common s-belief that the expected value of 0 is
greater than x* is given precisely by those types with signals greater than or equal
to x1 , where x1 solves:
~(xi-
(xx*)(1.4)
i)~
This is because xi believes there is a fraction s of types with signal at least
x1 , and those types believe there is at least a fraction s who received signal at
least x*. Any other x' would decrease either the first or the second order beliefs
on E[0] > x*, and therefore it would have a lower common s-belief on E[0] > x*.
Taking into account that 4b is injective, the solution to Equation 1.4 is given by
Xi - X1 = XI - x*, that is x 1 = (xi - x*)/2.
Next we proceed to show the result for any k. By an argument similar as the
one expressed above, the largest s such that it is common s-belief that E[0] > x*
happens when
62
(
- Xk-1) =
k-1 - Xk-2)
(
2
o- )
2-X
<D 1-
k2
x
= <
(X2- X1
k2
(
1
X
-
(i
-*
Again, taking into account that <D is injective, we need to solve the system of
linear equations:
=k-=
Xk-1
Xi -
Xk-1
~
X2
Xk-2
-
=
X=X
Xk-2
-
- Xk-2,
-
Xk-3,
X*
To solve the system, I follow Lugon (2008). First, we need to find two independent solutions to the equation Xk+2
-2Xk+ 1
+k
=
0. The first solution will be
rk, which solves the equation whenever rk(r 2 - 2r + 1) = 0. This happens when
r2 -
2r + 1, and solving the second-order equation, we find r = 1. We need a
second solution to the equation, and this one will be k, which indeed solves the
equation, since k + 2 - 2(k + 1) - k = 0. Taking together these two independent
63
solutions, we have that the general solution to the equation is given by A + Bk.
Using the initial and final conditions we find A and B:
B=
->
Xi=Xk=A+Bk
=x=x*,
k
k
.
A
This implies that
xi - x*
,
k
Therefore, the s such that it is common s-belief that
x1x*)-
EIIO]
> x*, is given by
(xi - x*)
Nf2o-
k v'1-
Attacking is q-dominant for player with signal x*, where q is such that
qx* + (1 - q)(x* - 1) = 0 =>t q = I - x*.
x* E arg min
,
Therefore, when x* is chosen as to minimize
we have that type xi will engage in cooperative reasoning, and choose Attack,
whenever
64
1-*
When this condition does not hold, Player i engages in cooperative reasoning.
Following a similar reasoning as in the proof of Theorem 4, we find that Player
i Attacks if and only if x > 1/2, what concludes the proof for the case with a
private signal.
Private and common signal. The proof will follow closely the case with a
private signal. Player i who received signal xi believes that the signal for Player j
is distributed
.
2
4
+
T
2u
xi
yT(
+r22
3
c0
2
2r2
+T
2
Therefore, type xi believes that the probability xj is larger than a certain x' is
' - a2
1-2+-
2y+
(X, - X'
oxy - X)
+2(
Tt
X 1 (O-, T)
=
2
2,22+r4
2
ar+r
where N(u, r)
=
v(. 2 +T
2
)(2U 2
2
+
4 ).
The same argument can be applied
to the beliefs of xj, who believes that the probability the other players has signal
at least x' is given by
.2 (y - X') + r 2 (xj -X'
R(5 T)
65
As in the proof of the case with a private signal, we solve the system:
4
+
Xk_1)
07(Y -
2
Xkl)
T(X-
1 (O-, T)
T 2 (X2
.2
xi)
)
X1) +
T 2 (k-l
-
Xk-2)
t(U, T)
f
-
-
(y
(
4b
_ Xk-2) +
(Y
x*) + T2 (X 1
N(o-, r)
1(o-, r)
Because 1< is injective, this is equivalent to solving the system of linear equations:
9.(Y
-
Xk-1) + T2 (X
c. 2 (y -
-
Xk1)
x 1 ) + T 2 (x2 -
2(y -
Xk-2)
+
2
(k-1
-
Xk-2)
X1 ) = u.2 (y - x*) + 72 (X 1 _ X*)
Simplifying, we need to solve the second-order linear equation
T2 Xk
(u 2
+ 2T 2 )xk-1 + (,
2
+ r 2 )Xk-2 = 0,
with initial conditions Xk = xi, and xo = x*. Following Lugon (2008), we look
for a solution of the form rk, therefore we need to solve
2r-2
(U 2 +2T 2 )r + (U 2 + T 2 )
66
=
0.
Solving, we find:
+2T
2
+(2+
2
2
)-47 2 (- 2 +.2
2
2T
2
2
+
>
)
a2
2
27 2
2,T
therefore we have the solutions r1 = 1 and r2 = 1+ I, and the general solution
to the equation is
Xk =
A + B (1
Using the initial conditions, we have
x* =xO=A+B, and
(1+
(1
(1+
-B+B
+
T
1 +x*
-2
=->
B=
.
k
2
(+ RT2)k-
/
A+B
Xi= Xk =
Therefore,
T2
and hence x1 - x*
(y'2Y
- x*)
-
472(+,2) -
+ r 2 (x1
**.
-1
2
+a2) =x*
)
72
1
+
A +B
-
x1=
T(l +)k
1'
Using this, we find
x*)
a.2T
)
(o,
(Y
-
x*
Xi -
x*
(+ a2)k-1
Therefore, type xi engages in cooperative reasoning and chooses Attack whenever
67
> min
X
y -X*
2,.
Finally, taking into account that
+
i
"2
1
*
<
(1.5)
we find the desired ex-
pression.
If the condition in Equation 1.5 does not hold, then type xi engages in competitive reasoning. She believes that E[O]
=
2
,
YT
2
x
,
and therefore using the by now
standard argument of competitive reasoning, we find that she Attacks whenever
a 2y7-2T
> 1/2.
2
U2 + T
PROOF OF THEOREM 8.
1. The fact that players undominated threshold strategies is straightforward
from the conditions we found in Proposition 7.
2. In Equation 1.2, note that in the denominator of the right hand side, the
argument of 4b is a combination of y - x* and xi - x*. However, the second
argument is divided by (1+
)k
-
1, which is always greater than or equal
than 1, for a = r. Therefore, y - x* has more weight than xi - x*, and the
player gives more importance to y than to xi when solving the optimization
problem.
68
3. With one or two common signals, all players with the same k choose the same
action, and therefore coordination is perfect (and identical in both cases)
4. With a private and a common signal, some players might receive signals such
that Equation 1.2 holds, whereas others might receive signals such that it
does not hold. Therefore it is possible that different players choose different
actions, and therefore there is less than perfect coordination (which happens
with one or two common signals).
5. Note that y ~ Jr(O, Or2 ), and M.L2 ~K(G, 2). Because of that, the probability that players Attack with one signal is
that players Attack with two signals is<
0> 1-
-(
(4D
and the probability
(1-)I), which is larger when
0.
6. Note that when
U2
=
T2 ,
ex-ante both signals are equally informative about
0. First, let's show that a player with xi
y Attacks less with both signals
than if she had only received the public signal. This is because she must
choose x* < xi K y, and therefore
1 - y < min
because in the right-hand side, the numerator is larger than y, and the denominator smaller than 1. However, note that ex-ante, half of the individuals
will receive a signal xi
y. Therefore, when 9 > 1 - 0, on expectation, the
69
population Attacks less when receiving a public and private signal, than
when receiving only a public signal.
PROOF OF PROPOSITION 9. Let's consider a player with signal x*. Such player
is indifferent between playing Attack and Not attack whenever
P(q >1 -
) - c=
<-* P( > 1 -q)
=c,
where q is the fraction of individuals who Attack. In order to find the q that
makes the player indifferent, we have
P(O > 1 - q) = D (
therefore q = 1- x* + o
x* -
4
1q)
(
= C =
x* - 1+ q
-
= 4- 1(c),
(c). Hence, Attack is q-dominant for all players with
signal at least x*. Following a reasoning similar to that in the proof of Proposition
7, we find that the x* that maximizes common s-belief that E[0] > x* is given by
(D (x-X*)
Therefore by Assumption 2, Player i engages in cooperative reasoning
and chooses Attack, if and only if
>mn1 - x* + 04-'(C)
On the other hand, when this condition does not hold, the player engages in
competitive reasoning. Players of level 0 randomize uniformly, therefore a player
70
of level 1 chooses Attack if and only if
P( > 1/2) = 1
xi
- 1/2
o-
> c ==> xi
1/2 + o-<b~l(c).
PROOF OF PROPOSITION 10. Let's look at the government problem. From
Equation 1.3, we can find a threshold :t such that Player i Attacks whenever
xi < t; the threshold t is defined implicitely by
mn1 - X* + 0-<-1(c)(16
X*=
mi
.
(1.6)
The payoff for the government is 1-0 - q when the Attack fails, and must incur
cost C(a) for manipulating the signals. The proportion of players who Attack q is
),
therefore the problem for the government is
maxl 1 -06>4D
a<
O~-t
o
1-0 <a 4 -. C(a).
0+
(
given by
For 0 E [0, 1], as a- -+ 0, the problem for the government becomes
max 1[0+a<.] (1-0-1[0+a
a<o
Therefore, the solution for the government is:
71
]) -C(a).
a=
if 0 <Oor 1 -0 < C(| - 1),
0
0
if I - 0 > C|(:t - 0|).
This determines a threshold 9, given by 1- 0 = C(
0 -). When 0 < 0, there is
no successful attack: either because 9 is so low that the government does not need
to jam the signal (a = 0 because 9 < t), or 9 is low enough that the government
can prevent the attack by jamming the signal (a = t - 9). On the other hand,
when 9 > 0, the preventing the attack would require a too costly jamming, and
therefore the government chooses a = 0 and allows the attack to succeed.
72
El
Chapter 2
You Are Just Like Me: Bounded
Reasoning and Recursive Beliefs
2.1
Introduction
Models of bounded rationality in game theory have been successful at predicting
behavior in situations where the Nash equilibrium is not adequate, Camerer and
Smith (2011). The best representative of such models is the Level-k model, which
assume that players perform a discrete number of thinking steps, Stahl and Wilson
(1995, 1994), Nagel (1995). In this model, each player has a level k: players of
level 0 randomize uniformly, and players of level k +1 believe others are of level k,
and behave accordingly. A more general model, called Cognitive Hierarchy (CH),
inherits the bounded rationality of Level-k models but allows more flexibility in
players' beliefs, Carnerer et al. (2004). Despite their simplicity, these models have
73
been very successful at predicting behavior in scenarios where traditional game
theory could not. The best known example is the p-beauty contest, where players
choose an integer between 0 and 100, and the winner is closest number to p times
the average: whereas the Nash equilibrium is to play 0, Nagel (1995) found that
a large fraction of people behave precisely as either level 1 or level 2, who play
33 and 22 respectively. This result has been replicated many times with different populations, always with similar results (see Nagel (2008) for a review). The
success of Level-k and Cognitive Hierarchy models extends to several applications.
Brown et al. (2012) argue that the model can explain the failure of spectators to
infer the low quality of a movie from its "cold opening" (i.e. without showing it to
the critics before the premiere), and Ostling et al. (2011) apply it to the "lowest
unique positive integer lottery" played in Sweden. In the context of mechanism
design, Crawford and Iriberri (2007) argue that Level-k agents exhibit the winner's
curse in common-value auctions, and Crawford et al. (2009) found that in such
chase the optimal mechanism differs from the standard case.
Despite the success these models have achieved, one of the mayor problems in
their application is their disconnection with standard models of incomplete information. There is no criterion which informs us ex-ante about when we should use
standard game theory vs. Level-k models. Therefore, it is difficult to compare
the results obtained from standard game theoretic reasoning, from the results of
Level-k models, as there have been few attempts to analyze both models using a
74
unified approach (Strzalecki (2014) and Murayama (2014) are notabe exceptions).
A theoretical reason why Level-k and CH models have been difficult to analyze
using standard tools, is due to the fact that these models impose two simultaneous constraints on players: agents with higher level can have more complex
beliefs, and they can also perform more computations (in the form of iterations of
the best-response correspondence).
Building on Strzalecki (2014), I develop a model of cognitive type spaces,
which are a class of Harsanyi type spaces that incorporate Level-k and CH models
into standard games of incomplete information. I maintain the assumption that
the level of a player determines the complexity of her beliefs, but players are fully
rational, as it is standard in the literature of games of incomplete information.
In particular, the model allows players of level k to believe others have level k
as well, as long as they have the exact same beliefs. This assumption is based
on the important psychological phenomenon known as the false consensus effect: a pervasive cognitive bias by which people tend to think that others are like
themselves, in the sense of sharing their beliefs, convictions and preferences, Ross
et al. (1977), Engelmann and Strobel (2012). Because a player of level k cannot
explicitly model other of the same level (for she would need to be of level k + 1),
she instead assumes that others of level k share her beliefs, as predicted by the
false consensus effect. This in turn implies that all players have a well-defined
hierarchy of beliefs, and therefore Level-k and CH models can be analyzed using
75
standard game theoretic tools.
The standard way of measuring how similar players' beliefs are is using the
product topology: two players are close in the product topology if their first-order
beliefs are similar, their second-order beliefs are similar, etc. for all possible orders.
On the other hand, behavior is measured in the strategic topology: two
players are close in the strategic topology, roughly speaking, if they have the same
rationalizable actions. However, players who are close in the product topology
are not necessarily close in the strategic topology, Ely and Peski (2011). This has
dramatic (and unfortunate) implications for game theory: even if we are able to
identify a player's beliefs with great accuracy, we are never certain that we can
predict their behavior accurately. The main result of the paper states that, when
restricted to cognitive type spaces, the product topology and the strategic topology
coincide (Theorem 13).2 This means that behavior is robust to small perturbations
in beliefs when restricted to cognitive type spaces: therefore, if players' beliefs can
be represented by a cognitive type space, the researcher can be confident that,
conditional on identifying players' beliefs, the predictions of the model in terms of
behavior are robust to small specifications of beliefs.
This result is important because it shows that some of the problems of "discon'Formally, Level-k and CH models can be embedded in the Mertens-Zamir universal type
space; and there exists a universal cognitive type space, where any cognitive type space can be
continuously embedded (Proposition II).
2
Theorem 13 makes the stronger claim that the product topology and the uniform strategic
topology coincide, when restricted to cognitive type spaces.
76
tinuity" in game theory may be the result of assuming too much rationality on the
part of players. As Weinstein and Yildiz (2007) and Ely and Peski (2011) showed,
these problems are pervasive in game theory. However, by imposing assumptions
about players' reasoning abilities which are more realistic, we are able to show
that the predictions of the model are robust.
I apply the model to the well known Email game from Rubinstein (1989): two
players play a coordination game, where they have to choose to either Attack or
Not attack; but with some probability the game is such that N is a dominant action. When the actual game is the coordination one, players exchange confirmation
emails back and forth until an email is lost. Rubinstein (1989) famously showed
that irrespectively of the number of messages received, players never attack. His
result, however, relies on players following an arbitrarily long chain of reasoning.
In contrast, in cognitive type spaces, when players receive enough messages both
actions become rationalizable: because players' beliefs have a bounded complexity, upon receiving enough messages they believe they are playing the complete
information version of the coordination game.
We have incorporated two psychological phenomena as assumptions into the
model: players' beliefs have a bounded complexity (Kinderman et al., 1998), which
is also standard in Level-k and CH models; and players are subject to the false
consensus effect. There is ample evidence that these two psychological facts are
77
pervasive and relevant for human decision-making, and we show that they also
have implications for game theory. This paper gives a step into incorporating
psychologically and cognitively grounded assumptions into game theory and, in
doing so, offers new results that show how our results can be more robust than we
previously thought.
Strzalecki (2014) showed that there exists an universal type space for CH models, and how behavior in the Email game can be different when one assumes
bounded rationality. In this paper, I extend Strzalecki's model in two directions.
First, Alice has beliefs not just about the "cognitive type" of Bob, but also about
some fundamental 0 GE, what Bob things about 9, etc. This allows for a unified
analysis of bounded rationality and incomplete information. Second, I allow Alice
to think that Bob has the same level, in which case she assigns to him her own
beliefs. CH models have the undesirable feature that "every agent believes she is
the smartest in town", what precludes any sort of common knowledge. By allowing
agents to believe that others are like themselves, common knowledge of events can
be generated.
Common knowledge plays a crucial role in game theory as well as many economic situations, Chwe (2001). By allowing agents to have recursive beliefs, the
model connects Level-k and CH models with traditional incomplete information
models A la Harsanyi. It is also related to Lewis (1969), Chwe (2001) and Kets
78
(2014), who pointed out that common knowledge can be generated recursively.
For example, consider two friends, Alice and Bob, who make eye contact: Alice
sees Bob, Alice sees that Bob sees her, and Alice knows that Bob can reason as
she does. In Kets (2014), boundedly rational agents with finite hierarchies of beliefs can have common knowledge, because it arises from low order beliefs. In this
paper, agents might have common certainty of events because they believe others
are like themselves, and therefore mistakenly take their beliefs as representative of
the population.
The paper is also related to the literature in epistemic game theory. Mertens
and Zamir (1985), Brandenburger and Dekel (1993) showed the existence of a
universal type space for incomplete information games T*, and Strzalecki (2014)
showed an analogous result for CH models. I show the existence of a universal type
space for CH models with incomplete information, and prove that it is a subset
of the universal type space T* (Proposition 11). Moreover, a pervasive finding in
epistemic game theory is that players with similar beliefs can behave differently,
Weinstein and Yildiz (2007). In my model, this will not be the case: players with
similar beliefs will behave similarly. The reason is that agents' higher order beliefs
are generated recursively, what pins down the tail of the hierarchy, in such a way
that similar types have also similar tails of the hierarchy of beliefs. In proving this
results, I make use of the strategic topology defined by Dekel et al. (2006) and its
characterization in terms of the hierarchy of beliefs by Chen et al. (2014).
79
The rest of the paper is organized as follows. Section 3.2 presents the model.
Section 2.3 applies the model to the Email game. Section 2.4 presents the main
result of the paper. Section 3.6 concludes. All proofs are included in the Appendix.
2.2
The Model
Given a measurable space (S, E) we will write A(S) for the space of probability
measures on S, equipped with the --algebra generated by the sets of the form
c A(S) : v(E) > p}, where E C S is a E-measurable set and p
{v
consider games of incomplete information with two players, I =
E
[0, 1] . We
{ 1, 2}. We will use
the standard notation for players: i denotes an arbitrary player, and -i denotes
the other player. Let 9 be a finite space throughout the rest of the paper, which
is the set of payoff-relevant states of the world. A game is a profile G = (Ai, ui)iEI
where A, is a compact set of actions for player i E I and ui : E x Ai x A-i -+ R is
her utility function. A type space is given by a set of types T = ]Ji T and beliefs
vi(ti)
C A(E x Ti). I use the notation vi(O, Elti) for the probability type tj assigns
to the state being 0 C 9 and Player -i being of type t-i E E C T-j.
For any type space T, and for any tj
C T for i E I, we can define the hierarchies
of beliefs in the following way. 3 Let 7i = A(O), and
3
This description of beliefs is based on Chen et al. (2014).
80
1= {(ti, ... ti+ 1 ) E 7'
x
margg7-1 tm+ 1 = tm}.
-(
x 'T~):
(2.1)
The condition in Equation 2.1 states that the m + 1-th order coordinate of
beliefs coincides with the beliefs of order k < m.
showed that given T* =
H0'
1
Mertens and Zamir (1985)
3T m , any type space can be embedded in T*, by
defining beliefs as follows: first order beliefs are given by 7rx(ti) = marge vi(ti),
ir7+11 (0,
EIti) = vi(0, (r",)- 1 (E)|ti)
+
and (m + 1)-th order beliefs are given by
Given a type tj E T, we denote by hi(ti) the hierarchy of beliefs (ir (ti), rj (ti),...) E
Ti*.
Definition 5. We say ti
is a clone of type ti, denoted by ti
~ ti, if h-i(ti) =
hi(ti).
In other words, two types are clones whenever they have the exact same beliefs.
Note that from Equation 2.1 we have that T* = T*j, and therefore even if tj and ti
are types for different players, their beliefs are defined over the same set, namely
A(O x T*j) = A(O x Ti*). The concept of a clone is useful to extend the definition
of cognitive type space from Strzalecki (2014) to the present context, because it
will allow us to introduce Level-k reasoning into type spaces.
81
Definition 6. A cognitive type space T is a tuple (0, Ti, ki, vi, B)jE1 , where
B E N, and
* ki : T -+ {0, 1,
" vi : T -
...
, B} is the cognitive level
A (0 x T-j) are beliefs; such that
vi(G, EVti) > 0 ==
k-j(t-j) < ki(ti) or ti
~ tj for all t-i E E.
In Strzalecki (2014), a type of level k must believe other types are of level
0, 1 ...
, k - 1. Here, Player i is allowed to believe Player j is of the same level, but
only if she is like herself, that is, if j has her exact same beliefs. This is related
to the psychological phenomenon known as the false consensus effect: the fact
that people disproportionally believe others share their beliefs and preferences,
Ross et al. (1977).
Note also that all types have well defined hierarchies of beliefs, and therefore
can be embedded in the Mertens-Zamir universal type space T*.
This is what
allows us to study cognitive type spaces using the standard tools of game theory,
as can be appreciated in the following example.'
Example 5. Consider the following cognitive type space.
4Note also that we
have not imposed a common prior.
Indeed, except for the trivial case
where all agents are of level 0, the Common Prior Assumption needs to be dropped. For the
implications of dropping the common prior assumption in Economics, see Morris (1995).
82
E=
STi = {tit~, ti,1}, T2
{l,h}.
,
=
,
" The state space is
* The levels are given for all i E I by: ki(t9)
=
ki(t?) = 0; ki)(t= ki it)=
1; ki(t?) = 2.
9 Finally, the beliefs vi are defined as follows for all i:
vi (t?)
to
1
h
1
viti?)
Li1i
h
vi(tI)
v 2 (t2)
l h
toLi
Pi
vi (i)
1
h
to
1/2
0
1/2
0
tli
-i
1
1
h
0
In Example 5, the conditions in Definition 6 are satisfied: types put positive
probability either on types of lower level, or on their clones (t9 ~ ti0 , ji'
i' -
0 , and
~ ti
i). Note that T = T, x T2 is a regular Harsanyi type space: the definition of
cognitive type space simply imposes restrictions on the form that beliefs can take,
which in turn will affect behavior. We turn now to analyze how players behave,
given their beliefs.
Definition 7. Given a game G = (Aj,ui jE and e > 0, an action a G Ai is an
c-best response to a conjecture _E
A_ - x E), written a E BfIt(G, -i, e) if
83
u(ai, 77-i) ;> ui(a', q-i) - e for all a' c Aj.
When dealing with incomplete information, it is useful to have a solution concept that does not depend on the particular type space we have written, but only
on the beliefs that players have: Interim Correlated Rationalizability (ICR) fulfills
that requirement, Dekel et al. (2007).
Definition 8 (Interim Correlated Rationalizability). Given a type space (Ti, Vi)iE,
and a type tj E T define Ri,0 ,,E(ti) = Aj, and
there exists v E A(T_, x
Ri,k+1,E(ti) =
as E Ai
E
x Aj) such that
(1) v(sj,0,aj) > 0 -z* a-i E R-i,k,,(s-i),
(2) ai E BRi(G, margAjxe
1, C),
(3) margxT, v = vi(ti).
The 6-interim correlatedrationalizabilitycorrespondenceis Ri,,(ti) = n=R
Rk,,
For c
=
t).
0 we denote it as Ri,k(ti), and we define the interim. correlatedrationaliz-
ability (ICR) correspondence as Ri(ti) = noOR4,k(ti).
Dekel et al. (2007) showed that the ICR correspondece is uniquely determined
by the hierarchy of beliefs of a type. As we have showed above, each type in a
cognitive type space determines a hierarchy of beliefs in the Mertens-Zamir universal type space T*. Because of that, we can associate each t E T to the set of
rationalizable actions Ri(hi(ti)).
84
As we saw above, Mertens and Zamir (1985) showed that there exist a universal
type space T*
=
li:I Ti*, such that any
type space T could be embedded into T*.
Strzalecki (2011) showed a similar result, and the same result extends to my model
of cognitive type spaces.
Proposition 11. There exists a set CB
type space C =
{,
=
HiCI C
, such that for any cognitive
T, ki, vi}iE with bound B, there exist injective functions bi/:
Ti -- + Cr such that, for any ti E T, hi(ti) = hi(pi(ti)).
Proposition 11 shows that it is without loss of generality to analyze the behavior
of players using cognitive type spaces, because these types can be uniquely mapped
to Cr preserving the hierarchy of beliefs, and ICR depends only on that hierarchy.
We are now ready to apply the model.
2.3
Application: The Email Game
We turn now to analyze the Email game, popularized by Rubinstein (1989), and
studied in a Cognitive Hierarchy context by Strzalecki (2014). The description of
the game is as follows. There are two generals, and each of them in charge of a
division. Their objective is to conquer an enemy city and, in order to accomplish
that, each of them should attack from a different location. Because of that, they
can only communicate through email. Each general has two actions: to attack
(A), or not to attack (N). The enemy can be of two types: strong or weak. When
the enemy is strong there is no possibility of victory, and so the generals would
85
prefer not to attack. When the enemy is weak, victory is possible, but only if both
generals attack simultaneously. Both generals share a common prior: the enemy
is either weak or strong with probability 1/2 each. The payoffs are as follows."
When the enemy is strong:
A
N
A
-2, -2
-2, 0
N
0, -2
0,0
A
When the enemy is weak:
N
A
1
-2,0
N
0,-2
0,0
Because the generals cannot communicate directly, they have implemented a
communication system through email.
General 1, who is closer to the enemy,
learns whether the enemy is strong or weak. If the enemy is weak, an email is
sent automatically to general 2, who sends an email back with the confirmation
that 2 received the email from 1, etc. Each time an email is sent, it is lost with
probability c. This process of emailing back and forth is repeated until an email
is lost, in which case no more emails are sent.
The player who has sent m messages has the following posterior over how many
messages the other player has received
5Payoffs
as in Dekel et al. (2006) and Strzalecki (2014).
86
" Player 1 who sends 0 messages knows that Player 2 sent 0 messages with
probability 1.
" Player 2 who sends 0 messages assigns probability 1/2 to the enemy being
strong, and E/2 to the enemy being weak and the message lost. Hence his
posterior assigns probabilities 1/(1 + e) and c/(1 + e) to each event respectively.
" In any other situation, Player i who sent m messages places probability e
on the message being lost before Player -i received it, and (1 - e)e on the
message being lost after Player -i received. His posterior assigns probability
1 and 1
to each event respectively.
After receiving a large number of messages both generals have "almost common
knowledge" that the enemy is weak (that is, they have mutual knowledge up to
a high order). If there was common knowledge of the enemy being weak, then
both actions (A and N) would be rationalizable. As Rubinstein showed, when
dealing with common knowledge, "almost" is not enough. Indeed, Rubinstein
(1989) showed that the unique rationalizableaction for each player is not to attack,
regardless of how many messages they received. This result is due to a "contagion"
of behavior: players who send 0 messages do not Attack because they put too litte
probability on the enemy being weak; because of that, players who send 1 message
will not Attack because, even if they know the enemy is weak, they believe the
other player will Not attack with high probability. But then players who sent 2
messages face the same problem, and choose to Not attack, and indeed the situation
87
is identical for players who received any number of messages.
However, we can
see immediately that the theorem requires that players who received thousands of
messages, either perform an inductive argument, or reason explicitly a thousand
steps: and that is precisely where cognitive type spaces will prove their usefulness.
2.3.1
Analysis
Let's analyze this game using the framework of cognitive type spaces. Let M
{O, 1, 2,... }. We will use the cognitive type space (T x M, ki, vi, B)ic,, with the
following constraints:
1. The cognitive level of a type (ti, mi) depends only on the first coordinate:
ki(ti, mi) = ki(ti, mt) for all mi, m' c M.
2. All types have correct posteriors over the number of messages the other
player has received, as described above.
3. All types of level k > 1 believe others to be of level k - 1 (and types of level
0 believe others to be of level 0).
In the Cognitive Hierarchy literature, it is generally assumed that level-0 players
are irrational and they simply act randomly (usually following a uniform distribution over their actions). Of course, which action agents of level 0 play affects
the behavior of all agents of higher level. Rubinstein's original result depends on
a contagion from types who sent 0 messages and would always play N; Strzalecki
(2014) obtains the opposite result in a CH setting by assuming that level-0 agents
88
play A: If agents of level 0 always Attack, then a type of level k > 1 has Attack as
the unique rationalizable action when she observes at least k+li=l messages. This
means that enough messages should convince a player to Attack; how many messages are needed, depends precisely on the level of a player.
Theorem 12 below states that, upon receiving enough messages, players behave
as if they were playing the complete information version of the coordination game.
We see that, in a sense, the result is similar to Strzalecki's, in that there is a threshold after which players consider they have received enough messages. However,
because agents in cognitive type spaces behave according to Interim Correlated
Rationalizability, the behavior of agents' of level 0 is not exogenous and depends
on the information they have, what in turn affects the behavior of the rest of
players.
Theorem 12. For Player i of level k who sent m messages, if m > k+1i=1
both
actions are rationalizable. Otherwise, only Not attack is rationalizable.
The intuition behind this result is that when players observe "enough" messages,
they interpret that there is evidence that the enemy is weak, as if the players were
de facto in a game of complete information: in that game both Attack and Not
Attack are rationalizable. This is because a player of level k is unable to follow
a chain of reasoning that involves more than k orders of beliefs: instead, at that
point, she considers that the other player is like herself, and therefore that she
also believes the enemy to be weak. On the other hand, when players observe
89
few messages, they are able to explicitly consider the beliefs of other players, what
generates a contagion process similar to that of Rubinstein (1989). This is particularly appealing, because it coincides with the intuitive notion that almost common
knowledge is indeed quite similar to common knowledge.' This model is a step in
reconciling psychologically plausible behavior with a formal, game theoretic modeling.
Theorem 12 shows that players who receive a large amount of messages behave
in the same way as players who receive "infinite" messages: both behave as in the
complete information version of the coordination game. Therefore, behavior is
continuous in the limit, because players who receive many messages behave similarly to those who are in the game of complete information. The next section will
build upon this intuition, and show that this is a general result.
2.4
Main Result
Different topologies can be defined on type spaces. The product topology, is
defined in the following way: a sequence of types tn converges to tj in the product
topology if and only if the m-th order beliefs of tn converge to the m-th order beliefs
of t for all m. This is a reasonable topology, but as Rubinstein (1989) showed,
6
Thornas et al. (2014) performed an intriguing experiment where it seems that even low
levels (two to three) of mutual belief are enough to generate behavior consistent with common
knowledge.
90
convergence in the product topology does not imply that the types behave similarly.
To address this issue, Dekel et al. (2006) defined the strategic topology. I use
the equivalent definition by Chen et al. (2014), where the strategic topology is
defined as the topology of the strategic converge, which is defined as follows.
Definition 9. A sequence of types t' converges strategically to t* if for every game
G and every action ai G Aj, the following are equivalent:
1. ai is rationalizablefor tj in G
2. for every c > 0, there exists n4 such that for every n > nG, ai is erationalizablefor t7 in G.
Intuitively, types are close in the strategic topology when they behave similarly. We can make this even more strict, by imposing that nG not depend on the
game (i.e. n = nE for all games): that is the definition of the uniform strategic
topology.
In the Mertens-Zamir universal type space, convergence in the uniform strategic
topology implies convergence in the strategic topology, but the converse is not
true. Moreover, convergence in the strategic topology implies convergence in the
product topology, but the converse is not true. Therefore, in general, these are
three different topologies. Theorem 13 below, which is the main epistemological
result, states that, restricted to cognitive type spaces, the product topology and
the uniform strategic topology coincide. This implies that, unlike in the general
91
case (Weinstein and Yildiz (2007)), restricted to cognitive type spaces, behavior is
continuous in beliefs.
Theorem 13. Let T be a cognitive type space with bound B < oc. The product
.
topology, strategic topology and uniform strategic topology coincide on T
The proof of Theorem 13 uses the fact that, because types in cognitive type
spaces have restrictions on the complexity of their beliefs, if their low order beliefs are similar enough, so too must their higher order beliefs be similar. This in
turn implies that cognitive type spaces are well behaved. Consider the following
situation: The researcher thinks that a player (Alice) is well modeled as a type t.
However, because of small errors of perception done by the researcher, she allows
that Alice might actually be of type tE, were t, -4 t in the product topology, as
e -+ 0. This means that, as e -+ 0, types tE have hierarchies of beliefs that coincide with t up to increasing orders of beliefs. If the researcher is confident that
t, E T for some cognitive type space T and for small c, then Alice's rationalizable
actions, as considered by the researcher, will be very close to the actual rationalizable actions, i.e. they are c-rationalizable. That is, if the researcher thinks that
Alice's true type belongs to a cognitive type space with some margin of error, then
the actual type t will behave very similarly. Even more, it will be well approximated uniformly for all games. This stands in contrast to the usual results for the
Mertens-Zamir universal type space - Weinstein and Yildiz (2007) - where types
that are close in the product topology might behave very differently.
92
A fundamental restriction for the validity of the result is that the cognitive
type space T must have a finite bound B < oo. Otherwise, Theorem 13 does not
hold: we can construct a sequence of types tT in the Email game with m = k
so that, by Theorem 13, each of them has N as the unique rationalizable action. However, in the limit m = k = o0, players have common knowledge that
the game is a coordination game, and therefore both A and N are rationalizable.
This lack of continuity (in particular, of lower-hemicontinuity) cannot be avoided
if we allow players of unbounded k because, if we want to have continuity of behavior, we need that the tail of the beliefs be determined by the lower order beliefs.
2.5
Conclusion
Level-k and Cognitive Hierarchy models impose two restrictions simultaneously:
agents with higher level have more complex beliefs as well as more computational
power. Building on Strzalecki (2014), I develop cognitive type spaces where all
players have the same computational capacity, and where the level k of a player
determines the complexity of her beliefs. In particular, players are allowed to
believe others have the same level if they are "like themselves", an assumption inspired by the psychological phenomenon known as the false consensus effect. This
allows us to use the standard techniques of game theory for games of incomplete
information to analyze cognitive type spaces.
93
In the well-known Email game I showed that when players receive few messages, they behave as predicted by standard incomplete information game theory,
due to a contagion process similar to that in Rubinstein (1989). On the contrary,
when they receive enough messages, players behave as if there was complete information. This is appealing because it corresponds to the intuition that if players
observe a large amount of messages (say, thousands), then they would interpret
this as proof that the underlying game is common knowledge. The model offers a
formal way to express this intuition.
As it turns out, the intuition from the Email game is quite general. The main
theoretical result shows that in cognitive type spaces, the behavior of players is
continuous in their beliefs. If we, as researchers, infer agents' beliefs with small
errors, then we can predict agents' behavior with a small margin of error. Formally, restricted to cognitive type spaces, the product topology and the uniform
strategic topology coincide. This is remarkable because, as Weinstein and Yildiz
(2007) showed, for general type spaces the opposite happens: two players with
similar beliefs can have different behavior. By including the assumption that players' beliefs have a bounded complexity (and are generated recursively due to the
false consensus effect) it is possible to generate robust predictions.
Conceptually, this paper gives a step in closing the gap between models of
bounded rationality (in particular, Level-k and Cognitive Hierarchy models) and
94
standard game theory. Models of bounded rationality have been usually analyzed
outside of the scope of standard game theory. Incomplete information games, on
the other hand, assume too much rationality and common knowledge on the part
of the players, and can benefit from adding more realistic assumptions. My model
stands in the middle ground between these literatures, and shows how including
psychologically grounded assumptions into game theory can improve the accuracy
of our predictions.
Appendix
Universal Cognitive Type Space
I start by defining a universal cognitive type space, based on Strzalecki's. Define Zik
recursively in the following way, where ; should be interpreted as the probability
the other player is "like myself" (i.e. a clone):
Z,9 = {O} x A(E x{1}),
Zj = {1} x A(
Zj
x
Zoi U1}),
= {2} x A(e x ZojUZ 1 U{,}),
Z
Definition 10. Given tj
= {B} x
a
dX Ut - Z= U
ei})
(pti , ktj) and t _i = (ly,_, kt-_J, we define the relation
95
ti = ti
if kt, = kt
and pt, = pt_,.7
Definition 11. The Universal Cognitive Type Space (with bound B) CB iS given
by
{E, CP, kP, VP} iEI, where
B
cB=
Uzi.
k=O
Given t = (kt,, pit,) E C, we define kB(t,) = kt, and v0 as the measure that
extends the following pre-measure:
{it
(0, E )
vi (0, Elti)
i
if for all ti E E,tUi 0 tj and kB(tB
0
for all measurable E c CB , and if E n F = 0, vp(O, E U F ti)
k(ti)
)
= V.
(0, Elti)
+
pt(0,,S) if E = {ti} and t _ ti
if ki(Lt_) < kr(t ) for all ti E E,
71v (0, FIti).
The intuition for the definition is the following. A type of level k has beliefs
over types of level 0, 1, .
,
k - 1, i.e. beliefs over
Uk-1 Z'; and also
beliefs about
the other player being like herself, which I have denoted by 4. Hence, for each
possible state of the world 0, a player of level k has as beliefs a distribution on
Uk-J Z' U {
}, which includes all possible types with a lower level, as well as "her-
self".
7
Note that Zi, = Zki for all k, and therefore when k,= k=,, both distributions p, and pt
are over the same set, namely E x U-I
U {}.
96
PROOF OF PROPOSITION 11. Let CB
=
{E, Cf,
kr, vp, B}icE. For t E T, let
t~, and define 4 as follows. First, k,(t) = ki(ti). Then, define p4 recursively
as follows. If ki(ti) = 0, then
sti) (0, )= margev(|ti).
If ki(ti) > 0, then p/e(t,) is defined as the measure that extends the following
pre-measure:
J
[,(t)(0,E)
margqvj(jtj)
if E = {,
vi(0,?_-,(E)Iti)
for E such that t-i E E
0
if for all t-i E E, t-i /C tj and k_(t-j) ;> ki(ti).
==>
k-Mt0 < kj(tj),
for all measurable E C CB*, and if EnF = 0, p(ti)(9, EUFjtj) = p4 (ti)(0, Elti)+
A0(t) (0, FIti). Note that this is well defined, because tj assigns probability only to
her clones, or to types of lower cognitive level (in which case /i has already been defined). Because of the way we have defined p
o. is given by v (0, EI0i(ti))
v),
=
poi (4i)(0, E).
To prove that hi(ti)
of the hierarchy. For m
=
=
hi(0i(ti)), we will prove it by induction on the order
1, the marginals over
E
of t2 and Oi (ti) coincide by
definition. For m + 1 > 1, we want to prove that irr+'(tj) = -Rm+1(0i(ti)), where
7r, : T -+ Tm, *ir : C -+ T", and
iri
and -ri are defined as follows:
97
7r,+1(0, EIti)
r'+1 (0, ELI
=
vi (, (r"') 1(E)Iti).
i(ti)) = f(0, (ir") -(E)I
0i(ti)).
We will first show that -k'+' and 7r'+1 coincide on the sets such that t*
(*7r")- 1 (E). For t*_
(7rm.)-'(E), we have
-rf+1(0,E=i(ti))
=
='(=, (Tr')-1(E)|@0(t))
vi(O, 0 ((ir"L)~(E))It) =v(0, (r)- (E)It) = 7r4m+'(0, E t),
where the second identity is the definition of vP, and the third identity holds
by the induction hypothesis (i.e.
7rm =
jjf
o ob).
On the other hand, when
1
(7rm)-'(E) = {t*}, by the induction hypothesis we have that (ri)- (E) =
V/'j (t* ). Therefore
77+1 (0,Elti)
=
=
v (0, (7r")- 1 (E)Iti) = vj(0, t*ijti) = VB(0,
(0 , 4_i (t*_)j'(ti)) =
Moreover,
7rWm+1
(r") 1-'(,
(E)I i(ti)) = kr'+1(, EI40i(ti)).
and ftm1 agree trivially on sets such that (7r-)-1 (E) contains
types t-i with k-i(t-i) > ki(ti) and t
7r+1 (0,
('jg(ti)) =
9, ti, since
EIti) = vi (0, (F7r")- 1(E)jtj) =
(0=
(- ( 1(E)|(ti)) = fr+1(0, Ei (ti)).
98
Note that vi and vg are o--finite, because their values are in the interval [0, 11.
Therefore, by Caratheodory's Extension Theorem, if they coincide as pre-measures,
they must coincide as measures (because their extension is unique). Because of
that, we have that 7r(+1(9, Ejtj) = Frm+ 1(9,EjI(tj)), what concludes the proof,
since
Proof of Theorem 13.
I will use /i to denote beliefs of types in the Mertens-Zamir Universal type space
T*. For a type tj = (kt1 , pti) E Cf, we will use the notation /pi(0, EIti) := fit, (0, E).
The Prohorov distance is defined in T in the following way.i Let d -= 0, and
for each m > 0, and each si, tj E T* define:
dk+ 1 (s, t) = inf {6> 0:
#3(,Ejtj)< Ol(O,E k .s)
+ 6 for each measurable E C T
where
E~k
= {si E T-i : dk;(s_i, t_,) < J for some t-j such that t-i C E}.
Sequence tV converges to tV in the product topology, if d' (ti, t) -+ 0 for all
8
See Chen et al. (2014) for details on the Prohorov distance.
99
m. We can also define the following distance:
dw (si,ti) = sup di (si,ti).
Then, t7 converges to t* in the uniform weak topology if d"v(ti, t )
-*
0.
Note that the uniform weak topology is finer that the product topology.
In a spirit analogous to the Prohorov metric on T, we can define a distance
Df on each of the Z in the following way. Let D9 be given by the distance of the
supremum on Zi, that is, for si, tj E Zi,
DO(sj, ti) =sup 1pst(O, g-ti) - Pi(0, -;Isi)
For k > 1, let pk be defined on Z-
I
DI
pl (siti) =
1
(si,ti)
.
0e8
1
U {} by
if Si, tj E Z
k
if si
=
0
if si
=
Then, define D+1l on Z1
t or si
=
=
ti.
as
D&+1(s , ti) = inf {6 > 0 : pi(0, EIti) < pi(0, E~k
where
100
Isi) + 6 for each measurable E C Uk OZi U
Ek
=
{s_ EZi, U {} :pi(sti) < 6 for some tj such that t-i EE}.
Important observation.
Note that, by definition, the Mertens-Zamir uni-
versal type space is identical for both players: T* = T*i =
f1l1 T", where T"
is defined in Equation 2.1. This means that if a given type xi E Ti*, then it is also
the case that xi E Tj. Moreover, given any F c Ti*, then Fjr
= F
c T*j. In
the proofs below, I will use these facts several times throughout.
Lemma 14. Let k E N.
On Zk, convergence in D
implies convergence in the
uniform weak topology, that is, for all c > 0, there exists a 6 > 0, such that for all
E c Z?, measurable,
hi(E') C hi(E) j
for all m > 0.
Proof. The proof will be by "double induction", on the level k and on the order of
belief m. We define functions hi as the restriction of hi to
can always take
6
U1= 0 Z. Note that we
< e, as we will use that fact repeatedly.
Case k = 0. We will prove it by induction on m.
9
Recall that hi maps cognitive types to the Mertens-Zamir universal type space Ti*.
101
Case m = 1. Recall that for xi, yj E Ti*, d! is defined as:
d'(xi, yi)
=
inf {6> 0 : 3#(O, Fjyj) < #3(O,Fdo
Ixi)
+ 6 for each measurable F C T*
.
*
But F . = T*. for all 6 > 0, since d&i - 0. Because
#i3(0, T*j 1xi) = marge 4(01xi),
the condition in the definition of d. becomes
0i3(0, FIyi) < marge3i(0|xi) + 6.
Moreover, this condition must hold for all measurable F C T*., including
F = T*i. Hence,
d'(xi, yi) = inf { 6> 0 : marge i(01yi) < marge /3(Ojxj) + 6}.
Recall that for si, tj eEZi,
D'(si,ti) = sup jpj(0, 4si) - pj(O, gjtj)j.
For tj E Zi, we have that bLi(,,jti) = marge /3(9jhj(tj)). Therefore
D'(si, ti) < c
=
Imarge O3(Ohj(tj))-marge O3(Ohj(sj))| < c
102
=>
dj(hi(si), hi(ti)) < E.
* Induction step on m. Suppose that the induction hypothesis is true up to m,
and we want to prove it for m+1. Note that, for t E Zi, tj only puts positive
probability on t-i whenever hi(ti) = h-i(t-i). But, because T* = T*., if such
t-i belongs to a set F c T%, then tj E h 1 (F). Therefore:
S(0, FI hi (ti)) =
marge pi (0|1ti)
0
if tj E hi1 (F)
otherwise.
If tj V h71(F), it is trivially true that 0 = marge 8i3(01hi(ti)) <; marge (01hi (si))+
e. Otherwise, suppose that tj E h71 (F), and hence hi(ti) E F. By the induction hypothesis, for c > 0, there exists a 6 > 0, with e > 6, such that
D9(si, ti) < 6 implies dT (hi(ti), hi(si)) < c, and hence si E h- 1 (Fdrn).
Ol(0, Flhi(ti)) = marge pit(Oti) < marge pi(Olsi)+E =
/(0,
Fdm|hi(si))+c,
where the equalities are by definition, and the inequality comes from D9(si, ti) <
6 < e. Therefore dfl+l(hi(si), hi(ti)) < E, as we wanted to prove.
Induction step on k . The induction hypothesis is that the lemma is true
for up to k, and we want to prove it for k + 1. We will do it by induction on m.
* Case m = 1. Convergence in Dk+1 implies that the marginals over
converge, since
103
E
also
D'+1 (si, ti) < E ->Cti(O, Elti) < pi(O,
Ek
p-i
Isi)
+ e.
In particular, this is true for E = UkUO Z1_ U {}, in which case E =
E%+1,
ap-i
and
<
'ZIU
{}OIsi)
0
)
{}ti
L=0
+ E.
1=0
Taking into account that pi (0, UfkO ZL U{ } ti)
=
marge [ (01ti)= marge i3(01hi(ti)),
we have that
marge 3(Ojhj(tj)) < marge #3(O|hj(sj))+
6,
and so dj(hi(si), hi(ti)) < E.
Induction step on m. Suppose that the induction hypothesis is true up to
M, and we want to prove it for m + 1, that is, we want to show:
hi(Ek+l) C hi(E)-
=> hi(Ej k
l)
C hi(E)Emsi,
or equivalently that if Dk+1(s, ti) < 6, then d'+1(hi(si),hi(ti)) < E.
The induction hypothesis on k implies that for i E I, F c Tj* and for all m:
104
hk((hk)-1(F)") C
h ((h)-1 (F))"j
c
F
,
(2.2)
were the first inclusion comes from the induction hypothesis, and the last
one is a universal property of functions. Let si, ti E Z +1 , and F c T*.
Again, note that T* = T*j, and therefore if t_, is such that h-i(t-i) = hj(tj),
and h-i(t-i) C F, then hi(ti) E F.
- If hi(ti)
F, then (hk_i)-
1 (F)k+l
= (hki)-'(F)6 k . Let E:= (hkk)- 1(F).
Taking 6 < E:
3i3(0, Fhi(ti)) = pi(0, EIti) < Pi i(0, Ek |s) + 6 =
= 3i(0, hk-j(Egk)jhj(sj)) + 6 < Oj(0, FdT hi (si)) + c,
(2.3)
where the last inequality is a consequence of equation 2.2. Therefore,
di+1(hi (si), hi(ti)) < E.
- If hi(ti) E F, then by the induction hypothesis, for si we have that
Dk+ 1 (si, t,) < 6 =- h (si) E Fj, = F6 .
(2.4)
Let E := (hk )-(F). Rewriting equation 2.2:
hk~i (E
105
)
c F-,,
(2.5)
Taking equations 2.4 and 2.5 together, we have
i
U {hi(si)} C Fj ,
(Ek)
(2.6)
Hence
/3(OFjhj(tj)) = i((OEU {}Itj) < pi(O, [EU { }]k
= p1(O, Ek U{}|sj)+6
1si)
+6 =
3(O, h i(E(Uhisi)}Ihi(si))O+6 < 0i(0, Fd) +,
=
where the last inequality comes from equation 2.6 and the fact that
6 < 6. Therefore, dr3+1(hi(si),hi(ti)) <6.
.
Lemma 15. Zik is compact for all k > 0 in the topology associated to D
Proof. First, note that if X is compact and metric, then A(X) is compact and
metric with the Prohorov distance - Kets (2014). The proof will be by induction
on k.
E x {} is clearly compact,
as
e is finite.
Hence, Zi = {0} x A(E x {g}) is
compact, what proves the k = 0 case.
Now, if Zl is compact for 1 = 0, ...
compact sets are compact).
,k,
Therefore, Z+
compact.
106
so is U1oZf U
1
=
{k
{g}
(finite unions of
+ 1} x A(Uk OZki U
{})
is
Lemma 16. If tV E Cp, and h (t, ) -+ t in the product topology, then hi(t() -+ t!
in the uniform strategic topology.
Proof. Because k < B, there must exists a finite set K such that only the Zk with
k E K contain infinitely many elements of the sequence tV. Take an arbitrary
k E K, and consider the subsequence
tn"
of tn whose elements belong to Zk. By
Lemma 15, Zk is a metric and compact space in the distance Df. Hence
tn"
has
a convergent subsequence tn", in the topology associated to D , that converges
to an element i E Zk. By Lemma 14, convergence in D implies convergence the
uniform weak topology, hence hi (tn",)
-+
hi(ti) in dJw.
We now want to prove that hi(ti) = t*. Suppose not: there would exist a belief
order j C N such that 7ri(t*) # iri(hi(ii)), and so we would have that ri (hi(t!))
and 7ri (hi (tn",)) have different limits. However, because Tj is a metric space in the
Prohorov distance,' 0 a convergent sequence cannot have a convergent subsequence
with a different limit, thus obtaining a contradiction. Therefore, t* = hi(ti). Moreover, because by Lemma 14 we have hi(tn",)
-,
hi(fi) in dVw, then hi(tn,) +t
in dVw.
Note that since the k E K was arbitrary, we can repeat the previous reasoning
for all k E K: there exists a subsequence t!(k) C Zk of tV such that hi(t(k))
converges in the uniform weak topology to tI. Because there are only finitely
10 -i is defined in Equation 2.1.
107
many elements of tV that do not belong to UkEK Z, and IC is finite, the fact
that all subsequences hi (tn (k)) converge weak uniformly to t* implies that hi(tO)
converges weak uniformly to t*: for each e > 0 choose n, as the maximum of the
n such that to E
Ut Z.
and dV' (t;, hi(t(k))) < E for all k E IC. This proves
that convergence in the product topology implies convergence in the uniform weak
topology. Finally, the weak uniform topology and the uniform strategic topology
are equivalent, Chen et al. (2014).
D
PROOF OF THEOREM 13. Let Oi be the functions that embed T into CF; from
Proposition 11 we know that hi/(V(tn)) = hi(tO) for all n. Therefore, hj((ti))
-+
t* in the product topology. Because sequence ij(t?) belongs to CF, by Lemma 16
hj(t?) = hi(oi(t?)) -* t* in the uniform strategic topology. Therefore, convergence
in the product topology implies convergence in the uniform strategic topology;
because the latter is weakly finer, they must coincide.
D
Email game
PROOF OF THEOREM 12. The proof of the theorem is by induction, along the
lines of Strzalecki (2014). First, notice that for all types with m = 0 the only
rationalizable action is Not attack, irrespectively of k. This is so because t' is
certain the enemy is strong, and therefore Attack is dominated by Not attack; to
puts probability 1/2 on an enemy of each type, and therefore for this type Attack
is also dominated by Not Attack.
108
Next, consider the induction step for all players with k > 1 (we will prove the
case k = 0 below). Player 1 who sent m messages and is of level k believes that
Player 2 is of level k - 1. Therefore, by the induction hypothesis, type t~-1 will
have Not attack as the unique rationalizable action whenever
whenever k+1
<
-
<
M-1, that is,
m. But in that case, because t' puts probability 1/(2 - e) on t',
Not attack is also her unique rationalizable action. On the other hand, whenever
< m -1 (i.e.
k+1
< m), by the induction hypothesis player 2 has both actions
as rationalizable, and therefore so does tm. The proof for Player 2 is analogous.
Finally, types with k = 0 always have both actions as rationalizable whenever
they send at least 1 message, because in that case they believe there is common
certainty the enemy is weak. This concludes the proof.
109
110
Chapter 3
Social Pressure in Networks Induces
Public Good Provision
3.1
Introduction
Why does collective action take place? As Samuelson (1954) observed, given that
collective action and public goods are by definition non-excludable, most individuals have no incentive to participate. One possibility is that people participate out
of a fear of social sanctions. As (Silbey, 2005, p. 115-6) points out, large numbers
of young British enlisted in WWI out of social pressure. However, this possibility
has been largely neglected on theoretical studies of public good provision,' despite
the abundant evidence of its importance. If social pressure is effective in enlisting
people to go to war, which has an enormous private cost, why would it be any less
'An exception is Miguel and Gugerty (2005).
111
effective in the provision of other public goods? Indeed, as the empirical literature
in voting makes clear (Gerber et al., 2008; Funk, 2010; DellaVigna et al., 2014)
social pressure plays a large role in inducing people to vote.
We propose a model that is explicitly based on social pressure: individuals are
embedded in a network, and they receive disutility when their friends contribute to
the public good but they do not. Agents face a decision of whether to contribute
to a public good: the benefit b is public, but the cost c is private and larger than
the benefit, c > b. At the beginning of the game, nobody is contributing to the
public good; the game is dynamic, and every period the game ends with constant
probability. If it continues, an agent is selected randomly to revise her strategy.
This simple model captures two interesting features. The first one is that whenever an agent i contributes to the public good, there is an immediate contagion
in the social network. Initially, friends of i find optimal to contribute, as they
do not want to suffer social pressure. Crucially, they cannot revise their strategy until they are randomly chosen to do so. Friends of friends of i know that
friends of i will contribute as soon as they are able to revise their strategy. They
measure their potential expected disutility from social pressure (which increases in
the probability that the game continues), and they compare it with their certain
cost of contributing to the public good. If the former is greater than the latter,
they also decide to contribute whenever they get to revise their strategy. But then
112
agents who are at distance 3 in the network (that is, friends of friends of friends
of i) will make the same reasoning, and so will the rest of agents in the network,
generating a contagion in best responses by which most individuals in the network
end up contributing.
We call the first individual to contribute the leader, and we show that under
some conditions agents will decide to become leaders, generating a large contagion
and hence a large contribution in the network as an outcome. One such conditions
happens when the contagion generated by an agent i is so large that the expected
benefit from inducing others to contribute is larger than the private cost of contributing. The contribution to the public good by the leader herself is negligible,
but because she manages to spread the contagion to a large fraction of the population, she has incentives to contribute even when nobody has contributed before. It
can also happen that, when several agents are afraid of social sanctions by others,
those agents contribute (thereby punishing those who do not contribute); therefore
the fear of social sanctions becomes self-fulfilling.
Consider the following example. Rosa Parks became a spearhead of the civil
rights movement in the United States when, in 1955, she refused to give up her
seat on a bus to a white person. Under the law of Mongtomery (Alabama), black
people had to let their seats in a bus for white people when the "white section"
was full. When Rosa Parks refused to give up her seat, the bus driver called
113
the police, who arrested her. This event sparked a movement that eventually led
to major advances in civil rights in the United States: "the community outrage
around [Park's] arrest was rooted in her long history and political activism and
their trust in her", Theoharis (2013). There is also an important theoretical reason
why social pressure might play a role in collective action: otherwise, individuals
would need to obtain utility directly from participating in the revolts. 2 However,
there is one problem with that explanation. While the benefit of the the revolt
is a public good, each individual must bear a private cost of participation, and
there is no reason why individuals would be willing to participate in collective
action, as made clear by Samuelson (1954). In contrast, social pressure explains
why individuals have incentives to participate in collective action, even when the
benefit is public.
This paper is related to several literatures. In terms of the motivation, the
most relevant literature is that of revolutions and protests, modeled as collective
action. Granovetter (1973); Palfrey and Rosenthal (1984); Kuran (1987); Chwe
(2001); Medina (2007); Jimenez-Gomez (2014) all model collective action in dif-
ferent ways, but they have one common assumption: individuals obtain private
utility from participating in collective action. Hence, contrary to our paper, they
assume that b > c (at least under certain conditions).
They argue that this is
the case because individuals can loot or get a share of the benefits from revolting.
2
This is actually the case in most models of collective action, like Granovetter (1973); Palfrey
and Rosenthal (1984); Kuran (1987); Chwe (2001); Medina (2007).
114
For example, Kuran (1987) assumes that people obtain disutility from expressing
beliefs that are not consistent with their true beliefs, and hence when a dictatorial
regime begins to fall, more and more people express their true preferences, creating a wave of protest and bringing the regime down. However, his model equates
preference expression with political participation, which is an extreme simplification. Participating in a demonstration which is under gun fire is not the same
as expressing rejection of the regime within a small circle of friends. It has been
productive to assume that benefits from revolting are private, but we need to go
one step further if we are to to explain why people participate in very risky actions, which are not going to bring a substantial private outcome. One of the
contributions of this paper is to provide a mechanism that would give individuals
the incentive to contribute, even if the final outcome is a public good. McAdam
(1986) argues that the benefit b comes from the mobilization of personal ties in
preexisting social networks. By explicitely separating the public benefit b from the
utility of mobilizing (or keeping) personal ties, our paper makes clear when is it
that social pressure is effective in generating collective action.
The literature on public good provision has tried to analyze how different characteristics of the population affect the level of public good provided in equilibrium.
Alesina et al. (1999); Alesina and La Ferrara (2000); Vigdor (2004) analyze the
relationship between public good provision and ethnic, racial and socioeconomic
heterogeneity. Their results show that heterogeneity in the mentioned character115
istics is associated with lower public good provision. Even though our model does
not include heterogeneity along those dimensions, we believe that it points to a
causal mechanism: public good provision would indeed be lower if heterogeneity
across ethnicity, socioeconomic status, etc. generates social norms with less sanctions, or makes it more likely that the network is fragmented in cliques that do
not interact with each other (Example 7). Each of those cases would make social
pressure not operational, and hence public good provision to decline.
Another related literature is that on social capital. Miguel and Gugerty (2005)
have a model of public good provision with social sanctions, but unlike in our
model, social sanctions are a function of the share of contributors in the whole
population. Because of that, their model exhibits multiplicity of equilibria and
a threshold such that once there are enough contributors, everyone find optimal
to contribute. Our contribution here is to have a dynamic model where social
pressure is local, what generates a situation where contagion of contribution can
happen even when everybody starts not contributing. In support of this intuition,
Besley et al. (1993); Besley and Coate (1995) have documented the importance
of social pressure and social interactions in the context of savings associations.
On the theoretical side, Karlan et al. (2009) consider a model where individuals
that are not directly connected in the social network can nevertheless use links
with other people as "collateral". Our paper is similar in spirit to the idea that a
person can use her links in the social network as valuable assets. In our model,
116
by contributing to the public good, an agent credibly commits to "destroy shared
links" with non-contributors, hence creating incentives for others to contribute.
Our paper is also related to the literature in "leadership". Quoting from Hermalin (2012): "Leadership should be seen as a phenomenon distinct from authority
or exercise of some office or title. The defining feature of leadership is that a leader
is someone with voluntary followers. A central question is, therefore, why do they
follow? What is it, then, that leaders do that make followers want to follow?". The
question of leadership is rich and complex, and there is a large literature that has
addressed it in the context of teams (see Hermalin (2012) for a survey). Precisely
because of this complexity, there are only a handful of papers that have consider
leadership in a larger setting, where a single individual can tip the equilibrium of
the whole society. For example, in a paper of technology/behavior diffusion, Morris (2000) takes the innovators as exogenous. But why would those individuals
become innovators in the first place? Like us, Ellison (1997) considers the case in
which a single player can generate contagion to the whole population. In his case
a single "rational guy" can sometimes make the rest of the (myopic) players change
to a Pareto-superior equilibrium. Corsetti et al. (2004) also consider the importance of a single player. Both of these papers however take leaders as exogenous.
Acemoglu and Jackson (2012) consider a case where an individual can change a
social norm, and that social norm will stick to at least some future generations.
Leaders in their model are endogenous, but only a special class of individuals can
117
become leaders. A contribution of our paper is to analyze how and when an individual becomes a leader endogenously.
Finally, our paper is related to the literature in social dynamics: Kandori et al.
(1993); Young (1993); Morris (2000) among others. Unlike the previous literature,
we consider fully rational and forward-looking agents. This makes our paper different in several features that we have mentioned above, mainly: contagion is fast,
and it is started by leaders who do so in a calculated manner. A remarkable branch
of this literature, started by Matsui and Matsuyana (1995), and recetenly continued by Matsui and Oyaina (2006); Oyama et al. (2008, 2014), considers agents
who are forward-looking and can only revise their strategy as a Poisson process.
However, unlike in our model, they consider a setting in which the action of any
single agent does not affect the payoff for the rest of the population. Because of
that, there is no contagion or leadership considerations.
The rest of the paper is organized as follows. Section 3.2 introduces the model.
Section 3.3 analyzes the contagion that takes place in the network, once people
have started contributing. Section 3.4 analyzes under which conditions individuals
become leaders by contributing when nobody has contributed before. Section 3.5
analyzes the game when there is bounded visibility in the network. Section 3.6
concludes. An Appendix provides proofs and details.
118
3.2
The model
There are is a finite number n of agents in the society, represented by set N. The
structure of society is given by a friendship network g E {0, 1}NxN, where gij = 1
if agents i and j are connected, and gij = 0 otherwise.3 The network is undirected
(gij = gji), that is, friendship is reciprocal. Let Ni denote the set of i's friends:
Ni = {j E N : gij = 11.
Each agent faces a binary choice: to contribute or not to contribute to the public good. Time is discrete, t = 1,2,..., and each period only one agent (selected
at random) can revise her strategy. Therefore, every agent must play the same
strategy in every period until she has the opportunity to revise. The game can be
summarized every period by a state S c {0, 1 }N, where si denotes the action last
chosen by agent i, and si = 1 means the agent contributed.
Initially S = { 0 }N, i.e. every agent starts the game not contributing.
At
the beginning of each period, the game ends with exogenous probability 1 - q, in
which case payoffs are realized. If the game continues, an agent i E N is selected
at random to play, from a i.i.d. uniform distribution. Agent i chooses an action
3
Throughout the paper we use the terms friends instead of the more common terminology
neighbors for individuals who are connected.
119
from Ai(S), where
Ai(S) =
0,
{0,1}
if si
{1}
if si = 1.
=
This means that once an agent has contributed, she must stick to that action
for the rest of the game. We can think of this as the fact that once an individual has
made a payment to the government, or participated in a violent demonstration, it
is extremely difficult or impossible to undo such action. 4 Given this structure, the
game is such that agents play one at a time, selected at random with replacement
from the set N, until the game ends. When the game ends and the state is S,
payoffs are realized, according to the following utility function:
ui(S)-=b
E
sk- 2/(1- si) J
si,
jENi
(kcN
-csi
(3.1)
Agent i derives utility b from each agent who contributed and pays the private
cost c if she herself contributed. Moreover, agent i suffers social pressure whenever she does not contribute to the public good: in that case she incurs disutility
bVY for each friend who contributed. 5 To make this a game of public good provision, we assume that the private cost is larger than the benefit an agent enjoys by
4
In terms of the model, this prevents the usual punishment strategies in repeated games to
play a role. For example, players cannot play a grim trigger strategy, since they cannot undo
their contributions.
5This assumes that agents value relationships "per se", for some emotional
reason and not
for example because having friends allows and individual to be connected to other agents in the
network. This can be partially justified by the literature in psychology and behavioral economics
Gintis et al. (2003); Fehr and Gachter (2002); de Quervain et al. (2004); Giirerk et al. (2006).
Despite this evidencc-, it can also be considered as a "reduced form" utility from a more general
model, where having friends signals some underlying disposition, like being honest.
120
contributing.
Assumption 5. c > b.
With no social pressure (
=
), it is easy to see that nobody would ever con-
tribute in a subgame perfect equilibrium (SPE). Indeed, suppose that n -1
agents
have contributed, and the last agent has to decide whether to contribute. Because
all other agents need to contribute in the future, there is no reputation loss for
not contributing. Because c > b, she will choose not to. But that means that
previous actions of other agents cannot change what the last agent will do, and
so the second-to-last agent does not contribute either. By induction we find that
no agent will ever contribute. The key ingredient in the model that will generate
contribution is social pressure.6
In order to simplify the analysis, we assume that the cost of contributing is
smaller than the punishment by a single friend. This ensures that whenever a
friend of agent i contributes, it is optimal for agent i to contribute as well.
Assumption 6. b4 > c.
Moreover, given the probability q that the game continues, we can define
Q
as the probability that a given agent i will be able to play before the game ends.
'Social pressure and social sanctions can be modeled in various ways. In the Introduction
we discussed Miguel and Gugerty (2005), where social pressure is inflicted presumably by the
collectivity on the deviators, without taking into account the structure of society. We, on the
other hand, view social pressure as derived precisely from the local interactions of agents.
121
Because the game is symmetric, Q is the same for all agents. We will assume that
Q
remains constant for any population size n. If this was not true, the results
would depend artificially on the size of the population, as will become clear in
section 3.3.
Assumption 7. The probability that the game continues q, as a function of n, is
such that Q is constant:
q =
3.3
.(3.2)
Contagion
In this section we will study how the fact that an agent contributes can affect how
others behave in the network, and generate a "wave of contagion" that spreads
contribution by best-response dynamics to a large fraction of the network. A very
similar effect was analyzed in Morris (2000), in the context of myopic agents. The
conditions we will find will be such even when agents are forward-looking, they
have as a dominant strategy to contribute.
3.3.1
Simple Case: An Illustration
Consider the case where a given agent i has contributed to the public good. Because, by Assumption 6 we have bV) > c, the best response for any agent j E Ni
is to contribute.
However, even if j desires to contribute, she still has to wait
122
until she is selected to play. We define agents whose best response is to contribute
but have yet not done so, as predisposed agents. Hence, all friends of i become
predisposed after i contributes.
The next question is: what will an agent k V Ni who is friend of j E Ni do
when she gets a chance to play? If k decides to contribute, she obtains an extra
payoff of b - c +
, where
is a random variable that denotes payoff from further
contagion generated by k (it is not very important to know its particular form as
it will drop out of the equations). However, if k does not contribute, her payoff V
will depend on the different scenarios that could happen:
* When j plays, hence contributing, and k gets to play before the game ends.
In that case k will contribute (because bb > c) obtaining an extra payoff
bounded above by b - c+
"
.7
If k plays before j gets a chance to play, then k faces exactly the same choice,
and hence the extra payoff is simply V
* If j plays, hence contributing, and then k does not get a chance to play
before the game ends, she will incur extra disutility --b0
" Finally, it might happen that the game ends before either k or j get to play,
in which case k receives an extra payoff of 0
7
it is bounded above because k can only generate as much contagion as if it had originally
contributed, hence is an upper bound on the real contagion payoff at this point
123
Therefore,
j
will become predisposed when the certain cost of contributing is
smaller than the expected social punishment; and we have the following sufficient
condition.
Proposition 17. Agent k contributes when she has a predisposed friend j if
C
- < I + #0,
b
(3.3)
where 3 is the probability that j plays before k plays again, and before the game
ends.
The intuition for Equation 3.3 is that the ratio of cost to benefit of contributing
(which is higher than 1 by Assumption 5) cannot be too large in comparison with
the expected social punishment
#30.
If condition 3.3 holds, the best response for
agent k is to contribute, and so she becomes predisposed. But the argument only
uses the fact that k has a predisposed friend, and hence this same argument can
be repeated for every agent in the network who has a predisposed friend. This in
turn implies that after a single agent i contributes, the whole connected component where i is immediately becomes predisposed. 8 The reasoning is as follows.
We already saw that once agent i contributes, agents in Ni also become predisposed.
When equation 3.3 holds, all agents who have a friend in Ni will also
become predisposed. But then all agents who have a friend of a friend in Ni will
also become predisposed, etc. The contagion will spread, and after a single agent
8
A connected component of the network is a subset C of agents such that any agent i E C is
a friend of at least another agent in C.
124
contributes the whole connected component becomes predisposed. Note however
that this argument is not exlusive to the case where Equation 3.3 holds: we turn
now to analyze the more general case.
3.3.2
General Case
So far we only looked at those agents with a predisposed friend, but the same
reasoning can be generalized to the case when an agent k has m friends who are
predisposed (recall that an agent is predisposed if her best response is to contribute
when she has the option to revise her strategy).
Proposition 18. If agent k has m predisposed, she contributes if
1
C
b< 1 +
Q
:h - 0(h, m),
(3.4)
h=1
where 0(h, m) is the probability that h out of m individualsplay before the game
ends and before individual k gets to play.9
Note that the the left-hand side of equation 3.4 does not depend on m. The
distribution 0(., m) is increasing in m in the First Order Stochastic Dominance
(FOSD) order, and hence the right-hand side is increasing in m. 10 Hence there
exists a minimal m* that makes equation 3.4 hold, that is, such that for all m > m*
9
+mI-+1>
0(h, m) = 1
1+
Q(ML
h~
I
the derivation can be found in the Appendix.
10Given two distributions F and G, we say that F is greater than G in the FOSD sense if
F(x) ; G(x) for all x.
125
agents with at least m predisposed friends have as a unique best response to contribute when chosen to revise their strategy, therefore becoming themselves also
predisposed. We call m* the contagion threshold, defined as the minimal m that
makes equation 3.4 hold. Note that rn* is decreasing in b, , and
ing in c. Hence, greater values of b, ' or
Q,
Q, and
increas-
or lower values of c make contagion
more likely. These comparative statics of m* with respect to the parameters of
the model are intuitive: higher benefit, lower cost, or higher social pressure all
induce more contagion. The fact that a higher
Q
(higher probability of revising
before the game ends) induces more contagion is also clear: not only does this
generate more possibilities for agents to contribute, but those possibilities are in
turn anticipated by others, and that will make them also more likely to contribute.
Let's turn now to study how contagion spreads through the network. We use
the notation from Morris (2()00). Let S be a set of agents, we define operator
1H1(S) := S, and for k > 1 we define H
recursively:
17Jr(S) =HIr 1 (S) U fi E N : jNj n rr(S)j
m}.
It is clear that this is an increasing sequence of sets, and we define HO (S)
as the limit of that sequence. We can see then what happens when an agent i
contributes when the contagion threshold is m*. By Assumption 6 bW > c, so
immediately all agents in Ni become predisposed. Then, all agents who have at
least m* friends in Ni also become predisposed, so the contagion spreads to the
126
set H .(Ni). Proceeding recursively, the contagion spreads through the network
and in the end all agents in N. (Ni) are predisposed. This contagion happens in
the same period that agent i contributes, so when next period starts any agent in
N1 . (Ni) will contribute if selected to play. Hence we have the following result.
Proposition 19. When agent i contributes, all agents in H. (Ni) become predisposed.
Proof. We will prove it by induction. By Assumption 6, whenever i contributes all
her friends become predisposed, and because I' .(Ni) = Ni, this proves the case
k = 0. Now, suppose the statement is true for k, what means that all individuals
in
1- k(Ni)
are predisposed. Then, by the definition of m*, all agents with at
least m* friends in the set Hk.(Ni) will also become predisposed, what implies
that individuals in HM'(N*
)
=
fl.(N.) U {i c N : INi
n
I.
1k
;> m*} become
D
predisposed.
We have seen how a contagion of best responses takes place in the network. This
phenomenon is similar to the contagion in Morris (2000); Oyama and Takahashi
(2011), and many other models of contagion, with two remarkable differences. In
this paper, all agents are forward looking, and contagion takes place in anticipation
to the actions of others. This is a contribution with respect to the previous literature, where the individuals are either not forward looking (or only some individuals
are, as in Ellison (1997)), or they are unable to initiate a contagion, Matsui and
Matsuyama (1995). Second, contagion happens in the same period that agent i
127
contributes: because players are forward-looking, contagion is instantaneous. This
is in stark contrast with the papers in the literature of social dynamics, where
contagion is supposed to happen in the long run or as the limit of some updating
process." Interestingly, both aspects of the current model are related. It is precisely because agents are rational and forward-looking that contagion spreads so
fast: everyone anticipates that others will also contribute.
Note that despite the potential for multiple equilibria, agents who are affected
by the contagion have as a dominant strategy to contribute, irrespectively of what
non-predisposed agents do. This comes at the expense of obtaining a lower bound
on the actual contagion on the network.
The conditions we require here are
stronger than necessary for a contagion to happen, but if they do happen, we
can unambiguously claim that contagion takes place. As argued in the Introduction, we are interested in conditions that would generate collective action, hence
the focus on sufficient conditions. Let's analyze how (and if) contagion happens
in a few well-known networks.
Example 6 (Regular m-lattice).
1
Consider that agents belong to Zd, where d is
the dimension of the lattice. Agents are friends if they are next to each otheri.e
gij = 1 if
"See Ellison (2000); Kreindler and Young (2014), for a discussion about the speed of contagion.
"Example from Morris (2000).
128
d
xi-xk =1,
k=1
where x is the k-th coordinate for agent i. In a regular lattice, contagion
does not propagate for m* > 1. The reason is that friends of i are not friends
themselves, i.e. the clustering coefficient is
0.13
Therefore, even if the friends of i
become predisposed, theirfriends will not become predisposed for m* >
].
Example 7 (Cliquish network). Consider network g such that
1. g is composed by a collection of cliques (a clique c is a subset of g, such that
if i E c, then gij = 1 for all j E c). In other words, all agents in a clique are
friends with everybody
else in that clique
2. Cliques are only connected via a single link, as can be observed in the figure
below
In a cliquish network, contagion does not happen for m* > 1. Consider for
example m* = 3. As soon as an agent i contributes, all agents in the clique become
13The clustering coefficient is defined as the number of triads of agents who are linked to each
other, divided by the possible number of triads. Because no three agents are all linked to each
other in a regular lattice, the coefficient is 0.
129
predisposed by Assumption 6. However, because cliques are only connected by a
single link, contagion will not extend from a clique to the next one. Hence, in a
cliquish network, contagion extends to the clique, but no further.
Example 8 (Watts-Strogatz small world network). The Watts-Strogatz model is a
generalizationof the Erdds-Renyi (ER) random, networks, that allows for a greater
degree of clustering. Let n be the number of agents in the network, and 2K be the
mean degree of the network. The model is built in two steps:
1. Initially agents are placed in a ring, where each agent i is connected to the
K agents on each side of her
2. Then, every link is replaced with probability 3 by a random link
When 3 = 1, this corresponds to the ER model. However, for / < 1., this model
has higher clustering than the ER model.
Let's analyze the contagion that takes place in this type of networks, as a function of 3. In order to simplify the analysis, I will assume that with probability
/
the links are destroyed instead of redirected. Therefore, we are finding a lower
bound on the possible contagion. Suppose that agent i is the first to contribute.
By Assumption 6, all friends of i become predisposed. In particular, the K agents
situated at the right side of i in the ring, become predisposed as long as the link
between i and them is preserved. That means that the agent j situated at distance
130
K+1 in the ringfrom i, will become predisposed if at least m* of the agents between
i and j are friends of i, which happens with probability:
KKm
() :=
Note that when
#
(in)(1 -
#)mK-m.
(3.5)
= 0 and K > m*, contagion extends to the whole network
with probability 1. In general, the number of agents that are affected by contagion
on each side of the ring is distributed according to a geometric distribution. The
probability that there are at least n steps of contagion is given by $(3)fl. Therefore,
the expected number of contagion steps that happen in a direction of the ring is
and the total number of expected agents affected by contagion is
example, for m*
=
1 2
For
3 and K = 2, we can observe the number of total agents predis-
posed by contagion in the graph below (note that the Y-axis scale is logarithmic):
Contagion
500
50
-
100-
105
0.2
0.4
0.6
0.8
1.0
Until this point we have considered how contagion spreads when an agent
contributes. In some sense contagion exhibits strategic complements: the more
likely an agent is to contribute, the more likely others contribute because of the
131
fear of social pressure. However, why would an agent contribute in the first place?
By not contributing she could wait and free-ride on others' contributions. What
would then make an agent become a leader? We now turn to analyze this situation
with more detail.
3.4
Leadership
Hermalin (2012) defines a leader as someone with voluntary followers - as opposed
to someone invested with authority, whom people are somewhat forced to obey.
He suggests that leaders serve three main roles: they are judges, experts and coordinators. While the role of judges and experts are without a doubt important for
leadership, we are especially interested in the third role: leaders as coordinators.
There are some situations where the behavior of agents following early adopters of
a technology or behavior might not be optimal, as in the case of herding (Banerjec,
1992). However, very often there are multiplicity of equilibria (as in the case of
different conventions) and the optimal thing is to coordinate on which equilibrium
to choose. Herinalin (1998) recognized that even in this case, leaders might have
an incentive to select one equilibrium over another, potentially misleading her followers. He analyzed when is it possible to lead by example, so that the leader is
invested in the choice that she is advocating. In our model there is no ex-ante
conflict of interests between agents, as they all agree that more of the public good
is better. However, agents can still lead by example, because by contributing they
recruit their friends, and it is possible to generate a snowballing of social conta132
gion, as we saw in the previous section.
Consider the case when nobody has ever contributed so far in the game, and
i gets chosen to play. By contributing she spreads contagion to at least the set
HI0 (Ni), and she hedges herself against social pressure. However, she also incurs
the private cost c, and if the game ends soon, she does not reap the benefits of the
contagion she generated. If, on the other hand, i does not contribute, and someone else (say j) contributed afterwards, then i could obtain the public benefits of
the contagion generated by j, without having paid the cost c (of course, in this
case it could happen that i ends up suffering social pressure, if one of her friends
contributes).
We see that there are two forces in play. One is the contagion, which exhibits
strategic complementarities. But there is also a "volunteer's dilemma" of waiting
for others to contribute in the first place, that exhibits strategic substitutes. Which
force is stronger depends critically on the parameters of the game. Because of these
competing forces, it is difficult to obtain a characterization of the set of equilibria.
This phenomenon seems to happen in any game of strategic complements when it
is modeled as a dynamic game that ends with a certain probability each period.
For example, the literature on currency attacks, like Morris and Shin (1998), has
used global games, a model with strategic complements. In these models there are
no strategic substitutes; however, in real life situations like revolts and currency
133
attacks we can imagine that some individuals want to hold back in order to learn
information or simply not to expose themselves until it is more convenient to do
so, hence the volunteer's dilemma. We believe that this intuition is important in
order to understand collective action as well, and we are not aware of any paper
The concept we will use is that of Subgame Perfect Equilibrium (SPE):
-
analyzing this particular phenomenon.
is an SPE if it is a Nash equilibrium of every subgame; in other words, if whenever
an agent i is chosen to play, she best responds to -_i." There are two main classes
of equilibria which are useful to consider, because they represent the two extremes
of the spectrum. Spearheaded equilibria are such that there is at least one agent
i E N, such that i contributes even if nobody else would ever contribute. Agent
i does not contribute out of social pressure (after all, if she did not contribute,
it could happen that nobody ever would), but because she wants to generate a
contagion in the network, so that many individuals end up contributing, and i can
enjoy the public good generated by such contributions. Because of that, i is truly
a leader, since her reasons to act are to induce others do so so.
At the other extreme, we have social-pressure equilibria, where there is a
set of agents Y such that they all fear that others will contribute, and that if
they do not contribute they will be punished, and so they end up contributing as
well. Therefore, the fear of punishment becomes self-fulfilling, and it is the reason
"More details on SPE can be found in Fudenberg and Tirole (1991).
134
why everybody in Y contributes. In this case, agents in Y are not concerned with
inducing others to contribute; only with avoiding social punishment.
3.4.1
Spearheaded equilibria
We turn now to obtain sufficient conditions for a spearheaded equilibrium to occur.
Notice that the key requisite is that there is at least one individual i who has enough
incentives to contribute, even if nobody else would.
Proposition 20. If there exists an agent i such that
k-1
k-1
E H
< 1 + M=1
1=M
where k
=
- +
Q
,(3.6)
|Hg (Ni) 1, then any SPE is a spearheaded equilibrium.
The intuition behind this result is as follows. The left-hand side of Equation
3.6 is just the usual ratio of cost to benefit, and the right-hand side is the number
of agents who 1) are predisposed by i, and 2) are expected to contribute before
the game ends. When the benefit of mobilizing enough people is greater than the
private cost, i contributes even if nobody else would contribute otherwise - and
therefore in any SPE there must be at least one individual who contributes when
chosen to revise her strategy. The right-hand side of Equation 3.6 is between 1
and k: when Q --+ 0, so that the game ends immediately, only i gets to contribute;
when Q -+ 1, so that the game lasts indefinitely, all agents in Il' (Ni) get to
135
contribute. Therefore, for intermediate values of
viduals in I.
Q,
the expected number of indi-
(Ni) who get to contribute is somewhere between 1 and k: the more
likely the game is to continue, the higher the right hand side of Equation 3.6, and
the more likely a spearheaded equilibrium can exist. Crucially, the leader takes
into account the fact that others will contribute after she does so, and hence the
forward-looking behavior is fundamental to leadership.
As it is clear from Proposition 20 agent i will be more likely to be a leader,
the greater H* (Ni) is, that is, the farther agent i can extend contagion. It would
be interesting to characterize leaders in terms of characteristics usually considered
in the networks literature, like eigenvector centrality, or being a hub. However,
neither of these characteristics fully characterize leadership in our model.
* Hubs (degree centrality). A hub, in networks terminology, is an agent i
that is connected to a disproportionately large number of other agents (and,
when considering infinite networks, such that a positive fraction of agents
in the network are connected to i). It is easy to see why being a hub does
not necessarily imply being a leader. Consider the following example: the
contagion threshold is m* = 2, and node i is connected to a large number
of agents and each j E Ni is connected to only one other agent k, such that
k
Ny for j' E Ni, j' 5 j. In that case, when agent i plays 1, all agents in
Nj become predisposed, but contagion stops there, since by assumption no
agent in the network is connected to two agents in Ni. Hence, H17.(Ni) = Ni.
136
Even if i is connected to a large number of agents, that might not be enough
to have the condition in Proposition 20 hold.
* Eigenvector centrality assigns relative scores to all nodes in the network
recursively, such that if node i is connected to high-scoring node j, then
that connection contributes more to the score of i than equal connections to
low-scoring nodes. In particular, consider a network with a few hubs of the
type described in previous example, connected to each other. Those hubs
will have high eigenvector centrality, yet using the same reasoning, we can
conclude that for each of them IlO (Ni) = N, so they will not be leaders.
3.4.2
Social pressure equilibria: Supermodularity and MPE
Because the condition in Proposition 20 might be too strong to be satisfied in
certain networks, we will look at weaker conditions that still guarantee contribution. In order to do that, we will try to find conditions under which the game
becomes supermodular, that is, such that whenever an agent contributes, everybody is more likely to also contribute. In order to define supermodularity properly,
we need to define Markov Perfect Equilibria. A strategy o- is a Markov strategy
if o- only depends on the state S. Because we have made the assumption that
once a player contributes, she cannot undo her contribution, restricting behavior
to Markov strategies is quite intuitive. A Markov Perfect Equilibrium (MPE) a
is a Subgame Perfect Equilibrium in which players use Markov strategies. We say
that the game
137
" is supermodular if all - MPE are such that a(S) is weakly increasing (with
respect to the lattice order),
" is supermodular given state So if all a MPE are such that
-(S) is weakly
increasing for all S > So.
Matsui and Matsuyama (1995); Matsui and Oyama (2006); Oyama et al. (2008,
2014) consider a similar problem than we do. In their model, agents are forwardlooking and can only revise their strategy as a Poisson process. However, unlike in
our model, they consider a setting with an infinite number of agents, so that the
action of any single agent does not affect the payoff for the rest of the population.
Ovama et al. (2008) prove that in such a setting, the game will be supermodular
whenever the stage game is supermodular. That result does not carry over to our
setting, for the following reason: two agents might be willing to contribute if no
one has already done so, in order to start a contagion process; however once the
contagion has been started, neither of them has incentives to contribute. This
becomes clear in the following example: in a network such that m* = 2, agents i
and
j
can generate a contagion to a very large set of the population (so that it
is worth to become a leader), but i
#
H(Nj) and
j
=
I(Ni). In such a case,
once i becomes a leader and starts contributing, j has no incentives to contribute
anymore: contagion has already been initiated, and she is not likely to suffer from
social pressure.
138
Because the condition in Proposition 20 might be too strong to be satisfied in
certain networks, we will look at conditions that guarantee that there exists at
least one MPE with contribution. Note that proposition 20 does not use social
pressure; we can find weaker conditions by exploiting the fear of social sanctions.
In order to do that, we will find a set of agents Y where they all are willing to
become leaders, and each agent in Y contributes partly because of the fear of social
pressure from the rest of agents in Y. We formalize this intuition in the following
definition.
Definition 12. A set Y is rn-regularif for all i E Y, Y C I
(Ni).
That is, Y is rn-regular if all agents who belong to Y become predisposed
when an agent from Y contributes and the contagion threshold is rn: agents in Y
contribute (even when nobody has done so) because the rest of Y is also willing
to contribute, becoming a self-fulfilling prophecy.
Proposition 21. Let Y be a m*-regular set of agents, such that {b > F[N\Y].
Then, conditional on an agent from Y contributing, the game is supermodular.
Proof. Once agent j E Y contributes, contagion extends to all agents in Y at
least (by the assumption of Y being m*-regular). That means that all agents in Y
become predisposed, and have "contribute" as best response. Hence, to check for
the supermodularity of the game, we need to show that best responses of all players
not in Y are weakly increasing in the state S. But agents not in Y have no incentive
to become leaders, since br[N\Y < c, that is, even if they could predispose the
139
rest of the agents in the network, it would not be worth to contribute.
Hence,
the only reason why they would contribute is because of social pressure, which is
E
weakly increasing in the state.
Proposition 22. If there exists a m*-regular set Y, then there is a SPE where all
agents in Y contribute whenever they play.
When the cost of contribution c for a given agent i is low, and the expected
benefit bF[rL,* (Ni)] is high, then the prediction of the model is that we will have
spearheaded equilibria.
This would be the case for example with voting in a
democracy, where the cost of going to the polls for an individual is relatively low.
However, in the case of civil rights movements, the private cost of participating
in protests and revolutions can be very high. In these cases, it seems plausible
that the conditions in Proposition 20 are not met. However, the conditions in
Proposition 22 might still hold, what means that there might be some SPE where
contribution happens, even in these cases with large private cost. The reasons why
contribution happens sometimes and not some other times are difficult to pinpoint
(as in every case of multiple equilibria), and can very well be a matter of coordination. In Jimenez-Gomez (2014), I analyze the conditions under which players
are able to coordinate in revolting against a regime; however it is necessary to
assume that each individual has an incentive to participate. Therefore, our model
provides some conditions under which it is incentive compatible to participate in
collective action (due to social pressure), and we hope to contribute to this ongoing
fundamental question in the literature.
140
Let's consider now contagion from the point of view of a principal who desires
to prevent contribution from happening (for example, a dictator that wants to
stop a demonstration from happening, or the Montgomery establishment who is
against the bus boycott). The principal can impose an extra cost 6 > 0, only to the
first person to contribute. The intuition for this is that, while it is relatively easy
for a government or an organized minority to retaliate against a single person, it
is very hard to fight against a mass of individuals. For example, Rosa Parks and
her husband suffered greatly as a consequence of her actions: they lost their jobs,
developed health problems, and received hate call and mail persistently - until
they left Montgomery due to this persecution, Theoharis (2009).
The key fact to observe is that 6 affects the cost-to-benefit ratio for the first
agent to contribute, which now becomes
. Note that, for spearheaded equilib-
ria, the left hand side of Equation 3.6 increases in 6 and, for 6 large enough, no
individual finds it optimal to become the first to contribute. If the principal has
to pay an "intimidation cost" 6 per player to which it wants to increase the cost,
this is an inexpensive way to do so because, in equilibrium, the principal does not
even need to pay the cost: the threat of incurring the extra cost 6 is enough to
prevent an agent from becoming a leader.
141
The same holds true for social-pressure equilibria. Note that in Equation 3.4
the left hand side is also increasing in 6, and therefore for high enough 6, we have
that m* increases. For 6 large enough, Y stops being m*-regular, for the new value
of m*. In that case, agents in Y do not contribute, because the threat of social
pressure is not high enough as compared to the private cost c +6. Note that, once
again, the principal only needs the threat of the higher cost c +6, in order to stop
contagion before it even starts.
3.5
Bounded rationality and visibility
So far we have implicitly made a very stark assumption: individuals know the network, and moreover they can perfectly observe the actions taken by others, even
if they are far away in the network. Instead, we could think that there is bounded
visibility: players can only observe the actions taken by those who are at distance
less than a certain threshold k, where the distance is measured as the shortest
path between two players.
A path between agents i and j is an ordered set of agents such that the first
element is i, the last element is j, each element has a link to its successor; the
length of the path is defined as the number of nodes minus 1. The distance
d(i, j) between agent i and j in N is defined as the shortest path between i and j,
and as +oo if not such path exists. Moreover, define the k-neighborhood of player
i as
142
N = {j E N: d(i, j) < k}.
We implement k-bounded visibility by assuming that player i can only observe
actions taken by players in Nk. We can define a generalization of operator H:
given X C N, we define Hm(SjX) := S n X, and for k > 1 we define recursively
the following set:
i (SIX) = fl;-1(SIX) U {i E X: |Ni n Hl'
(SIX)j
> m}.
Proposition 23. When agent i contributes, all agents in fO.(NINjk) become
predisposed.
The intuition behind Proposition 23 is analogous to that of Proposition 19, with
the exception that contagion is restricted to agents who observe i contributing, i.e.
those individuals who are in Nk. As a consequence, we have the following result.
Corollary 24. Contagion with k-bounded visibility is increasing in k.
The proof of Corollary 24 is straightforward because, by definition, if X C Y
then 1r(SIX) c Hr(SIY) for all m and r. Therefore, the more visibility agents
have in the network, the more contagion that happens. This could be one of the
main reasons why governments and elites attempt to restrict the visibility of individuals in networks; for example, during the Arab Spring, Facebook and Twitter
were massively used by protestors, Bruns et al. (2014).
143
Suppose now that E < 1 + 80, so that whenever a friend i becomes predisposed, it is a best response for i to contribute. We say that an individual j has
public visibility if every agent in the network can observe the action taken by
j (independently of the degree k of bounded visibility). In a sense, we should
understand public visibility as a characteristic some individuals have, by virtue of
being famous or well-known, so that even individuals who are far from them in the
social network learn about their actions. When the network g is connected (i.e.
when there exists a path from any agent to any other agent following links in the
network), we have the following result.
Proposition 25. If the network is connected, when an individual with public visibility contributes, all agents contribute.
Proposition 25 offers a potential explanation for why some symbols are so
powerful: even if not all African Americans in Montgomery knew the fine details
of the social network, they all knew that what Parks had done (and knew that this
was common knowledge):1 5 because the public good was presumably very large
for the African American population, it is plausible that condition E < 1+ #
held
in such case, and therefore that a single prominent case could spark widespread
collective action.
"In Jimenez-Gornez (2014) I analyze the implications of individuals having (or not) common
knowledge of events in coordination.
144
3.6
Conclusion
We presented a model of collective action where the agents are forward looking
and have concerns for social sanctions. In the model, agents can only revise their
actions stochastically. We defined an agent as predisposed if she is willing to contribute to collective action whenever she can revise her strategy, and analyzed
a contagion process through the network, by which agents become predisposed
by best response dynamics. In particular, we found conditions under which an
agent with m predisposed friends becomes predisposed herself, and defined the
minimal such m* as the contagion threshold. Contagion in the literature of social
dynamics is usually a slow process, which requires several generations of agents
best-responding; in contrast, contagion in our model happens immediately, due to
the forward-looking nature of agents.
In the second part of the paper we studied leadership: when does an agent
choose to contribute even when nobody has contributed so far. One condition
that suffices for contribution happening in any SPE is that the expected gain from
contributions by those affected by the contagion is larger than the private cost
of contributing. This is the case when a single agent can use her influence on
her friends, who in turn use their influence on theirs, etc. to generate a wave
of contributions in the network, that eventually compensates for the private cost
of contributing in the first place. The forward-looking behavior is critical: it is
because agents are able to foresee the contagion that they are capable of gener145
ating that they decide to lead by contributing. We also found less demanding
conditions, under which at least some SPE exhibit contribution: in these cases,
agents start contributing not because of desire of generating contagion, but out of
a self-fulfilling fear of being socially sanctioned.
This paper contributes to the literature in social dynamics by considering a
population fully composed by rational agents. Because of the intrinsic difficulty
of such analysis, a number of restrictive assumptions had to be made. A promising avenue of future research consists on improving these assumptions and finding
weaker conditions under which still have meaningful results. Several extensions
could be developed for the baseline model. We have assumed homogeneity in the
benefits b, costs c and punishments bU. Introducing heterogeneity would generate
new interesting predictions; in particular, analyzing how heterogeneity in the parameters interacts with homophily, when both heterogeneity and homophily take
place along the same dimension (i.e. we could think that agents with lower cost
of contribution tend to be friends with each other). A related question is how dispersion in those variables, as measured by Second Order Stochastic Dominance,
affects contagion and leadership. It is not clear ex-ante that more dispersion in, for
example, cost c would lead unambiguously to more contagion, or more leadership.
In conclusion, this paper offers a simple explanation for why individuals participate in collective action in the presence of social pressure. The conclusions on this
146
paper can be incorporated to the literature on political revolutions as a justification of why (and how) it is incentive compatible, under certain circumstances, for
individuals to participate in collective action, giving validity to their reduced-form
models of revolutions.
Appendix
Computation of the probabilities of different events
=
(1) and 0 = 0(1,The
1), where
constants a, 3 and 0 are given by a = a(1),
(
functions a(-), 0(-) and 0(-, -) are defined below.
Let 0(h, m) be the probability that h out of m individuals play before the game
ends and before individual k gets to play. Then O(h, m) is given by
O(h,m) = q(n) [-n(h - 1,m-1)+ n-n n
n
(h, m)]
therefore
O(h, m) = O(h - 1, m - 1)
q(n)m/n
,_/
q
_
0(h
- 1, rn - 1)
Oh-17m-1
1+
1-q(n)
I
Hence, for h > 1,
(hm)
=
(h - 1, km - 1)
1 + Qm
1
1
1+ Qm
1
147
9(h-2,rm-2)
1+ Q(m-1)
1
=(1,m-h+1)J
+1
1=0
)
m-2
Q(m-1)
Moreover, we have that
0(1,m) = q(n)
-7(m) + n
In
1
-
nI
0(1,m) ]
0(1, m)
=
7y(m)
1
1+ 1
) ,
hence
where -y(m) is the probability that the game ends before any of m agents (in
this case, k and the other m - 1) gets to play:
n
n(mh) =-
q(n) + q(n)
-
m
=
and hence
7( ) - n-m = -q(n)
7(m)n + I/N - 11
M +
1
Therefore, O(h, m) is given by
1
+
h-i
m--I1)
_Q~m
l
Ih-
H
Q(m-
)
Q(
+
9(h, m )=
The probability that either of m players gets to play and player k also gets to
play is given by
148
a(m) = q(n)
[-Q +
n
n
m
-
a(m)]
n
Therefore
a(m) = l+Q
Q+
Finally, given z, the probability that player k gets to play before any of m
players do, and before the game ends, is
/3(m) = q(n)
- -
(n -
(
1+
or equivalently:
O(m)
-
q()nl
m)
n
Taking into account that q(n) =
,
=
q(n)-.
n
we find
n 1+
1
M
+
/3(m) =
Distribution of F. Define 17(k) as the random variable of the number of
players that get to play before the game ends, from k out of the n total players,
given that 1 has already played. To compute 17(k) observe that
F(k) = 1+ X(k - 1),
149
where X(k) is defined recursively by
k 1)+ n kn
(
-
X(k) = q(n)
(k)],
hence, for k > 1,
Sk[(k -
1) +1]
+
k- I
and Z(0) = 0. Therefore, -k is given by:
k
k
X(k) =Z11-1
m=1 I=m
The proof is by induction.
Z(1)
=
Q.
1
(3.7)
Q
For k = 1, we have from Equation 3,6 that
From equation 3.7, we find the same value.
Now, suppose that Equation 3.7 holds up to k: we will prove it for k + 1:
X(k + 1)=
k[X(k - 1) + 1]
k1
=
k 1 I
k-+Q
k+1
+k
m=1 I=M
Q
k+
+
k
Q
Em=1 f~m
k+1 k+1
'-1+
I
m=1 J=m+
Proofs of the results
PROOF OF PROPOSITION 17. Recall that we have the following four scenarios:
150
" With probability
#Q,
j plays, hence contributing, and then k gets to play
before the game ends. In that case k will contribute obtaining an extra payoff
+
bounded above by b - c
" With probability
#,
k plays before j gets a chance to play, and hence the
extra payoff is simply V
" With probability 0(1 - Q), j plays, hence contributing, and then k does not
get a chance to play before the game ends: she incurs extra disutility -bW.
" With probability -y( 2 ), the game ends before either k or j get to play, in
which case k receives an extra payoff of 0
Note that V as defined here is an upper bound on the actual payoff of player
k, as it does not take into account the fact that some of her friends who are not
predisposed might become so. Because of that, we are finding sufficient conditions
for the public good to be provided. Because of the argument above, V is given by
V = Q(b -c+
)+ OV -(1
- Q),bo.
Hence, a sufficient condition for individual k to decide to contribute is b-c+
V, what holds if
c- b b
<
111(1 - Q)
1
Note that
151
- -1-0
>
__0_(1 1 -8
Q)
_ _iQ)
-1-
I
(1+Q)-
_
-Q
1+Q
Therefore a sufficient condition for player j contributing is
C
b
K1+3
<
,
QQ
as we wanted to prove.
PROOF OF PROPOSITION 18. The payoff of contributing for an agent k remains the same independently of the number of predisposed friends m, namely
b - c + . The payoff for not contributing is bounded above by V(m), given by
considering the probabilities and payoffs of three different events.
9 If any of the m predisposed friends of k plays (hence contributing), but k gets
to play again before the game ends, she will contribute (because b4 > c),
and hence she obtains a payoff bounded above by b - c +
. This happens
with probability a(m).
e If k plays before any of her friends gets to play, then k is again in the same
situation, and hence she obtains payoff V(m). This happens with probability
* If h out of the m predisposed friends of k play (hence contributing), and k
does not get to play again before the game ends, she will suffer disutility
-hb. This happens with probability 0(h, m).
152
Solving for V(m), we find
- c + ) - bp E_, h - 0(h, m)
V(m) = ,-8rnh=(3.8) 1 -#(m)
_(m)(b
Agent k will contribute if b - c +
> V(m), which is exactly equation 3.4.
0
PROOF OF PROPOSITION 20. If i plays and nobody has contributed or would
ever do so, then the payoff for not contributing for i is simply 0. If she contributes,
her payoff is at least b(k) - c, where k = II* (Ni) 1. Hence, when this is larger
than 0, Player i prefers to contribute, and that is the definition of a spearheaded
equilibrium. The right hand side of Equation 3.6 is simply the expression for F(k),
D
what concludes the proof.
PROOF OF PROPOSITION 22. Let a be the strategy profile where agents in Y
always contribute (and all j V Y best respond to N\j). Consider an agent i E Y.
Because agents in Y are predisposed, and i E N. (Nj) for all j E Y, that means
that i has at least m* predisposed friends, and by definition of m*, it is a best
El
response for i to contribute.
PROOF OF PROPOSITION23. The proof is completely analogous to that of
Proposition 19, and included only for completeness. The proof is by induction:
by Assumption 6, whenever i contributes all her friends become predisposed, and
because 1I0 *(NiINJV) = Ni, this proves the case k = 0. Now, suppose the statement
153
is true for k, what means that all individuals in ['. (NiINk) are predisposed. Now,
notice that for an individual j to follow this chain of best responses, it must be
the case that j E Nk, in which case, by the definition of m*, agent j with at least
m* friends in the set Hk .(Ni) will also become predisposed. Hence, individuals in
fl*'(Ni)
=
Hi.(Ni) U {i E N: Ni
n
f.
> m*} become predisposed.
El
PROOF OF PROPOSITION 25. Note that because 2 < 1+30, that means that
whenever agent j has a predisposed friend, she also becomes predisposed. Moreover, this is common knowledge. When an agent i with public visibility contributes,
all agents know that friends of i will contribute. But, because the network is connected, this means that friends of friends of i will contribute, and hence friends
of friends of friends i, will contribute, etc. Therefore, following an inductive argument, all agents become predisposed.
El
154
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