A Two-Sided Global Game of Revolution
Higher Order Uncertainty and Mass Action
May 29, 2012
Abstract
We examine a political revolution in the context of a global game with two opposing groups - regime supporters and revolutionaries. Both citizens and regime
elites use public and private information of regime strength to judge the likely actions of others and hence decide whether or not to support the regime. The novel
aspect of our approach is that the actual size of the rebel force as well as the size
of the regime supporters, and therefore the likelihood of a successful revolution, is
endogenously determined by the equilibrium strategies of individuals. Our analysis suggests an elite focused approach better explains revolutionary success and
participation. Regime elites have more precise information about regime strength
than do citizens, which implies they better forecast the behavior of both groups.
As a consequence elite behavior, not citizen turnout, drives the flip between stable
regimes and successful revolutions.
1
Introduction
In 2011 a successful revolution swept Egypt’s President, Hosni Mubarak, from power
when the people massed on the streets of Cairo and overwhelmed the supporters of his
regime. Revolutions of this kind are commonly understood as tipping point events, in
that the rebels succeed only when there are enough of them acting in concert. The structure of such situations is simple: people want to support a revolution that will succeed,
yet if the revolution is to fail, then the people don’t want to participate because of the
1
associated costs. Revolutionary success requires that the number of participants reach
a tipping point - the minimum number of rebels needed to spark regime change. But
an individual’s best response relies on their expectation about the behavior of others.
If a citizen expects enough other people will challenge the regime, then he too wishes
to challenge the regime. When enough people expect the regime to fail, because of the
coordinated efforts of others, then these expecatations are self-fulfilling and the regime
falls.
While the basic tipping point argument regarding participation in mass political action is compelling, it is only half the story. In particular, why is it that some regimes
require a large number of rebels to topple them, while others require very few? What determines the tipping threshold? We contend that the willingness of regime elites to resist
the rebels is of particular importance in determining this threshold; yet this willingness is
a strategic concern, which has traditionally been incorporated into the “technology of revolt” (Atkeson, 2001; Hellwig, 2002; Bueno De Mesquita, 2010; Bueno de Mesquita, 2011;
Angeletos et al., 2007; Shadmehr and Bernhardt, 2011). By incorporating the strategic
competition between elites and rebels, we find that the abandonment of elites1 is crucial
in deciding whether a regime fails or succeeds. In the case of the Velvet Revolution,
“Dense masses chanted ‘Freedom’, ‘Resign’, and most strikingly, a phrase that might be
translated as ‘Now’s the time’ or ‘This is it’. And neither the white helmets nor the red
berets moved in.” (Garton Ash, 1990, p. 83)
The global games literature (Morris and Shin, 2003, 2002; Hellwig, 2002) has focused
on how combinations of public and private information regarding an underlying fundamental - an element which influences outcomes but is not endogenous to the system affects whether enough people expect enough other people to challenge the regime so
that the threshold to defeat the regime will be achieved. Unless a citizen is sufficiently
confident that enough others will participate, he will not; and if enough other citizens
share his belief, then the regime survives. Revolutionary success then depends on the
1
See Brinton (1965) for a discussion of the abandonment of intellectuals from regime support in the
context of the English, American, French, and Russian revolutions.
2
relative proportion of people who rebel and the relative proportion of elites who support
the regime. The central theme of our paper is explaining how these perceptions mutually
arise and determine the tipping point as an equilibrium phenomenon.
Citizens and elites face both fundamental uncertainty and strategic uncertainty. Fundamental uncertainty arises because of imperfect knowledge regarding the state of the
world. For instance, individuals are uncertain as to the exact amount of disposable money
available to the regime, the physical health of the leader, etc. - but this is not the only
type of uncertainty individuals face. In addition to not knowing the state of the world,
individuals can be uncertain as to how others will behave. And it is this strategic uncertainty that is critical in determining revolutionary outcomes. Strategic uncertainty arises
because of incomplete information regarding the behavior of individuals. A regime elite
decides her action based on her belief regarding her fellow elites and her belief regarding
citizen actions. Similarly, a citizen decides his action based on what he believes elite and
citizen actions will be. This interdependence of beliefs quickly becomes complicated.
Our analysis suggests that in unraveling these issues of strategic uncertainty it is the
elites which play the greatest role in deciding whether the regime survives or fails. An
individual citizen, based on his private information, infers the total proportion of citizens
who will rebel, as well as the total number of elites he expects to support the regime.
An elite makes a similar calculation. Since regime insiders are privy to more precise
information than citizens about the regime fundamental, they more accurately predict
the proportion of elites who support the regime and the proportion of citizens who rebel.
Often, before citizens realize they should take to the streets, the elites are well aware of
the outcome.
The revolutionary process is modeled as a global game with two distinct groups,
regime elites and citizens. Elites preserve the regime if they expect the regime to survive
and be strong; simultaneously citizens protest when they believe they can overwhelm
elite supression. The elites’ decisions to back the regime set the threshold which citizens
must exceed in order to overturn the regime. Simultaneously, the decision of whether or
3
not to rebel made by citizens sets the threshold level of support needed to insulate the
regime from failure. The processes of rebellion and regime support depend upon each
other, and we model this relationship in a setting where all actors receive private as well
as public information about some underlying regime fundamental.
The paper proceeds as follows, section 2 reviews the literature on regime change and
global games, in section 3 the model along with benchmark results are presented, section 4 presents the two-sided global game of regime change, section 5 analyzes empirical
implications of the model, section 6 concludes. Proofs are relegated to the appendix.
2
Revolutions
The causes of revolutions have generated a large literature which is only partially reviewed
here, a more complete but slightly dated review is Goldstone (2001). Most modern social
science explanations of revolutions begin with Gurr (1970) who developed what came
to be called, relative-deprivation theory. This theory explains that revolutions occur
because the citizen population is forced to endure a sufficient level of hardship, which
then causes a loss of legitimacy of the state. Once individuals are sufficiently unhappy
and no longer view their government as legitimate they rise up and overthrow the regime.
Relative-deprivation provides a rationale for why individuals might want regime change,
but it is a unilateral concept that does not handle the way individuals are judging the
behavior of others around them. In particular, it does not explain how individuals coordinate. In addition, Snyder and Tilly (1972) show that there is no empirical connection
between basic measures of well-being and the outbreak of revolutions across 130 years of
French history.
Skocpol (1979) argues that relative-deprivation as well as collective action based theories (Tilly, 1978; Tullock, 1971), rely too heavily on the intentions of individuals, and
ignores the spontaneity of revolutions which are rarely planned. She argues for a structuralist perspective, which examines the structural interactions of classes within a society
4
as well as the international context in which these class interactions are placed. Skocpol
(1979) argues that revolutions occur within an institutional and historical context, and
can be properly understood only by identifying the objective conditions determining a
state’s relation with other states as well as actions of elites.
The structural perspective, according to Skocpol (1979), allows the analyst to rise
above the viewpoints of the situation and identify the objective context in which revolutions occur. An immediate difficulty which stemmed from the structural perspective
was its inability to predict when a society would undergo dramatic social change. More
generally, successful revolutions have often come as a surprise to commentators and even
to those involved (Garton Ash, 1990).
In a series of papers (Kuran, 1989, 1991a,b), Timur Kuran posits that there is a difference between a public or expressive preference and a private or personal preference.
There is a tension between these “two selves” which can be effected by beliefs about
the relative proportion of individuals who are believed to share an individual’s private
preference. Individuals hide their disapproval of a regime because they fear they are the
only one with such a disposition. Revolutionary success depends upon achieving a critical
mass of protest. When enough individuals feel an unpleasant enough tension between
their personal preference for the regime and their public preference, a single individual
can set off a cascade of individuals who alter their public preference by withdrawing support for the regime.
By combining the intuitions of the structural and subjective approach, Lohmann
(1993, 1994), and Ginkel and Smith (1999), examine how weak regimes may signal weak
fundamentals to citizens by taking some actions which are only optimal for weak regimes.
These models focus on public signals and fundamental uncertainty but fail to account for
the effects of strategic uncertainty.
Global games provide a powerful tool for examining situations of coordination in the
presence of strategic uncertainty. Strategic uncertainty arises in the context of revolutions
because a citizen’s actions are influenced by his beliefs regarding the behavior of fellow
5
citizens and elites. An elite’s actions are likewise influenced by her beliefs regarding the
behavior of citizens and her fellow elites. This creates an infinite hierarchy of beliefs.
Consider for example some citizen: his willingness to rebel depends upon whether his
neighbors will also rebel, and how many elites will suppress this rebellion. Yet his calculation of his neighbors actions depends on their calculations of his actions, and so on...
The global game approach allows one to handle this infinite regress in a manageable way.2
Following Carlsson and Van Damme (1993), a global game is one in which there is an
unknown global parameter, the fundamental, which determines the exact game in which
players are involved. Individuals in the game receive noisy information regarding this
fundamental which serves to connect the hierarchies of beliefs by correlating first-order
posteriors, keeping belief hierarchies relatively close.3 In particular, we assume that a
state variable which influences the reward elite supporters can expect from continued
survival of the regime correlates expectations among all individuals in the society.
The global game approach has been fruitfully applied to currency crises (Morris and
Shin, 1998, 2001, 2003; Hellwig, 2002; Angeletos et al., 2006), as well as political contexts.
Atkeson (2001) uses the model of Morris and Shin (2001)4 as a model of riots. Rioters
are uncertain, but receive noisy information, regarding the exact number of police who
will show up. Rioters who believe few police will show up choose to riot, while those
who expect punishment do not. Angeletos et al. (2007) examine dynamics in the regime
change game where rebels need to amass an unknown number of individuals. Over time
failed revolts provide information as to the fixed value of the regime threshold. Shadmehr and Bernhardt (2011) look at a two-player regime change game where a revolution
succeeds when both individuals participate. They find that uncertainty regarding the
payoff of successful revolt leads to novel interactions between repression and incidence
of protest, which they term punishment dilemmas. Bueno de Mesquita (2011) compares
global game models which treat the fundamental as either a payoff or a threshold; he
2
This issue is particularly stressed by Hellwig (2002).
Under a reasonable topology.
4
See also Hellwig (2002).
3
6
finds these modeling assumptions effect the number of equilibria.
Using a similar motivation as in Lohmann (1993), Angeletos et al. (2006) examine the
signaling ability of a policy maker in the context of a global game of currency crises. They
find that this signaling ability of a policy maker reduces strategic uncertainty, introducing
a multiplicity of equilibria which would be absent without this signaling opportunity.
Bueno De Mesquita (2010) uses a clustered information structure similar to that used
in global game applications, to model the effect of a revolutionary vanguard’s terrorist
action on individuals who are trying to guage the anti-regime sentiment of the population.
Terrorist acts provide a crude signal regarding the overall level of anti-regime sentiment
because they are less likely to succeed when the population is in favor of the regime.
In complement to the existing literature, our model of revolutions as a two-sided
global game stresses the importance of elite behavior, which has been documented (Brinton, 1965; Garton Ash, 1990; Bueno de Mesquita and Smith, 2011), but not stressed as
the key deteminant of revolution success. From the starting point of a relatively deprived
citizenry who desire regime change, we examine how structural factors, modeled as the
regime fundamental, and heterogeneous information about this fundamental interact to
create the consistent, self-fulfilling beliefs of elites and citizens. These beliefs in conjunction with structural factors determine the relative turnouts of elites and citizens and
therefore the outcomes of revolutions.
3
The Model
We begin by introducing the benchmark model, which closely follows that of Hellwig
(2002), and is used as a model of riots (Atkeson, 2001), regime change (Angeletos et al.,
2006), and is denoted the canonical regime change game by Bueno de Mesquita (2011).
There is a unit mass of citizens with binary action sets, who can choose either to
support the regime or to rebel against it. The payoffs for citizens are as follows,
7
Citizen Payoff
Rebel(R)
No Rebel(NR)
Status Quo
Regime Change
−c
x
0
x−δ
Citizens want regime change, and they receive a reward worth x > 0 if the regime collapses. Their cost of protest in the event that the revolution fails is c > 0. If the regime
collapses, then we assume there is a benefit to having participated in its downfall. This
reflects the increased prospect of being priviledged by the successor regime, or avoiding
the risk of being accused of having collaborated with the previous regime.5 Hence if
regime change occurs and a citizen participated then his payoff is x; if he did not particpate then his payoff is x − δ > 0. For notational convenience, denote the cost ratio for
citizens as C =
c
.
c+δ
Each player observes, with some noise, the value of a global parameter, which we
call the regime fundamental. We parametrize this regime fundamental as θ ∈ R. For
convenience we assume this is drawn according to an improper uniform prior (DeGroot
1970)6 .
Denote the proportion of citizens who rebel by R. We assume that the regime fundamental determines the exact number of citizens required to overwhelm the regime through
the continuously differentiable function w(θ). In particular, if R ≥ w(θ), then the regime
fails; otherwise the regime survives. This formulation is standard (Atkeson, 2001; Hellwig, 2002).
We make the following assumption regarding the relationship between fundamentals
and outcomes,
Assumption 1 There exists a θ ∈ R in which, if θ ≤ θ the regime falls regardless of the
actions of individuals, and there exists a θ ∈ R in which, if θ ≥ θ the regime survives
regardless of the actions of individuals. Moreover, θ and θ are common knowledge.
5
This might simply be the psychological benefit of being able to say, “I was there.”
We do this for expositional convenience. If, for instance, we assumed a normal proper prior then
its mean and variance could be subsumed into the public signal but this would involve carrying extra
parameters through all calculations.
6
8
These are typically referred to as the dominance regions (Morris and Shin, 2003; Bueno de
Mesquita, 2011), because they induce dominant strategies for those who receive extreme
signals. Assumption 1 implies that it is possible for some regime to survive without
assistance from elites, e.g. by hiring mercenaries to quell any rebellion. Likewise, it is
also possible for a regime to fail from within, without a popular revolt.
The Information Structure
This section describes the information structure in the society. As is common in global
game applications (Morris and Shin, 2003), there is some information available which is
commonly known, and is formally represented by a public signal. There is also a significant amount of decentralized information, formally represented by private signals, which
are different for every individual, but are correlated through the regime fundamental.
All actors see a common public signal Q = θ + τ , where τ is independent of θ, and
normally distributed with mean zero and variance
1
.
α
As is common convention (DeG-
root, 1970), we refer to α as the precision of the public signal. In addition to the public
signal, citizen member j recieves a private signal ŷj = θ + υj where υj is independent of
θ and Q, and is normally distributed with mean zero and variance γ1 .
Actors update their beliefs in a straightforward way upon seeing the private and public
signals. These first-order beliefs, meaning they are beliefs about the regime fundamental,
follow directly from Bayes rule. These signals shape not only an actor’s belief regarding
the value of the regime fundamental, but more importantly, they shape what actors perceive about the actions of other actors.
Citizen j, after observing his private signal ŷj , and the public signal Q believes that θ
is normally distributed with mean yj and precision α + γ, where yj = (1 − η)Q + η ŷj and
η =
γ
α+γ
is the relative importance of a citizen’s private information in their posterior
belief regarding the regime fundamental. This is a consequence of Bayesian updating of
information generated from normal distributions (DeGroot, 1970).
Rx
For notational clarity let Φ(x) = −∞ φ(θ)dθ be the distribution function of a stan9
2
dard normal variate, where φ(x) =
x
√1 e− 2
2π
is the density function and let x = Φ−1 (p)
be its inverse function, where x is the pth quantile.
The Canonical Model of Regime Change
We now present the equilibrium results for the canonical regime change model employed
in Hellwig (2002); Angeletos et al. (2007); Bueno de Mesquita (2011). Although standard,
we present the canonical one-sided model because we build on this technology.
A strategy for citizen j is a measurable function which maps from an individual posterior expectation of θ to the binary action set of an individual. We consider symmetric
Bayesian Nash equilibria in strategies which are monotonic in posterior expectations, otherwise called monotone or cutoff strategies.7 This is a straightforward class of strategies
to focus on and is standard in global game applications (Angeletos et al., 2007; Bueno
De Mesquita, 2010; Shadmehr and Bernhardt, 2011).
For any state of regime fundamentals θ, if the citizens use a symmetric cutoff strategy,
y, then the proportion of citizens who choose to rebel is
Z
y
√
R(θ, y) =
φ(
=
−∞
√
γ
Φ( η (y
γ
(y
η
− ηθ − (1 − η)Q))dy
(3.1)
− ηθ − (1 − η)Q))
Since higher states increase the proportion of citizens who recieve a relatively high
posterior, there is a monotonicity between the state of the world, θ, and the number of
citizens who rebel. This implies that under a cutoff strategy profile there exists a critical
threshold state, θ̃, in which the mass of individuals whose posteriors are below the cutoff
7
In a more complete specification, a strategy would map from the public signal and the private signal
into the action set. Then beliefs would be required to be consistent with Bayes rule, which would restrict
the posterior to yj . Since we are exclusively interested in Bayesian Nash equilibria in which beliefs
are consistent, for ease of exposition we follow the common convention in global games and consider
strategies as a map from the posterior into the action set.
10
is exactly equal to the threshold value at that state, more precisely,
w(θ̃) = R(θ̃, y)
(3.2)
When citizens follow the symmetric cutoff strategy profile, characterized by y, the
expected payoff from not rebelling to a citizen who holds posterior yj , is,
Uj (N R, y|yj ) = (x − δ)P (θ ≤ θ̃|y, yj )
(3.3)
which is the payoff a citizen receives from regime change multiplied by the probability
the regime collapses. The expected payoff to rebelling, for a citizen who holds posterior
yj , is
Uj (R, y|yj ) = xP (θ ≤ θ̃|y, yj ) − cP (θ > θ̃|y, yj )
(3.4)
This is the payoff a rebel recieves when they participate in the revolution multiplied
by the probability the revolution succeeds minus the cost one pays when the revolution
fails multiplied by the probability of this event. The difference between these payoffs is
monotonic in the posterior yj ; therefore there exists some indifferent citizen who has a
posterior ỹ. Citizens with posteriors higher than ỹ prefer to support the regime; while
those with a posterior below ỹ prefer to rebel. From this, we are able to derive the
equilibrium for the canonical regime change game.
Proposition 1 The canonical regime change game has an equilibrium, characterized by
a cutoff, below which citizens rebel and above which citizens support the regime, (ỹ, θ̃),
which solves citizen indifference,
√
Φ( α + γ(θ̃ − ỹ)) = C
(3.5)
and a critical threshold condition
w(θ̃) = R(θ̃, ỹ)
11
(3.6)
Moreover if
w0 (θ̃)
α
<
min
√
γ θ̃∈(θ,θ) φ(Φ−1 (w(θ̃)))
(3.7)
then this equilibrium is the unique iterated dominance solution.
The proof follows by standard arguments(Atkeson, 2001; Hellwig, 2002), details can be
found in the appendix.
The canonical model is aesthetically pleasing as it provides a clean characterization of
a unique equilibrium, but the threshold which rebels must overcome is not strategically
chosen. Yet, elite behavior is critical in determining the outcome of many revolutionary
events. Rather than the people simply overwhelming the regime, as is captured by the
canonical model, Simon (1987) identifies dissent within the regime as the precipitating
factor in the overthrow of Ferdinand Marcos in the Philippines in 1986. Defense Minister,
Juan Ponce Enrile, and the army deputy chief of staff, General Fidel V. Ramos, deserted
the regime and called for Marcos’s resignation. With the expectation that the police and
army would not be there to stop them, the people flooded into the streets and Marcos
fell with little violence. All too often it is elites, or more precisely their defection, that
decide a regime’s fate.
In contrast, when elites remain loyal, regimes can withstand huge levels of protest as
was the case in Burma in 2007 (Larkin, 2010) or Iraq in 1991. Given the centrality of
elites, models of revolution should incorporate their strategic calculus. What is more,
since elites tend to be better informed about the regime’s fundamentals than the citizens,
their support is the driving factor in determining regime survival. Indeed, the regime
elites are more certain the revolution will succeed than the rebels who precipitate it. The
remainder of this paper addresses this with the two-sided global game of regime change.
12
4
Elite Participation
Elites
This section generalizes the canonical model of regime change by endogenizing the derivation of the function w above. In particular we allow there to be another group, which we
call regime elites, whose interests are opposed to those of the citizens. This introduces
another level of strategic uncertainty into the interaction. An individual, in addition to
making an inference regarding the behavior of individuals in their own group, now must
make inferences regarding the behavior of individuals in another group - the quality of
whose information is different.
The critical threshold which citizens must meet to spark regime change is not a deterministic function of the regime fundamental. Rather it is determined by the aggregate
actions of regime elites who can support or desert the regime. Formally, consider a unit
mass of regime elites,8 each with a binary action set.
Let W and R represent the proportion of elites who support the regime and proportion of citizens who rebel, respectively. If the proportion of the elites who support the
regime is greater than the proportion of the masses who protest, that is if W > R, then
the regime survives. Yet, should a bigger proportion of the citizens rebel than elites support the regime, R ≥ W, then the regime collapses. Revolutionary success then depends
upon the proportions of actors on each side rather than the turnout of a single group.
The payoff for the elites are given by:
Elite Payoff
Status Quo
Regime Change
θ
−k
θ−d
0
Support Regime(S)
Desert(D)
The payoff to being a loyal elite member of a regime that survives is the regime fundamental, θ. Should the elite member desert and the regime survive then the elite will
8
Technically each group will be a unit mass. One can adjust the size of these groups, e.g. assuming
the mass of elites is strictly smaller than the mass of citizens, but this does not change any of our results
so for simplicity we adopt a unit mass for both.
13
suffer a cost associated with her disloyalty. We assume this cost is d > 0, so an elite’s
payoff if she deserts and the regime survives is θ − d. If the regime falls, then an elite’s
payoff depends upon her actions. We normalize the payoff from deserting a failed regime
to zero. Should the elite have supported a failed regime, she pays a cost k > 0. This cost
reflects any potential retribution for having attempted to oppress the eventual victors.
Hence elites want to support a regime that will survive, but they want to desert if they
expect the regime to fail. Denote the cost ratio to elites as K =
k
.
k+d
We model the information of elites in a similar fashion to that of citizens. Each
elite member, in addition to observing the public signal Q, also recieves a private signal ẑi = θ + εi , where εi is independent of θ and Q, and is normally distributed with
mean zero and variance β1 . Since elites frequent the palace and directly benefit from the
regime, it seems responsible to assume their knowledge of the regime is the most precise,
so β > γ and β > α. Having seen their private signal, ẑi , and the public signal, Q, an
elite’s posterior beliefs about θ, are normally distributed with mean zi = κẑi + (1 − κ)Q,
and precision β +α, where κ =
β
α+β
is the relative importance of an elite member’s private
signal in their posterior.
A strategy for elite member i is a measurable function which maps from an elite’s posterior expectation of θ to her binary action set. We examine symmetric Bayesian Nash
equilibria in cutoff strategies. To contrast with the benchmark model another layer of
strategic uncertainty has been added. The threshold citizens must reach to spark regime
change now incorporates the elite’s behavior. Likewise, the threshold elites must reach
in order to save the regime incorporates expectations about citizen turnout.
Focusing on cutoff strategies, suppose that each citizen chooses to rebel if and only
if his posterior of the regime fundamental drops below some critical value, y ≤ y, and
suppose that each elite chooses to abandon the regime if and only if her posterior drops
below some critical value z ≤ z. The goal is to find values for which these cutoff strategies
constitute a Bayesian Nash equilibrium, which we characterize by a triple, (z, y, θ̂), which
consists of a cutoff value for elites, one for citizens, and the critical threshold, θ̂, in which
14
the regime survives if and only if θ > θ̂.
The two-sided aspect of the global game prevents one from precluding multiple equilibria; in particular we must consider two types of equilibria - for ease we examine these
seperately. We start by considering, what we regard as, the substantively more interesting case, where the equilibrium revolutionary success is determined by the self-fulfilling
expectations and actions of the players. We then more systematically examine monotone
equilibria of the two-sided global game of regime change, in particular we look at how
the value of the public signal, Q, effects the number of equilibria. Since which equilibria
exist depends upon the value of the public signal, the following definitions will be useful
in stating our results: First define the following parameters, which depend only on the
given precisions of information in the model,
p
β(α + γ)
λ1 = √
√ ,
α( β + γ)
p
γ(α + β)
λ2 = √
√ ,
α( β + γ)
√
βγ − α
λ3 = √
√
α( β + γ)
(4.1)
Definition 1 (Moderate Public Signal) A public signal, Q, is called moderate if Q ∈
[Q0 , Q0 ] where
Q0 = θ + λ2 Φ−1 (K) − λ1 Φ−1 (C)
(4.2)
Q0 = θ + λ2 Φ−1 (K) − λ1 Φ−1 (C)
(4.3)
and
Denote the set of moderate public signals as Q.
It should be noted that our definition of moderacy depends upon the dominance regions
- this relationship is examined below.
The Interior Equilibrium
We now turn to characterizing the behavior in which individuals are forecasting the
behavior of others and outcomes are determined by their higher order expectations. An
equilibrium must satisfy three criteria. The first criteria finds the posterior, y ∗ , that
15
makes citizens indifferent between rebelling and not; this is similar to the criteria derived
in the canonical model. The second criteria gives a parallel condition for elites which
determines the posterior, z ∗ , that makes an elite indifferent between supporting the regime
and deserting. The final criteria determines the state, as a function of these cutoffs, which
determine the critical threshold below which the regime fails, and above which the regime
survives.
As we have already seen in the discussion of citizen decision making in the canonical
game, a citizen who sees a signal that the regime is very weak believes that many other
citizens have seen similar signals. Such a citizen readily takes to the streets because he
anticipates many others will do the same. In contrast, a citizen who receives a signal
that the regime is very strong believes it likely that many other citizens received similar
signals and that relatively few received signals of regime weakness. Such a citizen is
reluctant to take to the streets because he anticipates the revolution will not succeed.
There is some intermediate signal between these extremes such that having received it,
a citizen believes the likelihood that the citizens will overwhelm the regime just balances
the cost and benefits of rebelling. The beliefs induced by this signal make the citizen
indifferent between rebelling and not rebelling. Citizens who see high signals such that
their posterior beliefs are y > y ∗ , stay home and those who see signals of regime weakness,
such that y ≤ y ∗ , revolt.
The same logic determining citizen behavior is applicable to elite decision making.
Elites who see signals of strong fundamentals, such that their posterior beliefs are z > z ∗ ,
infer a sufficiently large proportion of other elites saw similarly strong signals that the
regime is likely to survive. These elites support the regime. In contrast, elites who see
sufficiently weak signals, i.e. z ≤ z ∗ , desert. As above this leads to an elite indifference
condition. Both citizen indifference and elite indifference are similar to the indifference
condition, (3.5), derived in the canonical game, however the critical threshold condition
differs from the canonical game because it depends on the decisions made by both groups.
We now derive this condition.
16
Turnout by Citizens and Elites
RHΘ,yL and WHΘ,zL
1
Citizen Turnout, RHΘ,yL
Elite Turnout, WHΘ,zL
0.5
Θ
ΘHz,yL
Figure 1: Regime Fundamentals and Turnout by Citizens and Elites
For any given cutoff strategy profile, (z, y), we can calculate, as a function of the regime
fundamental θ, the proportion of citizens and elites who turnout, and hence whether the
regime survives or falls. Given θ, the citizen posteriors are normally distributed with
mean θ and precision α + γ. Since only those citizens whose posterior is y < y rebel, the
proportion of citizens who revolt given the state θ and strategy y is
√
R(θ, y) = Φ(
γ
(y
η
− ηθ − (1 − η)Q))
(4.4)
Similarly, given θ, elite posteriors are normally distributed with mean θ and precision
α + β so the proportion of elites who support the regime is
√
W (θ, z) = 1 − Φ(
β
(z
κ
√
− κθ − (1 − κ)Q)) = Φ(
β
(κθ
κ
+ (1 − κ)Q − z))
(4.5)
The state θ(z, y) for which R and W are the same is the critical threshold above
which the regime survives and below which the revolution succeeds. This can easily be
seen graphically; Figure 1 shows the turnout of citizens and elites as a function of the
regime fundamental θ. As the regime fundamental increases, a greater proportion of elites
support the regime, i.e. W(θ, z) increases in θ. In contrast, as the regime fundamental
increases there are fewer citizens who have posteriors over the threshold y and rebel; hence
17
R(θ, y) decreases in θ. Figure 1 illustrates why integrating regime elites into the strategic
environment is important. Since elites base their decisions on more precise information,
the W curve is more responsive to changes in the fundamental than the R curve, which
is relatively flat. Informally, this means changes in elite behavior are pivotal in the flip
from regime stability to revolutionary success.
The decision calculus of an elite is as follows: given the posterior, zi , if elite i supports
the regime then her expected payoff is
E[θ|θ > θ(z, y), z] − kP (θ ≤ θ(z, y)|z)
(4.6)
This is the expected value of the regime fundamental, θ, conditional on the regime survival
minus the cost of having suppressed demonstrators if the regime fails, multiplied by i’s
expectation that the regime will fail.
The expected payoff of desertion is
E[θ − d|θ > θ(z, y), z]
(4.7)
Equating these, elite i is indifferent between rebelling and not rebelling when
P (θ > θ(z, y)|zi ) = K
(4.8)
An equilibrium is characterized by the triple (z ∗ , y ∗ , θ∗ ), which are the values which
simultaneously solve the following equations,
P (θ ≤ θ∗ |y ∗ ) = C
P (θ > θ∗ |z ∗ ) = K
R(θ∗ , y ∗ ) = W(θ∗ , z ∗ )
18
Citizen Indifference
(4.9)
Elite Indifference
(4.10)
Critical Threshold
(4.11)
An equilibrium is interior to the nondominance region when public signals are moderate.
Proposition 2 If Q ∈ Q, then there exists a unique interior equilibrium in cutoff strategies characterized by (z ∗ , y ∗ , θ∗ ), where the survival of the regime is not dependent on the
dominance regions alone.
Proof: In Appendix.
It is straightforward to explicitly derive the equilibrium of Proposition 2. The information structure of the game implies the elite indifference condition, (4.9), can be written
as,
P (θ > θ∗ |z ∗ ) = Φ(
p
α + β(z ∗ − θ∗ )) = K
(4.12)
Similarly, the citizen indifference condition, (4.10), can be written as,
√
P (θ ≤ θ∗ |y ∗ ) = Φ( α + γ(θ∗ − y ∗ )) = C
(4.13)
The critical threshold condition, (4.11), can be written as
√
Φ(
γ
(y ∗
η
− ηθ∗ − (1 − η)Q)) = Φ(
√
β
(κθ∗
κ
+ (1 − κ)Q − z ∗ ))
(4.14)
This allows us to state the following corollary of Proposition 2, which follows by standard
linear algebra,
Corollary 1 The equilibrium values (z ∗ , y ∗ , θ∗ ) are
√
z ∗ = Q + λ1 Φ−1 (C) − λ3 κΦ−1 (K)
(4.15)
√
y ∗ = Q + λ3 ηΦ−1 (C) − λ2 Φ−1 (K)
(4.16)
θ∗ = Q + λ1 Φ−1 (C) − λ2 Φ−1 (K)
(4.17)
Corollary 1 highlights the exact relationship between the cost ratios, public information
and the critical threshold.
19
We now turn to characterizing all equilibria within the class of monotone, or cutoff
strategies.
Monotone Equilibria of Two-Sided Regime Change
In addition to the interior equilibrium characterized above, there are potentially two
other equilibria in monotone strategies. These additional equilibria depend critically on
the specification of the dominance regions. In this section we characterize these endpoint
equilibria and discuss the conditions under which each of the equilibria exist.
A citizen believes the revolution fails with at least probability p, regardless of the
actions of individuals, when
P (θ ≥ θ|y) ≥ p
(4.18)
and an elite believes the regime survives with at least probability p, regardless of the
actions of individuals9 , when
P (θ ≥ θ|z) ≥ p
(4.19)
From these beliefs, define the following pair of values, (z 0 , y 0 ), which solve the following
indifference conditions, one for citizens and one for elites,
P (θ ≥ θ|y 0 ) = C
(4.20)
P (θ ≥ θ|z 0 ) = K
(4.21)
These are the posterior expectations, relative to the cost ratios, at which individuals are
sufficiently confident in the strength of the regime that they need not consider what other
individuals are doing.
We similarly define when citizens and elites believe the regime fails with at least
9
These are called p-belief operators by Monderer and Samet (1989).
20
probability p when
P (θ < θ|y) ≥ p
(4.22)
P (θ < θ|z) ≥ p
From these beliefs, define the pair of values, (z0 , y0 ), which solve the following indifference
conditions, again one for citizens and one for elites,
P (θ < θ|y0 ) = 1 − C
(4.23)
P (θ < θ|z0 ) = 1 − K
(4.24)
These are the posterior expectations at which individuals are sufficiently confident the
regime fails that they need not consider the behavior of others.
These values characterize the endpoint equilibria of the two-sided global game of
regime change.
Lemma 1 If Q ≤ Q0 then there exists an equilibrium characterized by (z 0 , y 0 , θ) where
the regime surives when θ ≥ θ. Moreover,
z0 = θ +
√ 1 Φ−1 (K)
α+β
(4.25)
y0 = θ −
√ 1 Φ−1 (C)
α+γ
(4.26)
and
In this equilibrium the regime survives only when the regime fundamental exceeds θ,
which is the set of regime fundamentals where the regime survives independently of the
actions of individuals. Although some elites choose to support the regime, there is never
enough of them to suppress mass action unless the regime would survive without their
support anyway.
Lemma 2 If Q ≥ Q0 then there exists an equilibrium characterized by (z0 , y0 , θ) where
21
the regime survives when θ ≥ θ. Moreover,
z0 = θ +
√ 1 Φ−1 (1
α+β
− K)
(4.27)
y0 = θ −
√ 1 Φ−1 (1
α+γ
− C)
(4.28)
and
Proofs of these lemmata are in the appendix. In the latter equilibrium the revolution
succeeds only when the regime fails on its own, i.e. when the regime fundamental is below
θ. Although some citizens rebel, they only overwhelm the elite supporters whenever the
regime would collapse even in the absence of any protest.
The endpoint equilibria are effectively coordination equilibria. Coordination is imperfect because there is always some individuals on each side who recieve such extreme signals
that they have dominant strategies. However whenever the public signal is extreme, for
example whenever Q > Q0 , individuals are unable to coordinate on the equilibrium which
is pessimistic about the regime’s prospects.
Lemma 1 and 2 along with Proposition 2 fully characterize the set of equilibria in
monotone strategies which is summarized in the following proposition,
Proposition 3 (Monotone Equilibria in the Two-Sided Regime Change Game)
The Equilibria to the two-sided global game of regime change depend on the public signal
in the following manner,
1. If the public signal is moderate, Q ∈ Q, then there are three equilibria in monotone
strategies: (z0 , y0 , θ), (z 0 , y 0 , θ), and (z ∗ , y ∗ , θ∗ ).
2. If the public signal is extremely weak, Q ≤ Q0 , then there is a unique equilibrium,
(z 0 , y 0 , θ).
3. If the public signal is extremely strong, Q ≥ Q0 , then there is a unique equilibrium,
(z0 , y0 , θ).
22
Proof: In Appendix.
Graphically, Proposition 3 is illustrated in Figure 2. For moderate public information
there are three equilibria. Whenever public information is extremely pessimistic the only
equilibrium is coordination against the regime, namely (z 0 , y 0 , θ). Likewise whenever
public information is extremely optimistic the only equilibrium is coordination in favor
of the regime, namely (z0 , y0 , θ).
The endpoint equilibria, which were not present in the canonical model, are a consequence of the two-sided strategic uncertainty. Since outcomes depend on the behavior
of another group, the iterated dominance arguments of Morris and Shin (2004), which
inductively eliminate strategies, are unable to start - the base case fails.
The global game approach is typically used as a method of equilibrium selection (Carlsson and Van Damme, 1993; Morris and Shin, 2004) because strategic uncertainty typically
has the effect of reducing the set of equilibria - this is seen in the canonical game. However when we introduce competing groups, much of the strategic uncertainty can cancel
out, reintroducing multiplicity of equilibria. However, this is not globally true. In particular, when the magnitude of the public signal is very large, the strategic uncertainty
involved with the behavior of one side swamps out that of the other. This has the effect
that equilibrium multiplicity occurs only when public signals are moderate. This contrasts with the canonical game where the imprecision of public information was the key
to uniqueness.10
The usefulness of global games as an equilibrium selection criteria is generally very
limited. In particular it is shown that this approach is not useful for selection when
there is a large amount of public information (Hellwig, 2002), strategic information revelation (Angeletos et al., 2006), dynamics and learning (Angeletos et al., 2007), and the
failure of dominant strategies (Bueno de Mesquita, 2011). This paper establishes that the
global game approach can have limited application as an equilibrium selection criteria in
the context of two-sided strategic uncertainty.
10
Hellwig (2002) examines this specific point in depth.
23
Uniqueness
Uniqueness
Multiplicity
Hz0 ,y0 ,ΘL
Hz* ,y* ,Θ* L
Q0
Hz0 ,y0 ,ΘL
Q0
Q
Figure 2: Equilibrium and Public Signals
We next examine how the moderacy of public signals is dictated by the set of states
which are undominated,
Proposition 4 In terms of the undominated region, U = (θ, θ), define Q(U) in the
obvious way as the set of moderate public messages given the undominated region U, the
monotone equilibrium set of the two-sided global game of regime change is
1. As U approaches [θ, ∞), Q(U) approaches [Q0 , ∞), and the two-sided global game of
regime change generically has three equilibria, (z 0 , y 0 , θ), (z0 , y0 , θ), and (z ∗ , y ∗ , θ∗ ),
on Q and one, (z 0 , y 0 , θ), on Qc .
2. As U approaches (−∞, θ], Q(U) approaches (−∞, Q0 ], and the two-sided global
game of regime change generically has three equilibria, (z 0 , y 0 , θ), (z0 , y0 , θ), and
(z ∗ , y ∗ , θ∗ ), on Q and one, (z0 , y0 , θ), on Qc .
3. As U approaches R, Q(U) approaches R. and the two-sided global game of regime
change globally has three equilibria in monotone strategies: (z0 , y0 , θ), (z 0 , y 0 , θ),
and (z ∗ , y ∗ , θ∗ ).
Proof: In Appendix.
Proposition 4 captures the sense in which the endpoint equilibria are those in which
the coordination incentive completely determines behavior. In the limit the endpoint
equilibria become equilibria of pure coordination, and behavior is independent of the
information agents recieve. Also, in the limit the interior equilibrium always exists and
depends critically on the beliefs and behavior of agents.
24
Proposition 4 shows that the restrictions on behavior which follow by the dominance
regions determine how restrictive the moderacy of public signals condition is on the applicability of the interior equilibrium. One consequence of Proposition 4 is that as the
undominated region, U, approaches the real line, the set of moderate public signals, Q,
becomes less restrictive - in the limit all public signals are moderate.
Proposition 4 examines the role of the dominance regions, which have recently received
attention in political science by Bueno De Mesquita (2010); Shadmehr and Bernhardt
(2011) and Bueno de Mesquita (2011). Bueno de Mesquita (2011) shows that one can
expect two equilibria in the case where the dominance region is unbounded in one direction, otherwise called one-sided limit dominance. The first two items of Proposition 4
look at the equilibrium correspondance of the two-sided global game when it is a game
of one-sided limit dominance, and finds either three equilibria or one. This distinction
occurs because the two-sided strategic uncertainty leads to a more subtle role of public
information. More generally, the last item of Proposition 4 examines the equilibrium
correspondance when there are no dominance region assumptions, and finds there to be
generically three equilibria in montone strategies.
5
The Empirics of Revolutions
“We must, indeed, all hang together, or assuredly we shall all hang seperately.”
Benjamin Franklin, 1776.
In the next section we explore the comparative statics of the interior equilibrium, i.e. we
restrict attention to the case where public signals are moderate.
Transitions, Turnout, and Violence
The critical threshold for revolutionary success is characterized by the point at which the
R and W curves intersect. If these curves cross at a relatively high level of participation,
then the threshold rebellion will involve a large proportion of citizens who rebel and
25
a large proportion of elites who support the regime. Likewise, if R and W cross at a
relatively low level of participation, then a violent revolution cannot occur.
We define critical turnout, T (Q, C, K), as level of participation of both sides at the
equilibrium critical threshold, θ∗ ,
W(θ∗ , z ∗ , Q) = R(θ∗ , y ∗ , Q) = T (Q, C, K)
(5.1)
This is the minimum amount of participation on the part of citizens needed to spark a
change in regime, which the following proposition shows, depends upon cost ratios but
not public information.
Proposition 5 If Q ∈ Q, i.e. looking at the interior equilibrium, the minimum level of
participation required to spark regime change, T (Q, C, K), is independent of the value of
the public signal, Q. Moreover, the the function, T (C, K), is decreasing in the cost ratio
for citizens, C, and decreasing in the cost ratio for elites, K.
Proof: In Appendix.
These follow by direct introspection on equations (A.32) and (A.34) in the appendix.
There is also the direct implication of Corollary 1, which describes the relationship between cost ratios and the tipping point threshold.
Corollary 2 If Q ∈ Q, then the critical threshold, θ∗ , at which the regime fails is increasing in the cost ratio for citizens, C, and decreasing in the cost ratio for elites, K.
Figure 3 and 4 graphically illustrate the results in the above statements. These figures plot the turnout of elites, W (θ, z ∗ ), and citizens, R(θ, y ∗ ), as a function of the regime
fundamental θ. Figure 3 shows the consequences of a shift in the cost ratio that elites
face, K. The thick-solid and thick-dashed lines represent equilibrium turnout levels at
moderate costs for elites - this is a direct replication of Figure 1. Figure 3 holds the citizens’ cost ratio constant and shows how turnout changes as the elite’s cost ratio increases
26
Turnout by Citizens and Elites with Changes in Elite Costs HKL
RHΘ,y* L and WHΘ,z* L
KL
TKL
KM
TKM
TKH
KH
Θ * KH
Θ * KM
Θ
Θ * KL
Figure 3: Minimum Turnout varying Elite Cost
Turnout by Citizens and Elites with Changes in Citizen Costs HCL
RHΘ,y* L and WHΘ,z* L
TCL
CL
TCM
CM
CH
TCH
Θ * CL
Θ * CM
Θ * CH
Θ
Figure 4: Minimum Turnout varying Citizen Repression
27
or decreases.11
Increasing the elites’ cost ratio, K =
k
,
k+d
has two effects. First, it reduces the turnout
of citizens; second it increases elite turnout. This has two effects in terms equilibrium
∗
predictions. The critical threshold (labeled θK
) is shifted to the left and the turnout
H
at this critical threshold, T (C, KH ), is also reduced. In contrast, if the elite cost ratio
∗
is reduced, (labeled KL ), then both the critical threshold, θK
, and the critical turnout,
L
T (C, KL ), increase.
The results suggest important policy tradeoffs. If international organizations such as
Human Rights Courts increase the post transition punishment of elites for suppressing
citizens, then these elites hang together, a consequence of which is that the minimum
amount of violence required for a transtion is reduced but such a transition becomes less
∗
∗
and TKH < TKM . In contrast, if rebel
< θK
likely, this is illustrated in Figure 3 by θK
M
H
leaders could in some way commit to granting amnesty for former elites then a successful
revolution would become more likely, but any successful regime change would involve
higher citizen participation and is likely to be more violent, this is illustrated in Figure 3
∗
∗
and TKL > TKM .
< θK
by θK
M
L
Figure 4 holds the elite cost ratio constant and illustrates the effects of changing the
citizen cost ratio, C =
c
.
c+δ
Increasing the cost of participating in a failed rebellion, c,
increases citizen turnout, and likewise decreases elite turnout. When the consequences of
a failed revolution become more severe, rebels hang together. Increasing the citizen cost
ratio enhances the importance for the citizens of acting in concert, which has two important consequences. First it increases the likelihood of successful revolution, illustrated in
Figure 4 by θC∗ L < θC∗ M < θC∗ H . Second it reduces the minimum level of citizen turnout
needed to enact such a transition, illustrated by TCL > TCM > TCH .
It is reasonable to presume the expected cost of punishment for participating in a
failed revolution in countries with well developed bureaucracies is higher than in counFigure 3 and 4 are constructed assuming Q = 3, α = 15 , β = 5 and γ = 15 . The moderate cost
example is CM = 21 and KM = 12 . The high and low cost cases are constructed assuming CH = 32 ,
CL = 13 , KH = 23 and KL = 13 as appropriate.
11
28
tries lacking such administrative capacity. For instance, East Germany kept extensive
records on all citizens, and so could have readily identified and located individuals who
defied the regime. On the other hand, Myanmar lacked the capacity to document it’s
citizens.12 Our results suggest that when repressive capacity is high, as in East Germany,
transitions are comparatively nonviolent. In contrast, as events in Myanmar in 2007
showed, when states lack repressive capacity revolutions can be very violent.
Predicting Revolutions
Revolutions have been observed to come as a surprise, not only to casual observers, but
even to those on the streets (Kuran, 1989; Garton Ash, 1990). For instance, the RAND
Corporation study of the collapse of the Marcos regime in the Philippines (Simon, 1987)
examined six13 independent studies released shortly before that revolution. Although all
of these studies identified dissatisfaction with the regime, none of these studies predicted
imminent collapse. Our model explains the failure of analysts to predict revolutionary
success.
Analysts typically base their assessment on information which is publicly available. An
immediate implication of this publicity is that it is available to regime elites and citizens
alike, who incorporate this information into their strategic calculations. Unfortunately
this degrades the value of public information for predicting regime failure, which we now
show by computing the ex ante probability that the regime falls conditioned only on the
public signal, Q,
√
P (θ ≤ θ∗ |Q) = Φ( α(θ∗ − Q))
(5.2)
which depends on the public signal and the equilibrium critical threshold. This probability reflects the assessment of an individual who observes the public signal but receives
no private information.
12
In Wintrobe (1998)’s terms, we might refer to these as the totalitarian and the tinpot, respectively.
These were written by two congressional committees, two government agencies, and two private
consulting firms.
13
29
The critical threshold incorporates the effect of the public signal and the aggregate
effect of private signals through the best responses of all the actors involved, which from
Corollary 1 leads to the immediate implication,
Proposition 6 If Q ∈ Q, then the ex ante probability the regime fails conditional on
public information is
√
Φ( α(λ1 Φ−1 (C) − λ2 Φ−1 (K)))
(5.3)
which is independent of public information. Moreover, it is increasing in C and decreasing
in K.
Proof: In Appendix.
The ex ante likelihood of regime change is driven entirely by cost ratios and signal precisions. This occurs because the public signal is observed by all agents involved, and this
is common knowledge. In particular every citizen knows Q, and each citizen knows each
citizen knows Q, and so on... Moreover every elite knows every citizen knows Q, and they
know every citizen knows that every elite knows that every citizen knows Q, and so on...
Shifts in public information shift the actions of the players, which effects the conditions
under which revolutions succeed. At the same time, public information changes an analyst’s beliefs about the likely state. In terms of predicting revolution, without access to
the private signals, these factors fully offset.
Another focus of analysts and commentators is the size of protests. Yet, revolutionary
success is better predicted by elite behavior than by citizen behavior. As illustrated by
Figure 1, the turnout of elites is more sensitive to different levels of the regime fundamental than the turnout of citizens. This is formally established in the following proposition,
Proposition 7 If Q ∈ Q, then at the critical threshold, θ∗ , the turnout of elites changes
more quickly than the turnout of citizens, more precisely,
∂W(θ∗ ,z∗ ) ∂θ
∗
∗
∂R(θ ,y ) > 1
∂θ
30
(5.4)
Proof: In Appendix.
This proposition has implications for monitoring the progress of revolutions. Protest size
is fairly easy to monitor, which leads analysts to use it as an estimate of the regime
fundamental, which in turn often becomes the basis of a best guess for revolutionary
success. Yet, because citizen information is imprecise relative to that of elites, citizen
turnout is relatively insensitive to the regime fundamental. A more accurate estimate of
the regime fundamental can be guaged from elite behavior. Unfortunately observing this
information may be logistically difficult, as it generally takes place behind closed doors
rather than in public squares. As the RAND Corporation’s ex post study of Marcos’s
deposition (Simon, 1987) showed, elite defection was decisive, and yet the ex ante studies
focused on more readily observable but less informative signs of regime instability to
predict events in the Philippines. As a consequence none of these six studies foresaw
Marcos’s imminent demise.
6
Conclusion
The theoretical contribution of this paper is to elucidate in what way public and private information shape perceptions in a context where outcomes depend on the mass
participation of two groups whose interests are opposed. An elite’s assessment of their
payoff to support the regime is a good estimate of the payoff they expect other regime
elites to expect. Likewise, a citizen’s assessment of the average elite payoff for supporting
the regime is a reasonable, but less accurate, estimate of an elite’s assessment as well as
the assessment of fellow citizens. In the presence of strategic uncertainty, global games
provide a powerful tool to incorporate an individual’s belief hierarchy and the effect of
these beliefs on actions and outcomes in a tractable manner.
Revolutions are fundamentally about opposing interests. Which group prevails depends upon which group turns out the most members. To our knowledge, this paper is
the first to integrate multiple competing groups within a global game structure. However,
31
from a technical perspective the introduction of competing groups comes at a cost, as the
uniqueness of equilibria in the canonical game disappear. Angeletos et al. (2007, 2006),
and Bueno de Mesquita (2011) also show in other contexts, extensions of the basic global
game framework can lead to multiple equilibria.
Revolutions are typically thought of as events of mass participation. As a result,
much extant research has focused on the people coming together. Yet our model shows
elites play an essential role in revolutionary dynamics. The Russian Tsar fell in 1917
because the army chose not to stop the people from swarming his Winter Palace. The
Shah of Iran’s fate was similar in 1979. Elite desertions were equally critical in the collapse of communist regimes at the end of the Cold War, e.g. the Velvet Revolution, and
during the Arab Spring revolutions. Yet, when elites hang together they can suppress
even large uprisings as was the case when Saddam Hussien’s supporters remained loyal
and suppressed large Shia uprisings in the wake of the first Gulf War in 1991. Similarly,
elites loyal to Myanmar’s regime suppressed a massive uprising led by priests in August
2007 (Larkin, 2010).
Benjamin Franklin stressed the importance for revolutionaries to act in concert. However, it is elites who better coordinate to hang together because their knowledge of regime
strength is more precise, so their beliefs about behavior are closer together than those
of the citizens. Our model indicates that elite sentiment is a much better predictor of
revolutionary success than public signals or the behavior of citizens. Indeed, the model
indicated that because of equilibrium best responses to public signals, easily observed perceptions of regime strength have little effect on the judgements of regime collapse based
solely on public information. Observations of mass protest levels are also a relatively
weak gauge of underlying regime strength or weakness. The most accurate assessments
of regime fundamentals come from elite perceptions. Unfortunately compared to public
signals and the behavior in public squares, these factors are hardest to ascertain. But
to use an old adage, searching for your keys under the streetlight does not help if you
did not lose them there. The integration of elite behavior into a global game model of
32
revolutions suggests a refocusing of attention on elites rather than citizens.
A
Appendix
Proof of Proposition 1: The indifferent citizen is one whose posterior satisfies
P (θ ≤ θ̃|ỹ) = C
(A.1)
which using the normality assumptions,
√
Φ( α + γ(θ̃ − ỹ)) = C
(A.2)
and rearranging is
ỹCI = θ̃ − √
1
Φ−1 (C)
α+γ
(A.3)
Now the critical number of citizens needed for regime change is given by
w(θ̃) = R(θ̃, ỹ)
(A.4)
again using normality,
√
w(θ̃) = Φ(
γ
(ỹ
η
− η θ̃ − (1 − η)Q))
(A.5)
and rearranging, the critical threshold is
ỹCT = η θ̃ + (1 − η)Q +
η
−1
√ Φ (w(θ̃))
γ
(A.6)
First observe that
∂ ỹCT (θ̃, Q)
w0 (θ̃)
=η 1+ √
∂ θ̃
γφ(Φ−1 (w(θ̃)))
33
(A.7)
and the two limit conditions,
lim
θ→θ
lim
ỹCT (θ, Q) = −∞
(A.8)
ỹCT (θ, Q) = ∞
θ→θ
Since ỹCT these limit conditions along with continuity imply there exists a θ̃ such that
ỹCI (θ̃) = ỹCT (θ̃, Q) establishing that a symmetric cutoff equilibrium exists.
Combining the limit conditions, (A.8), with the corresponding limits for ỹCI which
are finite at both ends we note that uniqueness holds under the condition that
∂ ỹCT (θ̃)
,
∂ θ̃
which by (A.7), and since
∂ ỹCI (θ̃)
∂ θ̃
∂ ỹCI (θ̃)
∂ θ̃
<
= 1 yields,
α
w0 (θ̃)
√ < min
γ θ̃∈(θ,θ) φ(Φ−1 (w(θ̃)))
(A.9)
and then to establish that (ỹ, θ̃) is the unique iterated dominance solution,
Lemma 3 If the solution to the system of equations,
√
Φ( α + γ(θ̃ − ỹ)) = C
(A.10)
w(θ̃) = R(θ̃, ỹ)
is unique, then (ỹ, θ̃) is the unique strategy profile which survives iterated deletion of
strictly dominated strategies.
Proof: To begin we allow any strategy. We must consider the worst case for each citizen.
Assumption 1 ensures there exists a θ for which the regime fails and a θ for which the
regime survives regardless of the actions of citizens. Suppose we start with the strategy
profile in which no citizen rebels, which is the cutoff strategy, ỹ0 = −∞. In this case a
34
citizen has a dominant strategy to rebel only if
P (θ ≤ θ|ỹ0 , y) ≥ C
(A.11)
Take the maximum value for which this inequality holds, call it ỹ1 . Now suppose the
strategy profile is such that all citizens rebel, which is the cutoff strategy, ỹ 0 = ∞. In
this case a citizen has a dominant strategy to not rebel only if
P (θ ≤ θ|ỹ 0 , y) ≤ C
(A.12)
Take the minimum value for which this inequality holds, call it ỹ 1 . The values ỹ1 and ỹ 1
are finite, so in particular, ỹ0 < ỹ1 and ỹ 0 > ỹ 1 .
Given the deletions above, we want to look at the outcomes when this implies there
exists θ1 such that for all θ < θ1
R(θ, ỹ1 ) > w(θ)
(A.13)
and likewise there exists a θ1 such that for all θ > θ1
R(θ, ỹ 1 ) < w(θ)
(A.14)
n
∞
Continuing in this fashion generates two sequences, {ỹn , θn }∞
n=0 and {ỹ , θ n }n=0 . We
now establish that these sequences are monotonic. Assume for n−1 we have the following
ỹ0 < ỹ1 < · · · < ỹn−1
ỹ 0 > ỹ 1 > · · · > ỹ n−1
35
these then imply the following sequences,
θ0 < θ1 < θ2 < · · · < θn−1
(A.15)
θ0 > θ1 > θ2 > · · · > θn−1
We use θn−1 and θn−1 in the next iteration, suppose we start with the strategy profile
in which no citizen with posterior mean y > ỹn−1 rebels. Since θn−1 > θn−2 ,
P (θ ≤ θn−2 |ỹn−1 , y) < P (θ ≤ θn−1 |ỹn−1 , y)
(A.16)
thus by definition it must be that ỹn > ỹn−1 . Suppose we start with the strategy profile
in which every citizen with posterior mean y > ỹ n−1 rebels. Since θn−1 < θn−2 ,
P (θ ≤ θn−2 |ỹ n−1 , y) < P (θ ≤ θn−1 |ỹ n−1 , y)
(A.17)
thus by definition it must be that ỹ n < ỹ n−1 . These then imply there exists θn such that
for all θ < θn
R(θ, ỹn ) > w(θ)
(A.18)
and likewise there exists a θn such that for all θ > θn
R(θ, ỹ n ) < w(θ)
(A.19)
which by the monotonicity of R yield θn > θn−1 and θn < θn−1 . This establishes that
n
∞
{ỹn , θn }∞
n=0 is an increasing sequence and {ỹ , θ n }n=0 is a decreasing sequence, which by
(A.10) have a common limit, (ỹ, θ̃).
Proof of Proposition 2:
The indifferent citizen is characterized by the cutoff y ∗
36
and must be indifferent between supporting the regime and rebelling,
√
Φ( α + γ(y ∗ − θ∗ )) = C
(A.20)
rearranging we obtain
y ∗ = θ∗ −
√ 1 Φ−1 (C)
α+γ
(A.21)
Now the indifferent elite, z ∗ , is characterized by
p
Φ( α + β(θ∗ − z ∗ )) = K
(A.22)
rearranging
z ∗ = θ∗ +
√ 1 Φ−1 (K)
α+β
(A.23)
Setting W(θ∗ , z ∗ ) = R(θ∗ , y ∗ ) substitution and rearranging give,
θ∗ =
√
√ β√ z ∗
κ( β+ γ)
+
√
γ
√ √ y∗
η( β+ γ)
−
√α Q
γβ
(A.24)
Equations (A.21), (A.23), and (A.24) are an independent linear system, thus they have
a unique solution.
Proof of Proposition 3:
Proposition 3 is proven by proving Lemma 1 and 2. We
prove Lemma 1,
Consider the symmetric monotone strategy profile characterized by cutoffs (z 0 , y 0 ),
described in the text. It is derived in the following manner
√
Φ( α + γ(y 0 − θ)) = C
(A.25)
rearranging we obtain
y0 = θ −
√ 1 Φ−1 (C)
α+γ
37
(A.26)
and for elites
p
Φ( α + β(θ − z 0 )) = K
(A.27)
rearranging
z0 = θ +
√ 1 Φ−1 (K)
α+β
(A.28)
For (z 0 , y 0 , θ) to be an equilibrium it must be that no individual has an incentive to
unilaterally deviate, which is true if and only if the point at which R(θ, y 0 ) and W(θ, z 0 )
cross, call it θ1 , is greater than θ. This is true if and only if
Q ≤ θ + λ2 Φ−1 (K) − λ1 Φ−1 (C) = Q0
(A.29)
which establishes the lemma. Lemma 2 follows by an identical argument. Combining
Lemma 1 and Lemma 2 completes the proof and fully characterizes the equilibria in
monotone strategies.
Proof of Proposition 4:
Define two arbitrary sequences {θm }m∈N → −∞ and
n
{θn }n∈N → ∞, such that θm < θ , for all n and m. Then denote the set of undominated states as
n
Un,m = (θm , θ )
(A.30)
which has the corresponding sets of moderate public signals, Qn,m = Q(Un,m ). Properties
of Qn,m follow immediately by (4.2) and (4.3).
For the first part: Taking n → ∞, then Lemma 1 shows that there exists a sequence
of games in which {(z 0 , y 0 , θn )}n∈N is an equilibrium on {Qn,m }n∈N . The second part
follows by an indentical argument. The third part follows by combining the first and
second parts.
38
Proof of Proposition 5: Turnout can be written two ways,
√
T (Q, C, K) = Φ(
γ
(y ∗
η
− ηθ∗ − (1 − η)Q))
(A.31)
which by substitution from Corollary 1,
√
Φ(
γ √
( η(λ3
η
√
− λ1 η)Φ−1 (C)) − (1 − η)λ2 Φ−1 (K))
(A.32)
Inspection shows this is decreasing in K. Also
√
T (Q) = Φ(
β
(κθ∗
κ
+ (1 − κ)Q − z ∗ ))
(A.33)
which by substitution from Corollary 1,
β √
( κ(λ3
κ
√
Φ(
√
− λ2 κ)Φ−1 (K)) − (1 − κ)Φ−1 (C))
(A.34)
Inspection shows this is decreasing in C, and is independent of Q.
Proof of Proposition 6: By standard properties of the normal distribution
√
P (θ < θ∗ |Q) = Φ( α(θ∗ − Q))
(A.35)
by substitution from Corollary 1,
√
−1
−1
α λ1 Φ (C) − λ2 Φ (K)
Φ
Monotonicity properties follow by inspection.
39
(A.36)
Proof of Proposition 7: From (4.5)
W(θ, z ∗ ) = Φ(
√
+ (1 − κ)Q − z ∗ ))
β
(κθ
κ
(A.37)
so then
√
∂W(θ, z ∗ ) p
= βφ( κβ (κθ + (1 − κ)Q − z ∗ ))
∂θ
(A.38)
evaluating at the critical threshold
p
p
βφ
β(Q + λ1 Φ−1 (C) − λ2 Φ−1 (K)) +
√α Q
β
√
−
β
(Q
κ
−1
+ λ1 Φ (C) − λ3
√
κΦ (K))
−1
(A.39)
collecting terms
p
p
√
βφ ( α+β−(α+β)
)Q
+
(
β−
β
α+β
√ )λ1 Φ−1 (C)
β
p
p
−1
+ (λ2 β − λ3 α + β)Φ (K)
and simplifying
p
p
p
−1
−1
α
βφ ( √β )λ1 Φ (C) + (λ2 β − λ3 α + β)Φ (K)
(A.40)
and from (4.4),
R(θ, y ∗ ) = Φ(
√
γ
(y ∗
η
− ηθ − (1 − η)Q))
(A.41)
so then
√
∂R(θ, y ∗ )
√
γ
= − γφ( η (y ∗ − ηθ − (1 − η)Q))
∂θ
(A.42)
evaluating at the critical threshold
√
− γφ
√
γ
(Q
η
√
−1
−1
+ λ3 ηΦ (C) − λ2 Φ (K)) −
√
−1
−1
γ(Q + λ1 Φ (C) − λ2 Φ (K)) −
√α Q
γ
simplifying
√
√
√
−1
−1
α
− γφ ( √γ )λ2 Φ (K) + (λ1 γ − λ3 α + γ)Φ (C)
40
(A.43)
Now take the ratio of (A.40) and (A.43), which using properties of the exponential function is,
q
β ζ
e
γ
(A.44)
where
√
ζ = Φ−1 (K)(λ2 ( α−√γβγ ) + λ3
p
α + β) + Φ−1 (C)(λ1 (
√
βγ−α
√
)
β
√
− λ3 α + γ)
(A.45)
Note the following identity of the precisions,
√
√
βγ − α
βγ − α
λ2 p
= λ3 = λ1 p
γ(α + γ)
β(α + β)
(A.46)
which implies ζ = 0, hence since, β > γ,
∂W(θ∗ ,z∗ ) s ∂θ∗ β ∂R(θ∗ ,y∗ ) = γ > 1
(A.47)
∂θ∗
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