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Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/evolutionofperceOOacem
CEWEY
B31
^415
36
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
EVOLUTION OF PERCEPTIONS AND PLAY
Daron Acemoglu
Muhamet Yiidiz
Working Paper
01 -36
June 2001
RoomE52-251
50 Memorial Drive
Cambridge, MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssm.com/abstract=^ 290779
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
EVOLUTION OF PERCEPTIONS AND PLAY
Daron Acemoglu
Muhamet Yildiz
Working Paper 01 -36
June 2001
Room
E52-251
50 Memorial Drive
Cambridge,
MA
02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
httpi/Zpapers-Ssm-coni/abstract^ xxxxx
"MASSAciJypfnwsTm^
OFTECHMOLOGY
Evolution of Perceptions and Play^
Muhamet
Daron Acemoglu
MIT
Yildiz
MIT
June 2001
Abstract
^.
An
agent with a misperception
misperception and
its
may
have an evokitionary advantage when his
behavioral impUcations are recognized by others.
In such
situations evolutionary forces can lead to misperceptions, yielding "irrational behavior,"
such as the play of strictly dominated strategies.
this reasoning relies
sumed
We
on the assumption of subjective rationality
point out that
— agents
to choose the behavior that maximizes their perceived payoffs.
subjective rationality does not have solid evolutionary foundations:
are as-
However,
in the pres-
who do not maximize their perceived payoffs may
than those who do. We show that relaxing the subjective
ence of misperceptions, agents
have greater fitness
rationality requirement,
although agents
havior:
somewhat
paradoxically, leads to effectively rational be-
may have
systematic misperceptions, they will develop
other biases to undo these misperceptions, and will act as
a result, systematic biases in experimental settings
if
they are rational. As
not necessarily translate
demonstrate that the same evolutionary forces,
the long run, lead agents to play as if they have a common prior, even though
into irrational behavior.
in
We
may
also
each agent will have different and possibly incorrect perceptions of payoffs and the
rules of the
game.
Keywords: Common
prior, evolution, neutral stability, misperceptions, per-
ceptions, rationality.
JEL
*We thank
ticipants at
Classification: B40, C72, D84.
Abhijit Banerjee, Navin Kartik, Eric
MIT, SITE and Michigan
for useful
Van den
comments.
1
Steen, Jorgen Weibull,
and seminar par-
Introduction
1
Much
of economics
built
is
on the notion that agents
that maximize
will take actions
Alchian (1950), Friedman (1953), and Becker (1962) have
their payoff or fitness.
emphasized how "evolutionary" competition, either through the survival of the
who
social evolution, will tend to eliminate agents
The evolutionary game theory hterature
tion.
dynamics, evolutionary forces
rest point of
game
lying
(see, e.g.,
will
or
deviate substantially from maximiza-
formalizes these insights. Under" rephcator
will typically eliminate all
an evolutionary process
fittest
all
dominated
strategies,
and any
correspond to a Nash equilibrium of an under-
the recent textbooks on evolutionary
game
theory, WeibuU, 1997,
or Samuelson, 1997).^
The psychology
literature
and recent research
phasize a variety of biases and misperceptions
(e.g.,
over confidence, wishful thinking,
law of small numbers). Although some of these "nonrational" types of behavior
result
arise
there
from cognitive hmitations,
because
is
it
many
may
scholars argue that nonrational behavior can
confers no payoff disadvantage
an informal
em-
in behavioral economics, instead,
and may even have payoff benefits.
First,
theorem that, given the complexity of the econoinic situations
folk
that agents are involved in and the speed with which evolution works, there will not be
strong enough evolutionary forces to wipe out nonrational behavior (See MuUainathan
and Thaler, 2000,
for
a discussion and justification
for this view).
mists have argued that certain types of behavior that appear "irrational"
commitment
devices.
Robert Frank (1988),
tendency to seek revenge
may be
useful
for
many
econo-
may be
useful
Second,
example, points out
how
the irrational
by acting as a commitment to punish those who
break their promises. Robson (1990), Banerjee and Weibull (1993), Dekel, Ely and Yi(1999) similarly show how,
lankaya (1998), and Kockesen et
al
(types) are observable, evolution
may
to gain
when
agents' preferences
select preferences or types that act "irrationally"
commitment advantage.
The argument
that irrational behavior
agents' preferences to be (to
tions, including Dekel,
may be
some degree)
a useful commitment device requires
observable.
In fact, a
number
Ely and Yilankaya (1998), Ely and Yilankaya (1999) and
and Vega-Rodondo (2000) show that when preferences are not observed
'
of contribu-
at
all,
Ok
evolution
Other (non-monotonic) evolutionary dynamics may lead to strategies that are not Nash equilibria
in the underlying game. See Bendor et. al. (2001) and Stenneck (2000).
(and are even dominated)
.
Nash
takes us back to
equilibria. All of these papers, irrespective of
whether they allow
preferences to be observed or unobserved, rely on a strong notion of subjective rationality:
agents are assumed to always maximize their perceived utility or payoff. Other
game theory
contributions in
also start
from
different perceptions
subjective rationality to derive tight predictions
1996,
Van den
(e.g.,
and use a notion of
Harrison and Kreps, 1978, Morris,
Steen, 2001, Yildiz, 2000).
we argue that subjective
In this paper,
foundations (see also
Blume and
rationality does not have strong evolutionary
Easley, 1992). Subjectively irrational agents
tematically choose strategies that do not maximize their perceived payoffs
Somewhat
higher payoffs than subjectively rational agents.
—who
sys-
—may obtain
paradoxically, once
we
allow
mutations (deviations) that relax subjective rationality, there are strong forces pushing
agents towards effectively rational behavior (with limited information)."
sults
can be summarized
as:
Our main
re-
allowing subjective irrationality leads to effectively rational
behavior (with limited information) in any neutrally stable outcome of evolution.'^
Nevertheless,
we
find that there
is
no immediate
accurate perceptions. Evolution will select agents
but these agents
will
link
who
between rational behavior and
will act "effectively rationally",
have very different perceptions, and sometimes
will
make systematic
mistakes (yet their mistakes will cancel each other). In this sense, our analysis provides a
microfoundation
for
the connnon premise that agents will typically behave as
if
they are
rational (e.g., Friedman, 1953). This reasoning suggests that systematic misperceptions
in
experimental settings or in situations with
little
relevance to long-run fitness do not
necessarily translate into widespread "irrational" behavior.
We
also
show that
in
any neutrally stable outcome of evolution, agents
different perceptions, but the play will correspond to a
a game where players have a
^We
call
an agent as
common
is
of interest since there
eiiectively rational with full information.)
We refer
(A stronger notion of rationality will
to this type of behavior as "effectively
rational" since agents, despite their misperceptions, act as if they are rational.
type of behavior could be referred to as "objectively rational" to contrast
To minimize
'Neutral stability
perform
is
strictly better
""But they
may
is
information iff he maximizes his expected fitness
Such an agent may lack some information, but will never have any
bias nor play dominated strategies given his information set.
behavior.
of
effectively rational with limited
with respect to his information.
correspond to
have
Nash equilibrium outcome
This result
prior.''
may
terminology,
we
stick to the
term
a weaker form of evolutionary
than
all
mutations
it
Alternatively, this
with subjectively rational
"effectively rational"
stability,
where stable strategies do not need to
in post-entry population.
not be able to distinguish
some underlying games from each other. This common
games are played.
prior will simply correspond to the frequencies with which these underlying
disagreement about the plausibility of the
for
common prior assumption
game theory
in
(see
example, Gul, 1998, and Aumann, 1998). Since the common-prior assumption
crucial epistemological foundation for equilibrium behavior in
most games, the
regarding the
common
prior
assumption have important implications
analyses (see
Aumann,
1987,
and
of the
common
prior assumption,
equilibrium analysis).
prior assumption
beliefs are
It is
Our
Aumann
and Brandenburger, 1995,
and Dekel and Gul, 1997,
a
criticisms
for all equilibrium
for the
importance
for the related criticism of
results suggest that equilibrium analysis with the
may have good evolutionary foundations
is
—even when
common-
players' expressed
incompatible with each other.
important to note that the neutrally stable behavior
only with limited information; some misperceptions
rance" so agents will not necessarily play a
,
full-information game.
When
will
may remain
be effectively rational
in the
Nash equilibrium outcome
perceptions are observable, agents
form of "igno-
of the underlying
may
develop misper-
ceptions in order to suppress information that would otherwise reduce the payoff to
all
parties.^
The
rest of the
paper
is
organized as follows. In the next section,
of examples that illustrate our results,
and
we
give a
also discuss related literature in
number
more
detail.
Section 3 analyzes evolution in a single-agent decision problem (without strategic interactions)
more
formally. Section 4 analyzes a multi-agent/game-theoretic situation
contains the main results of the paper. Since our main point
is
and
that effectively rational
behavior emerges once we allow agents to be subjectively irrational, in this section,
we
allow perceptions to be observed, creating the most favorable environment for deviations
from rational behavior
evolution
(see, e.g.,
when perceptions
^This result
is
Dekel, Ely, and Yilankaya, 1998). Section 5 analyzes
are not observed. Section 6 concludes.
obtained because perceptions are more than "cheap talk": they also determine
individuals process data from the real world.
the same
way has
to play the
same strategy
A
in
player
who
both situations.
We
how
two different situations in exactly
show below that when perceptions
perceives
are simply "cheap talk" systematic misperceptions that affect behavior cannot arise.
Examples, Outline and Related Literature
2
Examples and Outline
2.1
Consider the following 2x2
game and
interpret the payoffs as fitness levels, so that
evolutionary dynamics will favor strategies that have higher payoffs:
Example
(Game
of underlying
game
,
r
This game
player
1
is
2.2
1,1
3,0
dominance solvable and hence has a unique Nash equilibrium
chooses r and player 2 plays L.
different perceptions of their payoffs
Now
observability:
let
which
—they may see the payoff matrix differently— and
of different perceptions in this evolutionary
the underlying game. Moreover,
in
imagine a meta-game in which players have
an evolutionary process determining the frequency of
1.
0,0
1):
us
different perceptions.
meta-game are
still
The
fitness
given by the payoffs in
make two important assumptions:
game
agents' perceptions are observable to others in the
ample, the perception of the payoff matrix that one holds
is
(for ex-
engraved on their
forehead).^
2.
subjective rationality: players choose whichever action
maximizes
their perceived
payoffs.
Then, consider a type
for player 1
who
Misperceiv
Misperceived
(Game
perceives this
1\2
game
over
r.
1
2,0
4,2
1,1
3,0
I
is
a dominant strategy, and given subjective
This will encourage his opponent to play R, and so
a player with this type of misperception
perception benefits player
R
2):
This type mistakenly perceives that
I
L
instead as
,
r
rationality, will choose
game
because
it
will receive
enables
him
to
the "true payoff" of
commit
2.
This mis-
to playing his dominated
strategy in the underlying game, thus enticing player 2 to change his play. Without the
"^See
Frank (1988)
for
a defense of the view that perceptions and preferences are, at least to some
degree, observable to others.
.
misperception, player
and player
To
could not have committed to playing his dominated strategy,
1
would have preferred to respond by
2
L.
we simply apply standard
see the forces shaping the evolution of perceptions,
evolutionary reasoning to a meta-game where
Game
perception of
1
or misperceive
as
it
we
Game
allow player
2 (in other words,
To
for player 2,
even though there
highlight the choices facing player 2,
is
we
we
allow mutations
we allow no misperceptions
over accurate and inaccurate perceptions). For simplicity,
for player 2.
to either have the correct
1
also write out the four strategies
no evolutionary selection over these actions (instead,
player 2 chooses his action optimally after observing the perceptions of player
players are subjectively rational given their perceptions, the payoffs in this
1).
Since
meta-game
are
^
oi
,
,
LL
„
.
,
for player 2
corresponds to choosing
The
has misperceptions.
the fact that
when he
L when
3,0
misperception
0,0
2,2
0,0
2,2
when he has accurate
strategy,
I,
1 is
L when
player
game (and
1 is
observed to have
observed to have misperceptions.
is
1
will
it
he
Game
1,
incorporating
choose
r,
and when he
/.
clearly to develop a misperception, inducing
(2,2),
Strategy
R when
has accurate perceptions, but
1
perceptions, player
in the underlying
LR. This generates the payoff
of the underlying
.
payoffs then follow immediately from
best play for player
dominated
RR
3,0
player
misperceives the payoffs, he will choose
The
RL
1,1
corresponds to choosing
accurate perceptions and also
LR
LR
1,1
,
,
,
the evolution oi perceptions:
Strategy
LL
accurate perception
1\2
Meta-game
him
game. The best response of player 2
which
is
is
to play a
to choose
obviously not a Nash equilibrium outcome
contains the play of a dominated strategy along the
equihbrium path). Replicator or similar monotonic dynamics
will therefore
encourage
the development of misperceptions.
Intuitively,
when
player 2 sees an opponent with accurate perceptions, he will choose
L, anticipating that his opponent will choose
when
nent
r,
giving both players (1,1). In contrast,
faced with a player with a misperception, he will correctly reason that his oppo-
will
play
I,
and he
will
choose R, leading to
(2,2).
This example therefore suggests
that evolutionary pressures, far from wiping out misperceptions,
may
fa\'or
apparently
"nonrational" behavior. Specifically, two types of nonrational behavior arise as
come
of the evolution of perceptions in this case:
first,
agents
may have
air
out-
systematic
misperceptions; second, these misperceptions lead to the play of strictly dominated (and
hence non-equilibrium) strategies in the underlying game.
The
1
role of the observability
were not observed,
assumption above
(deviation) without misperception
The
1 will
Is
the extreme case, agents
who
one that
equally important:
is
choose
is
B
B
is
A
die).
and B, where
—
it
guarantees that the
preferable to A.
Now
much
has
all
higher payoff
(in
agents have developed
imagine selection over two
subjectively rational and the other subjectively irrational
in fact, here
we take the
subjectively irrational type to play the action that
will
choose
B
since he perceives
to be preferable to A, while the subjectively irrational type will choose
fact that
he perceives
B
A
despite the
to yield a higher payoff. Evolution will favor the subjectively
irrational type, choosing A. In fact,
A
A
But, suppose that
minimizes his payoff). The subjectively rational type
will
at
the sense that he will not necessarily play the actions that maximize his perceived,
payoffs
it
a mutant
subjective rationality reasonable in an evolutionary setting?
a misperception and think that
(in
Specifically,
translate into a specific action, encouraging player 2 to
Consider the choice between two options,
different types,
the misperception of player
2.
misperception of player
'^
if
would choose r and receive the higher payoff of 3
subjective rationality assumption
change his behavior.
clear:
could not act as a commitment device.
it
the expense of player
is
a subjectively
irrational type
with this misperception
do as well as a subjectively rational agent with accurate perceptions, who perceives
to yield higher payoff
and chooses
it.
This simple example demonstrates that, in the
presence of misperceptions, there are no compelling reasons to expect evolution to favor
subjectively rational agents.
Let us
now
return to the analysis of
ceive payoffs, but also to deviate
that change
how a
Game
1
and allow players not only to misper-
from subjective rationality
given perception or plan
is
(Proposition 5) states that in this case where
mapped
we
(for
example, mutations
into behavior).
Our major
result
allow subjective irrationality, evolution-
ary forces will take us towards "effectively rational play"
—
i.e.,
towards behavior that
is
^To many economists and game theorists, subjective rationality is natural, almost tautological:
axiom of revealed preferences, simply represent choices. But in an evolutionary
model where preferences (perceptions) can be observable at the beginning of the game, this revealedpreference notion does not apply, since we cannot possibly observe the representation of the choice
before the choice has been made.
preferences, through the
indistinguishable from that resulting from rational play and does not feature dominated
strategies given the information set.
To
better understand this result,
4 different "strategies" for player
let
us
now
return to the above
meta-game and allow
accurate perception and subjective rationality; mis-
1:
perception and subjective rationality; accurate perception and subjective irrationaUty;
and misperception and subjective
player
ity
1
perceives
Game
we simply mean
when the
2
player
Here by misperception we mean that
irrationahty.
true
game
Game
is
while by subjective irrational-
1,
playing the dominated strategy in the perceived game.
1
allow no misperceptions for player
2.
Meta-game
of the evolution
of perceptions
and
play:
1/2
LL
LR RL RR
accurate perception and subjective rationality
1,1
1,1
3,0
3,0
misperception and subjective rationality
0,0
2,2
0,0
2.2
accurate perception and subjective irrationality
0,0
0,0
2,2
2,2
misperception and subjective irrationality
1,1
3,0
1,1
3,0
Again the
when he observes
player 2 denotes his action
first letter for
We
player
1
with
accurate perception, and the second letter gives his action following a misperception by
player. For example,
he will play
i?,
I,
when
so
the outcome
player
if
is
irrational (row 4),
player 2 chooses L, the
(2,2). In contrast, if
he perceives
chooses L, the outcome
see this,
first
tively rational
of player
1
is
player
1
and
if
outcome
and player 2 plays
(2,2) is
LR
(or
is
1
RR), leading
not observed.
player 2 plays
RL
curate perceptions.
(or
RR)
It is
1
plays
is
r.
subjectively
Now
if
player 2
(3,0).
to (2,2).
still
is
is
stable.
subjec-
Imagine a mutation
keeping the misperception.
seen by player
2,
Therefore, with both
taking the system away from
population in which player
still
is
misperceives the payoffs and
player 2 will continue to play R, but the mutant player
3,
and when he chooses
no longer (evolutionarily or neutrally)
has accurate perceptions
subjectively rational or not
(0,0)
has a misperception and
that leads to subjective irrationality, while
the higher true payoff
is
he chooses R, the outcome
consider the case that player
1
subjectively rational (row 2),
is
to give higher payoff, but
outcome
Recall that whether player
he
I
is (1,1),
In this meta-game, the
To
has misperceptions and
1
1 will
(2,2).
but whether
LR
and RR,
respond with r and obtain
Similarly, the
incumbent
has accurate perceptions but behaviorally irrational and
is
invaded by the behaviorally rational mutants with ac-
therefore straightforward to verify that any rest point of our
evolutionary process will yield the outcome (1,1)
—the Nash equihbrium outcome of the
underlying game. This can be achieved as a combination of player
developing an ac-
1
curate perception and subjective rationality and player 2 responding with LL, or player
1
developing a misperception and subjective irrationality and player 2 again responding
with LL.
In essence,
are
hmited to subjective
rationality, misperceptions provide
However, once we allow the evolution of "play" (how perceptions are
commitment.
mapped
when agents
into behavior) as well as perceptions, players can
undo
their misperceptions
by
modifying their degree of subjective rationality. This destroys the commitment value
of misperceptions.
The
crucial difference here
ceptions and the non-observabihty of play.
As
is
between the observability of misper-
is
standard in game theory, we do not
(and cannot) allow strategies to depend on the strategies of other players.
which action a player chooses can depend on
past actions,
it
his
and
So
wliile
his opponents' perceptions
and
cannot depend on the exact mapping that his opponents use to translate
these perceptions into actions
(e.g., it
cannot depend on their actions
in the future or
the actions they could have taken in other situations).
The
sense in which the resulting behavior
"rational" needs
is
some
elaboration. Evo-
lutionary forces take us back to equihbrium outcomes of the underlying game, but this
could be supported either by accurate perceptions and subjective rationality, or by misperceptions and subjective irrationality. This explains our choice of terminology: "effectively" rational behavior
may be a combination
irrationality. So, in practice
perceive the world or
of inaccurate perceptions
we may observe agents with systematic
how they make
decisions.
But these biases
will
and subjective
biases in
how they
be undone by other
biases they develop and will not shape their important decisions (their decisions affecting
long-run
fitness).
The notion
of "rational behavior" that emerges
than the usual definition of rationality as
it
therefore weaker
allows misperceptions.^
Yet agents are also rational in the stronger sense of
prior.
is
effectively having a
common
Although agents may appear to have systematic misperceptions, viewing the same
situation in different ways, they will behave as
if
they
all
share the same
beliefs.
For
example, suppose a certain behavior, say risk-taking, has a fitness benefit. Some agents
^It is also weaker than the usual definition because it requires behavior to be effectively rational
with limited information that is, rational given the information set of agents, which may not feature
—
full
information. See below.
may view
risk.
this
But
behavior as "too risky"
both groups
at the end,
,
will
while others view
it
as "fun"
undertake the activity as
if
and
well worth the
they had the same
(and preferences). In this sense, our analysis also provides a foundation for the
beliefs
"common
prior" assumption: even
will act as
if
they have effectively
though agents may have
common
they
different perceptions,
priors.
Consider an example to illustrate this point further. Suppose that each of two agents
have to take one of three actions,
B
world, A,
a, b
and
c,
and there are three possible
states of the
and C. For simphcity, ignore strategic interactions and assume that each
agent's payoff
is
only a function of his
receives a payoff of
when the
1
own
action and the state of the world. Each agent
action matches the state of the world, and a payoff of
otherwise. Clearly, one possible evolutionary equilibrium involves each agent recognizing
the underlying state of the world correctly, and choosing the action corresponding to
the state of the world he perceives
case,
both agents
will
share the
(e.g.,
a
if
the state of the world
is
A).
In this
same perception. However, under the assumption that
each agent's perceptions are observable, there are other evolutionary equilibria where
agents will have different (and inaccurate) perceptions, but act as
perceptions. For example, suppose that both agents perceive
agent
1
misperceives
B
B
and agent 2 misperceives
as A,
as
A
following behavior for both agents: play a
I's
is
perception
is
A
and agent
2's
if
perception
.
C, and play c
C
if
common
accurately, but
In this case, neither
differ.
agent 2's perception
is
they have
and
C
agent has accurate perceptions, and moreover, their perceptions
if
is
But consider the
A, play
agent
I's
b if
agent
perception
C. This perception-play combination will give the same evolutionary fitness as the
perception-play combination where each agent recognizes the underlying world correctly
and responds to
Therefore, in this simple example,
it.
have different (and incorrect) perceptions, but act as
obtain the
Finally,
maximum
it is
all
players.
players
1
To
they have
i.e,
an example where behavior
where
it
common
priors,
and
is
effective rational only
with
does not correspond to equilibrium play of the un-
derlying full-information game. This can
by
if
possible for both players to
payoffs.
useful to give
limited information,
it is
happen because
illustrate this, consider
and 2 play the following game:
of suppression of
infommtion
once again a simple example. Suppose that
1\2
Suppression of information
(Game
Here
R
3,3
-1,1
w',-5
0,0
3):
a random variable, which takes the value
w
Both players observe
1/2.
5,
if is
L
before making their decisions.
the unique Nash equihbrium outcome
Nash equilibrium with
misperception that
is
(0,0),
aB.
while
Now
players receiving (3,3).
w=
and
each with probabihty
5,
It is clear
when w —
0,
that
w =
when
there
is
also a
imagine both players develop the
the time. Then, the (true) payoffs of the perceived
game
become
1\2
and there
now an
is
equilibrium with
L
R
3,3
-1,1
2.5,-5
0,0
{I,
L)
all
the time.
So
a particular systematic misperception to suppress information
and
be eliminated by evolutionary
will not
forces.
in this
game developing
will benefit
Because the agents are
both players
still
playing
according to the equilibrium of some coarsend version of the underlying game, the out-
come
is still
effectively rational
with this limited information, but does not correspond
to the
Nash equilibrium of the underlying full-information game.^
2.2
Related Literature
This paper
is
related to a large
evolution will lead to
is
body
of research on evolutionary
Nash equilibrium
is
game
theory. \Vhether
a central question of this literature, and
it
well-known that any rest point of a monotonic evolutionary process over types pro-
grammed
to play different strategies corresponds to
a Nash equilibrium outcome
(see
Weibull, 1997, or Samuelson, 1997, for surveys).
To the
best of our knowledge, however, there has been
little
work on meta-games
where evolution occurs over perceptions and the mapping of perceptions
into actions.
The
who
notable exceptions are
Blume and Easley
(1992) and Sandroni (2000)
analyze
^The results regarding suppression of information rely on the assumption that there is a perfect
mapping between players' true perceptions and what others observe about the players. When percepstill be evolutionary forces leading to effectively rational behavior,
but we no longer obtain the suppression of information as an evolutionarily stable outcome.
tions are not observable, there will
10
the benefits of developing accurate perceptions in non-game-theoretic situations,^" and
who study
Dekel, Ely and Yilankaya (1998),
(implicit)
assumption of subjective
the evolution of preferences under the
rationality.
agents' preferences are observable, any "stable"
Dekel
et.
outcome
al.
(1998)
will yield
show
that,
when
the same average
payoff as that of the "eflicient strategy." For example, in the Prisoners'
Dilemma game,
the unique stable outcome will be mutual cooperation.^^
A
number
of contributions, including Banerjee
Yilankaya (1998), Ely and Yilankaya (1999), and
that in these types of models,
to
Nash
equilibria.
Our
when
and Weibull (1993), Dekel, Ely and
Ok and Vega-Rodondo
(2000)
show
preferences are not observed, evolution takes us back
results are related to these findings,
but enable us to analyze the
co-evolution of perceptions and play. Furthermore, they are not driven by the assumption
that preferences are unobserved. In fact, for the most part,
are perfectly observed.
rationality"
we assume that perceptions
Instead, our results follow because
assumption embedded
in
many
we
relax the "subjective
analyses of evolution of preferences.
result, evolution favors effectively rational play,
As a
but does not wipe out misperceptions
or systematic subjective irrationality. Agents can hold systematic misperceptions, and
in interviews
and experimental settings
will display
a variety of biases.
This type of
behavior would not be consistent with models of evolution of preferences, which leads
to the
We
same preferences
for all agents.
believe that an analysis of the evolution of perceptions
in part,
and play
is
because the whole discipline of psychology and the recent emerging
behavioral economics are explicitly concerned with agents' perceptions and
affect behavior.
ity
interesting.
field of
how
these
Moreover, such an analysis enables us to relax the subjective rational-
assumption, which does not have good evolutionary foundations in the presence of
make forecasts and saving
no further game-theoretic interaction. Bhnne and
Easley (1992) compute agents' wealth in the long-run and drive a corresponding fitness function. They
show that, when the preferences do not coincide with this fitness function, rational agents may lose their
wealth and "irrational" agents may determine the prices in the long run, consistent with our argument
that subjective rationality does not have an evolutionary basis when there are misperceptions. In the
same vein, in an overlapping-generation model with risk averse agents, De Long et al. (1991) suggest
that agents with inaccurate beliefs about the future prices may drive the agents with accurate beliefs
out when the strategies imitated on the basis of ex-post wealth they generate rather than the ex-ante
expected utility level.
'^ Similarly, Kockesen
et al., 1999, analyze a class of games in which evolution rewaxds "negatively
'°For example, in
decisions,
which
Blume and Easley
affect
market
prices,
(1992) and Sandroni (2000) agents
but there
is
interdependent preferences".
11
,
misperceptions.
Whether
cations regarding
subjective irrationahty
what types
of behavior
is
imposed or not has important impli-
wiU be evolutionarily
common
framework, we can investigate whether players have a
importance to the game theory
Our
analysis
most
is
work by Samuelson
which
prior,
Waldman
Waldman
(1994) analyzes evolution in a world without
He shows
two characteristics
that in the presence of sexual inheritance, which
imphes that the offsprings of successful agents are not their perfect
may
replicas, evolution
may
not be the
may be
stuck with
pick characteristics that are "best responses" to each other, but
global maximrmi.
utility functions
Using this framework, he
illustrates
how
agents
that do not maximize their fitness, and compensate for this by making
other systematic mistakes.
Samelson (2001)
is
closely related to
he illustrates that in the presence of deviations from Bayes'
his evolutionary fitness
consumption
of central
(1994) and to recent independent
strategic interactions (similar to the single agent case below), but with
and sexual inheritance.
is
literature.
closely related to
(2001).
stable. Also, using this
by distorting
own
levels to his
his utility function (e.g.,
utility function)
.
of "effective rationality despite misperceptions"
and Waldman's notions. To see
distorting his utility function.
this,
rule,
It is
is
Waldman
(1994), as
an agent can increase
by including the others'
worthwhile to note that our notion
somewhat
consider an agent
different
from Samuelson's
who maximizes
According to our terminology,
this agent
by
his fitness
has "accurate
perceptions," since he will identify his preferred actions with those that
maximize
his
fitness. In contrast,
agents with inaccurate perceptions in our setup will report to prefer
actions that do not
maximize
good
are
for their
long-run
their fitness,
but nevertheless end up taking actions that
fitness.
Finally, our results are also related to analyses of evolutionary
play communication
(e.g.,
Warneyrd, 1993,
Kim and
dynamics with
Sobel, 1995, Banerjee
and Weibull,
2000) because perceptions are similar to signals sent by players before the
played. In these models,
when observed
behavior, neutral stability leads to
(see
signals
do not
pre-
game
is
restrict agents' characteristics or
Nash equilibrium outcomes of the imderlying game
Banerjee and Weibull, 2000).^^ Nevertheless, perceptions are different from cheap
talk because they also encapsulate
When
what players know about the world, and whether
observed signals restrict agents' behavior, evolution may lead to non-Nash outcomes, as
strictly dominated strategies as in Banerjee and Weibull (1993).
Robson (1990), and to
12
in
they can distinguish between different worlds. In particular, when the underlying game
many
consists of
different worlds or
will restrict his possible
at
subgames
—as we have here, a player's perceptions
him
plays by confining
same (mixed) sub-strategy
to play the
any two worlds that are perceived to be the same. This
results, in particular, for the result that jointly beneficial
is
important
in
some
of our
suppression of information
may
survive evolutionary pressure.
The
3
We
start
problem
some
Single-agent case
by analyzing the evolution of perceptions and play
—
in
i.e.,
in
a single-agent decision
a setting without strategic interactions. This analysis
of the key concepts
and
illustrate the reasoning
Throughout the paper we consider a large
ary selection, and assume that there
we use
will introduce
in the multi-agent case.
set of agent types subject to evolution-
a process of replicator dynamics that relates
is
the frequency of different agent types to a monotonic fimction of their fitness.
interested in the stable
we
tionary analyses,
to
Lyapunov
We
are
outcomes of the replicator dynamics. Like many other evolu-
limit ourselves to Neutrally Stable Strategies,
stable points of (strictly) monotonic replicator
which correspond
dynamics
(see
Bomze and
Weibull, 1995, or Weibull, 1997).
Assume a world where we have a
(actions) s
:
C
^^
VL
possible "worlds"
Intuitively, different worlds
individual
is
for
We
finite
and denoted by
same action may have
Q
g^,
on
fl.
The
measures the frequency of the events and
consequence
c.
We
different payoff consequences in
W by Q, with Q {w)
be denoted by Eq. For each
C
:
^
R
(s;
w)
G
From a
strict
u^^ (c) is
=
Eg^,
13
{u^ o
W,
on the consequences, and
on
evolutionary point of view,
the evolutionary fitness following
summarize the base preferences of the agent by
V
w
>
pair (uu,,g,o) represents the base preferences
actions, or the agent's evolutionary fitness.
g^,
will
pair (u,^,^^,) of a utility function Uu,
a probability distribution
set of
W with a generic member w.
denote the probability distribution over
each w. The expectation with respect to
we consider a
some state-space O. The
for
correspond to different types of decision problems that the
faced with, and the
different worlds.
C of consequences and a finite set S of strategies
with uncertain consequences
assumed to be
is
set
s)
.
E S and
at each s
w
G W, where Eg denotes the expectation with respect to
This
q.
notation will be used extensively in the next section.
Analogously to the base preferences of the agent,
and preferences
we
u,
any probability distribution q
for
define the payoff function
=
V (s)
Eg
{u o s)
This function gives the ("perceived") payoff of choosing action s given preferences u
and probability distribution
V
S
:
—^ M.
We
We
q.
consider a finite set
Intuitively, a perception
w'.
The
tt
{lu) is
set of actual
:W -^V.
an interpretation of a world w as a (possibly
worlds that can be realized
of possible worlds that the agent can perceive
We
call
M'^ of payoff functions
define a perception as a function
TT
world
V C
a perception n* with
tt*
(w)
=
(i.e.,
is
VD
different)
naturally contained in the set
{v (,
v{-,w) at each
u;)
w
;
tu
G W}).
as accurate.
Intuitively,
an accurate perception enables the decision-maker to correctly recognize the world he
is
IT
confronted with.
(w)
7^
v{-,w).
Given any n and
uj,
from the underlying frequencies,
differing
from
fitness function,
Every time a world
w
is
u
i.e.,
^
is
i.e.,
g
^
w
if
or the perceived utility function
g^,,
Uyj.^^
V
—^
said to be subjectively rational
if
denoted by
Definition 1 A play a
E=
Any
function
a
:
S
is
tt
[w] and takes an
called a play,
and the
5^.
a
A
a misperception at
drawn, the agent perceives the world as
action s E S, determining his fitness.
is
is
This corresponds to either to the perceived probability distribution
differing
set of all plays
we say that there
{v)
G arg max v
and only
if
(s)
subjectively rational play maximizes the agent's perceived payoff.
standard definition of rationahty in
Game
Theory. For many, this
is
This
is
the
a tautology, for the
'^Accurate perceptions may also correspond to a pair {u,q) ^ {uw,<lw) of inaccurate probability
distribution and an inaccurate utility function, such that Eq^, {uu, o s) = Eq(uos). For an example,
see
Samuelson (2001).
14
perceptions (or beliefs) are merely expressions of the choice, derived from the choice
tion using the principle of revealed preferences. In this paper,
We
and choice independent.
Definition 2 A pair
(with
(tt,
information)
full
and accurate
rr*
,
if
will take
if
a
we take the perceptions
as a generic subjectively rational play.
cr*
a) of a perception and play
and only
f\inc-
o
n
=
is
said to be effectively rational
a* o n* for some subjectively rational a*
i.e.,
a
(tt
{w)) G arg
max v
(s;
w)
ses
at each
w
That
W.
G
is,
a pair
(tt,
and a play
a) of a perception
is
effectively rational
distinguish the resulting choice from that of an individual
who
if
we can never
perceives accurately and
behaves rationally. Here, we use a strong notion of accurate perception;
it
implies that
the agent distinguishes each state from the others (hence our qualification, with
information). In our next section,
we
will also define effective rationality
full
with limited
information, which only requires an agent to maximize his expected fitness, given the
information available.
of misperception
Exajnple
1
and subjectively
Consider the case
Suppose also that
V2 (6)
=
1.
Our next example
V=
illustrates that there will
be many combinations
irrational play that yield a rational choice.
W
—
{w},
S =
{vi,V2} such that vi (a)
=
{a,b} with v (a)
1
and
v^ (6)
=
0,
Therefore, Vi corresponds to accurate perception, while V2
=
1
and
is
and v
(b)
V2 (a)
=
=
0.
and
a misperception.
Then, there are four combinations of perceptions and play:
1.
agent perceives accurately plays subjectively rationally, and chooses
2.
he inaccurately perceives b better than
ing
3.
and plays subjectively
b,
but plays subjectively
rationall}-,
choos-
b;
he accurately perceives a better than
chooses
4.
a,
a;
and
b;
he inaccurately perceives b better than
chooses
irrationally,
a.
15
a,
but plays subjectively
irrationally, aiad
Combinations
fitness
1
and
and
4 are effectively rational with full information,
yield higher
than that of the other two perception-play combinations. With any monotonic
rephcator dynamics, effectively rational agents will survive, while the rest will die away.
Some
of these effectively rational players will hold misperceptions, they will
make
clearly
maximize
irrational choices given their perceptions, but the overall choice will
their
fitness.
More importantly, the concept
of subjective rationality loses
appeal in the presence
its
of misperceptions: the subjectively irrational agents in 4 do better than the subjectively
rational agents in
Therefore, in a world where the agents
2.
may have
misperceptions,
subjective rationality does not necessarily have an evolutionary foundation.
Now, given a play
we can
cr,
define effectively accurate perceptions, which are not
necessarily accurate.
Definition 3 Given any
and only
With
if
a
o
ir
-play a, a perception
= a o tt*
a,
we
said to be (effectively) a-accurate if
is
.
this terminology,
Given any
tt
(tt,
a)
is
effectively rational
define the one-person
if
and only
if
n
is
cr-accurate.
meta-game
where
g
at each
tt
e
11.
(tt:
a)
= Eq
[v
{a
{w))
(tt
Note that g{TT]a)
is
;
w)]
= Eq
[E^^,
[u^ {a
(tt
{w)))]]
the agent's expected payoff with respect to the
base preferences, given that, at each w, the agent's perceptions will be n {w) and he
will play
a and choose a
Even though
G^j is technically
since the perceptions
G
tt
11
game GV,
^^
(tt («'))•
cr
is
fixed but the perceptions
a game with a strategy space
When
g
(tt;
a)
<
evolutionary fitness will increase
We
also consider the
it
is
not a usual
vary.
game
are not chosen; the agent happens to have them, perhaps
because they inherited them or as a result of mutations. For
as a meta-game.
11,
can
(tt';
g
if
a), given that
his perceptions
meta-game
G = (nxE,p)
16
this reason,
we
refer to
it
he will choose according to a, his
happen
to change from
tt
to
tt'.
where g
(tt;
a)
is
defined as above. This
game
depicts a situation where both the per-
ceptions and the play evolve together.
Definition 4 Given any play a, a perception
g{-n\a)
>
g{7r';a) at each
g{-K';a') at each
G
(7r',(T')
G
tt'
Likewise,
FI.
G
tt
11 ?s
G
(tt, cr)
Neutrally Stable (NS) for
IT
x
E
is
NS for G
iff
G^
iff
>
giTT^a)
x E.
IT
This definition simply states that a perception, or a combination of perception and
play, will
be Neutrally Stable
if
no other alternative gives higher
require that there are no forces taking the system
away from
we
simpler version of the definition of Neutral Stability
strategic interactions.
fact that
Lyapunov
Our
(see, e.g.,
Proposition 1 Given any
iff it IS
outcomes
we
Therefore,
This
this configuration.
is
a
meta-games with
give below for
interest in Neutrally Stable
motivated by the
is
any monotonic replicator dynamics
stable outcomes with respect to
be Neutrally Stable
will
fitness.
Weibull, 1997).
subjectively rational
cr*
,
any perception
tt
^Yl
is
NS for G^.
a* -accurate.
The proof
states that
of Proposition
under subjective
1
is
combined with that of Proposition
rationality, the siuviving agents will
2.
Proposition
hold accrurate percep-
tions (and will choose effectively rational behavior). This result implies that the
of misperceptions
we found
Game
in
1
of Section 2
cannot happen in the single-agent case. This
is
is
1
example
due to strategic interactions and
intuitive, since as
noted in Section
2,
misperceptions are useful because they provide commitment, and in this single- agent
meta-game, commitment has no value.
Next,
we
see that
be some other
will
when
the requirement of subjective rationality
effectively rational agents
is
relaxed, there
who, despite their inisperceptions, do as
well as these agents.
Proposition 2 Any
G
(tt, cr)
11
x
E
is
Neutrally Stable for
G
iff it is effectively
rational
with full information.
Proof. Take any
by
g
definition,
(fr,
a)
= Eq
effectively rational (vr,^)
we have v
[v
{a
{tt
(u;))
{a
;
{tt
w)]
{w))
]
> Eq
w)
[v
—
{a
and any
niax^gs u
{tt
17
{iv))
;
(7r,cr)
(a; ly)
iv)]
=
g
>
(tt,
G
11
x E. At each
v {a
a),
{tt
{w))
;
w)
.
showing that
w
G
W,
Hence,
(tt,
a)
is
Neutrally Stable for G. Conversely,
and showing that
Hence, taking a
=a=
1,
not effectively rational, then the former
is
Q {ivq) >
observe that
a* above,
(tt,
a*)
is
rendering the latter inequahty
0,
not Neutrally Stable
(7r,cr) is
To prove Proposition
a)
some Wq with
inequality will be strict at
strict,
if (tt,
G.
for
effectively rational
we obtain Proposition
iff tt is
(T*-accurate.
1.
The Multi-person Case
4
In this section
we analyze
evolution
when
there are
many
players strategically interacting
with each other. ^"^ In that case, when the perceptions are observable, under subjective
may
rationahty, a player
Section 2 using
Game
gain from his misperception, a fact that
as an example.
1
presence of misperceptions, subjective rationality
we focus on a process
Therefore, here
is
established in
also seen that in the
no longer supported by evolution.
which both perceptions and play evolve together.
The environment
4.1
We consider
a set
For each
G
{v^ (s,
w)
profile s
lu
,v'^
E
=
A'^
{1, 2,
W, we
{s,w)
,
.
VC
finite set
,
i)^
.
.
.
,
n} of players and a
consider an underlying
.
Here
S.
.
the strategy profile s
be a
in
we have
Nevertheless,
we
(s, lu)
is
w)) G
{;" (s,
R"
is
game
lu.
The
YlieN
"^^
of strategy profiles.
{N, S,v {-^w)), where v{s,w)
—
the players' base payoiT vector at a strategy
t
E
N
set of all feasible payoff functions
is
taken to
measures the evolutionary
played at
=
5*
set
fitness of
a player
when
that contains
(R")
1
for
each non-empty
By
is,
TT
a perception,
W C W.
we mean any
once again, denoted by
=
(7ri,7r2,
.
.
j as perceived
perception
.
,7r„)
H =
function from
V^^\
By a
G H'^ of perceptions.
by player
we mean any
i
when strategy
it
:
VV
^
perception
We
write
tt^
V.
The
profile,
set of all perceptions
we mean any
vector
{s,w) for the payoff of player
profile s is played at
world w.
By
a mixed
probability distribution x on the set of perceptions. Here, x [n)
can be considered as the proportion of population
who
has perception n.
'"This corresponds to frequency-dependent evolution in the terminology of Maynard-Smith (1982).
18
>
We
TT,
{-.w) for all
observable.
i, is
other agents. This
That
motivated by the discussion in the introduction where we pointed
is
most
is
when perceptions/preferences
are observed.
A (Si), where A (S^)
any such strategy
With
this assumption, a player's strategy
it is
a
V^ — A (5)
+
:
game
the
will
be played given the perceptions.
in a
z
randomly selected imderlying game
plays role
{w).
tt
where
of perceived payoff functions
who
tt,
tt,
We
{w)
•
A (5",).
•
•
,
,
V" and any
also important
It is
S,v
s,
Since
G
S",,
A {Si)
G
7r„ {iv})
a^ (5j,w)
is
{,iu)).
randomly allocated to play
Each agent with perception
then have a profile
(w)
(tti
i
and the
profile
in the underlying
some
game
given any two strategies
—
A) y,
To
Aa +
(1
—
A)
(x,
a)
for
/_i)
and
player's
(y, //),
any A G
own
play a
[0, 1]
:
V^
-^
each profile v
A [S)
=
(tti
is
{N, S, v
•
,
•
•
,
7r„
(•;
s,
^""If
{w))
•
•
•
,
is
,7r„
{w)) €
V"
w)). Given any v
[v)
G
G
—
Finally,
play.
=
+
we consider the meta-game
G
+
(1
A) (y,
/j,)
also a strategy-
a pair {x,a) of a perception x G A(V'"
{v)
he
will play at
[w)) of perceived payoff functions.
the perception function were observed, players would
how he would have
,
according to at
of a player with (x, a) in a population with aggregate distribution
current world, but also
7r„
•
(Ax
the mixture A(x, a)
determining which strategy a^
[w)
•
perceptions, a inixed strategy can
formalize the evolution of perception and play,
where a mixed strategy of a player
•
,
(possibly mixed) strategy'
denotes the probabihty of playing
a already depends on the
(w)
(tti
be defined simply as a pair (x,a) of a mixed perception and a mixed
(1
tt,,
(w) denotes the payoff function perceived by the
Observing his role
i.
(A'^,
of perceived payoff functions, a selected player plays
CTj (tti
Notice that
5j.
(-jw), not the perception function,
at each date each player
perceives the payoff function as
player
V^ —
:
observable. ^^
is
We consider a process in which
role
cr^
can also be thought as a solution concept, since
the realizations of the perceptions,
that
itself
denotes the space of the probability distributions on
profile
how
determines
that
TT
an evolutionary stable outcome
likely to arise as
be a function that maps from the perception profiles into strategies:
will
player,
agents see the payoff matrix as perceived by
is, all
out that nonrational behavior
it
by each
realization of the payofT perceived
games where the
focus on
perceived each
lu
each role
W
.
i
The expected
(y, ji)
and a
and
at
payoff
of perception
know not only how an agent
£
)
perceives the
For our qualitative results,
it is
not
important whether the entire perception function or only the realization of this function is observable
(so long as a player does not observe more about his perception than the others do
such "informational
asymmetry'' may allow agents to always distinguish between all
€ U', see Section 5).
—
ft'
19
and play
is
where
=
s
(si,
-^^^Yl
=
U{{x,a),{y,fj,))
.
.
and
5„)
,
.
—
tt
(tti,
v'
.
.
with probabiUty
probability y
x(7r,),
(tTj).
Sj
with probability ^
be
(s,
ii''
some
(sj, tt
for each
e
A
pair (x, a)
>
tt
=
(w)
(tti
{w)
,
.
,
.
.
will
have perception
some perception
7r„
The
drawn
ttj
tTj
with
(w)), he will play
some
some strategy
agent's true contingent payoff for such a play will
E
e
dynamics.
is
G
said to be Neutrally Stable for
iff,
for each
[y, ji),
there
such that
{1
-
e)
{x,a))
> U ((y, //), e(y, i^) +
{1
-
e) [x,
a))
(0, e).
As noted above,
It is
neutral stability corresponds to the
also clear that
when
equilibrium in the meta-game G,
As
He
(w)), while every other player j will play
(sj, tt (w))-
U {{x, a), e{y, fi) +
(see for
(with probability 1/n) in a randomly
The
w).
Definition 5
exists
This equation states the following.
7r„).
while every other player j will have
with probability a,
G
Q {w) a,{s„TT {w))Y[ii{sj,n (iv)) X {TT,)Yly (tTj)
probability Q{w)).
Observing the profile
Sj
Si
i
game {N,S,v {]w)) (with
underlying
,
.
agent will be selected to play a role
is,w)
i.e.,
(.r, cr)
is
Lyapunov
stability in replicator
neutrally stable, {{x,a) ,{x,a))
U ((.r, a), {x, a)) > U {{y,
/j.), (.t,
is
a
Nash
a)) for each (y,
;x)
example Weibull, 1997).
in
the single-agent case,
we can
define a
meta-game to describe the evolution of
perceptions for a fixed solution concept and the evolution of the solution concept (or
play) given a fixed set of perceptions.
that
when
evolution
trally) stable
is
The
discussion of
Game
2 in Section 2 illustrates
only limited to perceptions, misperceptions can arise as (neu-
outcomes of evolutionary processes. Here we discuss the evolution of both
perceptions and play.
To
this end,
we
first
analyze evolution of play under a
(and fixed) perception, and then show that any combination of
actually
embed a common
perception.
Using
this insight,
perceptions
we then provide a
characterization of the simultaneous evolution of perceptions
20
uncommon
common
and
play.
general
Evolution of play under a
4.2
Consider the case where
we
perception
agents are restricted to have a
how they
payoff function, while
to the process
all
common
game
play the
common
perception of the
(given the perception) evolves according
defined above. In this case, the
common
prior (perception) assumption
holds by construction.
Consider a meta-game G^- where
all
some
a'^
where the strategy of an agent
he
7f
each role
will play at
=
(tt, TT,
.
.
players. Let also
let
II
That
^
V^
:
is,
a
V^
:
-^
A (S)
ii""
be such that
V
:
we can
=
=
for Gt,
U ((tt
V
=
/x)
,
,
(^,
(tt, tt,
e (tt
+
e^
for
iff,
(1
,
.
.
+
e)
/^i,
[1
a'"
{.,n [w))
a'^
and
there exists
—
=
[w)),
cr (., tt
/x^
when
wew
(., tt
e
G
The information incorporated
and
=
(u;))
similarly,
common
players have
all
all
/x(.,7r(w)).
meta-game U^
ses
as:
a^x
some
e
>
such that
U ((tt, a)
or alternatively such that
e) (tt, a)),
a) for each
Define
is v.
perception n to
that
ii^
{v)
common
Definition 5 immediately implies that a play
tt).
,
each
n)
-
.
and
^^5;]5^^;'(5,«;)Q(7i;)a,(5,,7^(u;))J];^^-(s,,7^(u-)),
ieN
tt
payoff function
define expected (true) fitness in the
f/((7f,a),(7f,/.))
tt,
A{S) determining which strategy a J
A{S) be such
-^
a and ^ induce the same behavior as
t/M'7^/i^)
ble
—^
when the commonly perceived
i
A{S) and
perception. Then,
where
V
:
as the perception profile that assigns the
tt)
,
.
is
agents share a fixed pure perception,
,
V^
a
is
Neutrally Sta-
e (tt, /i)
(a, e/i
.
+
+
—
(1
(1
—
e) (tt,
e) cr)
>
(0, e).
in a
common
perception
tt is
represented by the infor-
tt"^
{n{w)).
mation partition
r^
The
cell
that contains a given
can be distinguished from
w
w
{tt"^
is
{tt
{w)) \w e
W}.
denoted by /" {w)
using the knowledge of
tt.
—
tained in
function
/'^
P
tt
{w)
(lu).
is
is
in
/'^
(w)
=
G
lu'
Knowing the function
realization n{w), a Bayesian can infer that the true world
but when
No
tt
I'^
{w)
and the
vr"^ {tt{w)),
not a singleton, he cannot distinguish between different worlds con-
Therefore, the expected true fitness given the realization n
is
EQ[v\r{w)]^-
^-—
21
Yl
Hv^nQi^o).
{iv)
and the
a))
>
In the case where
v{-;w).
=
{n {w))
tt~^
and Eq[v\P
iv,
Bayesian players knew only the perception function
If
payoffs
an invertible function,
is
tt
{w)]
=
and the perceived
it
using this coarse information they could compute expected payoffs as
7r(tf;),
EQ[v\r{w)].
Definition 6 Given any
I""
play a
(tTjct),
^ff
a G arg max V" Eq
—
f'
for some belief
a
said to be effectively rational with respect to
is
is
[u' (5)
\r
{w)] a' (s^) /i~' (s_j)
i,
about the other players' play s_j for each
iJ,_j^
said to be effectively rational with limited information iff
with respect to some I^
iff it is effectively
.
A
a
play
is
and
w
G
W
.
A
play
effectively rational
it is
said to be effectively rational with full information
rational with respect to /"' where
Because a maximizes the agent's
N
E
i
utility
also rules out the play of
given
tt*
some
[w)
—w
at each
w.
regarding other play-,
belief, ^-_j,
dominated strategies given the information
ers'
behavior,
/'".
In this sense. Definition 6 generahzes Definition 2 from previous section to the case
it
When
with strategic interactions.
ers' actions,
with that
Eq
[v\P
is
the definition of effectively rational behavior with
independent of the oth-
full
information coincides
in Definition 2.
given
Next,
{N,S,Eq
the payoff to each agent
(w)],
P (w),
set
with a set of players
(w)])
[v\I'"
information
the
from the underlying game
A'',
(A'',
one
can
a
construct
information set /", which
may
establish that there
We
S, v).
game and
w
E
W
.
G„, then a
Therefore,
Proof.
that a
game
refer to this as
a coarsened game,
all
an arbitrary
underlying worlds from each other.
a close link between the Nash
ecjuilibria of this
derived
the neutrally stable outcomes of G„.
Proposition 3 Given any n
stable for
not distinguish
is
new
strategy spaces S, and payoff functions
since the payoff functions are the averages of the true payoff functions over
We now
set
We
(tt (u;))
will
is
b
(tt («;))
otx
is
:
a
W
V
-^
and any a
V
^>-
Nash equilibrium of game {N,
is effectively
a
A(S'), if
S,
Eq
[v\I'"
is
neutrally
[w)]) for each
rational with limited information.
prove the contrapositive of the
not a
:
Nash equilibrium
of {N, S,
22
first part.
Eq
Let a
[v\I" {w)]) for
:
V
—^
some
A (S) be such
w eW. Then,
,
there exists
some k e
INC = J2^Q
N and ak
[^' (^' ^"^ 1^" (^)] ^'^ (^^^
Now, consider a
{w)) G
(tt
:
J] E-Q
^ A (5)
F
_
,
_
\
[v'' (S,
A
"^
W)
such that
(5'fc)
("'))
n
"^
^"^^^
(Sfc, TT
(W))
'^^
^^J'
|/^ (W)] afc
<Jj
J]^
(Sj,
TT
(w))
>
0.
with
/
(^
^fc
if ^
('"^))
^ and v
— n {w)
otherwise.
^^^{v)
y
=
Using the definitions, we compute that
[/-(a,<j)
showing that
{a, a) is
=
f/'^(a,a)
n
not a
Nash equihbrium
equihbrium of G^ whenever a
That aoir
stable for G^.
follows
from the
+ /iVC--
is
fact that
is
y
Q(u/) >?7"(a,a),
^—^
of the
meta game
But
G-^.
a
neutrally stable for G^. Therefore,
effectively rational
{a, a) is
is
a Nash
not neutrally
with limited information then immediate^
no dominated strategy
is
played in a Nash equilibrium.
This proposition implies that only Nash equilibrium strategies in the "coarsened"
game
(A'',
S,
Eq
[vlP
are candidate neutrally stable outcomes of the
(«')])
The
ception meta-game, G.„.
can be neutrally stable
is
why Nash
intuition for
straightforward: suppose
equilibria of the underlying
all
players share the
perception. Then, standard arguments from evolutionary
equilibria of the underlying
game
will
even
The same reasoning immediately
if
players have misperceptions,
in the underlying
nally, to see
game,
why Nash
v,
is
what matters
their fitness,
is
and the information incorporated
to distinguish
between
be neutrally
stable.
When
all
players have the
different worlds,
We
will
that
Nash
with limited informa-
generalizes to the case where
Nash
tt is
not accurate:
determined
b}- payoff's
in these perceptions.
game {N,S,Eq
neutrally stable, recall the example in the introduction
beneficial.
game
same accurate
game theory imply
effectively rational
equilibria of the "coarsened"
was mutually
per-
be stable outcomes of the meta-game G^, and
therefore, evolution leads to behavior that
tion.
common
[v\I'^
Fi-
(w)]) can
be
where suppression of information
same perception not enabling them
equilibria of this coarsened
provide a more detailed analysis of
tliis
in
game can
Example
also
2 below.
Motivated by the idea that agents can adjust their behavior much faster than their
preferences, Dekel, Ely
and Yilankaya (1998) assume that agents
23
will
always play a
Nash equilibrium
game where the
of the
even though these preferences
may
payoffs are determined
not reflect their true
by
fitness.
their preferences,
Our
proposition, in
can be interpreted as showing that when players adjust their behavior
contrast,
than their preferences
(in
faster
the sense that their behavior can change while their preferences
are fixed), the play will approach a
Nash equilibrium
of the underlying
game, where the
payoffs are players' true fitness, not their perceived payoffs.
We next extend
a
common
pure perception).
X G A(n) as
perception
tt
perception.
f/'(CT,M)
A
at each realization
Now
nuxed common perception
is
perception (rather than
a probability distribution
Unlike in G, however, we require the agents to share the same
G.
in
common
Proposition 3 to the case of a mixed
common
Thus, by construction, players always share a
tt.
construct the meta-game C^; with payoff function
=
yU'(a.,j.)x(T,)
Tren
K&n i^N
where
=
7f
there exists
(tt,
.
.
some
.
s&S
]+i
Naturally, a play
,7r).
>
e
w^W
such that U^
(a,
e/i.
o
is
Neutrally Stable for
G^
+
(1
—
(1
> U^
e)a)
(^t, e//
+
iff
—
e)
each
for
/7.,
a) for each
ee(0,e).
Take any commonly perceived payoff function v
x{tt)
>
the profile of perceived payoffs
0;
{v,v,
is
functions that can lead to perceived payoff v by
.
—
.
.
11^,
i^
{lu) for
,v).
=
some
w
Denote the
{ttItt
{w)
=
G
W and
vr
with
set of perception
v for some
w
G W}.
Observing v and knowing the distribution x on perceptions, a Bayesian can update
beliefs
about the perception functions and compute the expected payoff functions given
Given any
this coarse information.
perception function
the
^
TT
rij,,
we have
commonly perceived
Bayesian) will be
given V
=
-k
{iv)
E[v\x,v]^
Eq
and x
Y^
tt
G
Fl,,,
the posterior probability that the
common
is tt is
^
When
his
,
2:
N
,
Pr(7r|;r,^;)
=
0.
X
(tt)
(tt)
Now, given any common perception
and
tt,
— k (lu), the expected true payoff (as perceived by a
= Eq [v\P (w)] Then, the expected payoff function
payoff v
\v\n~^ (v)]
will
.
be
Fr{Tr\x,v)EQ[v\TT-^{v)]
24
= J^
^r {tt\x,w {w))
Eq [v\r
{tu)]
.
Effective rationality
mth
respect to
x
is
=
tt
defined as in Definition
6, vising
E [v\x,v\
as the
payoff function.
For any perceived payoffs are f
Once
{N,S,E\v\x,v\).
fink
we
again,
(w),
we can now construct another derived game
game because
are interested in this
between the Nash equihbria of
game and the
this derived
there
is
a close
neutrally stable outcomes
resulting from evolution over play.
Proposition 4 Given any mixed common perception
(J
:
y — A {S)
>
{N, S,
is
neutrally stable for G^, then
E [v\x, n (w)]) for each w
E
W
.
a
Therefore,
(tt
{w))
a on
and any n with
x,
a
is
x(7r)
Nash equilibrium
is effectively
>
0, if
of
game
rational with limited
information.
This proposition extends Proposition 3 to the case of mixed
is
useful because
when we
mixed perceptions.
Proposition
4.3
We
consider evolution over perceptions and play,
omit the proof of this proposition which
we
encounter
will
similar to that of
is
3.
consider the
and show that
at
meta game G, where the perceptions and play evolve
common mixed
Towards
common
We will then show that,
for
is
at the
mixed common perception x embedded
neutrally stable for the
initial
common
{x,
meta-game Gx, where the
in x.
neutrally stable (x, a) for G, and for each realization
Nash equilibrium
uncommon
any neutrally stable
play a
a
we
first
derive the
perception x embedded in x will contain exactly the same
information regarding the underlying state as the
(tt («;)) is
this goal,
perception x embedded in (possibly non-common) mixed perceptions x.
Loosely speaking, the
profile X.
together,
any neutrally stable outcome the agents play a Nash equilibrium
of a coarsened version of the underlying game.
have a
perceptions. It
Co-evolution of perceptions and play
Now we
cr
common
of the coarsened
(mixed) perception
a) in the
meta-game G, the
agents' perceptions are fixed
Proposition 4 proves that for each
n
(w) of agents' perception profile,
game {N,S, E[v\x,n
(w)]),
where agents
perception (prior), and estimate their payoffs via Bayes' rule using
available information.
This
will establish
all
our claim in the introduction that despite
different perceptions, agents will play as if they share a
common
choose some effectively rational behavior with limited information.
25
perception, and will
We
VDV
take aset
common
with cardinality |y
embedded
perceptions
—
|
\V"'\-
any perception
in
The
profile
V will be the image of the
tt.^^ We also fix a one-to-one
set
and onto function
The
function
common
(/>
provide the isomorphism between the perception profiles and the
will
embedded
perceptions
Common
embedded
perceptions
=
profile 7r
(tti,
.
,
.
.
7r„)
them.
in
H^ we
G
,
in perception profiles
consider a (generalized) perception
TT
The
set of all
perception
common
tt
—
(f)
O
Fl,
we
perception
same information
tt
is
as the
is
have the perception
also
embedded
tt
:
W ^ V with
TT.
such (generalized) perceptions
G
Given any perception
n = V^ For each such
— 0"^ o We will say that the
denoted by
profile tt
.
jr.
in the perception profile n:
unconrmon perception
profile
That
tt.
it
contains exactly the
is,
at each
w
G W, we
have
/* (w)
=
=
n-^ {^ {w))
n"'
{(/)-'
(0(7r (w))))
Hence, the information partitions generated by n and
Pljg^
{w)
Z""'
—
=
{w'Itt, {w')
distinguished from world
w
TTi
{w)yi G
by looking
X on
n
x,
in
we
=
r (w)
are the same. Here,
tt
G H, where
in the
(l)~^
o
tt
=
= Pv{r'{^)\x) =
(tti,
.
.
.
,7r„)
=
tt.
mixed (uncommon) perception x
probability to the conunon perception
x assigns to the probability
/"^
=
{lu)
worlds, w', that cannot be
all
tt.
mixed (uncommon) perceptions are
define a
mixed coinmon perception
tt
=
o
tt
l[x{7T,)
(2)
The mixed common perception x
in the sense that
embedded
is
x assigns the same
in each perception profile
tt
profile tt itself.
""Note that this theory is useful when we approximate the contiunuum with grid V,
VI will have the same limit.
|y"| and
(1)
.
by setting
embedded
as the
n
at the realization of
mixed perception
S:{n)
at each
tt-\tt {w))
the set of
A''} is
The mixed common perceptions embedded
defined similarly. For each
=
I
26
in
which case
.
As was the case with
Given any
-n'
tt'
—
given X and
tt
and any
0o7r',
given x and
[w)
tt
[w),
and
it
Pr
i.e.,
X and
€
{tt'\x, tt
=
W
^
the probabihty that
Pr
J]
(tt'Ix, tt
(w))
Eq
partitions.
we have a common perception
we have perception
= Pr (7r'|x, tt (it;)).
(w))
'^PT{^'\i,n{w))EQ[i!\r
=
is,
w
x and x generate the same information
the same as the probabihty that
is
E[v\i,n{tu)]
That
it,
Hence, by
profile
tt'
we have
(1),
{w)]
[v\r (w)]
= E [v\x, tt
{w)]
the expected payoflr fmiction given the perceived payoffs wiU be the
.
same under
X.
The next lemma
the relationship between the meta-game G, where the per-
clarifies
ceptions and play evolve together, and the meta-game G^, where the play evolves under
the mixed
Lemma
common
1 Let
:
=a
a
be such that
d
perception
V
o
^
cf)
A{S),
and n
=
U''{a,il)
We
Proof.
sen
=
'eiv
:
Jio
V
-^
(f).
A (S),
a -.V^ -^ A{S), and
= U {{x, a
lemma
/x
:
V^
->
A (5)
Then,
U{{x,ao4,),{x,jlo(t)))
^
=U
{{x a)
,
,
{x
,
/j.))
5]]t'M5,^t;)(5(w)x(^)a,(s„^(u;))]^/ij(Sj,7r(u'))
w^iv ses
jyi
^EEEE^M5,-)QH
n
wew
TTcn^
This
—
}:l
have
= - J]^^
U^a,jl)
x.
o 0)
YlxiTT,)
ieN
ses
2fc/v
,
(x,
states that the
/i
o 0))
=
[/ {{x,
a)
,
(x, ^i))
outcome of the evolution of play
in
any situation with
widely differing beliefs and perceptions can be represented as the outcome of evolution of
play in a situation with
common
perceptions. Perhaps, the fact that there
perception that captures the same information as a set of
lemma will
surprising.
But
will evolve,
and the long run, precisely
this
uncommon
is
a
common
perceptions
is
not
enable us to next prove the stronger result that strategies
as if
all
perception.
27
agents are using this
common embedded
,
We can now
Evolution of perceptions
mon perception game G^, and
game
TT
(A'^,
5,
to
Nash
E [v\x, tt {w)]). We write
under a mixed perception
x,
Proposition. 5 For every neutrally
stable for
any
tt
{N S
,
,
d,
where x
>
with Pr(7r|x)
E [v\x
TT
,
(w)])
where
,
o
com-
for the probability of a perception profile
(7r|a:)
—
(7r|x)
w
=
it
to neutrally stable outcomes of the
YlieN-"^ (^«)-
stable (x, a) for
and any
0,
Pr
common
the mixed
is
G
result of the paper,
equilibria of a coarsened version of the underlying
Pr
i.e.,
and prove the main
state
which hnks the neutrally stable outcomes of
.
G
perception defined by
W, a {n
{w})
a
Therefore,
tt.
= ao (f)~^
G, the play a
o
neutrally
Therefore, given
(2).
Nash equilibrium of game
a
is
is
n
effectively rational with
is
limited information.
Proof. Take any neutrally stable {x,a)
a — a o (f)~^ and
such that
—
p,
each
for
e
€
U ((x, a)
But, by
Lemma
1,
Since (x, a)
fiocf).
is
Take any
G.
for
fi
:
^
V
^{S), and
some
neutrally stable for G, there exists
e
let
>
(0,e),
,
(x, e/i
+
-
(1
e)
a))
> U
((x, p)
(x,
,
ep+{l-
e)
a)}
we have
[/*(CT,e/i+
(1
-e)a)
-
[/
((x,a o
(/>)
,
(x, e/i o
+
(1
-
e)
a
o 0))
= U{{x,a),{x,ep + {l-e)a))
and
U"^ {p, ejl
+
{I
—
e)
a)
U^
showing that a
is
— U ((x, p)
(a,
e^
+
-
(1
,
(x, e//
e)
a)
follows
of
cr
o
7r
TT
> U^
(w)]) at each
wG
a
{tt
W whenever
e) cr)).
{~p, e/i
+
(1
Therefore, for each
-
e)
e
e
(0, e),
a)
agents having
since
common
.x
~ a
o <p~^
is
neutrally
— a {tt {w)) is a Nash equihbrium of
(tt) = Pr (7r|x) > 0. Effective rationality
will
result.
ensure the emergence of behavior
perceptions and playing the Nash equilib-
rium behavior of some underlying game. That
28
a
{w))
This proposition states that evolution of play
all
4:
Nash equilibrium
again immediately follows from the
corresponding to
—
from Proposition
stable for G^, by Proposition 4 (for G^),
E [v\x,
(1
neutrally stable for Gf.
The second statement
(A'', 5",
+
is,
evolution will lead agents to play as
they have a
if
This
librium).
and act
common
is
prior and are effectively rational (in fact, they play an equi-
may
despite the fact that each agent
hold different mlsperceptions
"irrationally" given their perceptions. Their behavior will effectively
undo these
perception differences, leading to effectively rational behavior and to a situation with
effectively
common
perceptions.
Notice that the proposition
Nash
ecjuilibria of
maps the
neutrally stable outcomes of evolution to the
some coarsened game, yielding
This
ited information.
may
effectively rational behavior
with hm-
not, however, correspond to the equilibrium of the full-
may
information game, because some suppression of information
emphasis on limited information). This
tions that suppress information that
some "insurance" opportunities
is
because
all
may
agents
would reduce the payoff to
— as in Game 3 of Section
We
2.
take place (hence our
develop mispercep-
all
parties, destroying
will
now
present such
a neutrally stable outcome of this game in more detail.
Excimple
2:
Suppression of information
Consider the same example as in
game
3 in the introduction, with true payoff function v for this
1\2
where x can take values
with
and
Q {wi) — Q {W2) =
L
R
3,3
-1,1
u',-5
0,0
1/2,
w
where
takes
is
We take W —
5 with equal probabilities.
wi and 5 at
at
{wi
'W2-
{v{-;wi) ,v [,W2) tV] as the set of possible payoff functions, where v
V
(•;
W2))/2, which corresponds, in this case, to
pure perception-play pair
(tt ,
a)
-7/
where
\
vf
f
(u;)
(1,0)
'''*'"''^'={(0:i)
for
or
each
i
^ N. Here, the
L when he
plays
1
entry
is
+
v at each
if
f1
=
i;2
5) /2
w
G
=
v,
=
2.5.
0,
uh
=
5}
We take V =
— ({)(•; Wi) +
We
consider the
W and
otherwise
the probability that a' assign to the strategy
each player perceives the game as the average game, and plays
When
everyone has perception- play pair
The only way a mutant may have a
he chooses
(0
—
1
plays rows or columns, respectively. According to this strategy, a player
(or L) if
otherwise.
first
=
w =
Game
r at 1^2,
{it,
higher payoff
is
R)
a), each player gets 3.
that
and the column player continues to play
29
r (or
when he
L, giving
is
the row player
him a payoff of
5.
Strategy &" (^1,^2) dictates that the column player will play
mutant
in this case, perceives the payoff function at
must play
will play
r at {y, v).
At w\,
if
W2-,
only
is
(tt,
opponent, the
the column player
v,
(when he plays
the highest our mutant can get
lower than his payoff of 3 under
if its
In that case, our mutant
v.
the mutant's perception differs from
R, in which case the highest our mutant gets
he distinguishes W\ and
If
W2 as
L
is
r).
Hence,
=
(0.5)(5)+(0.5)(0)
if
2.5,
a).
the mutant does not distinguish Wi and W2, he must perceive the payoff function
In that case, the profile of perceived payoff functions will always be
as v at wi, too.
(vjC*),
hence the incumbents
always play L, and the mutant
will
again obtain (0.5)(5)+(0.5)(0)
=
2.5,
lower than
lower payoff, unless his perception function
check whether these mutants,
who do
is tx
3.
will
In either case, a
and he plays
1
always play
r
and
mutant gets
strictly
Now we
need to
at {y, v).
as well as incumbents against incumbents, cannot
do any better when they meet each other. But, when any two such mutants meet, each
will get exactly 3,
not higher than the incumbents' payoff. Therefore,
[it,
a)
is
neutrally
stable.
game may be
neu-
does not correspond to a Nash equilibrium of the
full-
This example shows that some Nash equilibrium of a coarsened
(even
when
information game).
Not
trally stable
it
all
Nash
equilibria of all coarsened
games can be neutrally
stable outcomes, however. For example. Proposition 2 shows that in one-person games,
each neutrally stable outcome leads to effectively rational behavior with
tion, ruling
out
all
non-trivial coarsenings.
We
full
informa-
next investigate this issue in games with
strategic interactions.
4.4
A
further restriction
In deriving Proposition
5,
mutations of perceptions.
and derive a stronger
we considered only mutations
We
will
restriction
now
of play,
and did not consider
consider mutations of both perceptions and play,
on the set of neutrally stable outcomes.
Let us write
for the
expected value of having perception x and playing a, while
30
all
the other agents
'
.
do the same, given that the underlying world
IT = -Y^ min max
"'
i&N
as the
is
minimum expected
w and
utility
We
also define
w*)
]^
v' (s,
jeN
s£S
an agent gets
Gj (s,)
he recognizes that the underlying world
if
gives his best response. Finally, write
Vj:
=
for the set of payoff functions
Proposition 6 Let
at each
'
^
w.
is
w
=
{v
TT
(w)
>
\x (tt)
0,
w
W}
e
perceived with positive probability under x.
a) be neutrally stable for G. If Vx
{x,
^V
then
,
^W
Proof. Assume V^
a neutrally stable
^V and take any v G
(x, a)
V\Vi. To derive a contradiction consider
> U^'
such that IP"
{x,a) for
W
some w* G
Consider the
.
following mutant (x,(t):
1.
he recognizes w*: x
7f
2.
[w]
=
=
(tt)
n [w) whenever
x
(tt)
w^
for
^'
(-5'
^^)
c^i
tt
with x
>
(tt)
0,
tt
=
{w*)
v,
and
w*; and
he gives his best response at w*:
arg max„, Ylses
each n and
(s^)
each
for
FIj^z
-^j
& N, when
i
(sj, vi,
•
.
.
,
Vn),
and
—
Vi
a^
v, a,
(t'l,
.
.
.
(I'l,
,
t'„)
.
.
=
,u„)
.
a,
G
(I'l,
.
otherwise.
When w ^
w*, the mutant (x, a) does as well as the incumbents.
that he will do strictly better than the incumbents at w*. At
-
>
^ ^
— >^
n
\^
{v^^\x,
w*)
Pr
(v_j|x,
w*) min
J2 n^in iTiax 5]] v'
i6Af
max
Pr
ses
(s,
w*)
n
jeN
^
v' (s,
w*) a^
max N^i)^
^J (5j)
31
=
[s,)
(s, u'*)
^'^'
>
iv*
,
will
now show
the mutant gets
J]^
TT
We
Oj {Sj, {v, i'_,))
cTj
^"'*
O^'
(sj)
'^)
.
.
.
,
t'„)
.
where
the vector of perceived payoff functions where
{v, i;_,) is
the payoff function as v and
V-i) given that the
{v,
lu*
respectively,
Vj,
and any other
and Pr {v_^\x,w*)
j perceive
the probabihty of
is
incumbents have mixed perceptions x and the underlying world
(Note that X^„_ gyn-i Pr {v_i\x, w*)
.
i
—
This establishes that his expected payoff
contradicting our hypothesis that {x,a)
and
1;
will
is
at
be
w* each incumbent gets V"'
,
strictly better
is
(x, a).)
than the incumbents',
neutrally stable for G.
This proposition shows that in any Neutrally stable outcome an individual cannot
more information. This
increase his payoff by introducing
games whose Nash
We
have so
When
far limited
The previous
Perceptions Are Not Observable
the analysis to situations in which perceptions are observable.
literature has also investigated situations in
be only imperfectly observable
1999,
(e.g.,
We now
may
which agents' preferences
Dekel, Ely and Yilankaya, 1998, Ely
and Ok and Vega-Rodondo, 2000).
Suppose that
coarsened
equilibria can correspond to Neutrally stable behavior.
Evolution
5
restricts the set of
and Yilankaya,
extend our analysis to this case.
in addition to the observable perceived payoff function 7r(-,u;),
each
agent also has a private perception function n^. As a result, each agent will observe his
own
Ti-P
{w),
and can condition
his behavior
private perception of other players.
We
on
this perception,
define a perception profile for player in role j,
that each agent can condition his play on
We
(tt
(»;))
is
Therefore,
Proof.
a
a
o
Nash equilibrium of
n
is effectively
It will
all
tTj,
a
:
as tZj—
A (S).
1/^+^ -^
Similarly,
(tt^, tti, ..., Tr^).^^
observed perceptions and his
we
This implies
own
private
An immediate
stable (£, a) for
the underlying
game
G
and any
{N, S,v {,
tc
lu))
with x{Tt)
for each
w
>
G
0,
W
rational with full infoimation.
generalization of Proposition 5 implies that
stable outcomes correspond to
''
not observe the
can now state
Proposition 7 Given any neutrally
a
will
can incorporate this into our analysis by simply
redefining the play function to have a larger domain,
perception.
but he
Nash
turn out to be redundant that
equilibria of
(7rj,
...,
tTh)
some coarsened game.
are observable, given that
tt^ is
all
neutrally
We
only have
private information.
Nevertheless, this more general form highlights that adding a flexible piece of private information to
our setup so
far
removes many of the neutrally stable outcomes of Proposition
32
5.
to rule out
Nash
underlying
game
is
s'j
(A^,
S,v {,w)). Suppose that
a Nash equilibrium of a game {N, S,
{N,S,v {,w*))
Eq
(x,
a)
is
equilibria of the
neutrally stable,
\v\x, tt (w*)]),
and a [n {w"})
but not a Nash equilibriimi of
some w* G W. This imphes that there
for
a role
exist
j,
and a strategy
such that
y
Now
is
tt^
=
yr*,
where
his fitness. Hence, (x, a)
full
is
information game,
effectively rational
with
than Proposition
the accurate perception. This devi-
6:
of
when he
it
rules out the play of all
full
still
given by a^j
plays the role j at w*
not neutrally stable. Since
a
{n [w))
is
dominated
(tt
{w)).
increasing
,
a Nash equilibrium of
hence a o
strategies,
tt
information.
to be observed,
we obtain a stronger propo-
now only Nash equilibria of the underlying game
neutrally stable outcomes. This
Nash equilibrium
s'j
we allow perceptions not
Therefore, once
sition
vr* is
[v'\x,7t{w)]
not observed by other players, so their strategies are
This then enables the mutant to choose
the
> Eq
{s'^,a_j{7r{w)),w)
consider a mutant with
ation
is
games that are not Nash
equilibria of these coarsened
is
intuitive, since starting
from any outcome that
some coarsened game, but not a Nash equilibrium
game, a player can take a deviation to expand
are candidate
his information set
is
a
of the underlying
and increase
his payoff.
Concluding Remarks
6
There are many examples of potentially
benefits
by acting
may be a
useful
as a
commitment
commitment
we argue that a key
irrational behavior that
device. For example, the
to punish those
who break
may have
evolutionary
tendency to seek revenge
their promises.
In this paper
link in this reasoning, that of subjective rationality, does not
strong evolutionary foundations, undermining
much
have
of the appeal of this argimient.
Subjective rationality requires that agents always choose actions that maximize their
perceived payoffs. Although this appears almost tautological,
we show that when
there
are misperceptions, subjectively irrational agents can have greater evolutionary success
than subjectively rational agents.
Once we allow mutations to lead to subjective
tions,
we
irrationality as well as mispercep-
find that selection will lead to effectively rational behavior, albeit with limited
information:
perceptions or preferences act as
33
commitment
devices only
when they
.
and relaxing subjective
are informative about future behavior,
link.
Thus somewhat
rationality breaks this
paradoxically, subjective irrationality creates stronger evolution-
ary forces towards "rational behavior"
Interestingly, there
is
no immediate link between rational behavior and accurate
perceptions. Evolution will select agents
who
will act "rationally"
have very different perceptions, and sometimes
will
,
but these agents will
make systematic mistakes
(yet their
mistakes will cancel each other). This reasoning suggests that systematic misperceptions
in experimental settings or in situations with little relevance to long-run fitness
do not
necessarily translate into widespread "irrational" behavior.
We
also
show that even though,
an "evolutionary equilibrimn," agents
in
very different perceptions, they will possess effectively
and the
payoffs, in the sense that they will play as
This result
is
assumption
in
of interest since there
is
game theory has good
that this assumption
It is
common
may have good
if
perceptions of the
common
they have a
disagreement over whether the
theoretical foundations.
Our
all
game
prior.
common
evolutionary foundations.
situations. First, according to our results,
will be-
when perceptions
observable, behavior will be effectively rational only with limited information, and
deviate from the
Nash equilibrium
pression of information
prior
analysis suggests
important to emphasize that our analysis does not imply that agents
have "rationally" in
have
will
of the full-information game.
may be mutually
This
is
are
may
because sup-
beneficial for all agents. Second, our analysis
has been limited to the case where there are no cognitive restrictions on behavior, and
therefore should be interpreted as implying that there will not be systematic biases given
the cognitive resources available to the agents. This does not rule out boimdedly rational behavior because of cognitive limitations. Finally, as with
all
evolutionary analyses,
our results apply in the "long run", and adjustment, especially after important changes
in environments,
may
take a long time, during which behavior that
evolutionarily stable can be observed.
34
is
not neutrally or
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