Games and Economic Behavior 67 (2009) 351–362
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Games and Economic Behavior
www.elsevier.com/locate/geb
Categorizing others in a large game ✩
Yaron Azrieli
Department of Economics, The Ohio State University, 1945 North High Street, Columbus, OH 43210, USA
a r t i c l e
i n f o
Article history:
Received 9 April 2008
Available online 27 January 2009
JEL classification:
C72
Keywords:
Categorization
Large games
Conjectural equilibrium
a b s t r a c t
We study the efficiency of categorization of other agents as a way of saving cognitive
resources in the settings of a large normal-form game. We assume that, when an agent
categorizes (partitions) her opponents, she only has information about the average strategy
in each category. A strategy profile is a conjectural categorical equilibrium (CCE) with
respect to a given categorization profile if every player’s strategy is a best response
to some consistent conjecture about the strategies of her opponents. It is shown that,
for a wide family of games and for a particular categorization profile, every CCE is
an approximate Nash equilibrium when the number of players is large. This result
demonstrates the potential of categorization as an efficient way to store information
in complex environments. Although possessing a coarse description of their opponents’
strategies, agents behave as if they see the full picture.
© 2009 Elsevier Inc. All rights reserved.
1. Introduction
It is commonly accepted in the psychology literature that people categorize the world around them. Perhaps the main
purpose of categorization is to help the cognitively limited categorizer to deal with the huge amount of information she
obtains from her complex environment. In particular, information about other people’s behavior is often processed with the
aid of social categories.1
In the context of interactive decision making, the fact that agents think in a categorical way about others raises several
questions. From a normative perspective it is interesting to check whether, or in what situations, categorizing other agents
is an efficient way to store information about their behavior. In addition, the question of what is the right way to categorize
seems important. From the descriptive point of view it is natural to ask whether predictions of standard solution concepts
change significantly if one assumes that agents categorize their opponents in the game.
In this paper, we try to address these questions in the settings of a large (with many players) non-cooperative normalform game. Remembering the entire strategy profile in a large game requires a huge amount of cognitive effort. It is
therefore natural to study the performance and implications of categorization when the number of interacting players is
large.
We interpret a categorization of a player as a partition of the set of her opponents in the game. We assume that, when
some player i categorizes her opponents, she only possesses a coarse description of the strategies they play. Namely, she
only keeps track of the expected proportion of her opponents choosing each strategy in each one of the categories in her
✩
This work is based on a chapter from my Ph.D. dissertation written at the School of Mathematical Sciences of Tel-Aviv University under the supervision
of Prof. Ehud Lehrer. I am grateful to Ehud Lehrer as well as to Pierpaolo Battigalli, Yuval Heller, Philippe Jehiel, Ehud Kalai, Myrna Wooders, participants
of several seminars and conferences, three anonymous referees and an associate editor for very helpful comments and references.
E-mail address: [email protected].
1
Good surveys of the social psychology literature on categorization are Fiske (1998) and Macrae and Bodenhausen (2000).
0899-8256/$ – see front matter
doi:10.1016/j.geb.2009.01.004
© 2009 Elsevier Inc.
All rights reserved.
352
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
categorization.2 Notice that, keeping the number of strategies fixed, the memory size required to store this information is
linear in the number of categories in i’s categorization. Thus, if the number of categories is significantly smaller than the
number of players, i uses much less memory to describe her opponents strategies.3
According to our assumption, a player who categorizes has an imprecise description of the strategy profile of her opponents. It is therefore not clear how such a player chooses her strategy. Since we are interested in measuring the performance
of categorization we adopt here a “worst case approach.”4 That is, we allow the categorizer to hold any belief about the play
of her opponents, as long as it is consistent with her information (i.e., induces the same expected strategy in each category).
When the strategy of every player in the game is a best response to some (consistent) conjecture, we say that the strategy
profile is a conjectural categorical equilibrium (CCE) with respect to (w.r.t.) the given categorization profile.5
With this modeling choice, the above questions are naturally translated into well-defined analytic problems. The efficiency of a categorization of a player in a given game can be measured by the largest possible loss incurred by this player
due to her (possibly) wrong conjecture. We say that a categorization profile is ε -efficient if this maximal loss is no more
than ε for each one the players. Measuring the efficiency of a categorization allows also to compare different categorizations. From the descriptive point of view, efficiency of the categorization profile implies that predictions of the CCE concept
are the same as those of Nash equilibrium.
Our main result concerns the existence of efficient categorizations. Of course, the finest categorization in which every
category consists of only one agent is always efficient. The question, therefore, is whether coarser efficient categorizations
exist. In Theorem 1 we answer this question in the positive for a class of large games that satisfy certain conditions. We
point to a specific categorization profile which is endogenously determined by the payoff functions in the game, and show
that it is ε -efficient if the number of players is large enough.
This result is interesting for several reasons. First, it demonstrates the potential of categorization as an efficient way to
store information in complex environments. Although possessing a coarse description of their environment agents behave
as if they see the full picture. Second, our result can be seen as a recommendation for how one should categorize others
when involved in a game-like situation. Finally, this result also increases the plausibility of the Nash equilibrium concept in
large games since it shows that Nash equilibrium must emerge even if players have limited information about the strategies
of their opponents.
To formalize our result, we refer to the model of Kalai (2004, 2005). There is a finite universal set of actions S. Γ (S) is
a family of normal-form games such that, for every game G in Γ (S) and for every player i in G, the set of (pure) strategies
available to i is some subset of S. With a fixed family Γ (S) in hand, one can very naturally define notions of uniform
continuity and anonymity of the payoff functions in Γ (S). These are the key assumptions needed to obtain the asymptotic
existence of efficient categorization profiles. See the next subsection for more on the relation between our assumptions and
results and those of Kalai (2004, 2005).
1.1. Related literature
Our main result (Theorem 1) is inspired by the works of Kalai (2004, 2005) on the robustness of equilibria in large games.
There it is shown that, when the number of players is large, Nash equilibria of a wide family of games are immune to many
modifications of the game. These modifications include various extensive form versions of the game such as sequential play
(instead of simultaneous play) and versions in which players can revise their initial choices. The main difference between
the current paper and Kalai’s is that here we keep the game unchanged while allowing players’ beliefs about their opponents
strategies to be incorrect.
There are also differences in the model and assumptions used. First, while we study only complete information games
Kalai allows players to be of several types (though a key assumption in his paper is that types are drawn independently
from a universal finite set). Another difference is in the anonymity and continuity assumptions used. In Kalai’s paper the
payoff to a player depends on her own type and action and on the empirical distribution of type–action characters of the
other players. Since the set of types is finite, this means that the games are ‘semi-anonymous,’ which is not required for
our result (see however Definition 4). The Lipschitz condition we use (Definition 3(ii)) is also different from the uniform
equicontinuity condition of Kalai (2004, Definition 3). Neither of them implies the other. In a recent paper Deb and Kalai
(2008) study stability of equilibria in large games that satisfy a Lipschitz condition similar to ours. An alternative framework
for analyzing large finite games is that of Wooders et al. (2006) and Cartwright and Wooders (2003a, 2003b).
2
The scenario we have in mind is that of a repeated play with no strategic links among the repetitions. Players use past experiences in order to form
conjectures about their opponents play and best reply to their conjectures. The expected proportion of players choosing a certain strategy in some category
corresponds to the average (over time) proportion of players that choose this strategy in that category.
3
An alternative more informative statistics that i can store about the strategy profile is the distribution of the number of players choosing each strategy
in each category. However, the memory size required to store this information is proportional to the number of players in the game. Therefore, it seems
inappropriate to assume that this is the information available to i when one studies the performance of categorization as a way of reducing mental effort.
See also Section 1.1 for a discussion of a related result of Cartwright and Wooders (2007).
4
This is similar in spirit to the way in which the performance of an algorithm is often measured in theoretical computer science.
5
It is possible to refine the set of CCE’s by requiring that each player plays a best response to the (unique) consistent conjecture in which all the players
in every category play the same strategy. We discuss such stereotypical beliefs in Section 6.1.
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
353
Cartwright and Wooders (2007, Proposition 1) prove a result which is in the same spirit as our Theorem 1. A player is
said to have stereotypical beliefs w.r.t. a given partition of the set of players, if she believes that all the players in any set
of the partition play the average strategy of that set. They show that stereotypical beliefs of a player about the strategies of
her opponents result in a small lose of utility for this player, given that the partition is into sets of approximate substitutes.
The most significant difference between the result of Cartwright and Wooders (2007) and ours is in the information that
players have about the strategy profile and, therefore, in the conjectures they form. While in our model each player only
knows the expected proportion of players in each category that choose each action, in their model each player knows the
joint distribution of these random variables. Thus, a player in our model has a much coarser description of the information
relevant to her decision. This is the reason why our results only apply to Nash equilibria of large games while their results
apply to correlated equilibria of games with any number of players.
Our notion of CCE is a variant of Battigalli’s (1987)6 conjectural equilibrium (CE). While CE is defined for arbitrary signal
functions, we consider the special case where the signals are the average strategies in groups of players. Closely related
and better known solution concepts are Fudenberg and Levine’s (1993) self-confirming equilibrium and Kalai and Lehrer’s
(1993a, 1993b) subjective equilibrium. An interesting refinement of CE is Rubinstein and Wolinsky’s (1994) rationalizable
conjectural equilibrium, which we discuss further in Section 6.3.
Several recent papers study issues related to categorical thinking in the context of decision making. Fryer and Jackson
(2008) develop a model of how past experiences are sorted into categories, and show that certain biases in decision making
emerge from this process. Pȩski (2007) shows that in symmetric environments categorization is an optimal way for predicting properties of future instances based on past instances. Azrieli and Lehrer (2007) axiomatize categorizations that are
generated by proximity to a set of prototypical cases. In Azrieli (2007) it is shown that categorization can create correlation
in the strategies of players in a random matching environment.
1.2. Organization
The next section illustrates the CCE concept and the main result of the paper by means of a simple example. Formal
definitions are given in Section 3 and the main result is in Section 4. Section 5 suggests several ways to extend and
generalize the main theorem and Section 6 contains some remarks. All the proofs are in Section 7.
2. An example
To illustrate the model and the main result of the paper consider the following family of games. For each natural number
n 1 there is a game with n + 1 players. In each game, player 1 is a male and the rest of the players {2, . . . , n + 1} are
females. Each one of the players should choose between spending his/her Sunday at the village (v) or at the beach (b). The
payoff function of each female is constant. Let p be the proportion of females that are at the village and q = 1 − p be the
proportion of females that are at beach. The payoff to the male is 0.01 + p 3 if he chooses v and q3 if he chooses b. Thus,
most of the male’s utility comes from matching the choice of the females, but he also gets a small constant utility from
choosing v.
Consider first the case where n = 3. Assume that the strategies of the three females are as follows. Players 2 and 3 play
v with probability 3/4 and b with probability 1/4, and player 4 plays b with probability 1. A simple computation shows
1
6
9
13
· 03 + 16
· ( 13 )3 + 16
· ( 23 )3 + 0 · 13 = 0.01 + 72
. On the other
that if the male chooses v then his expected payoff is 0.01 + 16
1
6
9
14
hand, choosing b gives the male an expected payoff of 16
· 13 + 16
· ( 23 )3 + 16
· ( 13 )3 + 0 · 03 = 72
. Thus, the best response for
the male is b.
The expected proportion of females in each one of the locations according to the above strategy profile is 0.5. Now, if the
male puts the three females into one category, then our modeling assumption implies that these expected proportions are
his only information about the strategies of the females. That is, the male only knows that on average half of the females
play v and half play b. The simplest conjecture of the male that is consistent with his information in this case is that all
three females play either v or b with equal probabilities. According to this stereotypical conjecture the distribution of the
number of females in each one of the locations is the same. Thus, under this conjecture it is optimal for the male to play v.
The former analysis shows that, when the number of females is small, it is possible that the male will choose the wrong
action and lose utility due to his categorization. Consider a similar game where the number of females n is large, and fix
any strategy profile for the females such that the expected proportion of females in each location is 0.5. Then by the weak
law of large numbers the realized proportion of females in each location will be close to 0.5 with high probability. This
implies that, for sufficiently large n, a best response for the male is to play v. It follows that, even if the male categorizes
the females and plays a best response to a stereotypical conjecture (or to any other consistent conjecture), his choice will
be a best response also to the true strategy profile.
In general, it is not possible to guarantee that for sufficiently large games the strategy profile be a Nash equilibrium. For
instance, if the constant payoff to the male for choosing v in the above family of games decreases to zero as the number of
6
See also Battigalli and Guaitoli (1988, 1997) and Dekel et al. (2004).
354
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
players grows, then it is possible that the male keeps making suboptimal choices. However, the loss of utility must vanish
as larger games are considered. This is the content of our main result.
3. Definition of CCE
A game G in normal form is defined by a triplet G = ( N , { S i }i ∈ N , {u i }i ∈ N ). N = {1, . . . , n} is the set of players. For each
i ∈ N, S i is the finite set of pure strategies (actions) of player i. Denote by S the product S =
i ∈ N S i , and for every player
7
S
u
i ∈ N let S −i =
S
.
A
typical
element
of
(
S
,
S
)
is
denoted
by
s
(
s
,
s
)
.
:
S
→
R is the utility function
i
−i
i
−i
i
j =i j
of player i ∈ N. Each player i may use a mixed strategy which is an element of ( S i ), the set of probability distributions
over S i , usually denoted σi . Thus, for every si ∈ S i , σi (si ) is the probability of player i choosing the action si according to
the mixed strategy σi . If σ = (σ1 , . . . , σn ) is a profile of strategies then σ−i denotes the strategies of players other than i.
As usual, u i is also used to denote expected utility whenever players use mixed strategies.
Assume that each player i ∈ N categorizes the rest of the players according to some criteria. Formally, for every i ∈ N, let
of the set N \ {i }. That is, C i = { B 1 , . . . , B m } where each B j is a non-empty subset of N \ {i }, j = k implies
C i be a partition m
B j ∩ B k = ∅, and
j =1 B j = N \ {i }. A categorization profile is a vector C = (C 1 , . . . , C n ), where each C i is a partition of N \ {i }.
For two categorization profiles C = (C 1 , . . . , C n ) and C = (C 1 , . . . , C n ), we say that C is finer than C if C i is finer than C i
for every i ∈ N.
Assume that there is a finite universal set of actions S (not to be confused with the product set S) such that S i ⊆ S
for every i ∈ N. Every profile of (possibly mixed) strategies σ = (σ1 , . . . , σn ) ∈
a non-empty set of players
i ∈ N ( S i ) and B ⊆ N induce a probability distribution over S, denoted σ B , which is defined by σ B (s) = | 1B | i ∈ B σi (s) for every s ∈ S.
×
×
×
Here we mean that σi (s) = 0 whenever s ∈ S \ S i . Thus, σ B (s) is the expected proportion of players choosing s in the set B
according to the profile of strategies {σi }i ∈ B .
Given a player i ∈ N, a categorization C i of N \ {i } and a profile of strategies σ = (σ1 , . . . , σn ), let F C i (σ−i ) = {τ−i : τ−Bi =
σ−Bi for every B ∈ C i } be the set of all strategy profiles of players other than i which induce the same distribution over S
like σ in every set B ∈ C i . Elements of F C i (σ−i ) are called consistent conjectures of player i at σ−i .
Definition 1. σ = (σ1 , . . . , σn ) is a conjectural categorical equilibrium (CCE) w.r.t. the categorization profile C = (C 1 , . . . , C n ) if, for
every i ∈ N, there exists a profile of strategies τ−i ∈ F C i (σ−i ) such that σi is a best response to τ−i .
Assuming that a categorization profile is exogenously given, a profile of strategies constitutes a CCE (w.r.t. the given
categorization profile), if every player best responds to some conjecture about the strategies of the others. However, the
conjecture of each player must be consistent with what she knows about the strategies of others, i.e., within the set F C i (σ−i ).
The set of all CCE in a game G w.r.t. a given categorization profile C is denoted by CCE G (C ). NEG is the set of Nash
equilibria of the game G. The following observation is simple but important (the proof is omitted).
Lemma 1. For every game G,
(i) If C refines C then CCEG (C ) ⊆ CCE G (C ).
(ii) If C is the finest categorization profile in G (every cell of every categorization contains only one player) then CCE G (C ) = NE G .
Corollary 1. Every Nash equilibrium is a CCE w.r.t. any categorization profile.
4. Efficient categorization profiles
This section contains the main result of the paper. It deals with a property of categorization profiles which we call
efficiency. A categorization profile is efficient if a best response to every consistent conjecture of every player is also a best
response to the actual strategy profile. When an agent categorizes her opponents according to an efficient categorization
she maintains her utility level with significantly less mental effort. Exact and approximated efficiency are formally defined
as follows.
Definition 2. Fix a game G and let ε 0. A categorization profile C is
rium8 of G. A categorization profile is efficient if it is 0-efficient.
ε -efficient if every CCE w.r.t. C is an ε -Nash equilib-
The rest of this section discusses sufficient conditions for the existence of efficient categorization profiles. Of course, the
finest categorization in which every category consists of only one agent is always efficient. What we show, however, is that
in many cases there are also coarser efficient categorization profiles.
7
8
The underline is used to emphasize that this is a vector of actions. The index s will be used for another purpose in the sequel.
A strategy profile constitute an ε -Nash equilibrium if no player can gain more than ε by deviating.
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
355
We will need the following notation. Fix a game G. For a profile of actions s = (s1 , . . . , sn ) ∈ S and two players j , k ∈ N
with S j = S k , let s jk be the profile of actions in which every player other than j and k plays the same as in s and players j
and k exchange their choices. That is, player j plays sk , player k plays s j and every player l ∈ N \ { j , k} plays sl . For a player
i ∈ N, we say that the players j , k ∈ N \ {i } are exchangeable for i (denoted j ∼i k) if S j = S k and u i (s) = u i (s jk ) for every
s ∈ S.
If j ∼i k, then player i only cares about the pair of actions taken by players j and k. She is not concerned with who
plays what. Thus, since we assume that i observes the expected distribution of actions in each cell of her categorization, it
is natural for her to put j and k in the same category.
It is easy to verify that ∼i is transitive and symmetric over N \ {i }. Let Ĉ i be the partition of N \ {i } to the equivalence
classes of ∼i and let Ĉ = (Ĉ 1 , . . . , Ĉ n ). The element of Ĉ i which contains player j is denoted by Ĉ i ( j ). Notice that our
notation neglects the dependence of the categorization profile Ĉ on the game G. This is so since it will always be clear what
is the relevant game. Notice also that Ĉ is endogenous: Each Ĉ i is uniquely determined by player i’s utility function u i .
If players were only allowed to play pure strategies and, in addition, players would always conjecture that their opponents play pure strategies then Ĉ would have been efficient (see Lemma 5 in Section 7.1). However, since players may
randomize, some conditions on the game must be added in order to maintain the sufficiency of Ĉ . Although restricting the
generality of our discussion, these conditions are valid for a wide family of games.
Definition 3. Fix a finite set of actions S. Let Γ (S) denote a family of normal form games such that, for every game G ∈ Γ (S)
and for every9 i ∈ N, S i ⊆ S.
(i) Γ (S) is uniformly bounded if there is M > 0 such that |u i | M for every G ∈ Γ (S) and for every utility function u i ∈ G.
for every G ∈ Γ (S), every two players
(ii) Γ (S) is uniformly Lipschitz if there is M > 0 such that |u i (s) − u i (sj ; s− j )| |M
N|
i , j ∈ N, every s ∈ S and every sj ∈ S j .
The uniform boundness condition is standard. The Lipschitz condition states that the effect of some player j changing
her action on the payoff of another player i should be inversely proportional to the number of players in the game. To justify
the name of this condition, consider the set S −i in a given game G as a metric space, where the distance between s−i , s−i
is the proportion of players that play different actions in the two profiles. Then the family Γ (S) is uniformly Lipschitz iff all
the functions u i (si ; ·) are Lipschitz continuous with the same constant M.
Finally, we also need to impose a restriction on the categorizations Ĉ i in the family Γ (S). Namely, most of i’s opponents
should be in relatively large categories. This will allow us to use the law of large numbers in order to conclude that the
realized distribution of actions in each category is close to the expected distribution with high probability. The following
definition provides a general condition which turns out to be sufficient for our purposes.
Definition 4. Let Γ (S) be as in Definition 3. Say that Γ (S) is sufficiently anonymous if there is a sequence of positive numbers
{ρn }n such that
lim
n→∞
e −nρn
ρn
=0
(1)
and
|{ j ∈ N \ {i }: |Ĉ i ( j )| < ρ| N | · | N |}|
−→ 0 uniformly in Γ (S).
|N |
(2)
To illustrate Definition 4 consider first the case where ρn = ρ for every n. Condition (1) is trivially satisfied in this case.
Condition (2) states that the proportion of players which are in categories of size less than a ρ -fraction of the total number
of players should vanish as the number of players grows. Definition 4 is weaker since it allows for ρ to decrease as larger
games are considered. For instance, one can take
ρn =
√
√1 , in which case condition (2) becomes |{ j ∈ N \{i }: |Ĉ i ( j )|< | N |}|
|N |
n
−→ 0.
1
n
In general, the faster the sequence ρn goes to zero the weaker condition (2) is. If ρn = then (2) is always satisfied.
However, (1) is not satisfied in this case. Thus, condition (1) bounds the rate in which the sequence ρn can go to zero. As
can be seen in the proof of our main theorem, the bound in (1) is obtained from a standard large deviations bound on the
sum of independent binary random variables.
While Definition 4 provides a rather weak condition that is sufficient to prove our main theorem, there are more natural
and transparent conditions which imply that Γ (S) is sufficiently anonymous. One such condition is that the number of categories in each of the categorizations Ĉ i in the family Γ (S) is uniformly bounded.10 In particular, if the game is anonymous
9
Throughout the paper, a quantifier of the form “for every game G ∈ Γ (S) and for every i ∈ N” should be understood as “for every game G =
( N , { S i }i ∈N , {u i }i ∈N ) ∈ Γ (S) and for every i ∈ N.”
10
Formally, there is M > 0 such that |Ĉ i | < M for every G ∈ Γ (S) and for every i ∈ N.
356
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
(any two players are exchangeable for any third player) then Γ (S) is sufficiently anonymous. We note that most kinds of
large games studied in the literature (e.g., market games, congestion games) are sufficiently anonymous.
We can now state the main result of the paper.
Theorem 1. Consider a family Γ (S) of normal form games which is uniformly bounded, uniformly Lipschitz and sufficiently anonymous. For every ε > 0 there exists n0 = n0 (Γ (S), ε ) such that if G ∈ Γ (S) has more than n0 players then the categorization profile Ĉ
in G is ε -efficient.
Corollary 2. If Γ (S) satisfies the conditions of Theorem 1 then, by Lemma 1, every categorization profile which is finer than Ĉ is also
ε -efficient.
We illustrate Theorem 1 and the importance of the various conditions with the following examples.
Example 1. Consider a family of games Γ (S) as in the example of Section 2, where the common set of actions is S = { v , b},
and where the utility function of every player i > 1 in every game G ∈ Γ (S) is constant. Player 1 (the male) gets a payoff
of 1 if he matches the choices of at least half of the females and 0 otherwise.
Clearly, a family of games as above is uniformly bounded and sufficiently anonymous (the categorization Ĉ 1 lumps
together all of 1’s opponents). However, it is not uniformly Lipschitz since a change in the choice of a single female may
have a large influence on the payoff of the male.
To see that Theorem 1 fails in this case consider the strategy profile in which players 2 and 3 play v with probability
3/4 and b with probability 1/4. For every i > 3, player i plays b with certainty if i is even and v with certainty if i is odd.
Restricting attention to games with an odd number of females, we have that the probability that the majority of females
choose v is 9/16. Thus, a best response for the male is to choose v and his payoff in this case is 9/16.
The expected proportion of females in each location according to the above strategy profile is 0.5. A consistent conjecture
for player 1 is that players 2 and 3 choose v with probability 1/4 and b with probability 3/4 and, for every i > 3, player i
plays v with certainty if i is even and b with certainty if i is odd. If this is 1’s conjecture then a best response for him is to
choose b. The actual expected payoff for this choice is 7/16. Thus, the loss of player 1 is 2/16, independently of the number
of players in the game.
Example 2. Let S = { v , b} as in the previous example and consider a family Γ (S) such that, for every positive integer n,
Γ (S) contains a game with3n + 1 players defined as follows. For every k = 1, 2, . . . , n, denote A k = {3k − 1, 3k, 3k + 1}.
a
n
The payoff to player 1 is n1 k=1 ( 3k )3 where ak is the number of players from the set A k which player 1’s choice matches
(k = 1, 2, . . . , n). The payoff functions of all other players are constant.
The purpose of this last example is to show that the uniform boundness and uniform Lipschitz conditions are not
sufficient for Theorem 1 to hold. Notice first that the payoffs in Γ (S) are uniformly bounded by 1. Also, the maximal
8
19
= 27n
< |10
, which means that Γ (S) is
difference in player 1’s payoff when some player j = 1 changes her action is 1 − 27n
N|
uniformly Lipschitz. However, Γ (S) is not sufficiently anonymous since, for every n, Ĉ 1 = { A 1 , A 2 , . . . , A n } and | A k | = 3 for
every 1 k n.
Now, consider the following strategy profile. For every k = 1, 2, . . . , n, players number 3k − 1 and 3k play v with probability 34 and b with probability 14 and player number 3k + 1 plays b with probability 1. The expected payoff to player 1
1
n
1 3
· (3)
9
6
1
13
· n · [0 · 13 + 16
· ( 23 )3 + 16
· ( 13 )3 + 16
· 03 ] = 72
, while choosing b yields an expected payoff of
6
2 3
1
14
3
· n · [0 · 0 +
+ 16 · ( 3 ) + 16 · 1 ] = 72 . However, a consistent conjecture for player 1 is that, for every 1 k n,
players 3k − 1 and 3k play v with probability 1/4 and b with probability 3/4, and player 3k + 1 plays v. By symmetry, the
if she chooses v is
1
n
3
9
16
expected payoff to player 1 according to this conjecture is
player 1 does not vanish as the number of players grows.
14
72
for playing v and
13
72
for playing b. Thus, the loss of utility for
5. Extensions
5.1. A generalized Lipschitz condition
According to the Lipschitz condition of Definition 3, the influence of some player j on the payoff of player i is bounded
by |M
. A natural question is whether one can obtain a similar result to that of Theorem 1 when this bound is weakened.
N|
A famous generalization of Lipschitz continuity is Hölder continuity.
Definition 5. A family of games Γ (S) is uniformly Hölder with exponent α (α > 0) if there is a constant M > 0 such that
M
| N |α for every G ∈ Γ (S), every two players i , j ∈ N, every s ∈ S and every s j ∈ S j .
|u i (s) − u i (sj ; s− j )| Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
357
The smaller the exponent α , the weaker this condition. If Γ (S) is uniformly Hölder with exponent α > 1 then the payoff
functions become independent of the choices of the opponents as the number of players in the game increases. In this
case, Theorem 1 is trivially true. However, if Γ (S) is uniformly Hölder with exponent α < 1 then a stronger version of the
anonymity condition is required in order to prove Theorem 1. Such a generalized version makes it clear that there is a
trade-off between the degrees of continuity and anonymity required.
Definition 6. A family of games Γ (S) is
{ρn }n such that
lim
n→∞
for some
e −n
2α −1
·ρn
ρn
α -sufficiently anonymous (0 < α < 1) if there is a sequence of positive numbers
=0
α < α , and
|{ j ∈ N \ {i }: |Ĉ i ( j )| < ρ| N | · | N |}|
−→ 0 uniformly in Γ (S).
| N |α
Corollary 3. Consider a family Γ (S) of normal form games which is uniformly bounded, uniformly Hölder with exponent α
(0 < α < 1) and α -sufficiently anonymous. For every ε > 0 there exists n0 = n0 (Γ (S), ε ) such that if G ∈ Γ (S) has more than n0
players then the categorization profile Ĉ in G is ε -efficient.
5.2. Approximately exchangeable players
Our main result is limited to the categorization profile Ĉ (or, by Corollary 2, to any finer categorization profile), where
each players lumps together players that have symmetric influence on her payoffs. It is possible however to formulate a
similar result when the players in each category have approximately symmetric influence on the utility of the categorizer.11
We first need the following definition.
Definition 7. Fix a game G and a categorization profile C in G. The diameter of C is diam(C ) = max |u i (s) − u i (s jk )|, where
the maximum is taken over all action profiles s, all players i ∈ N and all pairs of players j , k that belong to the same
category according to C i .
Corollary 4. Consider a family Γ (S) of normal form games which is uniformly bounded and uniformly Lipschitz. Assume that in every
game G ∈ Γ (S) there is a categorization profile C such that the sufficient anonymity condition of Definition 4 is satisfied when C
replaces Ĉ and, in addition, | N | · diam(C ) → 0 uniformly in Γ (S). Then for every ε > 0 there exists n0 = n0 (Γ (S), ε ) such that if
G ∈ Γ (S) has more than n0 players then the categorization profile C in G is ε -efficient.
5.3. Incomplete information
Throughout the paper we restricted attention to complete information games. Adding a move by nature before the start
of the game significantly complicates the presentation and analysis. First, it is not clear at what stage of the game a player
categorizes her opponents. It is possible to define ex-ante categorization profile, where each player categorizes before she
learned her type, or interim categorization profile where the categorization of each player is a function of her realized type.
Second, it is not clear how to define the set of consistent conjectures of a player at a given strategy profile. There are several
possible definitions of consistency that reflect different information that the categorizer store about the average behavior in
each category. Third, the definition of exchangeable players is ambiguous and, as a consequence, so is the definition of the
categorization profile Ĉ . Due to these complications we prefer not to deal with incomplete information games in this work.
5.4. Infinite action space
A crucial assumption in our model is that the universal set of actions S is finite. This restricts the applicability of our
results. In a recent paper, Deb and Kalai (2008) analyze the ex-post stability of Nash equilibria in large games using a similar
model to the one here. In their work, the sets of actions available to players are arbitrary compact sets in some metric space.
They define a metric over the set of action profiles in a given game and show that, if the utility functions satisfy a Lipschitz
condition similar to the one introduced here w.r.t. this metric, then Nash equilibria have the ex-post stability property when
the number of players is large. We conjecture that a similar result to that of Theorem 1 can be obtained when players’
strategy sets are compact given Deb and Kalai’s condition. This direction is beyond the scope of this paper.
11
I thank a referee for this suggestion.
358
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
6. Final remarks
6.1. Simple conjectures: A refinement
In a CCE the conjecture of an agent is limited only by the signal she observes (and by the restriction that agents play
independently of each other). One may want to restrict the belief that an agent can have even more by requiring that it will
be simple in some sense.12 By so doing, a refinement of CCE can be obtained.
Among all of a player’s possible conjectures, there is one which can quite naturally be considered as the simplest, namely,
the conjecture in which all players in each cell of her partition play the same strategy. A player holding this belief can be
seen as having a prototypical agent for each set in her partition. All the players in each set are playing the same as their
representing prototype. The common strategy in each cell is then uniquely determined by the signal the player observes.
Such a refinement leads to a special case of Jehiel’s (2005) analogy based expectation equilibrium (ABEE). The scope of
ABEE is much wider since it is defined for extensive form games and for any partition of opponents’ nodes in the game
tree. However, in the special case discussed here, our result implies that any ABEE is an approximate Nash equilibrium if
the number of players is large.
6.2. Correlated conjectures
Theorem 1 relies heavily on the assumption that a player takes into account only independent profiles of strategies of
her opponents. That is, we rule out the possibility that some player thinks that other players correlate their strategies,
even though this correlated strategy might be consistent with the signal that this player observes. The fact that correlated
conjectures are not allowed enables us to use the power of the laws of large numbers, which otherwise fail.
To emphasize this point consider once again a family of games as in Example 1, but where the payoff for the male
depends on the proportion p of females his choice matches as follows. If 0 p 13 then his payoff is 3p, while if 13 p 1
then he gets
4
3
− p. If correlated conjectures were allowed, then the following profile of strategies would constitute a
CCE w.r.t. the categorization profile Ĉ . The male plays v with probability 1, and every female plays v with probability
and b with probability
probability
2
3
2
)
3
· (4/3 − 1) +
1
.
3
Indeed, a consistent (correlated) conjecture of the male is that either all females play v (with
or all females play b (with probability
1
3
·3·0>
2
3
2
3
·3·0+
1
3
1
).
3
For this conjecture the best response for the male is v since
· (4/3 − 1). However, this profile of strategies does not become approximately Nash
as the number of players increases. This is because the payoff to the male will be close to 43 − 23 = 23 while deviating to b
would result in a payoff close to 3 · 13 = 1.
6.3. Rationalizable CCE
Rubinstein and Wolinsky (1994) introduced the notion of rationalizable conjectural equilibrium (RCE). This is a refinement
of the CE concept, which requires that conjectures of players be consistent with common knowledge of rationality and of
the signal functions of other players. Similar refinements of the CE concept are studied in Dekel et al. (1999), Gilli (1999)
and Battigalli (1999).
One may ask why we use CE and not RCE as our solution concept. Two reasons justify this choice. The first is that the
games we analyze have many players. It is natural to assume that in this case players do not “get into the heads” of their
opponents and draw conclusions which change their beliefs, simply because it is too complicated to do so. Second, every
RCE is also a CE. Thus, our main results would not change had we defined CCE using RCE and not CE.
6.4. Self-categorization
Throughout the paper, the categorization of player i is of the set N \ {i }. Thus, i does not include herself in any of the
groups of her categorization. The reason for this modeling choice is the common assumption that every agent knows the
action she plays. Inserting i into one of the cells of her partition C i (say B) can create a situation in which i’s conjecture
(about what the players in B are playing) is consistent with her signal (the average behavior within B) but not with the
action which she actually plays. Leaving i out of her own partition prevents such an awkward situation.
It should be noted, however, that individuals do not exclude themselves from their categorical perception of the society.
In fact, self-categorization and identity are among the most studied subjects in social psychology (for references see Ellemers
et al., 2002). The social categories to which one belongs and the way these categories are seen by the society can have
significant implications on one’s choices. This important issue is not addressed by the current paper.
12
This idea is certainly not new. Eliaz (2003) and Spiegler (2002, 2004) are examples of papers in which the solution concept takes into account the
complexity of the belief of an agent about what others do.
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
359
7. Proofs
7.1. Preliminary lemmas
Fix a family of games Γ (S) which is uniformly bounded (by a constant M > 0) and uniformly Lipschitz (w.l.o.g. with the
same constant M).
For the following Lemmas 2–6 and 8 fix a game G = ( N , { S i }i ∈ N , {u i }i ∈ N ) ∈ Γ (S), a player i ∈ N, a profile of strategies σ
and two positive numbers δ, ρ > 0. A typical element of the categorization Ĉ i is denoted by B. Let
E ρ = B ∈ Ĉ i : | B | ρ | N |
and
H ρ = Ĉ i \ E ρ = B ∈ Ĉ i : | B | < ρ | N | .
For every B ∈ Ĉ i and for every s ∈ S, let
#{ j ∈ B: s j = s}
− σ B (s) δ ,
|B|
D δ ( B , s) = s−i ∈ S −i : and denote
D δ,ρ =
D δ ( B , s).
B ∈ E ρ s∈S
Let Pσ−i be the probability measure on S −i induced by σ−i .
The following lemma uses a standard large deviations inequality to bound the probability that the realized proportion of
players in some category B choosing some strategy s is far from the expected proportion.
Lemma 2. Pσ−i ( D δ ( B , s)) 2e −2| B |δ , for every B ∈ Ĉ i and for every s ∈ S.
2
Proof. For a given B ∈ Ĉ i and s ∈ S, consider the sequence of independent random variables ( X j ) j ∈ B defined by X j = 1
X
B
if player j’s realized strategy is s and X j = 0 otherwise. Let X =
j ∈ B X j . We have D δ ( B , s) = {| | B | − σ (s)| δ} where
σ B (s) is the expected value of the random variable
X
| B | . By a classical bound of Hoeffding (e.g. Petrov, 1975, chapter III) the
probability of this event is not greater than 2e −2| B |δ .
2
Lemma 3. Pσ−i ( D δ,ρ ) 2|S|e −2| N |ρ δ
ρ
2
2
.
Proof. Using the previous lemma, and by the definition of E ρ
Pσ−i ( D δ,ρ ) B ∈ E ρ s∈S
2|S|
B∈Eρ
2e −2| B |δ = 2|S|
2
e −2| B |δ
2
B∈Eρ
2|S|e −2| N |ρ δ
2
e −2ρ δ | N | .
2
ρ
2
Lemma 4. If τ−i ∈ F Cˆ (σ−i ) then the bound of Lemma 3 holds when Pσ−i is replaced with Pτ−i . In particular, |Pσ−i ( D δ,ρ ) −
i
Pτ−i ( D δ,ρ )| Proof.
2|S|e −2| N |ρ δ
ρ
2
.
τ−i ∈ F Cˆi (σ−i ) means that σ B (s) = τ B (s) for every B ∈ Ĉ i and for every s ∈ S. Thus, Lemma 3 may be applied without
any change to Pτ−i . It follows that both Pτ−i ( D δ,ρ ) and Pσ−i ( D δ,ρ ) are in the interval [0,
2|S|e −2| N |ρ δ
ρ
2
]. 2
The following simple lemma uses the particular structure of the categorization Ĉ i to show that the payoffs to i under
a pure strategy profile and under a pure consistent conjecture (in that profile) are the same. This result is then combined
with the Lipschitz condition in the next lemma to bound the difference in payoffs to i against two pure strategy profiles in
which the proportion of players in each large category choosing each strategy is close to the expected.
Lemma 5. Fix two profiles of actions s, s ∈ S. If s−i ∈ F Ĉ (s−i ) then u i (s) = u i (s ).
i
Proof. s−i ∈ F Ĉ (s−i ) means that #{ j ∈ B: s j = s} = #{ j ∈ B: 3sj = s} for every s ∈ S and for every B ∈ Ĉ i . Thus, for every
i
B ∈ Ĉ i , there is a permutation of players’ names in the set B which transforms the restriction of s to B to the restriction of
s to B. However, every such permutation of players in the set B can be achieved by a sequence of exchanges of pairs of
players. By the definition of the partition Ĉ i , no such exchange affects the payoff of player i. The assertion follows. 2
360
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
Lemma 6. If s−i , s−i ∈ S −i \ D δ,ρ then |u i (si ; s−i ) − u i (si ; s−i )| M (δ|S| +
B∈Hρ
|B|
|N |
) for every si ∈ S i .
Proof. Since both s−i and s−i are not in D δ,ρ it follows that |#{ j ∈ B: s j = s} − #{ j ∈ B: sj = s}| 2δ| B | for every B ∈ E ρ
and for every s ∈ S. Thus, there is a profile of actions s−i ∈ F Ĉ (s−i ) such that s−i is obtained from s−i by no more than
i
+ B ∈ H ρ | B | δ|S|(| N | − 1) + B ∈ H ρ | B | changes in players’ actions. By the previous lemma, we have
that |u i (si ; s−i ) − u i (si ; s−i )| = |u i (si ; s−i ) − u i (si ; s−i )|. By the Lipschitz assumption, the influence of some player j = i
. It follows that |u i (si ; s−i ) − u i (si ; s−i )| (δ|S|(| N | −
changing her action on the utility function u i is not greater than |M
N|
|
B
|
B
∈
H
ρ
1) + B ∈ H ρ | B |) |M
M (δ|S| +
). 2
N|
|N |
B∈Eρ
s∈S δ| B |
Lemma 7. Let Ω be a finite set, P , Q two probability measures on Ω , X : Ω → R a random variable and ε > 0. Let
A ⊆ Ω be an event
such that | P ( A ) − Q ( A )| ε , and assume that r X (ω) R for every ω ∈ A (r R are two constants). Then | ω∈ A X (ω)( P (ω) −
Q (ω))| R − r + ε max(|r |, | R |).
Proof. Denoting E P (E Q ) the expectation operator w.r.t. to the measure P ( Q ), one has
= P ( A )E P ( X | A ) − Q ( A )E Q ( X | A )
X
(
ω
)
P
(
ω
)
−
Q
(
ω
)
ω∈ A
P ( A )E P ( X | A ) − Q ( A )E P ( X | A ) + Q ( A )E P ( X | A ) − Q ( A )E Q ( X | A )
= P ( A ) − Q ( A ) E P ( X | A ) + Q ( A ) E P ( X | A ) − E Q ( X | A )
ε max | R |, |r | + R − r .
2
Finally, the next lemma uses all the previous ones to bound the expected loss of i’s due to her possibly wrong conjecture.
This bound is then used to prove Theorem 1.
Lemma 8. If τ−i ∈ F Cˆ (σ−i ) then
i
| B | 6|S|e −2| N |ρ δ 2 u i (si ; σ−i ) − u i (si ; τ−i ) M δ S| + B ∈ H ρ
+
|N |
ρ
for every si ∈ S i .
Proof.
u i (si ; σ−i ) − u i (si ; τ−i ) u
(
s
;
s
)
P
(
s
)
−
P
(
s
)
σ−i −i
τ−i −i i i −i
s−i ∈ D δ,ρ
+ s−i ∈ S −i \ D δ,ρ
u i (si ; s−i ) Pσ−i (s−i ) − Pτ−i (s−i ) .
The first sum can be estimated by
M·
Pσ (s−i ) − Pτ (s−i )
u
(
s
;
s
)
P
(
s
)
−
P
(
s
)
σ
τ
i
i
−
i
−
i
−
i
−i
−i
−i
−i
s−i ∈ D δ,ρ
s−i ∈ D δ,ρ
M · Pσ−i ( D δ,ρ ) + Pτ−i ( D δ,ρ )
4M |S|e −2| N |ρ δ
ρ
2
,
where the first inequality is due to the fact that Γ (S) is uniformly bounded by M, and the third inequality is by Lemmas 3
and 4.
The second sum is estimated by Lemma 7 with Ω = S −i , P = Pσ−i , Q = Pτ−i , X (ω) = u i (si ; ω), A = S −i \ D δ,ρ and
ε=
2|S|e −2| N |ρ δ
B∈Hρ
|N |
|B|
2
ρ
. Notice that, by Lemma 6, the utility u i (si ; ·) is bounded in an interval of length not larger than M (δ|S| +
). Thus,
s−i ∈ S −i \ D δ,ρ
2 2|S|e −2| N |ρ δ
B∈Hρ | B |
+
u i (si ; s−i ) Pσ−i (s−i ) − Pτ−i (s−i ) M δ|S| +
.
|N |
ρ
Summing up the two inequalities gives the desired bound.
2
Y. Azrieli / Games and Economic Behavior 67 (2009) 351–362
361
7.2. Proof of Theorem 1
Assume that Γ (S) satisfies the conditions of the theorem and fix
tion 4. Let n0 be large enough such that
e −nρn
ρn
ε
< ( 36M |S| )
1
2δ 2
ε > 0. Fix 0 < δ <
ε
6M |S|
and let {ρn }n be as in Defini-
for every n > n0 and such that
|{ j ∈ N \ {i }: |Ĉ i ( j )| < ρ| N | · | N |}|
ε
<
|N |
6M
for every game G ∈ Γ (S) with at least n0 players and for every player i in G.
Fix G ∈ Γ (S) with | N | > n0 and let σ be a CCE w.r.t. Ĉ in G. Then, for every i ∈ N, there is a strategy profile τ−i ∈
F Ĉ (σ−i ) such that σi is a best response to τ−i . It follows from Lemma 8 that, for every i ∈ N and for every si ∈ S i ,
i
|u i (si ; σ−i ) − u i (si ; τ−i )| ε2 . Thus,
u i (si ; σ−i ) u i (si ; τ−i ) +
ε
2
u i (σi ; τ−i ) +
ε
2
u i (σi ; σ−i ) + ε .
7.3. Proof of Corollary 3
The proof follows the footsteps of the proof of Theorem 1. The first difference is in Lemma 6, where the bound becomes
|B| u i (si ; s−i ) − u i (si ; s ) M δ|S|| N |1−α + B ∈ H ρ
.
−i
| N |α
As a consequence, the bound in Lemma 8 becomes
| B | 6|S|e −2| N |ρ δ 2
u i (si ; σ−i ) − u i (si ; τ−i ) M δ|S|| N |1−α + B ∈ H ρ
.
+
| N |α
ρ
Define δn =
1
n1−α
, where
α < α is as in Definition 6. Let {ρn }n be as in Definition 6. Then, substituting δ|N | for δ and
ρ|N | for ρ , the right-hand side of the last inequality goes to zero uniformly in Γ (S). The result follows.
7.4. Proof of Corollary 4
Repeat Lemmas 2–4 with C i instead of Ĉ i everywhere. In Lemma 5, when Ĉ i is replaced by C i , the equality u i (s) = u i (s )
is not valid anymore. However, one has |u i (s) − u i (s )| diam(C ) · | N |. Thus, in Lemma 6 the bound becomes
|B|
u i (si ; s−i ) − u i (si ; s ) diam(C ) · | N | + M δ|S| + B ∈ H ρ
.
−i
|N |
The latter implies that in Lemma 8 the bound is
| B | 6|S|e −2| N |ρ δ 2
u i (si ; σ−i ) − u i (si ; τ−i ) M δ|S| + B ∈ H ρ
+
+ diam(C ) · | N |.
|N |
ρ
By assumption, one can choose δ and a sequence {ρn }n such that the right-hand side of the last inequality goes to
zero uniformly in Γ (S). This implies that, in any sufficiently large game in Γ (S), every CCE w.r.t. C is approximately Nash
equilibrium.
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