Journal of Mathematical Economics 46 (2010) 303–310
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Journal of Mathematical Economics
journal homepage: www.elsevier.com/locate/jmateco
Categorization and correlation in a random-matching game夽
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 25 October 2007
Received in revised form
25 November 2009
Accepted 25 November 2009
Available online 3 December 2009
a b s t r a c t
We consider a random-matching model in which every agent has a categorization (partition) of his potential opponents. In equilibrium, the strategy of each player i is a best
response to the distribution of strategies of his opponents (when they face i) in each category
of his categorization. We provide equivalence theorems between distributions generated
by equilibrium profiles and correlated equilibria of the underlying game.
© 2009 Elsevier B.V. All rights reserved.
JEL classification:
C72
D82
Keywords:
Random-matching
Categorization
Correlated equilibrium
1. Introduction
Mailath et al. (1997) consider a model in which agents are randomly matched to play a fixed game. They show that if
the pattern of matchings can be correlated across individuals (“local interactions”) then equilibria of the random-matching
environment correspond to correlated equilibria (Aumann, 1974) of the underlying game. MSS conclude that
“The results suggest that when working with matching models, we should be interested in correlated, rather than simply Nash, equilibria. When interpreting the correlation device involved in a correlated equilibrium, we should thus add
the possibility of local interactions to the assortment of existing interpretations: a referee, pre-play communication,
and observations of correlated signals.”
The vast majority of random-matching models in the literature (e.g. Kandori et al., 1993; Young, 1993; Fudenberg and
Levine, 1993), however, assume that the interactions are independent and uniform across agents. Furthermore, many natural
economic environments are better described by “global matching” than by “local interactions”. Consider for instance potential
buyers of a certain product that search the internet for sellers. Every day one of the buyers selects a seller and the pair then
bargain over the price. If buyers have similar preferences then it seems very natural to assume that the probability of a
certain seller being selected is independent of the identity of the buyer. This implies that the process of matchings between
夽 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.
∗ Tel.: +1 972 364 05386; fax: +1 972 364 09357.
E-mail address: [email protected].
0304-4068/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmateco.2009.11.015
304
Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
buyers and sellers is probably close to being uncorrelated.1 Another example is experiments in the lab, where the pattern
of matchings is controlled by the designer and is almost always uniform.
The aim of this paper is to show that a similar result to that of MSS can be obtained with interactions that are independent
and uniform across agents. However, unlike standard random-matching models (including the one in MSS) where each
player’s strategy is independent of his matched opponent, the players in our model may play opponent-contingent strategies
to a certain extent. We assume that each player is equipped with an exogenously given categorization, which is a partition
of his potential opponents.2 A player must use the same action against all members of each category in his categorization,
but may use different actions against opponents from different categories. This reflects a situation where an agent possesses
some information about his realized opponent. For instance, a prospective employer can see the skin color of a job candidate;
and a seller of some product may know the gender of a potential buyer. The information may be completely payoff irrelevant,
meaning that the underlying game is unchanged. But the very fact that such information is available allows agents to condition
their strategy on the category to which their opponent belongs.
With this simple modification, we are able to provide equivalence theorems between distributions generated by equilibrium profiles and correlated equilibria of the underlying game. To illustrate our results consider the following version of
“chicken”.
It is well known that the probability distribution which places a weight of 1/3 on each of the outcomes (B, L), (T, L)
and (T, R) is a correlated equilibrium. Now, assume that there are two populations N1 = {1, 2, 3} and N2 = {1 , 2 , 3 }, and
that an agent from each population is randomly selected to play the game (probability of 1/9 to each encounter). Each
player categorizes the set of his potential opponents as follows. The partition of player 1 is C1 = {(1 , 2 ), (3 )}. Similarly,
C2 = {(1 ), (2 , 3 )}, C3 = {(1 , 3 ), (2 )}, C1 = {(1, 2), (3)}, C2 = {(1), (2, 3)} and C3 = {(1, 3), (2)}. The strategies of the players
are given in the following matrix.
Notice that each agent’s strategy is measurable with respect to his categorization. Moreover, the strategy of each agent
maximizes (among all measurable strategies) his expected payoff. Therefore, this strategy profile is an equilibrium in the
above specified environment. Now, given that player 2 was chosen from N2 , the induced distribution on the set of action
profiles places probability 1/3 on each of the outcomes (T, R), (T, L) and (B, L). The same is true for any one of the 6 agents.
In particular, this implies that the total distribution of action profiles induced by this equilibrium is the same as the above
mentioned correlated equilibrium distribution.
In Theorem 1 below we prove that, for any correlated equilibrium with rational probabilities in any (finite) two-player
normal-form game, there is a finite two-populations random-matching environment and an equilibrium in this environment
which induces3 the given correlated equilibrium. In Theorem 2 we show that the same holds for symmetric correlated
equilibria (with rational probabilities) when ordered pairs are randomly drawn from a single population to play a symmetric
game. The restriction to rational probabilities can be eliminated at the cost of allowing for populations with a continuum of
agents.4 The proofs of these theorems are based on a previous result of Lehrer and Sorin (1997), as well as on an extension
of this result due to Di-Tillio (2004).
Referring to the above quote from MSS, our results strengthen the conclusion that correlated equilibrium rather than
Nash should be studied when one is concerned with random matching. Moreover, the results provide a new interpretation
for the correlated equilibrium concept: It is the result of an independent matching process where different agents are using
different categorizations of the set of their potential opponents.5
1
In this example the assumption that the distribution of matchings is uniform seems quite strong. However, the main difference between this paper
and that of MSS is that here the matching process is independent. The uniform distribution is merely a normalization.
2
There are several recent papers that study categorical thinking in economic models. Examples include Azrieli (2009); Fryer and Jackson (2008); Jehiel
(2005) and Pȩski (2006).
3
When we say that a certain strategy profile in the random matching environment induces a certain distribution we mean that, conditional on any given
player being chosen, the proportion of times that a certain action profile is played is equal to the given probability of that profile. In particular, it means that
all the players of each population have the same expected payoff. This is stronger than the result of MSS that only show that the average across all agents
(in both populations) is equal to the given correlated equilibrium.
4
We discuss this issue in Section 3.3.
5
The results no longer hold if all agents must use the same categorization. See Section 6.2.
Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
305
The rest of the paper is organized as follows. Section 2 formally defines the random-matching environments and the
equilibrium concept. In Section 3 we state our main results. The proofs of the theorems are in Sections 4 and 5. Section 6
contains some additional remarks.
2. The model
Let G = (A1 , A2 , u1 , u2 ) be a 2-player normal-form game, where Ai is player’s i finite set of (pure) actions and ui : A1 × A2 →
R is player’s i utility function, i = 1, 2. Throughout this paper a subscript −i should be read 3 − i. As usual, we denote
A = A1 × A2 and, with abuse of notation, a = (ai , a−i ) denotes the element of A in which ai ∈ Ai and a
−i ∈ A−i (for i = 1, 2). The
(ai , a−i ) is the
set of probability distributions over some finite set X is denoted by (X). If ∈ (A) then (ai ) =
a ∈A
−i
−i
marginal probability of ai ∈ Ai according to and, for every ai ∈ Ai with (ai ) > 0, (·|ai ) is the conditional of on A−i given ai .
2.1. The two-populations case
We consider a random-matching environment with two finite and non-empty populations N1 and N2 , where |N1 | = n1 and
|N2 | = n2 . Let N = N1 × N2 . Following Mailath et al. (1997), we call a meeting a selection of an agent ki from each population,
who then play the game G. The vector k = (k1 , k2 ) ∈ N is called the cast of the meeting. The cast of the meeting is drawn
uniformly within each population and independently across populations. That is, the probability that the cast of the meeting
will be k is n 1·n , for every k ∈ N.
1 2
To the above standard description we now add a new ingredient. This is the exogenously given categorization profile
which specifies how every agent partitions the set of his potential opponents. Formally, for every ki ∈ Ni let Cki be a partition
of N−i . For k−i ∈ N−i we denote by Cki (k−i ) the element of Cki which contains k−i . For i = 1, 2, let Ci = (Cki )k ∈ N be the profile
i
i
of categorizations used by agents in the population Ni , and let C = (C1 , C2 ). A complete description of the environment,
therefore, is given by (G, N1 , N2 , C).
A (pure) strategy for agent ki ∈ Ni is a function ki : N−i → Ai specifying the action that ki plays when confronted with
each agent from the other population. For i = 1, 2 we denote i = (ki )k ∈ N and = (1 , 2 ). We say that a strategy ki is
i
i
) whenever C (k ) = C (k ). The strategy
adapted to the categorization Cki if ki is Cki -measurable, that is ki (k−i ) = ki (k−i
ki −i
ki −i
profile is adapted to the categorization profile C if ki is adapted to Cki for every ki ∈ Ni and for i = 1, 2. Let i be a strategy
profile of population Ni and let D ⊆ Ni be a non-empty set. We denote by D the average strategy of the agents in D. That is,
for every k−i ∈ N−i and ai ∈ Ai , D (k−i ) ∈ (Ai ) is defined by D (k−i )(ai ) = (|{ki ∈ D : ki (k−i ) = ai }|/|D|).
Definition 1.
A strategy profile is an equilibrium in (G, N1 , N2 , C) if the following two conditions hold:
(i) is adapted to C.
(ii) For i = 1, 2, for every ki ∈ Ni and for every k−i ∈ N−i , ki (k−i ) is a best response to Ck
ui ki (k−i ), Ck
i
(k−i ) (ki )
≥ ui ai , Ck
i
(k−i ) (ki )
i
(k−i ) (ki ),
that is
for every ai ∈ Ai .
2.2. The single-population case
For this section we assume that G is a symmetric game. It will be convenient to denote the (common) set of actions by
B. Thus, A1 = A2 = B, A = B2 and u1 (b, b ) = u2 (b , b) for every (b, b ) ∈ A. There is only one population of agents denoted by
N = {1, 2, . . . , n}, where n ≥ 2. In this case, a meeting involves a drawing of an ordered pair from N. The first player takes the
role of player 1 while the second that of player 2. The drawing is uniform among all ordered pairs of agents, meaning that
the probability that the cast of the meeting is (i, j) (in this order) is (1/n(n − 1)) for every (i, j) ∈ N 2 , i =
/ j.
Each agent categorizes the rest of the agents in the population. Thus, for every i ∈ N let Ci be a partition of N \ {i}, and let
C = (C1 , . . . , Cn ). As before, Ci (j) is the element of Ci which contains agent j. A single population environment is characterized
by the triple (G, N, C).
A (pure) strategy for player i is a function i : N \ {i} → B. Notice that, both the categorization and the strategy of a player,
are independent of the role of this player in the chosen pair. Let = (1 , . . . , n ) denote the strategy profile of the population.
A strategy i is adapted to the categorization Ci if i is Ci -measurable. The strategy profile is adapted to the categorization
profile C if i is adapted to Ci for every i ∈ N.
For i ∈ N and D ⊆ N \ {i}, let D (i) be the average strategy of the agents in the set D when they face i. That is, for every b ∈ B,
D (i) ∈ (B) is defined by D (i)(b) = (|{j ∈ D : j (i) = b}|/|D|).
Definition 2.
A strategy profile is an equilibrium in (G, N, C) if the following two conditions hold:
(i) is adapted to C.
(ii) For every two different agents i, j ∈ N, i (j) is a best response to Ci (j) (i).
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Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
3. Main results
A correlated equilibrium in G (Aumann, 1974) is any distribution ∈ (A) such that
a−i ∈ A−i
ui (ai , a−i )(a−i |ai ) for every ai ∈ Ai with (ai ) > 0 and for every ai ∈ Ai , i = 1, 2.
u (a , a−i )(a−i |ai )
a−i ∈ A−i i i
≥
3.1. The two-populations case
Consider a random-matching environment with two populations (G, N1 , N2 , C). Let P denote the distribution over A
induced by the strategy profile , and for every ki ∈ Ni let P|ki denote the distribution over A induced by given that ki was
chosen from the population Ni . That is, for (a1 , a2 ) ∈ A,
P (a1 , a2 ) =
{(k1 , k2 ) ∈ N : k (k2 ) = a1 , k (k1 ) = a2 }
1
2
n1 · n2
and
P|ki (a1 , a2 ) =
|{k−i ∈ N−i : ki (k−i ) = ai , k−i (ki ) = a−i }|
n−i
.
Theorem 1.
(i) If is an equilibrium of a two-populations environment (G, N1 , N2 , C) then P is a correlated equilibrium of G.
(ii) For every correlated equilibrium of G with rational probabilities there exists an environment (G, N1 , N2 , C) and an equilibrium
in this environment such that P|ki = for every ki ∈ Ni , i = 1, 2.
Remark 1.
Since, P = (1/ni )
P
ki ∈ Ni |ki
(i = 1, 2), part (ii) of Theorem 1 remains valid if P|ki is replaced with P .
3.2. The single-population case
Fix a random-matching environment with one population (G, N, C). For a given strategy profile and for every (b, b ) ∈ B2
denote
P (b, b ) =
/ j, i (j) = b, j (i) = b }|
|{(i, j) ∈ N 2 : i =
n(n − 1)
,
and for every i ∈ N let
P|i (b, b ) =
|{j ∈ N \ {i} : i (j) = b, j (i) = b }|
n−1
.
Theorem 2.
(i) If is an equilibrium of a one-population environment (G, N, C) then P is a symmetric 6 correlated equilibrium in G.
(ii) For every symmetric correlated equilibrium of G with rational probabilities there exists an environment (G, N, C) and an
equilibrium in this environment such that P|i = for every i ∈ N.
Remark 2.
Since P =
1
n
P ,
i ∈ N |i
part (ii) of Theorem 2 remains valid if P|i is replaced with P .
3.3. Irrational probabilities and a continuum of agents
The restriction to correlated equilibria with rational probabilities in Theorems 1 and 2 may seem strange, and is hard to
interpret from an economic point of view. The mathematical reason for this restriction is simple: Since we deal with uniform
matching in finite populations the induced distribution cannot have irrational probabilities. In a model with populations
consisting of a continuum of agents, it is possible to obtain any correlated equilibrium as an equilibrium of a random matching
environment with (finite) categorizations.
We now briefly describe the model with two continuous populations. Let N1 and N2 be two copies of the interval [0, 1]. A
meeting is a random selection of a point in the square N := [0, 1]2 . We denote by the uniform (Lebesgue) measure over N
and by i the uniform measure over Ni . For i = 1, 2, every ki ∈ Ni has a finite (Borel measurable) partition Cki of N−i into sets
6
A distribution ∈ (A) is symmetric if (b, b ) = (b , b) for every (b, b ) ∈ B2 .
Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
307
of positive measure. A strategy for player ki is a (measurable) function ki : N−i → Ai . The definition of adapted strategies is
the same as in the finite case.
In order to be able to compute the average strategy of players in some element D of player ki ’s categorization (when they
face ki ) we must impose the additional assumption that k−i (ki ) is measurable as a function of k−i . Thus, we restrict attention to
strategy profiles such that the mapping k1 (k2 ), k2 (k1 ) : N → A is Borel measurable. This allows us to define the average
strategy of a set D ⊆ N−i (with positive measure) when they face ki by D (ki )(a−i ) = (−i {k−i ∈ D : k−i (ki ) = a−i } /−i (D)).
Definition 3.
A strategy profile is an equilibrium in (G, N1 , N2 , C) if the following two conditions hold:
(i) is adapted to C and satisfies the above measurability condition.
(ii) For i = 1, 2, for every ki ∈ Ni and for every k−i ∈ N−i , ki (k−i ) is a best response to Ck
i
(k−i ) (ki ).
Finally, let
P (a1 , a2 ) = {(k1 , k2 ) ∈ N : k1 (k2 ) = a1 , k2 (k1 ) = a2 }
and
P|ki (a1 , a2 ) = −i {k−i ∈ N−i : ki (k−i ) = ai , k−i (ki ) = a−i } .
Theorem 3. (i) If is an equilibrium of a continuous two-populations environment (G, N1 , N2 , C) then P is a correlated
equilibrium of G.
(ii) For every correlated equilibrium of G (not necessarily with rational probabilities) there exists a continuous environment
(G, N1 , N2 , C) and an equilibrium in this environment such that P|ki = for every ki ∈ Ni , i = 1, 2.
The proof of this theorem parallels the proof of Theorem 1, and is based on a result of Di-Tillio (2004) that extends the
Lehrer and Sorin (1997) model to allow a continuum of messages for each player. To keep the paper short we omit the
details. We note that a similar result holds for the single-population case. The proof of Theorem 2 can be easily adapted
to the continuous case to show that any symmetric correlated equilibrium can be induced as an equilibrium of a single
population environment.
4. Proof of Theorem 1
4.1. Part (i)
This part is a special case of the main theorem in Aumann (1987). Indeed, fix a two-populations environment (G, N1 , N2 , C)
and consider the two-players incomplete information game where the set of possible states of the world is = N, the
common prior is p(k) = n 1·n for every k ∈ and, for every k ∈ , the game being played at k is G (states of the world are
1 2
payoff-irrelevant). For i = 1, 2, the information partition of player i is given by Pi = {ki × D : ki ∈ Ni , D ∈ Cki }.
It is straightforward to see that any Bayes–Nash equilibrium of this incomplete information game is an equilibrium of
the random-matching environment and vice versa. By the main theorem in Aumann (1987), the distribution generated by
any equilibrium is a correlated equilibrium of G.
4.2. The result of Lehrer–Sorin
Before proving part (ii) of the theorem we take a small detour to recall an earlier result of Lehrer and Sorin (1997).7 We
will then see that (ii) is nothing but a reinterpretation of their result.
A public mediated talk is defined by two finite sets of messages S1 , S2 , and an announcement map f : S = S1 × S2 → X
where X is a finite set of public announcements. In a public mediated talk mechanism each one of two players chooses
(independently) a message si ∈ Si according to a distribution i ∈ (Si ), and the public announcement f (s1 , s2 ) is made. Then,
each player chooses an action ai ∈ Ai according to a decoding map i : Si × X → Ai .
Every public mediated talk mechanism induces a probability distribution P, over A in an obvious way. For a given
∈ (A), say that a public mediated talk mechanism simulates if
(1) P, (a|si ) = (a) for every a ∈ A, every si ∈ Si and i = 1, 2; and
(2) P, (a−i |si , x) = (a−i |i (si , x)) for every a−i ∈ A−i , every si ∈ Si and x ∈ X having positive probability under P, and i = 1, 2.
7
We do not present here the Lehrer–Sorin result in its full generality. Rather, we present a simpler version which is enough for our needs.
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Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
Theorem 4. (Lehrer and Sorin, 1997) Let ∈ (A) be a distribution with rational probabilities. Then there exists a public
mediated talk mechanism that simulates . Moreover, the mechanism can be constructed such that both 1 and 2 are the uniform
distributions over S1 and S2 respectively.
4.3. Part (ii)
Let be a correlated equilibrium of G with rational entries. By Theorem 4 there is a public mediated talk (S1 , S2 , f, X)
and decoding maps 1 , 2 such that if each player i chooses his message uniformly within Si then the public mediated talk
mechanism simulates . We now need to define a random-matching environment (G, N1 , N2 , C) and an equilibrium in
this environment such that P|ki = for every ki ∈ Ni , i = 1, 2.
The two populations will be N1 = S1 and N2 = S2 . For i = 1, 2 and for each ki ∈ Ni , the partition Cki will be the one generated
are in the same category in the partition C iff f (k , k ) = f (k , k ). The strategy profile will be
by f (ki , ·). That is, k−i and k−i
ki
i
−i
i
−i
defined by ki (k−i ) = i (ki , f (ki , k−i )). It is now straightforward to check that P|ki = for every ki ∈ Ni , i = 1, 2. Furthermore,
since is a correlated equilibrium, it is easy to show that is an equilibrium in (G, N1 , N2 , C). This concludes the proof of
the theorem.
5. Proof of Theorem 2
To prove part (i), notice first that P is always a symmetric distribution. To show that P is a correlated equilibrium repeat
the argument of Section 4.1 with the necessary changes.
In contrast to Theorem 1, here we cannot apply the result of Lehrer–Sorin in order to prove part (ii). Instead, we provide
a direct proof which uses similar ideas to those in the proof of Lehrer–Sorin but is somewhat simpler. Let ∈ (A) be a
symmetric correlated equilibrium
Assume that the (common) set of actions is B = {b1 , b2 , . . . , bm }.
with rational probabilities.
m
⊆ N, where
c = d and ckl = clk for every 1 ≤ k, l ≤ m, such that is
Then there exist d ∈ N and ckl
k,l=1 kl
1≤k≤m,1≤l≤m
characterized by the matrix ckl /d
1≤k≤m,1≤l≤m
. We denote ck =
m
c .
l=1 kl
We will need the following notation. If (x1 , x2 , . . . , xs ) is a string of s symbols then the Latin square corresponding to this
string is the s × s matrix
⎛
x1
⎜ xs
⎜ .
⎜ .
⎜ .
⎝x
3
x2
x2
x1
..
.
x4
x3
...
...
xs−1
xs−2
..
.
x1
xs
...
...
...
xs
⎞
xs−1 ⎟
.. ⎟
⎟
. ⎟,
x2 ⎠
x1
whose lines are successive right shifts of the string.
Now, consider the string of length 2d + 1 defined as follows (a diagram of this string appears below). The first symbol of
the string is 0. The next c1 symbols are b1 and following them are c2 symbols of b2 . We continue in this way until we reach
cm symbols of bm . This defines the first d + 1 symbols of the string. The last d symbols of the string are defined by (in this
order) cmm symbols of bm , cmm−1 symbols of bm−1 , . . ., cm1 symbols of b1 , cm−1m symbols of bm , cm−1m−1 symbols of bm−1 , . . .,
cm−11 symbols of b1 , . . ., c1m symbols of bm , c1m−1 symbols of bm−1 , . . ., c11 symbols of b1 . We call it the string generated by .
0
||b1 , . . . , b1 |
|b , . . . , b2 |
...
|bm , . . . , bm |
|b |
...
c1
c2
...
cm
cmm
...
2 m
|b |
...
cm1
...
1
|b |
...
c2m
...
m
|b |
|b |
...
c21
c1m
...
m
1
|b ||
1
c11
To illustrate the above construction consider the case where m = 3 and the (symmetric) distribution is given
by
Thus, c11 = 0, c12 = 1, c13 = 0, c22 = 1, c23 = 1, c33 = 2 and d = 7. The generated string of length 2d + 1 = 15 is
(0; b1 , b2 , b2 , b2 , b3 , b3 , b3 ; b3 , b3 , b2 , b3 , b2 , b1 , b2 ).
We now describe the environment (G, N, C) and the strategy profile . The population is taken to be N = {1, 2, . . . , 2d + 1}.
The action i (j) that player i plays when he meets player j is the action that stands in the ij place of the Latin square
corresponding to the string generated by . Finally, the categorization Ci is according to the strategy i , that is Ci (j) = Ci (j )
if and only if i (j) = i (j ).
Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
309
Let us return to the last example and consider the Latin square corresponding to the generated string. To make the reading
easier we write x = b1 , y = b2 and z = b3 .
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
0
y
x
y
z
y
z
z
z
z
z
y
y
y
x
2 3 4
x y y
0 x y
y 0 x
x y 0
y x y
z y x
y z y
z y z
z z y
z z z
z z z
z z z
y z z
y y z
y y y
5
y
y
y
x
0
y
x
y
z
y
z
z
z
z
z
6
z
y
y
y
x
0
y
x
y
z
y
z
z
z
z
7
z
z
y
y
y
x
0
y
x
y
z
y
z
z
z
8
z
z
z
y
y
y
x
0
y
x
y
z
y
z
z
9
z
z
z
z
y
y
y
x
0
y
x
y
z
y
z
10
z
z
z
z
z
y
y
y
x
0
y
x
y
z
y
11
y
z
z
z
z
z
y
y
y
x
0
y
x
y
z
12
z
y
z
z
z
z
z
y
y
y
x
0
y
x
y
13
y
z
y
z
z
z
z
z
y
y
y
x
0
y
x
14
x
y
z
y
z
z
z
z
z
y
y
y
x
0
y
15
y
x
y
z
y
z
z
z
z
z
y
y
y
x
0
Thus,
when players 3 and 5 are the cast of the meeting,
player 3 plays b2 while 5 plays b1 . The categorization of player 9
is C9 = {7, 10}, {4, 6, 8, 11, 12, 13}, {1, 2, 3, 5, 14, 15} .
We now show that P|1 = . A key observation which follows directly from the definition of a Latin square is that
j (1) = 1 (2d + 3 − j) for every j = 2, . . . , 2d + 1. In words, the action that player j plays against 1 is the same as the one that
player 1 plays against 2d + 3 − j.
Fix two actions bk , bl ∈ B (possibly bk = bl ). By the last observation
{j ∈ N \ {1} : 1 (j) = bk , j (1) = bl } = {j ∈ N \ {1} : 1 (j) = bk , 1 (2d + 3 − j) = bl }.
We claim that the construction of the generated string is such that the cardinality of this set is 2ckl . To avoid complex
notation, we leave this fact for the reader to convince herself (the diagram is helpful). Thus,
P|1 (bk , bl ) =
|{j ∈ N \ {1} : 1 (j) = bk , j (1) = bl }|
2d
=
2ckl
= (bk , bl ).
2d
Next, we show that P|i = for every i ∈ N. Notice that, for every 1 ≤ i ≤ 2d and every bk , bl ∈ B,
j ∈ N \ {i} : i (j) = bk , j (i) = bl
+1=
j ∈ N \ {i + 1} : i+1 (j) = bk , j (i + 1) = bl ,
where the summation means adding 1 to each element of the set, and with the convention that 2d + 2 = 1. Thus,
{j ∈ N \ {i} : i (j) = bk , j (i) = bl } = {j ∈ N \ {i + 1} : i+1 (j) = bk , j (i + 1) = bl } .
Since we already showed that P|1 = , it follows by induction that P|i = for every i ∈ N.
To finish the proof it should be shown that is an equilibrium in (G, N, C). First, it is clear that is adapted to C. Second,
fix i ∈ N and D ∈ Ci . By construction, there is b ∈ B such that D = {j ∈ N \ {i} : i (j) = b}. Thus, D (i) = P|i (·|b) = (·|b). Since is a correlated equilibrium, a best response to (·|b) is b. Thus, i plays a best response to D (i).
6. Final comments
6.1. Games with more than two players
Throughout the paper we only consider games G with two players. One can, however, generalize the model to allow
for games with any number
m m ∈ N of players. Assume that there
m are m different populations N1 , . . . , Nm , and that every
k = (k1 , . . . , km ) ∈ N := i=1 Ni is chosen with probability (1/ i=1 ni )(ni := |Ni |).
It is not clear in this case what should be the definition ofa categorization of a player. There are two options. The first is
that a categorization of ki ∈ Ni is any partition of the product j =/ i Nj . The interpretation of such definition is that each player
categorizes the set of possible compositions of opponents. The second option is that only product partitions are allowed.
/ i and Cki is the product of these partitions. Here, player ki categorizes
That is, player ki has a partition of every Nj , j =
each population separately and categorizes the composition of opponents accordingly. In my view, the second option is
behaviorally more plausible.
If the first option is used to define a categorization then the result of Theorem 1 holds in this generalized setting. The
proof of this uses a more general form of the Lehrer and Sorin (1997) result. We do not know if the theorem is true when only
product partitions are allowed. It is also not clear how to generalize the definition of categorization in the single-population
model.
310
Y. Azrieli / Journal of Mathematical Economics 46 (2010) 303–310
6.2. Common categorization
One may want to consider the case where all agents are using the same categorization. A natural question is whether any
correlated equilibrium can be achieved with this additional restriction. The negative answer to this question is a consequence
of the following argument.
Assume that in a two-populations environment (G, N1 , N2 , C) the categorization profile C satisfies Cki = Ck for every
i
ki , ki ∈ Ni and for i = 1, 2. Denote the (common) categorization of Ni by Ri (i = 1, 2). Fix any two sets D1 ∈ R1 , D2 ∈ R2 . Now,
given that (k1 , k2 ) ∈ D1 × D2 , the induced distribution over A is a product measure and, moreover, the marginal distributions
over A1 and A2 constitute a Nash equilibrium of G. It follows that the induced distribution P is in the convex hull of the
set of Nash equilibria of G. Since in general there may be correlated equilibria outside this convex hull, not any correlated
equilibrium can be obtained.
6.3. Endogenizing the categorization profile
In our model the categorization profile is part of the exogenously given environment. It would be interesting to further
develop this type of model by relaxing this assumption. One possible approach8 is to introduce a first stage to the game
where each player chooses his categorization, and to consider (sub-game perfect) equilibria of the extended two-stage game.
This direction is beyond the scope of the paper.
Acknowledgements
I am grateful to Ehud Lehrer as well as to Paul Healy, Yuval Heller, Philippe Jehiel, Jim Peck, Roee Teper, and especially to
two anonymous referees and an associate editor for helpful comments and references that significantly improved the paper.
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8
This idea is taken from Jehiel (2005, Section 6).