3. Rev. Correlated equilibrium - uniroma1.it · 2016-04-18 · Correlated equilibrium 3.1...

D. Tosato – Game Theory – Lecture Notes – a.y. 2012.2013 I.1

Sapienza University of Rome. Ph.D. Program in Economics a.y. 2015-2016

Microeconomics 2 – Game Theory - Lecture notes

3. Rev. Correlated equilibrium

3.1 Introduction: from independent to correlated randomization of strategies

3.2 Correlated equilibrium using a public signaling device

3.2.1 Correlating device

3,2,2 Correlated equilibrium

3.3 Correlated equilibrium using a partially private signaling device


3.3.2 Correlated equilibrium

3.4 Formal definitions of correlated equilibrium

3.4.1 Correlated equilibrium as a profile of strategies adapted to the information structure

3.4.2 Correlated equilibrium as a distribution over the set of pure strategies

Whereas a mixed strategy Nash equilibrium is based on the assumption that each player chooses

his mixed strategy independently of the choice made by the other players, correlated

equilibrium, proposed by Aumann (1974, 1987), admit the possibility of correlation between the

players’ randomization. This is a generalization of Nash equilibrium, directly relevant to games

with multiple pure strategy equilibria. We may think of correlation as the outcome of the

working of a random “signaling device”, that sends signals to the players, prior to the play of the

game, consisting in a set of recommendations as to the strategies to play. The specific interest in

correlated equilibrium is explained by the circumstance that payers can do better than in the

mixed strategy Nash equilibria (MSNE).

This Lecture Note is organized in the following way. Section 3.1 introduces the idea of

correlated equilibrium by comparing the set of Nash equilibria of two games - Battle of the Sexes

and Hawk-Dove - with those obtaining for the same games assuming that players may correlate

their strategy choices. In the case of the game Battle of the Sexes correlation of the strategies is

the rather obvious result of a private pre play agreement as to the strategy to follow depending on

the outcome of a random variable; 1 in the game Hawk-Dove, representing, for instance, a

situation of traffic regulation, the signaling mechanism can be associated to a public device

giving the right of way to cars coming alternatively from left and right of the road. The example

shows that the set of correlated equilibria is the convex hull of the Nash equilibria. The example

further shows that the set of correlated equilibria can be defined both in terms of the strategies

1 As a tool of pre-play communication among agents, this type of game is also classified in the category of games with

communication.


that players will adopt as well as in terms of probabilities that support a given set of payoff

profiles.

Concentrating the attention on the definition of correlated equilibrium in terms of the strategies

the players adopt, the subsequent two paragraphs are dedicated, following Fudenberg and Tirole

(1991, pp. 53-59) and Osborne-Rubinstein (1994, pp. 44-48), to the description of a correlating

device and its conceptual implications. The more structured Hawk-Dove game is taken as

example of the approach. We suppose first (Section 3.2) that players dispose of a purely public

correlating device and then (Section 3.3) that they receive a different but correlated signal. The

focus is on the definition of the correlating device, on the determination of players’ conditional

probabilities and on the proof that the players cannot gain by disobeying the recommendation

they receive. We show in Section 3.3 that, when the players have access to a sophisticated

correlating device that sends different but correlated signals, a larger set of correlated equilibria

can be reached including payoff vectors outside the convex hull of the Nash equilibria.

The final Section offers a formal definition of correlated equilibrium in terms of strategies as

well as in terms of probabilities. Following the definition and the description of the modus

operandi of a correlating device exemplified in Sections 3.2 and 3.3, we present first the

definition of correlated equilibrium in terms of strategies In this presentation we introduce the

distinction, following Fudenberg and Tirole, between an ex ante and an ex post approach

depending on the information players receive: the existence of a random correlating device with

the probabilities of the different outcomes or, in addition, also an information as to the outcome

that has occurred, as in the examples dealt with. We turn subsequently to the more complex

definition of correlated equilibrium in terms of a ‘universal’ correlating device, which requires a

different interpretation of the space of the outcomes of the possible correlating devices in terms

of pure strategy profiles of all players. The equivalence of the two definitions is shown.

3.1 Introduction: from independent to correlated randomization of strategies

Let 1, , ,i i iI S u be the normal form representation of a simultaneous-move finite

game in mixed strategies. A Nash equilibrium of the game is a profile of mixed strategies

1*, * Ii i i iS S , such that *, * , *i i i i for all i and all i iS .

The existence of a mixed strategy equilibrium, which analytically follows from the application of

Brouwer’s or Kakutani’s fixed point theorems,2 is based on the idea that each player chooses his

mixed strategy independently of the choice made by the other players. These independent

choices induce a probability distribution over the payoffs of the pure strategy profiles ,i i iu s s .

2 See Lecture Note 2.


Consider, by way of example, a 2x2 matrix game in which the strategy sets are 1 ,S T B 3 and

2 ,S L R ; denote player’s 1 mixed strategy over 1 ,S T B as 1 1 1,1S and

player’s 2 mixed strategy over 2 ,S L R as 2 2 2,1S . Independent randomization

by the players means that the probability distribution induced over the cells of the payoff matrix

is the product of the probabilities that each player assigns to the other players’ possible choices

of randomization over the set of their pure strategies. The induced probability distribution is

shown in Fig. 3.1 As indicated in Lecture Note 1, this is the Harsanyi-Aumann interpretation of

mixed strategies as the result of each player’s uncertainty as to the possible choices of the other

players (Harsanyi, 1973 and Aumann, 1974 and 1987).

Figure 3.1 – Probability distribution induced by independent mixed strategies

To gain an intuition of the notion of correlated equilibrium and of the working of a signaling

device, consider the Battle of the Sexes game of Fig. 2.

Figure 3.2 – Payoff matrix of the game Battle of the Sexes

3 ,T B respectively for Top and Bottom; ( , )L R for Left and Right.

1 2 Left Right

Top 1 2 1 21

Bottom 1 21 1 21 1

1 2 Hockey Ballet

Hockey 2, 1 0, 0

Ballet 0, 0 1, 2


The game has three Nash equilibria: two of them in pure strategies - ,H H and ,B B - and

one in mixed strategies - 1 22 1,

3 3 with expected payoffs equal to 2

3 - resulting

from the independent randomization of player 1 in response to his conjectures/beliefs regarding

the possible mixed strategies of player 2, and vice versa. The Nash equilibrium configuration is,

in any case, the outcome of a consistent alignment of the reciprocal conjectures of the two

players.

Suppose now that the random signaling device is based on the flip of a fair coin, with known one

half probabilities of both Heads and Tails and that the signal to both players is to play strategy H

if Heads shows and strategy B if, on the contrary, Tails shows. If the players comply with the

recommendation of the signal, the expected payoff of player 1, as well as of player 2 is

1 131 1, ,

2 2 2u H H u B B . Assume that the signal is based on the outcome of the flip of

an unfair coin – say, 0.7p H and 0.3p T - with unchanged recommendations. The

expected payoffs to the two players would then be respectively 1.7 and 1.3. We will later show

that the players have no incentive to deviate from the recommendation of the signal.

Figure 3.3 – Correlated equilibrium payoffs

It is easily seen that these two correlated payoffs are obtained as a convex combination of the

two pure strategy Nash equilibria. Fig. 3.3 shows that the set of correlated payoffs, that can be

reached following the recommendation of the signaling device, is the convex hull of the three


Nash equilibria. 4 Fig. 3.4 shows that the set of probabilities, that constitute a correlated

equilibrium, is the convex hull of the mixed strategies, degenerate and non degenerate, that

constitute the set of Nash equilibria.5

A final observation: Nash equilibria are a subset of the set of correlated equilibria: those

obtaining when the signaling device does not offer to the players any suggestion as to the

strategy to play. Given the existence of Nash equilibria, we can conclude that the set of

correlated equilibria is non empty.

Figure 3.4 – Correlated equilibrium strategies

2.1 Correlated equilibrium using a public signaling device

In the sequel of this Lecture Note we will consider the following Hawk-Dove game – also termed

the chicken game or the traffic intersection game – in which the players have two strategies: the

4 The convex hull of a finite number of vectors (points) 1,..., N

kx x is defined by the expression

1

1 1

conv ,..., , 0, 1

K KN

K k k k k

k k

x x x x x

. Recommendations of adopting complex mixed strategies

would be required to attain the different points of the convex hull. 5 It will later be shown how these probabilities can be determined.


aggressive strategy of the hawk H and the accommodating strategy of the dove D .6 Fig. 3.5

represents the payoff matrix of this 2x2 game. The aggressive strategy obtains a high payoff of 5

when meeting an accommodating strategy, whose payoff is only 1, and a zero payoff when faced

by an equally aggressive strategy by the opponent; when both players adopt an accommodating

strategy they both receive a payoff of 4.

Figure 3.5 – Payoff matrix of the game Hawk-Dove

Figure 3.6 – Convex hull of the Nash equilibrium payoffs of the game Hawk-Dove

6 In the traffic intersection version of the game the Hawk strategy could be better termed “go” and the Dove strategy

“stop”.

1 2 Dove Hawk

Hawk 5, 1 0, 0

Dove 4, 4 1, 5


Under independent randomization, the game has three Nash equilibria, which stated in terms of

mixed strategies and corresponding expected payoffs are

(3.1) 1 21 1, 1,1 , 0,0 , ,

2 2 ;

(3.2) 1 1 2 2 1 2, , , 5,1 , 1,5 , 2.5,2.5u u

The convex hull of these equilibria, expressed in terms of payoffs, is shown in Fig. 3.6.


A correlating device consists of:

1/ an information structure, which describes the possible outcomes of the random device taken

into consideration (for example, flip of a coin, the throw of a pair of dice, the extraction of a

colored ball from an urn of known composition);

2/ a signal representing the recommendations issued to the players as to the pure strategy to play

in response to the outcome of the random device.

1/ Formally, an information structure is a triple , ,iH p :

, with elements n 1,...,n N , is a finite state space, i.e. the set of the possible outcomes of

the random correlating device. In the preceding introductory Section 3.1, is the space of the

possible outcomes of the flip of a coin. We accordingly have 1 2, ,Heads Tails , for

short, H or T .

1 2,p p p is a probability measure on ; two hypotheses were made previously: 0.5p H

and 0.7p H .

iH is the information partition which describes player i ’s information concerning the outcome

of the random device. More specifically, when the state of the world, i.e. the result of the random

device, is a given outcome , player i observes – equivalently, is informed of, or knows –

the element of his information partition containing . Note carefully: this does not mean that

player i knows the state of the world that has occurred, but rather that he knows the information

subset of his information partition containing . Two examples help clarifying the limits of the

knowledge conveyed by the information partitions as well as the basic distinction between

informative and non informative partitions.


With regard to the notion of knowledge, suppose that contains the four states 1 2 3 4, , ,

and that the information partitions of the two players are 1 21 1 2 3 4, , , ,i iH h h and

1 22 2 2 1 4 2 3, , , ,H h h respectively. Assume that 2 is the “true” outcome of the

random device. Player 1 is informed that the true state belongs to 11h ; it could then be either 1

or 2 . In this case knowledge of the partition containing the true state does not imply that player

1 knows what state has occurred. The implications concerning 2’s knowledge of the state of the

world and his possible strategy choices are quite different. If 1 , player 1 would conclude that it

belongs to partition 12h of player 2; if 2 , to partition

22h . In this case, player 1 would be ignorant

of what player 2 knows.7

With regard to the distinction between informative and non informative information partitions,

suppose, as in the previous Battle of the Sexes and Hawk-Dove games, that the correlating device

has only two possible outcomes. The information partition may contain just one element

,iH H T or may contain the two separate elements 1 2, ,i i iH h h H T . The former is

a trivial partition, that offers no information to which no possible sensible recommendation can

be attached as to the strategy to play; the latter is, on the contrary, fully informative: the players

are told which outcome has occurred and can follow the possible recommendations separately

associated to the two events. If the information partition is non informative, the only possible

equilibria of the game are the Nash equilibria. If the information partition distinguishes between

the two outcomes of the correlating device, the profile of possible payoffs of the players is one of

the vectors in the convex hull of Figg. 3.3 and 3.6 depending on the type of recommendation

issued to the players. We shall fix the attention on a set of recommendations that result in the

determination of a correlated equilibrium, which is an improvement for both players with respect

to the non degenerate mixed strategies payoffs of the Nash equilibrium.

2/ A recommendation is an instruction concerning the pure strategy to play; formally, a mapping

:i i iH Ss , that is a function is that maps elements i ih H to pure strategies i is S with the

proviso that all the elements belonging to the same partition i ih H are mapped to the same pure

strategy i is S . The strategy is is a pure strategy adapted to the information structure:. We will

denote by i iss the strategy thus adapted – for instance, for player 1: play Hawk if the outcome

is Heads, play Dove if the outcome is Tails.

Let us see how these notions can be applied to the Hawk-Dove game. The correlating device 1s

defined by the following elements and the related structure of knowledge of the players:

7 See Aumann (1987, pp. 9-10).


1/ the probabilities 12

p H p T are common knowledge;

2/ given the assumption of a fully informative information partition, each player is informed of

the outcome that has occurred; more generally, each player is informed of which of the

information subsets of his information partition contains the true state;

3/ the information partitions of the two players are common knowledge;

4/ the recommendations issued to the players are:

i) to player 1, play Hawk if you are told that H has occurred, i. e. if 11h , and play Dove

otherwise, i. e. if 21h ;

ii) to player 2, play Dove if you are told that H has occurred, i. e. if 12h , and play Hawk


5/ the recommendations are mutual knowledge.

A word of comment on these elements. 1/ defines the assumption of common priors, which holds

whenever i jp p for any two players i and j.8 This assumption implies that people ascribe

different probabilities to different events only if they receive a different information, i.e. if they

have different information partitions as exemplified in the case of a state space containing four

possible outcomes.

2/ describes the players’ knowledge.

3/ excludes that there may be uncertainty on the part of a player about the information partition

of the other player. iH is part of the description of the model; it does not enter the description of

any particular outcome ; thus, differently from , it cannot be an object of uncertainty. Notice

that, while player 1 cannot be ignorant of the information partition of player 2, he may well not

know to which partition of 2H a given state may belong, as illustrated in the case previously

considered of a state space containing four possible outcomes.

4/ defines the strategies of the expanded game adapted to the information structure, for short the

strategies recommended by the correlating device.

5/ states that the players know the recommendations issue by the correlating device to each one

of them. Here the term “mutual” knowledge marks a distinction with respect to the notion of

common knowledge that presupposes a possibly infinite regress. Formally, a pure adapted

8 The assumption of common priors leads to the notion of objective correlated equilibrium, as opposed to the notion of

subjective correlated equilibrium when the common priors assumption is relaxed to allow different players to have

different subjective probabilities regarding the outcomes of the correlating device.


strategy can be viewed as a function is that maps elements i ih H to pure strategies i is S :

:i i iH Ss . We will denote by i iss the strategy thus adapted – for instance, for player 1:

play Hawk if the outcome is Heads. We can, therefore, define the game in pure strategies thus

expanded as *

1, , , , , ,

I

i i i i i i ii

G I H p S u s s

s s .9

The correlating device – described by points 1/ to 5/ above - induces the probability distribution

over the cells of the payoff matrix of the Hawk-Dove game indicated in Fig. 3.7.

Figure 3.7 – Probability distribution induced by the correlating device

We can conclude with the following definition of a pure strategy game adapted to the

information structure.

Definition 3.1 Given the pure strategy game 1

, , ,I

i i i i iG I S u s s and its mixed extension

1, , ,i i iI S u , the pure strategy game adapted to the information structure

, ,iH p is the game

(3.3) *

1, , , , , ,

I

i i i i i i ii

G I H p S u s s

s s

in which and i i i is s s s are the pure strategies adapted to the information structure,

i,e, the strategies recommended to the players by the correlating device.


9 As explained by Fudenberg and Titole (p. 56), there is no need to extend the definition of the game to include the

recommendation to play a mixed strategy, such as i is , provided that the outcome space is sufficiently large so as to

associate to each possible outcome of the correlating device a pure strategy.

1 2 Dove Hawk

Hawk 12

0

Dove 0 12


In line with the property of Nash equilibrium that each player’s strategy should be a best

responses to each other’s choices so that no player should have an incentive to deviate,

correlated equilibrium must have the same property. It could not be otherwise since the Nash

equilibria obviously belong to the convex hull of the payoffs associated to these equilibria. It is

then crucial to show that players cannot gain by deviating from the recommendation issued by

the correlating device described at point 4/ of the previous description of a game adapted to the

information structure.

Let the outcome of the correlating device be Heads. Both players know that 11h and

12h and can, therefore, determine the probabilities, conditional on the information received,

of the strategy choices of the other player on the assumption that the recommendations of the

correlating device are obeyed by the players. Let 2p be player’s 1 conditional probabilities

concerning the strategy choices of player 2 and 1p the conditional probabilities of player 2.

Remembering that recommendations are mutual knowledge, we have, if Heads

(3.4) 12 1 1p D h and 1

2 1 0p H h

(3.5) 11 2 1p H h and 1

1 2 0p D h

and, if Tails

(3.6) 22 1 0p D h and 1

2 2 1p H h

(3.7) 21 2 0p H h and 2

1 2 1p D h

Let Heads . The expected payoffs of player 1 following the recommendation and deviating

from it are respectively

(3.8) 1 1 11 1 2 1 1 2 1 1, , 5Eu H h p D h u H D p H h u H H

(3.9) 1 1 11 1 2 1 1 2 1 1, , 4Eu D h p D h u D D p H h u D H

We conclude that player 1 has no incentive to deviate from the recommendation of playing

Hawk.

Turning to player 2, we similarly have

(3.10) 1 1 12 2 1 2 1 1 2 1, , 1Eu D h p H h u H D p D h u D D


(3.11) 2 2 22 2 1 2 1 1 2 1, , 0Eu H h p H h u H H p D h u D D

and conclude that also player 2 has no incentive to deviate from the recommendation he receives

of playing Dove.

Suppose now Tails . Given the symmetry of the game, the expected payoffs of player 1 from

playing the recommended strategy Dove and from deviating to Hawk are those indicated by

(3.10) and (3.11) respectively, while player 2’s expected payoffs from playing the recommended

strategy Hawk and from deviating to Dove are those indicated by (3.8) and (3.9) respectively.

We conclude, therefore, that also in this case the players have no incentive to deviate from the

strategy suggested by the correlating device. The strategies recommended by the correlating

device are, therefore, best response to each other.

The players’ expected payoffs conforming to the signal are

(3.12) , , for , 1,2i i i j j j i i i j jEu s s p u s s i j

s s s s

In the Hawk-Dove game and for the specific recommendation we have taken into consideration,

we obtain

(3.13)

1 2

,

1 1, ,

2 2

1 1 55 + 1 = for , 1, 2

2 2 2

i i i j j

i i i j j i i i i j j i

Eu s s

u s s h u s s h

i j

s s

s s s s

These payoffs are in the convex hull of the Nash equilibria of the game, as indicated in Fig. 3.6.

3.3. Correlated equilibrium using a partially private signaling device

Whereas in the preceding Section we have considered a correlating device that sends a public

information, that is an identical information to both players, we suppose now to have a device

sending different but correlated signals to each of the players. We need for this purpose to have

a larger space of outcomes: we can think, for instance, to a correlating device defined with

respect to the color of a ball extracted from an urn containing an equal number of Black, Yellow

and White balls. The state pace now contains three possible outcomes - , ,B Y W - with


prior probabilities 13

p B p Y p W . As we will later see, this induce the probability

distribution indicated in Fig. 3.8 on the cells of the payoff matrix..

Figure 3.8 – Probability distribution induced by the correlating device


Suppose that the different but correlated signals received by the players are captured by a

different partition of their information sets: player 1 is informed if the ball is Back or Non Black;

player 2 if the ball is White or Non White. The subsets of the information partition of the players

are therefore

(3.14) 1 21 1, ,h B h Y W and 1 2

2 2, ,h B Y h W

Note that this is just one of the five possible information sets of player 1 and, similarly, of player

2 that can be formed from a state space containing three outcomes, namely: 1 , ,H B Y W non

informative; 1 , ,H B Y W fully informative; and the partially informative sets

1 , ,H Y B W , 1 , ,H B Y W .

Suppose, analogously with the previous description, that the correlating device is defined by the

following terms:

1/ the prior probabilities 13

p B p Y p W are common knowledge;

2/ the players are informed of the outcome in the terms of the information partitions described by

(3.14);

3/ the information partitions of the two players are common knowledge;

1 2 Dove Hawk

Hawk 13

0

Dove 13

13


4/ the recommendations players receive are:

i) to player 1, play Hawk if you are told that B has occurred, i. e. if 11h , and play Dove


ii) to player 2, play Dove if you are told that B or Y has occurred, i. e. if 12h , and play

Hawk otherwise, i. e. if 22h ;

5/ the recommendations are mutual knowledge.

Let 2p and 1p be players’ 1 and 2 probabilities, conditional on the outcome of the

correlating device and the associated adapted strategies, concerning the strategy choices of the

other player. These conditional probabilities are the players’ posterior beliefs about obtained

applying Bayes’ rule. For player 1, we have:

- if 11B h , player 1 knows with certainty that

12h and the recommendation received by

player 2 is to play Dove; player’s 1 conditional probabilities are, therefore,

(3.15) 12 1 1p D h and 1

2 1 0p H h

- if player 1 is informed that 21h , he knows that the outcome may have been, with equal

probability, Y or W ; and concludes that, with equal probability, player 2 may choose

strategy Dove or strategy Hawk; his conditional probabilities are, therefore,10

(3.16) 22 1

1

2p D h and 2

2 1

1

2p H h

With analogous reasoning we can determine player’s 2 conditional probabilities 1p :

- if 12h , he knows that the outcome may have been, with equal probability, B or Y ;

and concludes that, with equal probability, player 1 may choose strategy Hawk or strategy Dove;

his conditional probabilities are, therefore,

(3.17) 11 2

1

2p H h and 1

1 2

1

2p D h

- if 22W h , his conditional probabilities are

10 In Bayesian terms 22 1

1 2 1/

3 3 2p D h .


(3.18) 21 2 0p H h and 2

1 2 1p D h


We must know check that, for each possible outcome of the correlating device and associated

information structure, players complying with the recommendations received have no incentive

to deviate. Suppose that the outcome of the correlating device is 11B h . Player 1’s expected

payoffs following the recommendation and deviating from it are respectively

(3.19) 1 1 11 1 2 1 1 2 1 1, , 5Eu H h p D h u H D p H h u H H

(3.20) 1 1 11 1 2 1 1 2 1 1, , 4Eu D h p D h u D D p H h u D H

Notice that in these relations the conditional probabilities are the same, because they depend on

the same information namely 11h , but the expected payoffs are different. We conclude that

player 1 has no incentive to deviate from the recommendation of playing Hawk.

Turning to player 2, we similarly have

(3.21) 1 1 12 2 1 2 2 1 2 2, , 2.5Eu D h p H h u H D p D h u D D

(3.22) 1 1 12 2 1 2 2 1 2 2, , 2Eu H h p H h u H H p D h u D D

and conclude that also player 2 has no incentive to deviate from the recommendation of playing

Dove.

Suppose now Y with the implication 21h and

12h . Players’ 1 and 2 conditional

probabilities are again defined by (3.16) and (3.17). Player’s 1 expected payoffs from obeying

the recommendation - 1Eu D Y - and disobeying it - 1Eu H Y - are both equal to 2.5

and there is no incentive to deviate. Player’s 2 corresponding expected payoffs are those

indicated in (3.21) and (3.22), because his information partition does not allow him to distinguish

between B and Y , he assigns, therefore, conditional probabilities equal to 1

2 to player

1 playing Hawk and Dove.


Suppose, finally, W with the implication 21h and

22h . This is a situation directly

opposite to that already considered when the outcome is B , simply reversing the positions of

players 1 and 2.

This completes the proof that no player gains from disobeying the recommendations of the

correlating device.

Note that the recommendations of the correlation device result in only three possible strategy

profiles being played, namely the profiles , , ,H D D D and ,D H . The strategy profile

,H H will never be played. This shows that the probability distribution induced by the

correlating device considered is as indicated in Fig. 3.8. The expected payoff of player i , when

both players follow the signals they receive, is therefore

(3.23)

,

, , ,

1 1 1 105 + 4 + 1 = for , 1, 2

3 3 3 3

i i i j j

i i i j j i i i j j i i i j j

Eu s s

p B u s s B p Y u s s Y p W u s s W

i j

s s

s s s s s s

This expected payoff is outside the convex hull of the Nash equilibria, as depicted in Fig. 3.6.

The result thus obtained deserves to be duly underlined; it confirms the initial claim that with a

well thought partially private but correlated signaling mechanism players can reach payoff

profiles that Pareto dominate the payoffs of the convex hull of the Nash equilibria of the original

game with independent randomization.

3.4 Formal definitions of correlated equilibrium

There are two equivalent ways to formulate the definition of a correlated equilibrium (Aumann,

1987). The first reflects the approach followed in the preceding Sections, in which we have

constructed two examples of correlated equilibrium based on different correlation devices. Given

a correlation device, the “expanded game” *

1, , , , , ,

I

i i i i i i ii

G I H p S u s s

s s ,

displaying the assumed information structure and the related recommendations as to the

strategies that players ought to choose, is defined. A correlated equilibrium is then formulated in

terms of strategies adapted to the information structure, whose expected payoff cannot be

improved by deviating to a different strategy.


The second definition moves, so to speak, in the opposite direction. It assumes the existence of a

“universal” finite state space , p of possible probability distributions of a random variable

representing the outcomes of a random mechanism. In terms of the approach of epistemic game

theory, the random variable represents the set of correlated strategy profiles to be taken into

consideration.11 In the examples of Sections 3.2 and 3.3 only the four pure strategy profiles of

the Battle of the Sexes and of the Hawk-Dove games were considered; but the set of outcomes of

the state variable may well include mixed strategies. The probability measure p is then

directly defined on the set of strategy profiles of the original game 1

, , ,I

i i i i iG I S u s s . We

can think of these probability distribution as representing possible correlation devices. A

correlated equilibrium is then defined as any probability distribution over the set of pure strategy

profiles such that there is no incentive for any player to deviate from playing his component of

these strategy profiles.12

Alternatively, and in terms closer to the previous denomination of strategies adapted to the

information structure, we could define correlated strategies as random variables determined by a

mapping from the state space to the space of pure strategy profiles of the game. This mapping

determines the probability distribution over the cells of the payoff matrix. In the definition of

equilibrium we will follow this alternative definition of the notion of correlated strategies. The

distinction between the state space and the space of correlated strategies has the advantage of

defining the probabilities on the space of strategy profiles in terms of events (the sum of a subset

of possible outcomes) rather in terms of single outcomes.

It is worth remarking the conceptual difference between these two approaches based, the first, on

the notion of strategies adapted to the information structure and, the second, on the notion of

strategies correlated to the universal probability space: for short, the strategies adapted and the

strategies correlated approaches. Suppose that the probability space of the first approach

coincides with the universal probability space of the second.13 The former approach focuses on

the role of the information structure and the associated signaling device, the recommendations

that players ought to follow; the latter directly correlates the possible strategy profiles to the

outcome space and the derived probability distributions over the possible strategy profiles of the

game, the various cells of the payoff matrix. A differentiated public signal is needed to trigger

off the strategies adapted approach; the strategies correlated approach requires that the players be

able to establish, in a reciprocally consistent way, the mapping from the state space to the

11 Alternatively, and in terms closer to the previous denomination of strategies adapted to the information structure, we

could define correlated strategies as random variables determined by a mapping from the state space to the space of

pure strategy profiles of the game. This mapping determines the probability distribution over the cells of the payoff

matrix. 12 This would exclude possible distributions and related recommendation that would suggest a player to play a strictly

dominated strategy 13 This is a necessary assumption for the possibility of a comparison of the results.


strategy profiles of the game. Any possible probability distribution, just as any possible

information structure, would lead to a correlated equilibrium provided that the players know/are

informed/agree on the same probability distribution. The critical issue is then how this is

achieved, absent a public signal.

Auimann (1987) proves the equivalence between the strategies adapted and the strategies

correlated approach to the definition of correlated equilibrium he shows as well the connection

between the two alternative definitions of a correlated equilibrium in the correlated strategies

approach.

A further remark. In both cases the definition of correlated equilibrium can be presented from an

ex ante or from an ex post point of view. In the ex ante approach the equilibrium is defined as the

maximization of the expected payoff in terms of the possible information that the players can

receive. In the ex post approach – which Fudenberg and Tirole call the interim approach - the

definition is stated in terms of the maximization of the expected payoff conditional on the

information (recommendation) received by the player.14 The same distinction between the ex

ante and the ex post definitions of a correlated equilibrium applies to the correlated strategies

approach. The same terminology – ex ante vs ex post; or ex ante vs interim in Fudenberg and

Tirole’s - is used in the study of Bayesian Nash equilibrium.

3.4.1 Correlated equilibrium as a profile of strategies adapted to the information structure

It is convenient to start from the ex post definition which follows directly from the examples

examined in the previous Sections and turn afterward the ex ante definition.

Definition 3.2 (Ex post) Given the game *

1, , , , , ,

I

i i i i i i ii

G I H p S u s s

s s a

strategy profile 1,..., Is= s s is a Nash equilibrium in strategies adapted to the information

structure ,iH p if for all players i , all information sets with 0ip h and all i is S

(3.24)

, ,

i i i i

i i i i i i i i i i i i i i

h h h h

p s h u p s h u s

s s s s s

where the i i i ip s h s stand for the conditional probabilities that player i assigns, on the

basis of his information partition, to the strategy choices of all the other players.

14 The term interim approach may be justified by the consideration that in any case each player knows only a part of the

information, the part to him directly relevant and is ignorant of the information received to by other players. Personally

I prefer to use the term ex post approach since I believe that the relevant aspect of the information structure of the

players is the knowledge that enables everyone to act according to the principle of Bayesian rationality, that is, to

maximize the expected payoff given the conditional probabilities assigned to the other players choice of strategy.


We say the player i is Bayes rational at if his expected payoffs, given his information, satisfy

the inequality (3.24).

Definition 3.3 (Ex ante) Given the game *

1, , , , , ,

I

i i i i i i ii

G I H p S u s s

s s a

strategy profile 1,..., Is= s s is a Nash equilibrium in strategies adapted to the information

structure ,iH p if for all players i and every adapted strategy is

(3.25) 1, ,i i i i ip u p u

s s s s

Remembering that the conditional probabilities i i i ip s h s are Bayesian posterior

probabilities we can write

(3.26)

1

i i

i i i i i i i

i

h H

pp s h p h h

p h

s s

Substituting in (3.24), we obtain (3.25).

Note that the ex-ante Definition 3.3 requires that is maximizes player i ’s expected payoff

before knowing which partition ih of his information set iH contains the true state: hence the

ex-ante denomination. In the ex-post Definition 3.2 we require that is maximizes player i ’s

expected payoff conditional on his knowledge of which partition ih of his information set iH

contains the true state: hence the ex-post denomination.

3.4.2 Correlated equilibrium as a distribution over the set of pure strategies

Turning to the definition of correlated equilibrium in terms of probability distribution, we will

follow Aumann’s approach which, in the terminology here adopted, is formulated in terms of an

ex ante equilibrium condition. The ex post definition follows immediately reversing the order

just used to show the equivalence between these two approaches regarding the players’

knowledge of the true state. The introduction of conditional probabilities leads finally to the

proof that the each definition implies the other.

The starting point of the approach is the definition of a “universal” state space , p and a

mapping : i iS S S s from the universal state space to the set of pure strategy

profiles of the game 1

, , ,I

i i i i iG I S u s s . The essential motivation for this approach is to be


found in the goal of avoiding the shortcomings of the “strategy adapted to the information

structure” approach. There are two such shortcomings. The first derives from the circumstance

that the resulting correlated equilibrium depends on the specific correlation device taken into

consideration. As shown in Section 3.3, given a set of outcomes, there are several alternative

constructions of the players’ information partitions. To each such partition there corresponds a

different correlated equilibrium. The second points to the fact that the set of possible outcomes

of the random device, and thus the dimension of the state space, can be very large. This would

make computing the possible correlating equilibria unfeasible.

Let be a finite set of possible outcomes and the prior probability of occurrence of

each outcome with 0 and 1

.15 Let us further define the strategy profile

correlated with the possible outcomes of the universal state space by means of the mapping

: i iS S S s . Using the notation ,i is s s for the mapping, we can indicate player

i ‘s correlated strategy as i is s . The probability associated with the strategy profile s is

therefore

(3.27) s

p s s

s

s s

In words, the probability attached to the pure strategy profile s is the sum of the prior

probabilities of the outcomes that are mapped into that strategy profile.

Player i ‘s expected payoff, given the correlated strategy profile ,i is s s , is then

(3.28)

, ,i i i i i i i

s S s S s

Eu p s u s s u s s

s

s s

Definition 3.3 (Ex ante definition of correlated equilibrium). Given the game in strategic

form 1

, , ,I

i i i i iG I S u s s a correlated strategy profile * s is a correlated equilibrium

if and only if for all players i and all pure strategies *i i s s

(3.29) * * * * *, ,i i i i i i i i i iEu s Eu s s s s s s s

15 We can disregard outcomes with a zero probability of occurrence, since they would have no effect on the

determination of the correlated strategies and their probabilities of occurrence.


In words, the correlated strategy profile * s is a correlated equilibrium of the pure strategy

game G if and only if for every player i his component *i s of the strategy profile * s is a

best response to the other players components *i s of the strategy profile * s .

The corresponding ex-post definition is obtained introducing the conditional probabilities

* *i ip s s . We have

(3.30)

* * * * * * *, ,

i i i i

i i i i i i i i i i

S S

p u p u

s s

s s s s s s s s

An example using the 2x2 Hawk-Dove game illustrates the approach and paves the way to a

different approach to the proof of the existence of a correlated equilibrium.

Let , , ,P q r s t be a probability distribution over the cells of the game, as in Fig. 3.9, with the

property that these probabilities are non negative and sum to one.

Figure 3.9 – A probability distribution over the cells of the payoff matrix

A distribution P is a correlated equilibrium of the Hawk-Dove game if and only if it satisfies the

following weak inequalities:16

(2.26)

5 0 4 1

4 1 5 0

1 4 0 5

0 5 1 4

q r q r

s t s t

q s q s

r t r t

The first two inequalities in (2.26) refer to player’s 1 choices. Suppose that player 1 intends to

play Hawk. The probabilities that are relevant to the determination of the expected payoffs of

strategy Hawk versus strategy Dove are q and r . Then the first inequality establishes the

16 The presentation of the inequalities follows Brandenburger’s (1992) approach which is intuitively clearer than

Aumann’s.

1 2 Dove Hawk

Hawk q r

Dove s t


condition that the expected payoff of Hawk is not less than the expected payoff of Dove.

Dividing both sides of the first inequality by q r , we have

(2.27) 5 0 4 1q r q r

q r q r q r q r

It is immediately seen that q

q r and

r

q r are player’s 1 conditional probabilities that player 2

plays Dove and Hawk respectively.17 The second line in (2.26) imposes the same condition for

Dove to be preferred to Hawk, given that player 2 plays Dove with probability s and Hawk with

probability t . The last two inequalities in (2.26) are obtained reasoning in a similar way with

regard to the choices of player 2.

One can easily verify that both 1 , 02

q t r s and 1 , 03

q s t r are solution of

(2.26) as well as the distributions corresponding to the three Nash equilibria.

In more general terms, let the two players be indexed , 1,2i j , the strategies ik is S for all i

and k and the distribution ikp P for all i and k . The following is Aumann’s (1987, p. 6)

definition.

Definition 4.3 A distribution ikp P is a correlated equilibrium distribution if and only if

(2.28)

, , 0 for all ,

, , 0 for all j,

j

i

i ik jk i ik jk ik i

k S

j ik jk i ik jk jk j

k S

u s s u s s p s i k S

u s s u s s p s k S

Notice first that relations (2.28) are just an equivalent way to write relations (2.26). To prove that

the distribution defined in (2.28) is a correlated equilibrium distribution Aumann shows that

these inequalities can be reduced to those of definition (2.23) which is based on the introduction

of a correlating device and strategies adapted to it. As shown in the example, it is sufficient to

divide, for all i , the first line of (2.28) by ik ikp p . The resulting ratios ik

ik ik

p

p p are the

conditional probabilities of player 1 when the correlating device recommends to play either

Hawk or Dove according to the information concerning the outcome of the correlating device.

17 The same approach is used here as in the determination of the expected payoffs of equations (2.18) and (2,19).



References

Aumann, R.J. (1974), “Subjectivity and Correlation in Randomized Strategies”, Journal of

Mathematical Economics, vol 1, pp. 67-96

--------- (1987), “Correlated Equilibrium as Expression of Bayesian Rationality”, Econometrica,

vol. 55, pp. 1-18

Brandenburger, A. (1992), “Knowledge and Equilibrium in Games”. Journal of Economic

Perspectives, vol. 6, no. 4, pp. 83-101

Fudenberg, D. and J. Tirole (1991), Game Theory, the MIT Press, Cambridge, Mass, USA

Grrrnwald, A. (2007), “General-Sum Games: Correlated Equilibria”, Lecture Notes, Internet

Harsanyi, J.C. (1973), “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed

Strategy Equilibrium Points”, International Journal of Game Theory, vol. 2, pp. 1-23

Osborne, M.J. and A. Rubinstein (1994), A Course in Game Theory, the MIT Press, Cambridge,

Mass, USA

Sunanda Roy, A.C. (2015), “Correlated Equilibrium”, Lecture Notes for Econ. 618 (Game

Theory), Iowa State University, Internet

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