Correlated equilibrium under uncertainty

Mathematical Social Sciences 44 (2002) 183–209 www.elsevier.com / locate / econbase

Correlated equilibrium under uncertainty Kin Chung Lo* Department of Economics, York University, Toronto, Ontario, Canada M3 J 1 P3 Received 31 January 2001; received in revised form 30 April 2002; accepted 15 May 2002

Abstract We study strategic games in which each player lacks perfect understanding of opponents’ information structures, and as a result, does not know precisely the probability distribution governing opponents’ action choices. The Ellsberg Paradox demonstrates that when people act on unknown probabilities, their behavior may be inconsistent with the expected utility model. Motivated by the Ellsberg Paradox and related experimental findings, we assume that players’ preferences conform to the Choquet expected utility model of Schmeidler [Econometrica 57 (1989) 571] and the multiple priors model of Gilboa and Schmeidler [Journal of Mathematical Economics 18 (1989) 141]. A generalization of correlated equilibrium is proposed.  2002 Elsevier Science B.V. All rights reserved. Keywords: Correlated equilibrium; Knightian uncertainty; Non-additive probabilities; Multiple priors; Belief functions JEL classification: C72; D81

1. Introduction

1.1. Uncertainty aversion Due to its simplicity and tractability, the expected utility model axiomatized by Savage (1954) has been the most popular model of preference under uncertainty. According to this model, the beliefs of a decision maker are represented by a probability measure. Using the model to represent players’ preferences, many solution concepts for

*Tel.: 11-416-736-2100, ext. 77032; fax: 11-416-736-5987. E-mail address: [email protected] (K.C. Lo). 0165-4896 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0165-4896( 02 )00025-2

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strategic games have been developed. Important instances include Nash equilibrium (Nash, 1951), rationalizability (Pearce, 1984; Bernheim, 1984) and correlated equilibrium (Aumann, 1974, 1987). On the other hand, the descriptive validity of the expected utility model is challenged by the Ellsberg Paradox (Ellsberg, 1961), a version of which is as follows. An urn contains 90 balls, identical except in color. A decision maker knows that 30 of the balls are red. Each of the remaining 60 balls is either black or yellow, but the relative proportions are unknown. Consider the following four acts. (The act f1 yields $100 if the decision maker draws a red ball from the urn, and yields $0 otherwise. Similar interpretations are given to the other three acts).

f1 f2 f3 f4

Red

Black

Yellow

$100 $0 $100 $0

$0 $100 $0 $100

$0 $0 $100 $100

For each of the two acts f1 and f4 , the probability of winning $100 is precisely known. However, this is not the case for f2 and f3 . Using Frank Knight’s (1921) terminology, f1 and f4 involve only risk, but f2 and f3 involve ambiguity or uncertainty. Ellsberg argues that the typical preferences for the acts are f1 s f2 and f4 s f3 . These preferences reflect an aversion to uncertainty and are inconsistent with the expected utility model. In fact, they contradict any model of preference in which underlying beliefs are represented by a probability measure. (Machina and Schmeidler, 1992 call this property probabilistic sophistication). Subsequent experimental findings (summarized by Camerer and Weber, 1992) generally support that people are uncertainty averse. Motivated by the Ellsberg Paradox and related experiments, models of non-probabilistically sophisticated preference have been proposed. In the Choquet expected utility model of Schmeidler (1989), the decision maker’s beliefs are represented by a capacity (that is, a non-additive probability measure). In the multiple priors model of Gilboa and Schmeidler (1989), beliefs are represented by a set of probability measures. Following Gilboa and Schmeidler (1993), we reserve the term ambiguous beliefs for beliefs that are representable by either a capacity or a set of probability measures. Although the Ellsberg Paradox only involves a single decision maker facing an exogenously specified environment, it is natural to think that uncertainty aversion is also common in multi-person strategic situations. So far, two solution concepts, Nash equilibrium and rationalizability, have been generalized to allow for preferences to be representable by the Choquet expected utility and multiple priors models. See Dow and Werlang (1994), Eichberger and Kelsey (2000), Epstein (1997), Ghirardato and Le Breton (2000), Groes et al. (1998), Klibanoff (1993), Lo (1996, 1999a,b, 2000), Marinacci (2000) and Mukerji (1995). Correlated equilibrium is more tractable than Nash equilibrium, and is much sharper

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than rationalizability (see, for instance, Aumann, 1987, pp. 5–6 and pp. 14–15).1 Given the recent success in generalizing Nash equilibrium and rationalizability to allow for ambiguous beliefs, it is natural to ask whether analogous generalizations of correlated equilibrium can be developed. As far as we know, this question has not been addressed before.

1.2. Generalization of correlated equilibrium To study a situation of asymmetric information, game theorists typically start with a set V of states that is common to all the players. One of the states in V is the true state, but the players may not know which one. The information structure of each player i is represented by a function ai : V → Qi , where Qi is a set of signals. According to ai , if the true state is v, player i will receive the signal ai (v ). Reflecting player i’s perfect understanding of ai , after receiving a signal ui [ Qi , i knows that the true state lies in the event hv [ V : ai (v ) 5 ui j, and updates his beliefs about V from a prior m using Bayes rule.2 Numerous interesting game theoretic concepts have been formulated using the above model. An important instance, that is the focus of this paper, is correlated equilibrium. Roughly speaking, a correlated equilibrium of a strategic game consists of a set V of states, a common prior m on V, and for every player i, an information structure ai as well as a strategy such that the following is satisfied: For every player i, in response to every signal ui that occurs with positive probability, his strategy specifies an action that maximizes conditional expected payoff. An obvious scenario of ambiguity is that the probability measure m governing the state space V is unknown to the players, and as a consequence, beliefs about V are represented by a capacity or a set of probability measures. Unfortunately, analyzing this scenario involves updating of ambiguous beliefs, which is a controversial issue (see, for example, Gilboa and Schmeidler (1993) and Epstein and Le Breton (1993)). In this paper, we look at a different scenario in which this issue does not arise. We consider a situation in which players have more general information structures. Formally, each player i’s information structure is represented by a function bi : V → D(Qi ), where D(Qi ) is the set of all probability measures on Qi . The interpretation of bi is as follows. If the true state is v, then for each signal ui [ Qi , player i will receive ui with probability bi (v )(ui ). In other words, conditional on each v, the signal received by player i is a random variable distributed according to bi (v ). Our key assumption is that, while player i has perfect understanding of bi , he only knows, for each v [ V, the support of bj (v ) (for j ± i), and all of this is common 1

By ‘correlated equilibrium’ we mean ‘objective correlated equilibrium.’ In fact, rationalizability and subjective correlated equilibrium (more precisely, correlated rationalizability and a posteriori equilibrium, respectively) are equivalent. See Brandenburger and Dekel (1987). 2 In the literature, it is customary to suppress Qi and represent player i’s information structure by the induced partition hhv [ V : ai (v ) 5 ui jjui [ Qi of V. It is more convenient for this paper not to follow the customary practice.

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knowledge. Without knowing precisely the probability distribution bj (v ), player i is unable to pin down the distribution governing j’s action choice at v. This is how ambiguity enters the picture. To analyze the situation, we propose a new solution concept, generalized correlated equilibrium (henceforth GCE), that generalizes correlated equilibrium to accommodate preferences conforming to (an interesting subclass of) the Choquet expected utility and multiple priors models.3 Since we maintain the assumption that prior beliefs over V are represented by a probability measure, and each player knows his own information structure, no updating of ambiguous beliefs is involved.

1.3. Organization and notational conventions To make the exposition as clear as possible, we present in Section 2 the essential framework of the paper in the single agent setting. In Section 3, we first provide a brief review of correlated equilibrium. Then we adapt the framework in Section 2 to the context of strategic games, and provide two equivalent definitions of GCE. While the decision theoretic interpretation of GCE can be clearly seen from the first definition, the second definition is more concise and has pragmatic significance. We demonstrate the tractability of the second definition by showing that the set of GCEs of a strategic game is ‘convex.’ Section 4 contains a series of examples demonstrating how GCE is different from various existing solution concepts. Before we proceed further, we establish the following notation that applies throughout the paper. For any finite set Y, let D(Y) be the set of all probability measures on Y, 2 Y the set of all subsets of Y, and 3 (Y) the set of all non-empty subsets of Y. For any Cartesian set 3 nj 51 Yj and for any i 5 1, . . . , n, use 3 j ±i Yj to denote the set Y1 3 ? ? ? 3 Yi 21 3 Yi 11 3 ? ? ? 3 Yn .

2. Decision theory

2.1. Information Let V be a finite set of states, m a probability measure on V, and b : V → D(Q ) an information structure. If the true state in V is v, then for each signal u [ Q, the decision maker will receive u with probability b (v )(u ). Define

Q 1 5 hu [ Q : 'v [ V such that m (v ) . 0 and b (v )(u ) . 0j.

(1)

In words, given m and b, Q 1 is the set of signals that occur with positive probability. 3

Brandenburger et al. (1992) also consider more general information structures and propose a generalized correlated equilibrium concept. Their focus is not on whether a player knows the information structures of opponents. Rather, they assume that a player does not understand his own information structure. Their generalized correlated equilibrium is equivalent to subjective correlated equilibrium, but ours is not. See Section 4 for details.

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After receiving a signal u [ Q 1 , the decision maker’s beliefs over V are updated by Bayes rule. That is, he assigns posterior probability

m (v )b (v )(u ) m (v uu ) 5 ]]]]]] m ( vˆ )b ( vˆ )(u )

O

(2)

ˆ [V v

to each state v [ V. Let S be a finite set of payoff relevant states, and X a set of consequences. Objects of choice are acts from S to X. Let ^ be the set of all acts. The decision maker has a preference ordering K over ^, which will be our focus in Section 2.2. Although V is not (directly) payoff relevant, it is related to S. Following Dempster (1967), the decision maker knows that the two state spaces are linked by a function G : V → 3 (S), which is interpreted as follows. If the true state in V is v, then the true state in S must lie in G (v ). However, there is no further information on which element in G (v ) is more likely to be the true state. We illustrate our framework using a simple example. Consider a producer who manufactures a product using a standard procedure. The procedure is imperfect in the sense that product quality is high with probability 0.9, and low with probability 0.1. The producer has a quality control device for testing the product. Unsurprisingly, the test result delivered by the device is also not completely accurate. If the quality of the product is high, then the product will definitely pass the test; if the quality of the product is low, the product will fail the test with probability 0.8, and pass with probability 0.2. The profitability of the company depends ultimately on the level of demand for the product, which could be either high or low. Demand is related to (but is not totally determined by) product quality. If product quality is low, then demand will definitely be low; if product quality is high, then demand could be high or low, depending on various external market conditions about which the producer is ignorant. After the result of the quality control test is known (but before the external market conditions are realized), the producer has to decide whether to sell the company, or continue running it. The above situation can be formally described as follows: • The set V of states.

V 5 hv h , v l j, where v h (v l , respectively) denotes the state at which product quality is high (low, respectively). • The probability measure m on V.

m (v h) 5 0.9, m (v l ) 5 0.1. • The set Q of signals.

Q 5 hu p , u f j, where the producer receives the signal u p (u f , respectively) if the product passes (fails, respectively) the test.

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• The information structure b : V → D(Q ).

b (v h)(u p ) 5 1, b (v h)(u f ) 5 0, b (v l )(u p ) 5 0.2, b (v l )(u f ) 5 0.8. • The set S of payoff relevant states. S 5 hs h , s l j, where s h (s l , respectively) denotes the state at which demand is high (low, respectively). • The function G : V → 3 (S).

G (v h) 5 hs h , s l j, G (v l ) 5 hs l j. To illustrate updating of beliefs, suppose the producer receives the signal u p . According to Eq. (2), the posterior probability assigned to the state v h is 0.9 m (v h uu p ) 5 ]]]]]. 0.9 1 (0.1)(0.2) Note that after receiving the signal u p , the producer still has insufficient information to pin down the probability law governing the state space S, and is therefore facing a choice problem under uncertainty.

2.2. Preference We now turn to the decision maker’s preference ordering K over ^ (the set of acts from S to X). Several models of preference are relevant. The first is expected utility. According to this model, the decision maker behaves as if his beliefs are represented by a probability measure. More precisely, the utility representation U : ^ → R for K is given by U( f ) 5

O u( f(s))p(s),

(3)

s [S

where u: X → R is a von Neuman Morgenstern (vNM) index, and p is a subjective probability measure on S. As we explained in the Introduction, the Ellsberg Paradox and related findings demonstrate that when people act on unknown probabilities, their behavior may be inconsistent with the expected utility model, and more generally probabilistic sophistication. Two important models of non-probabilistically sophisticated preference have been proposed. In the multiple priors model, the decision maker’s beliefs are represented by a closed and convex set # of probability measures, and the decision maker is uncertainty averse in the sense that he evaluates an act by computing the minimum expected utility over the probability measures in #. That is, the representation for K is given by U( f ) 5min

O u( f(s))p(s).

p [ # s [S

(4)

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If the set # is a singleton, then Eq. (4) collapses to Eq. (3), confirming that this model generalizes expected utility. The Choquet expected utility model is a closely related alternative. Its exact functional form is omitted here. Roughly speaking, it resembles Eq. (3), with the probability measure p replaced by a capacity, and summation is undertaken in a specific manner. The Choquet expected utility and multiple priors models have a non-empty intersection (see Gilboa and Schmeidler, 1993, p. 40, Propositions 2.1 and 2.2). We are ultimately interested in a subset of that intersection. As the example in Section 2.1 demonstrates, after receiving a signal u, the probability law governing S may still be unknown to the decision maker. Nevertheless, the posterior probability measure m ( ? uu ) on V and the function G : V → 3 (S) can be used to reach the following conclusion: For any event E [ 2 S , the probability that the true state lies in E is at least b(E) ] ; m (hv [ V : G (v ) # Ejuu ), and at most ] b(E) ; m (hv [ V : G (v ) > E ± 5juu ).

(5)

(6) ] In words, b(E) is the probability that the random subset G (v ) is contained in E, and b(E) ] is the probability that G (v ) hits E. S The function b: ] 2 →4[0, 1] defined in Eq. (5) is called a belief function (Dempster, 1967 and Shafer, 1976). It is a special case of capacity, and also corresponds naturally to the following non-empty, closed and convex set # (b) ] of probability measures on S: ] # (b) # p(E) #b(E) ;E [ 2 S j. (7) ] 5 h p [ D(S): b(E) ] That is, # (b) ] contains all the probability measures that are] compatible with the information conveyed by the lower bound b] and the upper bound b. The notation # (b) ] is used because Eq. (7) is typically written as

# (b) ;E [ 2 S j, (8) ] 5 h p [ D(S): p(E) $b(E) ] which is known as the core of ]b. The equivalence of Eqs. (7) and (8) follows ] immediately from the fact that for all E [ 2 S , ]b(E) 5 1 2b(S\E). We impose the following assumption on the preference ordering K over ^. It is represented by the multiple priors model as defined in Eq. (4), where the underlying set # of probability measures is equal to # (b) ] as defined in Eq. (8). Equivalently, it is represented by the Choquet expected utility model, where the underlying capacity is b. ] In fact, both the multiple priors and Choquet expected utility models can be left behind the picture, since the utility function can also be stated in terms of m ( ? uu ) and G as U( f ) 5

O m(v uu ) F min u( f(s)) G.

v [V

4

(9)

s [ G (v )

Due to its intuitive appeal and tractability, the class of belief functions has proved useful in artificial intelligence, robust statistics and decision theory. See, for instance, Ghirardato (2001), Huber (1981), Jaffray and Wakker (1994), Mukerji (1997) and Wasserman (1990).

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The details can be found in Gilboa and Schmeidler (1994, Section 4, pp. 49–51).5 We find the functional form in Eq. (9) more tractable, and will use it for the rest of the paper.

3. Game theory

3.1. Correlated equilibrium In this section, we provide a review of correlated equilibrium. This is intended to help the reader see that the generalization undertaken in Section 3.2 is apparent. Let (V, m ) be a finite probability space that is common to a finite number n of players. For every i 5 1, . . . , n, player i’s information structure is represented by a function ai : V → Qi , where Qi is a finite set of signals. If the true state is v, then player i will receive the signal ai (v ). Define

Q i1 5 hui [ Qi : 'v [ V such that m (v ) . 0 and ai (v ) 5 ui j

(10)

to be the set containing all the signals that occur with positive probability. After receiving a signal ui [ Q 1 i , player i assigns posterior probability

m (v ) m (v uui ) 5 ]]]]]] m ( vˆ )

O

ˆ [ V : ai ( v ˆ )5ui j hv

to each state v [ hvˆ [ V : ai ( vˆ ) 5 ui j. The players are involved in a strategic game G 5 k(Si , u i ) ni51 l, where for each i 5 1, . . . , n, Si is a finite set of actions for player i, and u i : 3 nj 51 Sj → R is i’s vNM payoff function. Let gi : Qi → Si be a strategy of player i. According to gi , if i receives a signal ui , then i takes the action gi (ui ). The information structure ai and strategy gi together imply that i takes the action gi (ai (v )) at state v. For each v [ V, denote the (n 2 1)-tuple (g1 (a1 (v )), . . . , gi 21 (ai 21 (v )), gi 11 (ai 11 (v )), . . . , gn (an (v ))) of actions by g2i (a2i (v )). Definition 1. A correlated equilibrium k(V, m ), (Qi , ai , gi ) in51 l of a strategic game G 5 k(Si , u i ) ni 51 l consists of • a finite probability space (V, m ) 5

In Gilboa and Schmeidler (1994), Eq. (9) is written as U( f ) 5

O m(E) fmin u( f(s)) g,

E [2 S

s [E

S ¨ inverse of b. where m(E) 5 m (hv [ V : G (v ) 5 Ejuu ). The function m: 2 → [0, 1] is called the Mobius ]

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• for each player i 5 1, . . . , n, a finite set Qi of signals • for each player i 5 1, . . . , n, an information structure ai : V → Qi • for each player i 5 1, . . . , n, a strategy gi : Qi → Si 1

such that for every i 5 1, . . . , n, and every ui [ Q i ,

O m(v uu )u (g (u ), g (a (v))) $ O m (v uu )u (s , g (a (v ))) ;s [ S . i

i

i

i

i

i

i

2i

2i

v [V

2i

2i

i

i

(11)

v [V

Eq. (11) says that after receiving a signal ui , player i finds that the conditional payoff of taking the action gi (ui ) is at least as great as that of taking any other action s i . If this holds for every player i and every ui that occurs with positive probability, then k(V, m ), (Qi , ai , gi ) ni 51 l is a correlated equilibrium of G. In a correlated equilibrium, the prior m over V is the same for all the players. If Definition 1 is weakened by allowing the players to have different prior beliefs over V, then it will become the definition of a subjective correlated equilibrium. Using Definition 1 to describe the set of correlated equilibria of a strategic game would involve consideration of an infinite number of possible state spaces and information structures. The following alternative definition of correlated equilibrium does not suffer from this problem, and is therefore very useful. Imagine that there is a mediator who can communicate separately and confidentially with each player. The mediator chooses randomly an action profile (s *1 , . . . , s n* ) from 3 jn51 Sj according to a probability measure mˇ on 3 jn51 Sj . Each player i is recommended to take the action s *i , but he is not told the recommendations for any other players. Suppose that the above is common knowledge. After receiving a recommendation s *i [ Si with mˇ (hs *i j 3 3 j ±i Sj ) . 0, player i’s updated beliefs on 3 j ±i Sj are represented by the posterior probability measure mˇ ( ? us *i ). n Definition 2. A correlated equilibrium mˇ of a strategic game G 5 k(Si , u i ) i51 l is a n probability measure on 3 j 51 Sj such that for every i 5 1, . . . ,n, and every s *i [ Si with mˇ (hs *i j 3 3 j ±i Sj ) . 0,

O

mˇ (s 2i us i* )u i (s i* , s 2i ) $

s 2i [3 j ±i S j

O

mˇ (s 2i us i* )ui (s i , s 2i ) ;si [ Si .

s 2i [3 j ±i S j

According to Definition 2, mˇ is a correlated equilibrium of G if the players always find it worthwhile to obey the recommendations of the mediator. Definitions 1 and 2 are equivalent in the following sense. Given a correlated equilibrium mˇ of a strategic game G in terms of Definition 2, if we set

V 5 3 nj 51 Sj , m ((s 1 , . . . ,sn )) 5 mˇ ((s 1 , . . . ,s n )) ;(s 1 , . . . ,s n ) [ V, Qi 5 Si

;i,

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192

ai ((s 1 , . . . ,sn )) 5 s i

;i

;(s 1 , . . . ,s n ) [ V,

and

gi (s i ) 5 si

;s i [ Qi .

;i

n

Then k(V, m ), (Qi , ai , gi ) i 51 l constitutes a correlated equilibrium of G in terms of Definition 1. Conversely, given a correlated equilibrium k(V, m ), (Qi , ai , gi ) ni 51 l of G in terms of Definition 1, if we define, for every (s 1 , . . . ,s n ) [ 3 jn51 Sj ,

mˇ ((s 1 , . . . ,sn )) 5

O

m (v ),

h v [ V : g j (aj (v ))5s j ; j 51, . . . ,nj

then mˇ is a correlated equilibrium of G in terms of Definition 2, and it is an ‘incentive compatible direct mechanism’ of k(V, m ), (Qi , ai , gi ) ni 51 l. (See, for instance, Fudenberg and Tirole, 1991, pp. 57–58 for details). Definition 2 reveals immediately two facts about correlated equilibrium. First, it is a generalization of Nash equilibrium: Given a correlated equilibrium mˇ of G, suppose for every i 5 1, . . . ,n, there exists a probability measure mˇ i on Si such that mˇ 5 mˇ 1 3 ? ? ? 3 mˇ n . Then ( mˇ i ) ni 51 is a Nash equilibrium of G. Second, the set of correlated equilibria of a strategic game G is convex: Given any two correlated equilibria mˇ and mˇ 9 of G, and any real number l [ [0, 1], lmˇ 1 (1 2 l) mˇ 9 is also a correlated equilibrium of G.

3.2. Generalized correlated equilibrium Before we define GCE, we first explain how the scenario in Section 2 arises in the context of strategic games. We start with a finite probability space (V, m ) that is common to a finite number n of players. For every i 5 1, . . . ,n, player i is regarded as a decision maker as described in Section 2, and endowed with an information structure bi : V → D(Qi ). Again, define

Q1 i 5 hui [ Qi : 'v [ V such that m (v ) . 0 and bi (v )(ui ) . 0j

(12)

to be the set of signals that occur with positive probability. Reflecting the assumption that player i has perfect understanding of bi , after receiving a signal ui [ Q i1 , i assigns posterior probability

m (v )bi (v )(ui ) m (v uui ) 5 ]]]]]] m ( vˆ )bi ( vˆ )(ui )

O

(13)

ˆ [V v

to each state v [ V. n The players are involved in a strategic game G 5 k(Si , u i ) i51 l. Choosing an action in a game can be viewed as a choice problem under uncertainty. Since player i’s payoff depends directly on the action choices of all the players, and i is uncertain about the actions of his opponents, the payoff relevant state space for i is 3 j ±i Sj . Every action s i [ Si can be formally identified as an act over 3 j ±i Sj as follows. If i chooses s i and the true state is s 2i [ 3 j ±i Sj , then i receives the payoff u i (s i , s 2i ). Let gi : Qi → Si be a

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strategy of i. According to gi , in response to a signal ui , i chooses the action gi (ui ). Define the function Gi : V → 3 (Si ), where for all v [ V,

Gi (v ) 5 hsi [ Si : 'ui [ Qi such that bi (v )(ui ) . 0 and gi (ui ) 5 s i j.

(14)

That is, given player i’s information structure bi and strategy gi , the set Gi (v ) contains all the actions that player i may end up taking at v. We assume that player i knows gj and the support of bj (v ) (for all j ± i), implying that he knows Gj (v ). Finally, we explain how the two state spaces V and 3 j ±i Sj are related. From the perspective of player i, V is linked to 3 j ±i Sj by the function G2i : V → 3 ( 3 j ±i Sj ), where G2i (v ) ; 3 j ±i Gj (v ) for all v [ V. If the true state is v, then the action profile chosen by i’s opponents must lie in G2i (v ). Nevertheless, there is no further information on which profile in G2i (v ) is more likely to happen. According to Savage (1954), even under such a situation of ambiguity, player i behaves as if his beliefs over each G2i (v ) are represented by a subjective probability measure. However, the Ellsberg Paradox motivates us to deviate from Savage’s theory, and allow player i’s preference to conform to the utility function as specified in Eq. (9). That is, after receiving a signal ui [ Q i1 , i’s conditional payoff of choosing an action s i [ Si is equal to

O m(v uu ) F i

v [V

min s 2i [ G2i (v )

G

u i (s i , s 2i ) .

(15)

This completes the adaptation of the framework from the single agent setting to the context of games, and we are now ready to present our equilibrium concept. Definition 3. A generalized correlated equilibrium k(V, m ), (Qi , bi , gi ) in51 l of a strategic game G 5 k(Si ,u i ) ni 51 l consists of • • • •

a finite probability space (V, m ) for each player i 5 1, . . . ,n, a finite set Qi of signals for each player i 5 1, . . . ,n, an information structure bi : V → D(Qi ) for each player i 5 1, . . . ,n, a strategy gi : Qi → Si 1

such that for every i 5 1, . . . ,n, and every ui [ Q i ,

O m(v uu ) F min u (g (u ), s ) G $ O m (v uu ) F min u (s , s ) G ;s [ S . i

s 2i [ G2i (v )

v [V

i

v [V

i

s 2i [ G2i (v )

i

i

i

i

2i

2i

i

i

(16)

According to Eq. (16), after receiving a signal ui , player i finds that the conditional payoff of taking the action gi (ui ) is at least as great as that of taking any other action s i . GCE requires that this holds for every player i and every ui that occurs with positive probability. Clearly, if for every i and every v, bi (v ) is a degenerate probability

194

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

measure, then Definition 3 collapses to Definition 1, confirming that GCE is a generalization of correlated equilibrium.6 Although the decision theoretic interpretation of GCE can be clearly seen from Definition 3, describing the set of GCEs of a strategic game would involve consideration of an infinite number of possible state spaces and information structures. A more tractable definition (counterpart of Definition 2) is needed. Again, imagine that there is a mediator who can make recommendation separately and confidentially to each player. The mediator determines his recommendations in two stages. In the first stage, the mediator chooses randomly an element in 3 nj 51 3 (Sj ) according to m˜ , where m˜ is a probability measure on 3 nj 51 3 (Sj ). After an element 3 jn51 Zj [ 3 jn51 3 (Sj ) is chosen, the second stage randomization begins. For every i 5 1, . . . ,n, let b˜ i : 3 nj 51 3 (Sj ) → D(Si ) be a function with the property that for each 3 nj 51 Zj [ 3 nj 51 3 (Sj ) and for each s i [ Si ,

b˜ i ( 3 nj 51 Zj )(si ) . 0⇔s i [ Zi . n n n That is, for each 3 j 51 Zj [ 3 j 51 3 (Sj ), b˜ i ( 3 j 51 Zj ) is a probability measure on Si with n support Zi . Given the result 3 j 51 Zj of the first stage randomization, the second stage randomization is carried out as follows. For every i 5 1, . . . ,n, the mediator chooses randomly an action s *i [ Zi according to b˜ i ( 3 jn51 Zj ). Finally, s i* is recommended to player i. In the sequel, we call k m˜ , ( b˜ i ) ni 51 l defined in the preceding paragraphs a mediated communication system, m˜ first stage random device, and ( b˜ i ) in51 second stage random devices. In order to use a mediated communication system to simulate a GCE, assume that every player i knows m˜ and b˜ i , but not b˜ j (for all j ± i). That is, player i is told the probability law governing the first stage randomization and those governing how a recommendation for him is generated in the second stage. However, i is not given a clue regarding the second stage random devices for his opponents. Suppose that the above is common knowledge among the players. Define

n n n ˜ S1 i 5 hs i [ Si : ' 3 j 51 Zj [ 3 j51 3 (Sj ) such that m( 3 j 51 Zj ) . 0 and s i [ Zi j

(17) to be the set containing all the actions for player i that are recommended with positive probability. After receiving a recommendation s *i [ S 1 i , player i will update his beliefs regarding the first stage randomization result for his opponents. Reflecting player i’s understanding of k m˜ , ( b˜ i ) ni 51 l, i’s updated beliefs on 3 j ±i 3 (Sj ) are represented by the posterior probability measure m˜ ( ? us *i ), where for each 3 j ±i Zj [ 3 j ±i 3 (Sj ), 6

Hence existence of GCE is ensured. Examples in Section 4 suggest that there often exist GCEs that are not correlated equilibria.

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

O

195

m˜ (Zˆ i 3 3 j ±i Zj ) b˜ i (Zˆ i 3 3 j ±i Zj )(s *i )

Zˆ i [ 3 (S i )

O

m˜ ( 3 j ±i Zj us i* ) 5 ]]]]]]]]]]]]] . m˜ ( 3 nj 51 Zˆ j ) b˜ i ( 3 nj 51 Zˆ j )(s *i )

(18)

3 nj 51 Zˆ j [3 jn51 3 (S j )

Note that there is no structural difference between m ( ? uui ) as defined in Eq. (13) and m˜ (?us *i ) as defined in Eq. (18), even though the latter looks more cumbersome. Conditional on each 3 j ±i Zj , player i has no further information on which action profile s 2i [ 3 j ±i Zj is recommended to his opponents in the second stage. Parallel to Eq. (15), after receiving a recommendation s *i [ S i1 , i’s conditional payoff of taking an action s i [ Si is

O

m˜ ( 3 j ±i Zj us i* )

3 j ±i Z j [3 j ±i 3 (S j )

F

min

s 2i [3 j ±i Z j

G

u i (s i , s 2i ) .

(19)

Definition 4. A generalized correlated equilibrium k m˜ , ( b˜ i ) ni51 l of a strategic game G 5 k(Si , u i ) ni 51 l consists of • a probability measure m˜ on 3 nj 51 3 (Sj ) • for every i 5 1, . . . ,n, a function b˜ i : 3 nj 51 3 (Sj ) → D(Si ) with the property that for each 3 nj 51 Zj [ 3 nj 51 3 (Sj ) and for each s i [ Si , b˜ i ( 3 nj 51 Zj )(s i ) . 0 if and only if s i [ Zi such that for every i 5 1, . . . ,n, and every s i* [ S i1 ,

O O

m˜ ( 3 j ±i Zj us i* )

3 j ±i Z j [3 j ±i 3 (S j )

$

3 j ±i Z j [3 j ±i 3 (S j )

F

min

s 2i [3 j ±i Z j

m˜ ( 3 j ±i Zj us i* )

F

min

u i (s i* , s 2i )

s 2i [3 j ±i Z j

G )G

u i (s i , s 2i

;s i [ Si .

(20)

The interpretation of Definition 4 is very similar to that of Definition 2. A mediated communication system k m˜ , ( b˜ i ) ni 51 l is a GCE of G if the players always find it optimal to obey the recommendations of the mediator. Definition 4 generalizes Definition 2. To see this, observe that the former collapses to the latter if for each 3 nj 51 Zj [ 3 nj 51 n 3 (Sj ), m˜ ( 3 j51 Zj ) . 0 implies that 3 jn51 Zj consists of only one action profile; that is, the first stage random device m˜ can be identified as a probability measure on 3 j ±i Sj . Definitions 3 and 4 are equivalent. Any GCE k m˜ , ( b˜ i ) ni 51 l of a game G can be restated as a GCE k(V, m ), (Qi , bi , gi ) ni 51 l of G. Conversely, given any GCE k(V, m ), (Qi , bi , gi )in51 l of G, there exists a GCE k m˜ , ( b˜ i )in51 l of G which is an ‘incentive compatible direct mechanism’ of k(V, m ), (Qi , bi , gi ) ni 51 l. The precise statements and proofs can be found in Appendix A. When correlation is absent, GCE reduces to a generalized Nash equilibrium concept. To be precise, suppose k m˜ , ( b˜ i ) ni 51 l is a GCE of a strategic game G with the following two additional properties: • For every i 5 1 . . . ,n, there exists a probability measure m˜ i on 3 (Si ) such that

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

196

m˜ 5 m˜ 1 3 ? ? ? 3 m˜ n .

(21)

• For every i 5 1, . . . ,n, and for every 3 nj 51 Zj , 3 nj 51 Z 9j [ 3 nj51 3 (Sj ), n n [Zi 5 Z 9i ]⇔[ b˜ i ( 3 j 51 Zj ) 5 b˜ i ( 3 j 51 Z j9 )].

(22)

Then Eqs. (17) and (18) become ˜ S1 i 5 hs i [ Si : 'Zi [ 3 (Si ) such that m i (Zi ) . 0 and s i [ Zi j and

P m˜ (Z ),

m˜ ( 3 j ±i Zj us i* ) 5

j

j ±i

j

respectively, implying that the GCE can be restated solely in terms of ( m˜ i ) ni 51 as follows. n Definition 5. A generalized Nash equilibrium ( m˜ i ) i 51 of a strategic game G 5 k(Si , n u i ) i 51 l consists of, for every i 5 1 . . . ,n, a probability measure m˜ i on 3 (Si ) such that

O P m˜ (Z ) F min u (s*, s ) G O P m˜ (Z ) F min u (s , s ) G

3 j ±i Z j [3 j ±i 3 (S j ) j ±i

$

3 j ±i Z j [3 j ±i 3 (S j )

j ±i

j

j

j

s 2i [3 j ±i Z j

j

s 2i [3 j ±i Z j

i

2i

i

i

i

2i

;s *i [ S 1 i

;s i [ Si

;i. (23)

Generalized Nash equilibrium (henceforth GNE) is essentially the same as multiple priors Nash equilibrium defined in Lo (1999a, p. 261).7 Obviously, a GNE ( m˜ i ) ni51 of G reduces further to a Nash equilibrium of G if for every i, m˜ i can be identified as a probability measure on Si . The next section contains examples comparing GNE with GCE, demonstrating how correlation makes a difference. Finally, the set of GCEs of a strategic game is ‘convex’ in the following sense. Let n k m˜ , ( b˜ i ) ni 51 l, k m˜ 9, ( b˜ i9 ) i51 l and k m˜ 0, ( b˜ i99 ) ni 51 l be mediated communication systems of a strategic game G. Suppose there exists a real number l [ [0, 1] such that

m˜ 0 5 lm˜ 1 (1 2 l) m˜ 9,

(24)

and for each 3 nj 51 Zj [ 3 jn51 3 (Sj ) with m˜ 0( 3 jn51 Zj ) . 0,

lm˜ ( 3 nj 51 Zj ) n ˜ 99 ]]]] b i ( 3 j 51 Zj ) 5 b˜ ( 3 jn51 Zj ) m˜ 0( 3 nj 51 Zj ) i (1 2 l) m˜ 9( 3 nj 51 Zj ) 1 ]]]]]] b˜ i9 ( 3 jn51 Zj ) ;i. m˜ 0( 3 nj 51 Zj ) 7

(25)

Since multiple priors Nash equilibrium allows for extensive form games and all preferences conforming to the multiple priors model, it includes GNE as a special case.

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

197

Fig. 1. The prisoners dilemma.

If k m˜ , ( b˜ i ) ni 51 l and k m˜ 9, ( b˜ i9 ) in51 l are GCEs of G, then k m˜ 0, ( b˜ i99 ) ni 51 l is also a GCE of G. The proof of this result can be found in Appendix B. Observe that given the weight l that is used to carry out convex combination of the first stage random devices in Eq. (24), the weights for combining the second stage random devices in Eq. (25) depend on both l and the first stage random devices. This feature is completely natural. For instance, suppose m˜ ( 3 nj 51 Zj ) 5 0 and m˜ 9( 3 nj 51 n Zj ) . 0. Then b˜ i ( 3 jn51 Zj ) is arbitrary, but b˜ 9i ( 3 j51 Zj ) is not. In this case, all the n ˜ weight is attached to b 9i ( 3 j51 Zj ).

4. Examples In this section, we use several examples to illustrate GCE (as defined in Definition 4). Example 1. Fig. 1 is the prisoner’s dilemma. According to Eq. (19), after receiving a recommendation s 1* , player 1’s conditional payoff of choosing an action s 1 [ S1 is

O

Z 2 [ 3 (S 2 )

F

G

m˜ (Z2 us *1 ) min u 1 (s 1 , s 2 ) 5 m˜ (hCjus 1* )u 1 (s 1 , C) 1 m˜ (hDjus 1* )u 1 (s 1 , D) s 2 [Z 2

1 m˜ (hC, Djus *1 )u 1 (s 1 , D).

(26)

For all s *1 [ S1 , the expression in Eq. (26) is maximized only if s 1 5 D. The situation of player 2 is exactly the same as that of player 1. Thus, the only GCE coincides with the unique Nash equilibrium, which predicts that both players choose the action D. Example 2. The game in Fig. 2 is taken from Aumann (1974, p. 69, Example 2.3). It

Fig. 2. A three-player game.

198

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

involves three players. Player 1 chooses the row, player 2 chooses the column, and player 3 the matrix. In any correlated equilibrium of this game, player 1 chooses D and player 2 chooses L, leading to the payoff vector (1, 1, 1). There exists a subjective correlated equilibrium in which player 1 chooses U and player 2 chooses R, leading to the payoff vector (3, 3, 3). We now construct a GCE in which player 1 chooses U and player 2 chooses R, leading to the payoff vector (3, 3, 3). Let the first stage random device m˜ be

m˜ (hU j 3 hRj 3 hA, Bj) 5 1.

(27)

That is, the mediator recommends for sure the actions U and R to players 1 and 2, respectively. However, the second stage random device b˜ 3 (hU j 3 hRj 3 hA, Bj) governing the recommendation for player 3 is unknown to players 1 and 2.8 Given m˜ as defined in Eq. (27), we have

m˜ (hRj 3 hA, BjuU ) 5 1 and

m˜ (hU j 3 hA, BjuR) 5 1.

Therefore, player 1’s conditional payoff of choosing D is 0 and that of choosing U is 3; player 2’s conditional payoff of choosing L is 0 and that of choosing R is 3. As for player 3, regardless of b˜ 3 (hU j 3 hRj 3 hA, Bj), his conditional payoff of choosing A is the same as that of choosing B. This confirms that all the players are willing to obey the mediator’s recommendations. In this example, both the payoffs and predictions associated with a subjective correlated equilibrium can be replicated by a GCE. However, this is not a general phenomenon, as Examples 3 and 4 below demonstrate.9 Example 3. Consider the game of matching pennies in Fig. 3. While it is easy to construct a subjective correlated equilibrium in which the conditional payoff to every player is always strictly bigger than zero, it is impossible to construct a GCE with such a property. To see this, let k m˜ , ( b˜ i ) ni 51 l be a GCE of this game. The first stage random

Fig. 3. Matching pennies.

8

Since m˜ defined in Eq. (27) satisfies Eq. (21), and the second stage random devices can be made to satisfy Eq. (22), this GCE is equivalent to a GNE. 9 It follows that GCE is also different from the equilibrium concept of Brandenburger et al. (1992), which is equivalent to subjective correlated equilibrium, and that of Epstein (1997), which generalizes subjective correlated equilibrium in terms of preference.

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199

Fig. 4. A first stage random device.

device m˜ is depicted in Fig. 4. (The symbol ‘ul’ in the box hU j 3 hLj denotes the probability that m˜ attaches to the set hU j 3 hLj. Similar interpretations are given to the symbols in the other boxes). Suppose player 1 receives the recommendation s *1 5 U. Player 1’s conditional payoff of choosing U is the highest when 10

b˜ 1 (hU, Dj 3 hLj)(U ) 5 1, b˜ 1 (hU, Dj 3 hRj)(D) 5 1, b˜ 1 (hU, Dj 3 hL, Rj)(D) 5 1. (28) Given Eq. (28), player 1’s conditional payoff of choosing U is

m˜ (hLjuU )u 1 (U, L) 1 m˜ (hRjuU )u 1 (U, R) 1 m˜ (hL, RjuU )u 1 (U, R) ul 1 udl 2 ur 2 ulr 5 ]]]]]], ul 1 ur 1 ulr 1 udl

(29)

which is strictly positive only if ul 1 udl . ur 1 ulr.

(30)

By exactly the same reasoning, if player 1 receives the recommendation s 1* 5 D, his conditional payoff of choosing D is strictly positive only if dr 1 udr . dl 1 dlr;

(31)

if player 2 receives the recommendation s *2 5 L, his conditional payoff of choosing L is strictly positive only if dl 1 dlr . ul 1 udl;

(32)

if player 2 receives the recommendation s *2 5 R, his conditional payoff of choosing R is strictly positive only if ur 1 ulr . dr 1 udr.

(33)

The conditions in Eqs. (30)–(33) together imply ul 1 udl . ur 1 ulr . dr 1 udr . dl 1 dlr . ul 1 udl, 10

In fact, Eq. (28) violates the condition in the definition of GCE that for each 3 nj 51 Zj [ 3 nj 51 P(Sj ) and for each s i [ Si , b˜ i ( 3 nj 51 Zj )(s i ) . 0 if and only if s i [ Zi .

200

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

Fig. 5. A first stage random device.

a contradiction. It is also readily verified that in every subjective correlated equilibrium of the matching pennies game, the conditional payoff to every player is at least 0. We construct below a GCE in which the conditional payoff to each player is strictly less than zero. Let the first stage random device m˜ be the one as depicted in Fig. 5. In addition, require the second stage random devices b˜ 1 and b˜ 2 to satisfy

b˜ 1 (hU, Dj 3 hLj)(U ) 5 b˜ 1 (hU, Dj 3 hRj)(U ) 5 b˜ 2 (hU j 3 hL, Rj)(L) 1 5 b˜ 2 (hDj 3 hL, Rj)(L) 5 ]. 2 In this GCE, the conditional payoff to each player is always equal to 2 1 / 2. For the purpose of illustration, suppose player 1 receives the recommendation s *1 5 U. Given m˜ and b˜ 1 as specified above, we have 1 1 1 m˜ 1 (hLjuU ) 5 ], m˜ 1 (hRjuU ) 5 ], m˜ 1 (hL, RjuU ) 5 ]. 4 4 2 Therefore, the conditional payoff of choosing the action U is equal to 1 m˜ 1 (hLjuU )u 1 (U, L) 1 m˜ 1 (hRjuU )u 1 (U, R) 1 m˜ 1 (hL, RjuU )u 1 (U, R) 5 2 ], 2 and similarly, his conditional payoff of choosing D is also 2 1 / 2. Therefore, player 1 is just willing to obey the mediator’s recommendation. Example 4. The game in Fig. 6 is the game of matching pennies ‘with an outside

Fig. 6. Matching pennies with outside option.

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

201

option.’ There is an action O available to each player. If a player chooses O, then his payoff is e . 0, regardless of his opponent’s action choice. If e is small enough, there is still a subjective correlated equilibrium in which player 1 chooses either U or D, and player 2 chooses either L or R. However, it follows from arguments similar to those in Example 3 that in a GCE, players 1 and 2 always choose O. This shows that predictions generated by a subjective correlated equilibrium may not be replicated by any GCE. Finally, Examples 5 and 6 below show that while GNE is capable of generating predictions that are incompatible with expected utility maximization, GCE could make even further difference. Example 5. Consider the game in Fig. 7. It follows from standard dominance argument (Pearce, 1984) that if the players are expected utility maximizers, then player 1 will not choose the action U, and player 2 will not choose the action L. There exist GNEs in which the action profile (U, L) is chosen with positive probability. The pair ( m˜ 1 , m˜ 2 ), where 2 ] 3 m˜ 1 (Z1 ) 5 1 ] 3 0

5

if Z1 5 hU j if Z1 5 hM, Dj otherwise

and 2 ] 3 m˜ 2 (Z2 ) 5 1 ] 3 0

5

if Z2 5 hLj if Z2 5 hC, Rj otherwise

constitutes an example. In this GNE, every action profile is chosen with positive 1 probability (in the sense that S 1 1 5 S 1 and S 2 5 S 2 ), and each player’s conditional payoff (calculated using Eq. (23)) is 16 / 3. It can be verified that among all the GNEs with the property that (U, L) is chosen with positive probability, every other action profile is also chosen with positive probability, and 16 / 3 is the maximum conditional payoff that can be attained.

Fig. 7. A two-player game.

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

202

In contrast, there exist GCEs in which the action profile (U, L) is chosen with positive probability, and the profiles (M, C), (M, R), (D, C), (D, R) chosen with zero probability. For instance, let 1 1 1 m˜ (hU j 3 hLj) 5 ], m˜ (hU j 3 hC, Rj) 5 ], m˜ (hM, Dj 3 hLj) 5 ]. 2 4 4 This first stage random device m˜ and any second stage random devices ( b˜ 1 , b˜ 2 ) constitute a GCE. To see this, first suppose that player 1 receives the recommendation s 1* 5 U. Then his beliefs are updated to 2 ] 3 m˜ (Z2 uU ) 5 1 ] 3 0

5

if Z2 5 hLj if Z2 5 hC, Rj otherwise,

which is exactly m˜ 2 in the above GNE. Player 1’s conditional payoff of choosing any action is 16 / 3, and he is just willing to obey the mediator’s recommendation. Now suppose player 1 receives the recommendation s *1 5 M (or s *1 5 D). Then player 1 knows for sure that player 2 will choose L, and therefore, it is strictly better for him to follow the recommendation. In this case, player 1’s payoff is equal to 8. The situation of player 2 is exactly the same as that of player 1. Example 6. The game in Fig. 8 involves players 1, 2, and 3, choosing the row, column, and matrix, respectively. If player 3 is an expected utility maximizer, he will not choose the actions A and B, because his expected payoff associated with at least one of the other two actions is strictly above 2 (which is the maximum that can be achieved if A or B is chosen). The tuple ( m˜ 1 , m˜ 2 , m˜ 3 ), where

Fig. 8. A three-player game.

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

203

1 m˜ 1 (hU j) 5 m˜ 1 (hDj) 5 ], 2

m˜ 2 (hL, Rj) 5 1, 1 m˜ 3 (hAj) 5 m˜ 3 (hBj) 5 ], 2 is a GNE of this game with the property that player 3 chooses either A or B, and each player’s conditional payoff is equal to 1. It can be verified that among all the GNEs in which player 3 chooses either A or B, 1 is the maximum conditional payoff that can be attained. Any mediated communication system k m˜ , ( b˜ i ) 3i 51 l satisfying 1 m˜ (hU j 3 hL, Rj 3 hAj) 5 m˜ (hDj 3 hL, Rj 3 hBj) 5 ] 2

(34)

b˜ 2 (hU j 3 hL, Rj 3 hAj) 5 b˜ 2 (hDj 3 hL, Rj 3 hBj)

(35)

and

is a GCE. According to Eq. (34), the mediator recommends perfect coordination between players 1 and 3 on either (U, A) or (D, B), with probability 1 / 2 each. This represents player 2’s beliefs (even after he receives a recommendation, because of Eq. (35)). As a result, player 2 is indifferent between his two actions, and willing to obey any recommendation. Again, according to Eq. (34), the mediator generates ambiguity on player 2’s action choice. This makes player 3 avoid the actions C and D, and find it optimal to coordinate with player 1. As for player 1, it is obviously optimal for him to coordinate with player 3, regardless of player 2’s action choice. A distinctive feature of this GCE is that it prevents the undesirable outcome (0, 0, 0) from happening, and each player’s conditional payoff is equal to 2.

Acknowledgements I would like to thank a referee and especially an associate editor for valuable suggestions that led to substantial improvements, and the Social Sciences and Humanities Research Council of Canada for financial support.

Appendix A This appendix contains the proof of the equivalence of Definitions 3 and 4. We first show that every GCE k m˜ , ( b˜ i ) ni51 l as defined in Definition 4 can be restated as a GCE k(V, m ), (Qi , bi , gi ) ni 51 l as defined in Definition 3.

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

204

Proposition 1. Given a GCE k m˜ , ( b˜ i ) ni 51 l of a strategic game G 5 k(Si , u i ) ni 51 l, there is a GCE k(V, m ), (Qi , bi , gi ) ni 51 l of G such that for every i 5 1, . . . ,n,

Q i1 5 S i1 , ;s *i [ S 1 i ,

gi (s *i ) 5 s *i and

O m(v us* ) F min u (s , s O m˜ ( 3 Z us* ) F i

i

s 2i [ G2i (v )

v [V

j ±i

j

i

i

3 j ±i Z j [3 j ±i 3 (S j )

;s *i [ S 1 i

2i

)

G5

min

s 2i [3 j ±i Z j

(A.1) u i (s i , s 2i )

G

;s i [ Si .

(A.2)

Proof. Given k m˜ , ( b˜ i ) ni 51 l, define k(V, m ), (Qi , bi , gi ) in51 l as follows: • The set V of states.

V 5 3 nj 51 3 (Sj ).

(A.3)

• The probability measure m on V. For all 3 jn51 Zj [ V, n m ( 3 j51 Zj ) 5 m˜ ( 3 jn51 Zj ).

(A.4)

• The set Qi of signals.

Qi 5 Si .

(A.5)

• The information structure bi : V → D(Qi ). For all 3 nj 51 Zj [ V and for all s i [ Qi ,

bi ( 3 jn51 Zj )(si ) 5 b˜ i ( 3 jn51 Zj )(s i ).

(A.6)

• The strategy gi : Qi → Si . For all s i [ Qi ,

gi (s i ) 5 si .

(A.7) 1

1

Recall the definitions of Q i and S i as stated in Eqs. (12) and (17), respectively. The 1 fact that Q 1 follows from Eqs. (A.3)–(A.6). The fact that gi (s *i ) 5 s *i for all i 5 Si 1 s *i [ S i follows from Eq. (A.7). It remains to prove the equality between Eqs. (A.1) and (A.2). We begin by first establishing a relationship between the two probability measures m ( ? us *i ) and m˜ ( ? us i* ). For each 3 nj 51 Zj [ V,

m( 3

n j51

n m ( 3 jn51 Zj )bi ( 3 j51 Zj )(s i* ) Zj us i* ) 5 ]]]]]]]]]]] n m ( 3 j 51 Zˆ j )bi ( 3 nj 51 Zˆ j )(s i* )

O

3 nj 51 Zˆ j [ V

(A.8)

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

m˜ ( 3 nj 51 Zj ) b˜ i ( 3 nj 51 Zj )(s *i ) 5 ]]]]]]]]]]]] . m˜ ( 3 nj 51 Zˆ j ) b˜ i ( 3 nj 51 Zˆ j )(s *i )

O

205

(A.9)

3 nj 51 Zˆ j [3 nj 51 3 (S j )

The expression in Eq. (A.8) is the definition of m (v uui ) as stated in Eq. (13), with 3 jn51 Zj and s *i in place of v and ui , respectively. The equality between Eqs. (A.8) and (A.9) is due to Eqs. (A.3)–(A.6). Fix 3 j ±i Zj and sum up the expression in Eq. (A.9) over all the elements in 3 (Si ), we have

O

m ( 3 jn51 Zj us *i ) 5 m˜ ( 3 j ±i Zj us i* ),

(A.10)

Z i [ 3 (S i )

where the right hand side of Eq. (A.10) is defined in Eq. (18). Next, recall the definition of Gi as stated in Eq. (14). It follows from Eqs. (A.6) and (A.7) that for each 3 jn51 Zj [ V,

Gi ( 3 jn51 Zj ) 5 Zi .

(A.11)

We are now ready to establish that

O m( 3 Z us* ) F min u (s , s ) G 5 O m( 3 Z us * ) F min u (s , s ) G 5 O O m( 3 Z us* ) F min u (s , s ) G 5 O F min u (s , s ) G O m( 3 Z us* ) 5 O m˜ ( 3 Z us* ) F min u (s ,s ) G.

3 nj 51 Z j [ V

n j 51

3 nj 51 Z j [ V

j

i

n j 51

j

i

s 2i [3 j ±i Z j

j ±i

i

s 2i [3 j ±i Z j n j 51

j

i

2i

3 j ±i Z j [3 j ±i 3 (S j ) Z i [ 3 (S i )

3 j ±i Z j [3 j ±i 3 (S j )

i

s 2i [ G2i ( 3 nj 51 Z j )

j

i

i

2i

i

i

2i

s 2i [3 j ±i Z j

i

i

n j 51

2i

j

i

(A.12) (A.13) (A.14) (A.15)

Z i [ 3 (S i )

i

3 j ±i Z j [3 j ±i 3 (S j )

s 2i [3 j ±i Z j

i

i

2i

(A.16)

Note that Eq. (A.12) is Eq. (A.1), with 3 nj 51 Zj in place of v. The equality between Eqs. (A.12) and (A.13) is due to Eq. (A.11), and that between Eqs. (A.15) and (A.16) is due to Eq. (A.10). Finally, note that Eq. (A.16) is (A.2). h We now establish a converse to Proposition 1. Suppose the players are about to play a n GCE k(V, m ), (Qi , bi , gi ) i51 l of a strategic game G 5 k(Si , u i ) ni 51 l. A mediator approaches the players, and implements for them the following incentive compatible direct mechanism of k(V, m ), (Qi , bi , gi ) ni 51 l. The mediator, rather than player i, receives a signal ui [ Q 1 i , and then the mediator recommends player i to choose the action gi (ui ). Define, for all s i [ Si , the set

g i21 (s i ) 5 hui [ Qi : gi (ui ) 5 si j. After receiving a recommendation s *i , player i’s conditional payoff of choosing an action s i [ Si is

K.C. Lo / Mathematical Social Sciences 44 (2002) 183–209

206

O m(v us* ) F i

v [V

min s 2i [ G2i (v )

G

u i (s i , s 2i ) ,

(A.17)

where for all v [ V,

O b (v)(u ) m (v us * ) ; ]]]]]]]]] O m(vˆ ) O (s* )b (vˆ )(u ). m (v )

i

i

ui [g 21 i (s i* )

(A.18)

i

i

i

i

ui [g 21 i

ˆ [V v

It is immediate that Eq. (16) implies

O m(v us* ) F min u (s*, s ) G $ O m (v us * ) F min u (s , s ) G i

s 2i [ G2i (v )

v [V

i

i

s 2i [ G2i (v )

v [V

2i

i

i

2i

i

;s i [ Si .

That is, player i finds it worthwhile to obey the recommendation of the mediator. Proposition 2 below says that given any GCE k(V, m ), (Qi , bi , gi ) ni 51 l as defined in Definition 3, there is a GCE k m˜ , ( b˜ i ) ni 51 l as defined in Definition 4 such that k m˜ , ( b˜ i ) ni51 l is equivalent to the incentive compatible direct mechanism of k(V, m ), (Qi , bi , gi ) ni51 l. Proposition 2. Given a GCE k(V, m ), (Qi , bi , gi ) ni51 l of a strategic game G 5 k(Si , u i ) ni 51 l, there is a GCE k m˜ , ( b˜ i ) ni 51 l of G such that for every i 5 1, . . . ,n, 1 S1 such that gi (ui ) 5 s i j i 5 hs i [ Si : 'ui [ Q i

(A.19)

and

O

m˜ ( 3 j ±i Zj us i* )

3 j ±i Z j [3 j ±i 3 (S j )

5

O m(v us* ) F i

v [V

min s 2i [ G2i (v )

F

min

u i (s i , s 2i )

G

;s i* [ S i1

s 2i [3 j ±i Z j

u i (s i ,s 2i )

G

(A.20) ;s i [ Si .

(A.21)

Proof. For notational simplicity, define, for each 3 nj 51 Zj [ 3 nj 51 3 (Sj ), [ 3 nj 51 Zj ] 5 hv [ V : Gj (v ) 5 Zj

; j 5 1, . . . ,nj,

and similarly, for each 3 j ±i Zj [ 3 j ±i 3 (Sj ), [ 3 j ±i Zj ] 5 hv [ V : Gj (v ) 5 Zj

; j ± ij.

Given k(V, m ), (Qi , bi , gi ) ni 51 l, define k m˜ , ( b˜ i ) in51 l as follows: • The first stage random device m˜ . For each 3 nj 51 Zj [ 3 nj 51 3 (Sj ),

m˜ ( 3 nj 51 Zj ) 5

O

m (v ).

(A.22)

v [[3 nj 51 Z j ]

• The second stage random devices b˜ i : 3 nj 51 3 (Sj ) → D(Si ). For all 3 nj 51 Zj [ 3 nj 51

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3 (Sj ) such that m˜ ( 3 nj51 Zj ) 5 0, b˜ i ( 3 jn51 Zj ) is arbitrary. For each 3 nj 51 Zj such that m˜ ( 3 nj 51 Zj ) . 0, define

O

O

m (v ) bi (v )(ui ) v [[3 nj 51 Z j ] ui [g 21 i (s i ) n ˜ bi ( 3 j 51 Zj )(si ) 5 ]]]]]]]]]] m (v )

O

;s i [ Si .

(A.23)

v [[3 nj 51 Z j ]

By construction, the support of b˜ i ( 3 jn51 Zj ) is Zi . By construction of m˜ as stated in Eq. (A.22), s i [ S 1 i if and only if there exist v [ V and ui [ Qi such that m (v ) . 0, bi (v )(ui ) . 0, and gi (ui ) 5 s i . Recall the definition of Q 1 i as stated in Eq. (12), this fact can be more concisely written as Eq. (A.19). To establish the equality between Eqs. (A.20) and (A.21), it suffices to prove that for each 3 j ±i Zj [ 3 j ±i 3 (Sj ),

m˜ ( 3 j ±i Zj us i* ) 5

O

m (v us i* ).

v [[3 j ±i Z j ]

Recall that m˜ ( ? us i* ) is defined in Eq. (18). After substituting Eqs. (A.22) and (A.23) into Eq. (18), and after some tedious algebraic manipulations, Eq. (18) becomes

O

v [[3 j ±i Z j ]

O

m (v )

O

O

bi (v )(ui )

ui [g 21 i (s * i )

O

m˜ ( 3 j ±i Zj us i* ) 5 ]]]]]]]]]]]]]]. m (v ) bi (v )(ui ) 3 nj 51 Zˆ j [3 jn51 3 (S j ) v [[3 jn51 Zˆ j ]

(A.25)

ui [g i21 (s * i )

Recall that m ( ? us i* ) is defined in Eq. (A.18). Summing the expression in the right-hand side of Eq. (A.18) over all v [ [ 3 j ±i Zj ] will lead to the right-hand side of Eq. (A.25). h

Appendix B In this appendix, we establish that the set of GCEs of a strategic game is ‘convex.’ Proposition 3. Let k m˜ , ( b˜ i ) ni 51 l, k m˜ 9, ( b˜ i9 ) in51 l and k m˜ 0, ( b˜ i99 ) in51 l be mediated communication systems of a strategic game G 5 k(Si , u i ) ni 51 l. Suppose there exists a real number l [ [0, 1] such that

m˜ 0 5 lm˜ 1 (1 2 l) m˜ 9, and for each 3 nj 51 Zj [ 3 jn51 3 (Sj ) with m˜ 0( 3 jn51 Zj ) . 0,

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lm˜ ( 3 nj 51 Zj ) b˜ i99 ( 3 jn51 Zj ) 5 ]]]] b˜ ( 3 jn51 Zj ) m˜ 0( 3 nj 51 Zj ) i n

(1 2 l) m˜ 9( 3 j 51 Zj ) 1 ]]]]]] b˜ 9i ( 3 nj 51 Zj ) ;i 5 1, . . . ,n. m˜ 0( 3 nj 51 Zj ) If k m˜ , ( b˜ i ) ni 51 l and k m˜ 9, ( b˜ i9 ) ni 51 l are GCEs of G, then k m˜ 0, ( b˜ i99 ) ni 51 l is also a GCE of G. n Proof. Let k(V, m ), (Qi , bi , gi ) ni 51 l be the GCE of G which is derived from k m˜ , ( b˜ i ) i51 l n as in the proof of Proposition 1, and k(V 9, m 9), (Q i9 , b i9 ,g i9 ) i51 l the GCE of G derived from k m˜ 9, ( b˜ 9i ) in51 l in exactly the same fashion. Rename the elements in V, V 9, Qi , Q 9i so that V > V 9 5 [ and Qi > Q 9i 5 [. It is immediate that k(V 0, m 0), (Q 99 i , b i99 , g i99 )ni 51 l is also a GCE of G, where

• The set V 0 of states.

V 0 5 V < V 9. • The probability measure m 0 on V 0. For all v [ V 0,

m 0(v ) 5

H

lm (v ) if v [ V (1 2 l)m 9(v ) if v [ V 9.

• The set Q 99 i of signals.

Q i99 5 Qi < Q i9 . • The information structure b i99 : V 0 → D(Q i99 ). For all v [ V 0 and for all ui [ Q i99 ,

bi (v )(ui ) if v [ V and ui [ Qi b 99 ( v )( u ) 5 b i9 (v )(ui ) if v [ V 9 and ui [ Q i9 i i 0 otherwise.

5

• The strategy g i99 : Q 99 i → Si . For all ui [ Q 99 i ,

g 99 i (ui ) 5

H9

gi (ui ) if ui [ Qi g i (ui ) if ui [ Q i9 .

n Then it follows from Eqs. (A.22) and (A.23) that k m˜ 0, ( b˜ 99 i ) i 51 l is the GCE which is equivalent to the incentive compatible direct mechanism of k(V 0, m 0), (Q i99 , b i99 , g i99 )ni 51 l. h

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