Subgame Perfect Equilibria in Stage Games

Transcription

1 Subgame Perfect Equilibria in Stage Games Alejandro M. Manelli Department of Economics Arizona State University Tempe, Az August 2000 I am grateful to an associate editor and to two referees for valuable suggestions. Financial support from the National Science Foundation under Grant SBR is gratefully acknowledged. 1

2 Abstract It is well known that a stage game with infinite choice-sets, unless it contains a public coordination-device in each stage, may have no subgame perfect equilibria. We show that if a game with public coordination-devices has a subgame perfect equilibrium in which two players in each stage use non-atomic strategies, then the game without coordination devices also has a subgame perfect equilibrium. Key words: existence, subgame perfect equilibrium, infinite stage games, cheap talk, correlation. Journal of Economic Literature Classification Number(s): C6, C7, D8. 2

3 We consider the existence of subgame perfect equilibria in stage games where players have a continuum of actions. It is well known that even well-behaved stage games may have no subgame perfect equilibria. Harris, Reny, and Robson [1] provide a non-existence example and establish an important upper hemi-continuity result for stage games. 1 They prove that subgame perfect equilibria always exist in stage games where players in each stage observe a public signal to coordinate their actions. They also show that adding cheap talk to a stage game i.e., allowing players in one stage to send payoff-irrelevant messages to players in succeeding stages can replace public signals as coordination devices, therefore restoring the existence of subgame perfect equilibria. We prove that if the game with public signals has a subgame perfect equilibrium in which two players in each stage use non-atomic strategies, then the game without public signals also has a subgame perfect equilibrium. The basic intuition is that non-atomic mixing can replace public signals (and cheap talk) as a coordination device. We use the following setting to illustrate and expand on this intuition. Restrict attention, for simplicity, to two-stage games. In the first stage, each player i in a finite set I selects, in isolation, an action a i from a set A i. After observing the actions selected in the first stage, each player j in a finite set J selects an action b j from a set B j. Action spaces are compact metric spaces. We denote by A and B the Cartesian products i I A i and j J B j respectively. All payoff functions {U l } l I J are continuous, real valued, and have as arguments the actions selected by all players. The game described is summarized by Γ = [I, J, {A i } i I, {B j } j J, {U l } l I J ]. We consider two extensions of Γ. In the publicsignal extension of Γ, denoted by Γ ps, each second-stage player j observes before choosing his/her own action b j the actions chosen in the first stage {a i } i I, and the realization of a random variable Z uniformly distributed in [0, 1]. In the cheap-talk extension of Γ, denoted Γ ct, two first-stage players, i and i, choose in addition to their actions a i payoff-irrelevant message for second-stage players. and a i a We first illustrate how cheap talk can be used as a coordination device to replace a public signal. We then show, in the proof of the theorem, that a similar argument establishes that non-atomic mixing can also serve as a coordination device. Consider any subgame perfect equilibrium of Γ ps. Suppose first that in Γ ct players i and i choose cheap-talk signals X i and X i that are uniformly distributed in [0, 1]. Both signals are completely uninformative, and since selected by different players, are also independent. Define the real valued function S( ) on IR which will be used to coordinate second-stage actions by S(w) = w int(w), where int(w) denotes the integer part of w. Note that S( ) is a saw tooth function, and that for any k [0, 1], S(X i + k) and S(k + X i ) are both 3

4 uniformly distributed in [0, 1]. Consequently, each second-stage player in Γ ct is willing to adopt the equilibrium strategy used in Γ ps but with S(X i + X i ) instead of Z as the public signal. 2 Since first-stage players cannot unilaterally affect the distribution of S(X i +X i ), they are indifferent among all their cheap-talk signals. In particular, i and i have no incentive to deviate from X i and X i respectively. This establishes that the subgame perfect equilibrium guaranteed to exist in Γ ps, will have an analogue in Γ ct with cheap talk replacing the public signal. The argument provided above, except for the saw tooth function, is contained in the discussion following Theorem 45 in [1]. An associate editor suggested the use of the saw tooth function to simplify Harris, Reny, and Robson s argument, and the application of the simplified argument to prove the theorem below. (The original proof can be found in [3].) The theorem below relates the subgame perfect equilibrium outcomes of Γ ps with those of Γ. Its proof relies on the observation, that if first-stage players use non-atomic mixing, then virtual signals the analogues of X i and X i above can be directly constructed from the players payoff-relevant actions. Second-stage players can then use those signals to coordinate their actions. An outcome of Γ ps is a distribution λ over A B Z, that can be generated by strategies of Γ ps. The marginal distribution of λ on A B is denoted by λ A B. Theorem Let ({ˆα i } i I, { ˆβ j } j J ) be a subgame perfect equilibrium of Γ ps with outcome λ. If the strategies of any two first-stage agents, i, i I, are non-atomic, then λ A B is a subgame perfect equilibrium outcome of Γ. According to the limit distribution λ, any first-stage player i must choose a payoff relevant strategy according to the (marginal) distribution λ Ai. If at least two players exhibit sufficiently diffused behavior, then Γ has a subgame perfect equilibrium generating the outcome λ A B. Proof: To prove the theorem, we construct equilibrium strategies ({α i } i I, {β j } j J ) of Γ as follows. 1. According to the Lemma below, since α i and α i are non-atomic, there are random variables X i : A i [0, 1] and X i : A i [0, 1] such that both X i and X i are uniformly distributed on [0, 1]. These random variables form with the saw tooth function S( ) a virtual public signal S(X i + X i ). 2. The strategy ˆβ j of a second-stage player in Γ ps depends on the first-stage actions {a i } i I, and on the realizations of Z. In Γ second-stage players use the same strategy they use 4

5 in Γ ps but condition their actions on S(X i + X i ) instead of on Z: for each j, β j (a) = ˆβ j (a, S(X i (a i ) + X i (a i )). (1) Second-stage players had no incentive to deviate in Γ ps and nothing has changed for them. They use S(X i + X i ) instead of Z to coordinate their actions both random variables are uniformly distributed in [0, 1]. In Γ, first-stage players also adopt the same strategy they chose in Γ ps, i.e., α i = ˆα i for all i I. First-stage players have no incentive to deviate because no unilateral deviation can change the distribution of S(X i + X i ). To see this, suppose that player i has a profitable deviation a i. This deviation does not affect the distribution of S. If it is profitable in Γ, it must therefore be also profitable in Γ ps which is a contradiction. Q.E.D. Lemma Let α be an atomless probability measure on a separable metric space A. There is a measurable function X : A [0, 1] such that αx 1, its induced distribution, is the uniform on [0, 1]. Proof: The proof has two steps. The first step establishes the validity of the lemma when A is the unit interval. In this case the construction of X is straightforward, X is simply the cumulative distribution of the measure α. The second step extends the result using the fact that any uncountable, separable metric space A is isomorphic to the unit interval. 1. Suppose A = [0, 1] and define for any a A X(a) = α((0, a)). We now verify that αx 1 is the uniform on [0, 1]. To that purpose, pick any x [0, 1]. It then follows that αx 1 ((0, x )) = α[x 1 ((0, x ))] = α[{a : 0 X(a) x }] = α[(0, ā)], where ā = sup{a : α[(0, a)] x } = x where the last step follows because α is non-atomic, and therefore its cumulative distribution function α(0, a) is continuous and non-decreasing as a function of a. This establishes that X is distributed as a uniform in [0, 1]. 5

6 2. We show that the lemma holds for any separable metric space A. If α is non-atomic, then A is uncountable [2, page 44]. Then, there is a bijective function ϕ : A [0, 1] such that both ϕ and its inverse ϕ 1 are measurable [4, Theorem 2.12, page 14]. Let αϕ 1 be the distribution induced on [0, 1] by α and ϕ 1. Note that for all x [0, 1], αϕ 1 (x) = 0, and therefore αϕ 1 is non-atomic [2, page 45]. Then, we can apply the first step to the distribution αϕ 1 by defining X(a) = αϕ 1 [(0, a)]. Q.E.D. Remark 1: We have used two-stage games only for simplicity. The results and discussion apply to the general class of stage games considered in [1]. Remark 2: To replace a public signal with cheap talk, at least two players must have the ability to send cheap talk messages. If only one player had this ability, then such a player may have an incentive to manipulate the continuation of the game by varying his/her cheap talk signal. When two players send cheap-talk messages, no single player can unilaterally alter the continuation of the game. This is precisely the role of the saw tooth function discussed above. The same phenomenon occurs with non-atomic mixing. Remark 3: The following property can be established by combining the upper hemicontinuity of subgame perfect outcomes in games with public signals and the theorem above. Fix an infinite game and consider a sequence of finite games constructed by taking increasingly finer discretizations of the infinite game. Each finite game has a subgame perfect outcome, the outcome of a subgame perfect equilibrium. Outcomes are distributions and converge in a subsequence to a limit probability distribution. Our Theorem identifies conditions on the limit distribution that ensure the existence of a subgame perfect equilibrium: if the limit distribution prescribes non-atomic behavior for first-stage players, then the infinite game has a subgame perfect equilibrium. As an existence result, however, the theorem is not entirely satisfactory because it imposes conditions on limit distributions, not on primitives. 6

7 Footnotes 1. The subgame perfect outcomes of games with public signals are distributions on spaces that include payoff-irrelevant variables, the realization of the public signal. Harris, Reny, and Robson prove the upper hemi-continuity of the correspondence that maps games with public signals into the projection of subgame perfect outcomes on payoffrelevant variables. The interested reader should consult their paper for precise statements and for important related results. 2. A strategy for second-stage player j in Γ ps is a measurable function β j : A Z M(B j ), where M(B j ) represents the set of probability measures on B j. Similarly, a strategy for first-stage player i is a probability measure α i M(A i ). All spaces are endowed with their corresponding Borel sigma-fields. 7

8 1 References 1. C. J. Harris, P. J. Reny and A. J. Robson, The Existence of Subgame Perfect Equilibrium in Continuous Games with Almost Perfect Information: a Case for Extensive-Form Correlation, Econometrica 63 (1995), W. Hildenbrand, Core and Equilibria of a Large Economy, Princeton University Press, Princeton, A. M. Manelli, Subgame Perfect Equilibrium and Communication in Stage Games, working paper, ASU Department of Economics (1999). 4. K. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York,

9 List of Symbols α lowercase Greek letter alpha β lowercase Greek letter beta λ lowercase Greek letter lambda Γ uppercase Greek letter gamma ϕ variant of the lowercase Greek letter phi 9