Perfect Foresight Dynamics in Games with Linear Incentives and Time Symmetry

Transcription

1 Perfect Foresight Dynamics in Games with Linear Incentives and Time Symmetry Satoru Takahashi May 2, 2007 I am grateful to Drew Fudenberg, Aki Matsui, Fuhito Kojima, Daisuke Oyama, Takashi Ui, Michihiro Kandori, Ryo Ogawa, Ana B. Ania, Carlos Alós-Ferrer, Olivier Tercieux, the editor, William Thomson, and an anonymous referee for comments and discussions. Affiliation: Department of Economics, Harvard University. Address: Littauer Center, 1805 Cambridge Street, Cambridge, MA 02138, USA. address: stakahas@fas.harvard.edu 1

2 Abstract This paper investigates absorption and global accessibility under perfect foresight dynamics in games with linear incentives. An action distribution in the society is absorbing if there is no equilibrium path escaping from the distribution, and globally accessible if, from every initial distribution, there exists an equilibrium path which converges to the distribution. Using time symmetry of the dynamics, we show that every absorbing strict Nash equilibrium, if it exists, is globally accessible under zero rate of time preference. With the additional assumption of supermodularity, we prove that there generically exists an absorbing strict Nash equilibrium. Relations with a global game and a reaction-diffusion model also become clear. Keywords: perfect foresight dynamics, games with linear incentives, supermodular games, equilibrium selection 1 Introduction This paper investigates absorption and global accessibility under perfect foresight dynamics in games with linear incentives. An action distribution x in the society is absorbing if there is no equilibrium path escaping from x. If someone outside the society observes that the current action distribution is absorbing, then, without knowing the equilibrium belief of the members of the society, he can predict that the action distribution will be the same forever. 1 A distribution x is globally accessible if, from every initial distribution, there exists an equilibrium path which converges to x. If there exists a globally accessible distribution x, then the outside observer cannot exclude the possibility that the action distribution will be close to x in the future even if the current state is far from x. Since not all static Nash equilibria are absorbing or globally accessible, absorption and global accessibility provide equilibrium selection criteria for normal-form games. For example, Matsui and Matsuyama (1995) show that, in a 2 2 coordination game, the risk-dominant equilibrium is absorbing under every nonnegative rate of time preference, and globally accessible under every near-zero rate of time preference. Therefore, under every near-zero nonnegative rate of time preference, there exists an equilibrium path from the risk-dominated equilibrium to the risk-dominant one, but not vice versa. 1 The definition of absorption used in this paper is slightly different from the original one in Matsui and Matsuyama (1995). This difference is neglected in the introduction, and will be discussed in Subsection

3 Specializing in zero rate of time preference, this paper extends Matsui and Matsuyama s results in the following way. We show that, in a general symmetric game with linear incentives, every absorbing strict Nash equilibrium, if it exists, is globally accessible under zero rate of time preference. This is a nontrivial property of perfect foresight dynamics. Under a general binary relation on the set of action distributions, an absorbing state may not be globally accessible, although it is always true that no other states are globally accessible. We also show that, under the additional assumption of supermodularity, there generically exists a unique absorbing strict Nash equilibrium under zero rate of time preference, which is also a unique globally accessible state. In other words, both absorption and global accessibility are the same equilibrium selection criterion for generic supermodular games with linear incentives. The proofs rely on the following property which we call time symmetry. In perfect foresight dynamics in a game with linear incentives, an agent s incentive to choose one action over another depends only on pairwise matching with each of his opponents. Each agent who chooses his action in period t has a positive probability density p(s t) to match with another agent who chooses his action in period s. If the rate of time preference is zero, then p(s t) is a function of s t, which is independent of the sign of s t. In other words, each agent puts an equal weight on the past and the future agents as long as the distance from the present time is the same. Therefore, agents incentive structure is not affected by reversing the time axis. 2 Time symmetry depends on two assumptions: zero rate of time preference and linearity of incentives. Note that incentive linearity is satisfied in all two-player games, but not in most many-player games and in economic environments where agents instantaneous utilities are a nonlinear function of the current action distribution in the society. We also investigate relations among three different approaches to equilibrium selection: a dynamic approach via perfect foresight dynamics, a static approach via a global game, and a spatio-temporal approach via a reactiondiffusion model. We show that, in a generic supermodular game with linear incentives, the unique absorbing equilibrium in perfect foresight dynamics is also selected via the other two approaches. The previous literature has generalized the results in Matsui and Matsuyama (1995) in the following ways. Kim (1996) extends them to symmetric many-player coordination games with two actions and provides a condition 2 Dávila (2001) discusses a similar property in stochastic overlapping generations economies. 3

4 for a given static Nash equilibrium being absorbing and globally accessible under near-zero positive rate of time preference. Hofbauer and Sorger (1999, 2002) further extend Kim s results to symmetric and asymmetric potential games, respectively. They show that the potential maximizer is absorbing under every positive rate of time preference, and globally accessible under every near-zero positive rate of time preference. Oyama (2002) shows that if a symmetric two-player game has a strict p-dominant equilibrium with p<1/2, then the equilibrium is absorbing under every nonnegative rate of time preference, and globally accessible under every near-zero rate of time preference. Two different counterparts of Oyama s results in asymmetric games are proved by Hofbauer and Sorger (2002) and Kojima and Takahashi (2006). Applying the method of monotone potentials developed by Morris and Ui (2005), Oyama et al. (2003) unify and extend the potential method and the p-dominance method, showing that the monotone-potential maximizer is absorbing under every positive rate of time preference, and globally accessible under every near-zero positive rate of time preference. Kojima (2006) introduces the concept of uniform dominance and unifies Kim s and Oyama s results. Tercieux (2006) defines a set-valued generalization of strict p-dominance and proves a set-valued version of Oyama s results. In contrast to these extensions, we do not assume p-dominance or potentials. Instead, we use time symmetry of perfect foresight dynamics to exploit the relationship between absorption and global accessibility directly without knowing which state is actually absorbing. 2 Perfect Foresight Dynamics 2.1 Perfect Foresight Paths For N 2, consider a symmetric N-player game G =(N, A, u) in which players choose pure actions from a common action set A = {1, 2,...,Ā}, and all players have the same payoff function u : A N R. We denote by u i (j 2,...,j N ) the payoff of a player who chooses action i when the other players choose actions j 2,...,j N. For each i A, u i is permutation-invariant, i.e., u i (j 2,...,j N )=u i (j π(2),...,j π(n ) ) for every (j 2,...,j N ) A N 1 and every permutation π of {2,...,N}. The set of mixed actions is denoted by ={x =(x 1,...,x Ā ) R A x i 0 for all i A, i A x i =1}. We denote by e i the mixed action that assigns probability 1 on pure action i and probability 0 on the other pure actions. For each x, let û i (x) = u i (j 2,...j N )x j2 x jn (j 2,...,j N ) A N 1 4

5 be the payoff of a player who chooses pure action i when each of the other players chooses mixed action x. Let br(x) = arg max i A û i (x) be the set of best responses to symmetric mixed action profile (x,...,x) in pure actions. A mixed action x is a (symmetric) Nash equilibrium if x i > 0 implies i br(x); e i is a (symmetric) strict Nash equilibrium if {i} = br(e i ). We consider a single population of identical and rational agents from which N agents are randomly drawn to play the stage game. 3 We denote by φ(t) =(φ 1 (t),...,φ Ā (t)) the distribution of actions in the society at time t R. Agents are allowed to revise their actions only occasionally. Each agent s revision opportunities follow a Poisson process with arrival rate 1. 4 Revision opportunities are independent across agents, and, in each short interval [t, t +dt), the fraction dt of agents revise their actions. Agents who get revision opportunities in period t are called agents of type t. Denoting by α(t) =(α 1 (t),...,α Ā (t)) the distribution of new actions chosen by the agents of type t, we have the following relation between φ and α for each t R: φ(t) = t e s t α(s)ds. (1) The distribution φ(t) is the weighted average of α(s) for s<t, where e s t ds is the fraction of agents of type r [s, s +ds) who do not receive any revision opportunity between s +ds and t. Agents choices before period 0, α(t) for t<0, are given exogenously. In every period t 0, each agent of type t maximizes the expected total payoff he will obtain until he gets the next revision opportunity. If an agent of type t believes the future behavior of the society to be φ e, then, by taking action i, he expects to get payoff û i (φ e (s)) ds during the short interval from s to s +ds. Since the probability that he does not receive the next revision opportunity by time s t is e t s, his expected total payoff from choosing action i is V i (φ e,t)= t e t s û i (φ e (s)) ds. (2) Thus he chooses some pure action from BR(φ e,t) = arg max i V i (φ e,t). Note that we assume zero rate of time preference in (2). Since the interarrival time for revision opportunities has a finite mean (equal to 1), V i (φ e,t) 3 Although this paper uses the single-population model for notational simplicity, all the results in this paper can be extended to the setting in which N agents are drawn from N different populations to play an asymmetric stage game (Hofbauer and Sorger, 2002; Matsui and Matsuyama, 1995; Oyama et al., 2003). 4 By choosing the unit of time, we set the arrival rate to be 1 without loss of generality. 5

6 is well defined even under zero rate of time preference. If we added the rate of time preference, θ> 1, to this model, then the (normalized) total payoff of an agent of type t who chooses action i when φ e is expected to play would be V i (φ e,t,θ) = (1 + θ) e (1+θ)(t s) û i (φ e (s)) ds, (3) t where the effective discount rate is 1 + θ. Some papers in the literature assume θ>0in order to use the potential method, which relates perfect foresight dynamics with an optimal control problem which maximizes the total discounted value of potentials with discount rate θ (Hofbauer and Sorger, 1999, 2002; Oyama et al., 2003). This objective function is not well defined when θ 0. A positive rate of time preference is also necessary for the differential-game approach (Hofbauer and Sorger, 2002). Now we introduce the equilibrium condition. Namely, for every t 0, every agent of type t chooses a best response to his belief about the future action distribution, and the belief is the realized action distribution path. This is summarized as follows: Definition 1. A function φ: R isafeasible path if there exists a measurable function α: R satisfying (1). 5 We say that α induces φ. For each T [0, ], a feasible path φ induced by α is a T -perfect foresight path if, for every t [0,T] \ { }, α j (t) > 0 implies j BR(φ, t). An -perfect foresight path is called a perfect foresight path. We are mainly interested in behavior of perfect foresight paths, but the notion of a T -perfect foresight path for finite T has a technical advantage since it involves initial and terminal conditions symmetrically. We use the l -norm x = max i A x i for each x R A. The set of feasible paths, denoted by Φ, is compact in the topology of uniform convergence on compact intervals. For each T [0, ], the best response correspondence β T :Φ Φ given by β T (φ) ={φ Φ φ is induced by α, α i (t) > 0 i BR(φ, t) for every t [0,T]\{ }} is a nonempty-, convex-, and compact-valued upper hemicontinuous correspondence. Applying Kakutani s fixed point theorem to β T, we show the existence of T -perfect foresight paths that satisfy both initial and terminal conditions. (The terminal condition does not exit if T =.) 5 For each feasible path φ, a measurable function α that induces φ is unique up to values of α on a null set. 6

7 Proposition 1. For every T [0, ] and every measurable function α: R, there exists a T -perfect foresight path induced by α satisfying α (t) =α(t) for every t R \ [0,T]. Proof. Let Φ T,α be the set of feasible paths induced by α satisfying α (t) = α(t) for every t R \ [0,T]. This set is a nonempty, convex, and compact subset of Φ. Let β T,α :Φ T,α Φ T,α be the correspondence given by β T,α (φ) =β T (φ) Φ T,α. Since β T,α is a nonempty-, convex-, and compactvalued upper hemicontinuous correspondence, it follows from Kakutani s fixed point theorem that there exists a fixed point of β T,α, which is a T - perfect foresight path in Φ T,α. See Oyama (2002) and Oyama et al. (2003) for details. 2.2 Absorption and Global Accessibility A path φ is constant at x if φ(t) =x for all t 0. A state x is a static Nash equilibrium if and only if a constant path at x is a perfect foresight path. The following concept is equivalent to a stable state under perfect foresight (PF-stable state) (Matsui and Oyama, 2006). Definition 2. A state x is absorbing in Matsui and Oyama s sense (MOabsorbing, for short) if there is no nonconstant perfect foresight path from x. Every MO-absorbing state is a static Nash equilibrium because Proposition 1 and the above definition imply that there exists a constant perfect foresight path at the state. If an outside observer sees the society at an MOabsorbing state, then he can predict that the society will stay at the same state forever. In making this prediction, he does not need to know the equilibrium belief the agents will form about the future behavior of the society. If x is MO-absorbing, then, for every ε > 0 and every T [0, ), there exists δ>0 such that every perfect foresight path φ with φ (0) x <δsatisfies φ (t) x <εfor every t [0,T]. 6 Unlike in standard dynamical systems, MO-absorption is a strong stability requirement in perfect foresight dynamics. In a dynamical system that has a unique solution from every initial state, a state is MO-absorbing if and only if it is a rest point no matter whether it is stable or unstable. By contrast, in perfect foresight dynamics, since there may be multiple perfect 6 Otherwise, since the set of feasible paths is compact and the set of perfect foresight paths depends upper hemicontinuously on the initial state, there exists a perfect foresight path φ exactly from x such that φ (t) x ε for some t [0,T]. This violates the definition of MO-absorption. 7

8 foresight paths from an initial state, a static Nash equilibrium may not be MO-absorbing. For example, if the rate of time preference is sufficiently close to zero, then only the risk-dominant equilibrium is MO-absorbing in a 2 2 coordination game (Matsui and Matsuyama, 1995). Note that Matsui and Matsuyama (1995) use a slightly different definition of absorption. A state x is absorbing in Matsui and Matsuyama s sense (MMabsorbing, for short) if all perfect foresight paths from a neighborhood of x converge to x. There are two differences. First, MO-absorption only cares about paths that start exactly from x, while MM-absorption admits a small perturbation at the beginning of the dynamics. Second, MM-absorption focuses on the limit of each path, while MO-absorption does not allow any transient escape. Thus it is not clear whether and how these two absorption concepts are logically related. 7 Definition 3. A state x is globally accessible if, from every state, there exists a perfect foresight path which converges to x. Every globally accessible state is a static Nash equilibrium. 8 If a game has a globally accessible state x, then an outside observer cannot exclude the possibility that the action distribution will be close to x in the future even if the current state is far from x. Note that not all perfect foresight paths converge to x, so the observer cannot conclude that the society always moves to the globally accessible state. 2.3 Corresponding Global Game Fix a stage game G =(N, A, u) and an initial condition α(t) for t<0. Then each perfect foresight path from φ(0) = 0 es α(s)ds corresponds to a symmetric equilibrium in the following static N-player game with incomplete information. Each player n = 1,..., N has a type t n R, and player n of type t n has a posterior belief about the other players types t n = 7 We will see in Corollary 1 that the two absorption concepts are equivalent in supermodular games. In more general games, however, the relation between the two concepts is not clear. If a game has a strict p-dominant equilibrium with p<1/2, then the equilibrium is both MM- and MO-absorbing in perfect foresight dynamics (Oyama, 2002). The potential maximizing equilibrium in a potential game is MM-absorbing (under a positive rate of time preference) (Hofbauer and Sorger, 1999, 2002), but it is not known whether the equilibrium is MO-absorbing or not. 8 More generally, the limit of every convergent perfect foresight path is a static Nash equilibrium. 8

9 (t 1,...,t n 1,t n+1,...,t N ) with density p(t n t n )= max(t 1,...,t N ) e t 1 ω e t N ω dω = N 1 e t 1+ +t N N max(t 1,...,t N ). (4) Player n of type t n < 0 has a payoff function such that α(t n ) is his dominant strategy; a player of type t n 0 has payoff function u and chooses σ n (t n ) to maximize his expected payoff given his posterior belief p( t n ): [σ n (t n )] i > 0 i arg max u j (σ n (t n ))p(t n t n )dt n. j A t n If the other players follow a symmetric strategy σ 1 = = σ n 1 = σ n+1 = σ N = α and α induces φ as in (1), then the objective function for player n of type t n is u j (σ n (t n ))p(t n t n )dt n t n = u j (σ n (t n ))e t 1 ω e t N ω dω dt n t n ω=max(t 1,...,t N ) ω ω ω ω = u j (σ n (t n ))e t 1 ω e tn ω dt n dω ω=t n t 1 = t n 1 = t n+1 = t N = = û j (φ(ω))e tn ω dω = V j (φ, t n ). ω=t n Therefore, a symmetric strategy profile σ 1 = = σ N = α is a Bayesian equilibrium of this incomplete-information game if and only if α induces a perfect foresight path. The posterior belief system p( ) is derived from a common prior in the following way. A state of fundamentals ω and noises ε 1,..., ε N are stochastically independent random variables, s is uniformly distributed on R, and, for every n, ε n is exponentially distributed with mean 1. 9 Each player n cannot observe ω accurately, but observes t n = ω +ε n, which we call his type. Given t n, he forms a belief about other players observations. By a simple computation, we can see that this belief is equal to p( t n ) defined in (4). Following Carlsson and van Damme (1993), we call this game a global game of G. There may be multiple equilibrium paths from a given initial state in perfect foresight dynamics, and therefore, there may be multiple Bayesian 9 The uniform distribution on the whole line is an improper prior, but the posterior is well defined (Morris and Shin, 2003). 9

10 equilibria in the global game we constructed above. In contrast, the previous literature on global games has shown that each global game has an essentially unique equilibrium if it satisfies supermodularity conditions (Carlsson and van Damme, 1993; Frankel et al., 2003). The reason for this difference is that the previous literature assumes that there are dominance regions on both sides of the type space whereas our global game has a dominance region only on the negative side. 2.4 Linear Incentives and Time Symmetry A symmetric game G =(N, A, u) has linear incentives if there exists d: A 3 R such that u i (j 2,...j N ) u k (j 2,...,j N )=d ik (j 2 )+ + d ik (j N ) for every (i, j 2,...,j N,k) A N+1, or, equivalently, ˆd ij (x) =û i (x) û j (x) is an affine function of x for every (i, j) A 2. Every two-player game has linear incentives. Games with linear incentives are known to be well behaved for equilibrium selection (Selten, 1995). By (1), (2), and the linearity of ˆd ij, for every feasible path φ induced by α, we obtain V i (φ, t) V j (φ, t) = = = = t e t r ˆdij (φ(r)) dr e t r r t max(s,t) e s r ˆdij (α(s)) ds dr e s+t 2r ˆdij (α(s)) dr ds p(s t) ˆd ij (α(s)) ds, p(s t) = e s t. (5) 2 Thus total payoff difference V i (φ, t) V j (φ, t) is the weighted average of payoff differences ˆd ij (α(s)) under the weights of p(s t). Note that p(s t) depends only on s t. This means that each agent puts an equal weight on the past and the future agents as long as the distance from the present time is the same. We call this property time symmetry. We can interpret the symmetry of p(s t) in terms of the global game defined in Subsection 2.3. In the global game, p(s t) is the probability density that a player of type t believes that another player has type s, i.e., the marginal of p(s, t 3,...,t N t) defined in (4). Since the state of fundamentals ω is uniformly distributed on R and noises are symmetric among players, a 10

11 player of type t believes that another player s type is distributed symmetrically with respect to t. We call this property type symmetry. Note that if the static game has linear incentives, then the correlation structure of a player s belief about the other players types does not matter for his incentives. From this observation, we obtain the following lemma: Lemma 1. Suppose that a game has linear incentives and the rate of time preference is zero. (1) If φ and φ are feasible paths induced by α and α, respectively, and α(t) =α ( t) for every t R, then V i (φ, t) V j (φ, t) =V i (φ, t) V j (φ, t) for every (i, j) A 2 and every t R, and hence BR(φ, t) = BR(φ, t) for every t R. (2) For every T [0, ), ifα induces a T -perfect foresight path, then α defined by α (t) =α(t t) for every t R also induces a T -perfect foresight path. Proof. The first statement follows from the symmetry of p in (5). To prove the second statement, let φ and φ be paths induced by α and α, respectively. By the first statement, we have BR(φ, t) =BR(φ,T t). Since φ is a T -perfect foresight path, for every t [0,T], α i (t) =α i (T t) > 0 implies i BR(φ, T t) =BR(φ,t), thus φ is also a T -perfect foresight path. Note that it is not the path of distributions of actions in the society, φ, but the path of distributions of newly chosen actions, α, that is reversed with respect to the time axis. The assumption of zero rate of time preference is indispensable for Lemma 1. If the rate of time preference, θ, in (3) were nonzero, then p(s t) would depend on s t including its sign, and perfect foresight dynamics would no longer exhibit time symmetry. The linearity of incentives is also indispensable for (3). Therefore, we can apply Lemma 1 neither to general symmetric games with more than two players nor to economic environments such as one analyzed by Matsuyama (1991), where workers wages are given by nonlinear functions of the distribution of the aggregate labor supply between two sectors (agriculture and manufacturing). We illustrate Lemma 1 by an example. Let ( ) ( ) u1 (1) u 1 (2) a b = u 2 (1) u 2 (2) c d 11

12 be a payoff matrix for a 2 2 game. Consider the step function α with α(t) =e 1 for t<0 and α(t) =e 2 for t 0, which induces φ with φ(t) = e t e 1 +(1 e t )e 2 for t 0. We set α (t) =α( t) for every t R. α induces φ (t) =(1 e t )e 1 +e t e 2 for t 0. Then V 1 (φ, 0) = V 1 (φ, 0) = 0 0 e t [aφ 1 (t)+bφ 2 (t)] dt = e t [aφ 1(t)+bφ 2(t)] dt = 0 0 e t [ae t + b(1 e t )] dt = a + b 2, e t [a(1 e t )+be t ]dt = a + b 2, and hence V 1 (φ, 0) = V 1 (φ, 0). Similarly, we have V 2 (φ, 0) = (c + d)/2 = V 2 (φ, 0), and hence BR(φ, 0) = BR(φ, 0), which is a special case of time symmetry. From this example, we can rederive the main result of Matsui and Matsuyama (1995) for zero rate of time preference as follows. Suppose a + d b + c (supermodularity) and a + b>c+ d (e 1 risk-dominates e 2 ). Then BR(φ, 0) = BR(φ, 0) = {1}. On one hand, since BR(φ, 0) = {1} and every other feasible path φ from e 1 approaches e 2 more slowly than φ, the first agent who takes action 2 along φ does not have an incentive to do so. Therefore, e 1 is MO-absorbing. On the other hand, for each initial state x, since BR(φ, 0) = {1} and the linear feasible path φ x from x to e 1 (i.e., φ x (t) =(1 e t )e 1 +e t x for t 0) approaches e 1 faster than φ, every agent of type t 0 has an incentive to switch from action 2 to 1 along φ x. Therefore, φ x is a perfect foresight path and hence e 1 is globally accessible. Note that, in this example, we only need to consider incentives of agents of type t = 0 against φ and φ, the fastest feasible path from e 1 to e 2 and the slowest linear feasible path to e 1, respectively. This observation will be extended to more general supermodular games in the next section (Lemma 2 and Proposition 2). Using Lemma 1 and other lemmas, we have our first main result. Theorem 1. Suppose that the rate of time preference is zero. If a strict Nash equilibrium in a game with linear incentives is MO-absorbing, then it is also globally accessible. Therefore, every game with linear incentives has at most one MO-absorbing strict Nash equilibrium. Sketch of the Proof. Let e i be an MO-absorbing strict Nash equilibrium, and x be any state other than e i. To show that e i is globally accessible, we construct a perfect foresight path φ from x to e i in the following way. For every T [0, ), consider a T -perfect foresight path φ T induced by α T such that α T (t) =e i for every t<0 and α T (t) =x for every t>t(lemma 1). As T, φ T converges to a perfect foresight path from e i. Since e i 12

13 is MO-absorbing, the limit is a constant path at e i. Therefore, for every τ [0, ) and ε>0, we can choose T finite but sufficiently large so that φ T (t) e i <εfor every t [0,τ] (Lemma 3). Now we reverse the time axis with respect to period T/2, and define the path φ induced by α such that α (t) =α T (T t) for each t. By Lemma 1, φ is a T -perfect foresight path from x to e i. To show that φ is a perfect foresight path, we have to check i BR(φ,t) for every t>t. Since φ T is close to e i for a sufficiently long period, the reversed path φ is also close to e i in period T (Lemma 4). Since e i is a strict Nash equilibrium, i BR(φ,t) for every t>t. See Appendix A.1 for details. Theorem 1 does not hold for games without linear incentives. Consider the following game with a player s payoff function given by , 0 0 0, 0 0 0, (6) where he chooses a row and the other two players choose a column and a matrix, respectively. Both e 1 and e 3 are MO-absorbing, and hence neither is globally accessible. The proof is given in Appendix A.2, where we use Proposition 2 in the next section. 3 Supermodular Games The set A = {1, 2,...,Ā} of pure actions is considered to be totally ordered. The set of mixed actions is partially ordered with the stochastic dominance relation, i.e., x y if and only if Ā j=i x j Ā j=i y j for every i A. We write φ φ if φ(t) φ (t) for every t R; α α if α(t) α (t) for every t R. By (1), α α implies φ φ if α and α induce φ and φ, respectively. In this section, we assume that G = (N, A, u) is supermodular, i.e., u i (j 2,...,j N ) u k (j 2,...,j N ) is nondecreasing in (j 2,...,j N ) for every i>k. In this case, ˆdij (x) for i > j, max br(x), and min br(x) are nondecreasing in x. This monotonicity is preserved in perfect foresight dynamics: V i (φ, t) V j (φ, t) for i>j, max BR(φ, t), and min BR(φ, t) are nondecreasing in φ for every t R. In supermodular games, MO-absorption can be characterized by a simple condition. Note that, if there exists an MM-absorbing state in a supermodular game, then it is a strict Nash equilibrium of G, and hence a pure action (Oyama et al., 2003, Proposition 3.6). The same result holds for MO-absorption. Hence we focus on pure actions hereafter. 13

14 For i<jand T =(T i+1,...,t j ) with 0 = T i+1 T j <, we denote by φ T the feasible path induced by the following nondecreasing step function α T : e i for t<0, α T (t) = e k for i +1 k<jand T k t<t k+1, (7) e j for t T j. We call φ T an increasing path. An increasing path φ T is a subpath if max BR(φ T,T k ) k for every k with i +1 k j. (8) Decreasing superpaths are defined by reversing the order on A. An increasing subpath has the following simple structures: (i) α T (t) is a pure action for every t, (ii) α T (t) is nondecreasing in t, (iii) incentive constraints are imposed only in periods T i+1,...,t j, and (iv) in period T k, k is not necessarily a best response to φ T, but is smaller than or equal to some best response. 10 An increasing subpath φ T is not necessarily a perfect foresight path, but there exists a perfect foresight path larger than or equal to φ T. Using this result, we characterize MO-absorption in terms of increasing subpaths and decreasing superpaths. Lemma 2 (Oyama et al. (2003, Lemma 3.5)). In a supermodular game, if there exists an increasing subpath φ T from a state x, then, from every state y x, there exists a perfect foresight path φ φ T. Proposition 2. In a supermodular game, a state is MO-absorbing if and only if there is neither an increasing subpath nor a decreasing superpath from the state. Proof. The only if part follows from Lemma 2. The if part is proved by contradiction. Suppose that e i is not MO-absorbing. Let φ be a nonconstant perfect foresight path from e i induced by α. Since α k(t) > 0 for some t 0 and some k i, we assume without loss of generality that there exists such k with k>i. Let j be the maximum of all such k. Set T k = inf{t [0, ) :α l (t) > 0 for some l k} for each k with i +1 k j, and T k = T k T i+1. For T =(T i+1,...,t j ), we define the increasing path φ T as in (7). Since φ is a perfect foresight path and φ T (t) φ (t + T i+1 ) for every t 0, we have max BR(φ T,T k ) max BR(φ, T k ) k 10 Because of (ii) and the supermodularity, (iv) implies that max BR(φ T,t) max BR(φ T,T k ) k for every t T k. 14

15 for every k with i +1 k j. Hence φ T is a subpath from e i. We can obtain the same characterization for absorption in Matsui and Matsuyama s sense. Proposition 3. In a supermodular game, a state is MM-absorbing if and only if there is neither an increasing subpath nor a decreasing superpath from the state. Proof. See Appendix A.3. Corollary 1. In a supermodular game, MO-absorption is equivalent to MMabsorption. Proof. The claim follows from Propositions 2 and 3. As a corollary to Proposition 2, we can show a relationship between perfect foresight dynamics under different rates of time preference. Game G =(N, A, u) isstrictly supermodular if u i (j 2,...,j N ) u k (j 2,...,j N )is increasing in (j 2,...,j N ) for every i>k. Corollary 2. If e i is not MO-absorbing in a strictly supermodular game G = (N, A, u) under rate θ> 1 of time preference, then, for every θ ( 1,θ), there exists a neighborhood U of u such that, for every u U, e i is also not MO-absorbing in G =(N, A, u ) under rate θ of time preference. Proof. Note that Proposition 2 holds for every rate of time preference. By Proposition 2, without loss of generality, we assume that there exists an increasing subpath φ T from e i in G under θ. Since G is strictly supermodular, we have V j (φ T,t,θ ) V k (φ T,t,θ ) >V j (φ T,t,θ) V k (φ T,t,θ)inG for every j>k, every t 0, and every θ ( 1,θ). Since total payoffs V depend continuously on the static payoff function u, there exists a neighborhood U of u such that, for every u U, φ T is a subpath in G =(N, A, u ) under θ. By Proposition 2, e i is not MO-absorbing in G under θ. Note that Corollaries 1 and 2 are shown only for supermodular games. It is not known whether or not a non-supermodular game has such properties. Combining Proposition 2 with Lemma 1, we show our second main result, which is an extension of Oyama et al. (2003, Corollary 4.8). In its proof, we use the strengthened version of subpaths. For T =(T i+1,...,t j ), φ T defined by (7) is a strict subpath if min BR(φ T,T k ) k for every k with i+1 k j. Theorem 2. Every generic supermodular game with linear incentives has an MO-absorbing state under zero rate of time preference. 15

16 Sketch of the Proof. Fix a static game G. Let e i be the largest state accessible from e 1 by a strict subpath in G. Then there is no increasing strict subpath from e i by the maximality of e i (Lemma 5). Also, using time symmetry (Lemma 1), we show that there is no decreasing superpath from e i (Lemma 6). These observations imply that there is neither an increasing subpath nor a decreasing superpath from e i in an open set of nearby games G, and hence e i is MO-absorbing in G by Proposition 2. See Appendix A.4 for details. Remember that every MO-absorbing state is a strict Nash equilibrium in a supermodular game. Therefore, by Theorem 1, the MO-absorbing state is also globally accessible, and no other state is MO-absorbing or globally accessible. In other words, both MO-absorption and global accessibility in perfect foresight dynamics are equilibrium selection criteria which uniquely select the same equilibrium in every generic supermodular game with linear incentives. Both supermodularity and linearity of incentives are indispensable for Theorem 2. Without these assumptions, we can construct open sets of counterexamples as follows: the two-player non-supermodular game in Oyama (2002, Subsection 6.2) and its small payoff perturbations have no MO-absorbing state; the following supermodular game , , , (9) and its small payoff perturbations have no MO-absorbing state. The proof is given in Appendix A.5. There are also nongeneric counterexamples even if we maintain the assumptions in Theorem 2. Consider two 2 2 games with the following payoff matrices: O = ( ) ; I = ( ) In each game, there is no MO-absorbing state. In game (2, {1, 2}, I), however, we can recover the conclusion of Theorem 2 under a positive rate of time preference. Corollary 3. Every strictly supermodular game with linear incentives has an MO-absorbing state under a positive rate of time preference. Proof. Otherwise, it follows from Corollary 2 that there is no MO-absorbing state in an open set of supermodular games with linear incentives under zero rate of time preference. This contradicts Theorem

17 Since game (2, {1, 2},O) has no MO-absorbing state even under a positive rate of time preference, the assumption of strict supermodularity in Corollary 3 cannot be weakened to supermodularity. If we maintain the genericity assumption, we can extend Theorem 2 to a near-zero or positive rate of time preference through the following proposition. Proposition 4. In a supermodular game, if e i is MO-absorbing under rate θ> 1 of time preference, then there exists θ ( 1,θ) such that e i is also MO-absorbing under every rate θ > θ of time preference. Proof. If not, then, by Proposition 2, without loss of generality, we assume that there exists a sequence {φ Tm } m N of increasing subpaths from e i under θ m with θ m θ θ as m. Taking a subsequence if necessary, we can find the limit T k of {T m k } m N for each k i + 1. Letting j be the largest k such that T k <, we define φ T as in (7) with T =(T i+1,...,t j ). For every k with i +1 k j, since T m k T k and φ Tm φ T as m, we have max BR(φ T,T k ) k under θ. Since the game is supermodular and φ T is increasing, we also have max BR(φ T,T k ) k under θ. Hence φ T is a subpath from e i under θ. By Proposition 2, e i is not MO-absorbing under θ. 4 Relationship with Other Models Morris (1997, 2000) points out that local interaction and incomplete information games are analogous in that players are distributed and locally interact with each other. The only difference is that they are distributed in a space or in a type set. Here we add to this analogy perfect foresight dynamics, where agents are distributed in the time axis. This section explores formal relations among perfect foresight dynamics, a global game, and a spatiotemporal model with local interaction. These relations will help us understand time symmetry in perfect foresight dynamics, for the other two models have corresponding properties type symmetry and space symmetry. 4.1 Global Game As we saw in Subsections 2.3 and 2.4, perfect foresight dynamics corresponds to a certain global game, and the global game exhibits type symmetry in the same way that perfect foresight dynamics exhibits time symmetry. We can analyze symmetric equilibria in the global game based on this correspondence. In the next proposition, we modify the global game in two 17

18 respects. First, we introduce a parameter ν>0 to measure the level of noise. Each player n is assumed to observe ω + νε n, and we are interested in the limit when ν goes to 0. Second, we assume not only a player of type t n < 0, but also a player of type t n > 1 has a dominant strategy. These assumptions are standard in the literature of global games. Proposition 5. Suppose that G is a generic supermodular game with linear incentives and e i is the MO-absorbing state under perfect foresight dynamics with zero rate of time preference. Consider the global game such that player n of type t n chooses σn(t ν n ) to maximize his expected payoff induced by u for 0 t n 1, commits to e 1 for t n < 0, and commits to e Ā for t n > 1. Then, for every symmetric Bayesian equilibrium σ ν = σ1 ν = = σν N of the global game, σ ν (t) e i as ν 0 for every t (0, 1). Proof. See Appendix A.6. Perfect foresight dynamics corresponds to the global game with this particular noise structure. To find a correspondence to a global game with a general non-exponential noise structure, we need to extend perfect foresight dynamics so that revision opportunities follow non-poisson renewal processes. The limit equilibrium of a global game may depend on its noise structure (Carlsson, 1989; Frankel et al., 2003). Hence, it is probable that an MOabsorbing state in perfect foresight dynamics may depend on its revision structure as well. 11 By contrast, if a static game has a (generalized) potential function, then, independently of the noise structure, the limit equilibrium of a global game is given by the potential maximizer (Frankel et al., 2003). Thus it is also probable that the potential maximizer is selected in perfect foresight dynamics independently of the revision structure. 4.2 A Reaction-Diffusion Model Hofbauer (1999) proposes the following spatio-temporal model. Agents are 11 Frankel et al. (2003) provide the following 4 4 supermodular game: (10) and show that the limit of equilibria of the global game as the noise becomes small is the first action under some noise structure, and that it is the fourth action under another noise structure. In perfect foresight dynamics under zero rate of time preference with the Poisson revision structure, the fourth action is a unique MO-absorbing state. The proof is given in Appendix A.7. 18

19 distributed on R. The action distribution of agents at location ξ at time t is denoted by ρ(ξ,t). Agents change actions locally and myopically by best response dynamics à la Gilboa and Matsui (1991). The distribution also changes due to migration. This dynamics is described by the following reaction-diffusion system: ρ t (ξ,t) =α(ξ,t) ρ ρ(ξ,t)+ 2 (ξ,t), ξ2 (11) α i (ξ,t) > 0 i br(ρ(ξ,t)). (12) The reaction term α ρ models best response dynamics at each location, and the diffusion term 2 ρ/ ξ 2 models migration between locations. If the density of action i at location ξ is larger than the average of those in a neighborhood of ξ, then the excess density will diffuse to nearby locations through the diffusion term. 12 Note that (11) and (12) are space-symmetric, i.e., replacing ξ with ξ does not affect the dynamics. A Nash equilibrium x is spatially dominant if there exists a compact interval of the space such that if ρ(ξ,0) is sufficiently close to x for every ξ in this interval, then ρ(ξ,t) x as t uniformly in ξ on every compact interval (Hofbauer, 1999). In other words, a large but bounded region of a spatially dominant action can spread out unboundedly. By definition, each game has at most one spatially dominant equilibrium. Time t in perfect foresight dynamics is interpreted as location ξ in the reaction-diffusion model. In a game with linear incentives, a function α: R induces a perfect foresight path from x if and only if ψ(ξ) = p(s ξ)α(s)ds for all ξ 0 is a stationary solution to (11) and (12) with the boundary condition (dψ/dξ)(0) = ψ(0) x, where p is defined in (5). This link leads us to the following result, which is an extension of Oyama et al. (2003, Proposition 5.3.5). Proposition 6. In a generic supermodular game with linear incentives, the MO-absorbing state under perfect foresight dynamics with zero rate of time preference is spatially dominant under the reaction-diffusion model. Proof. See Appendix A.8. Theorem 2 and Proposition 6 imply that every generic supermodular game with linear incentives has a unique spatially dominant equilibrium. 12 See Fife (1979) for a rationale for the reaction-diffusion system. 19

20 5 Conclusion In perfect foresight dynamics for a game with linear incentives, an MOabsorbing strict Nash equilibrium, if it exists, is globally accessible under zero rate of time preference. Moreover, with the additional assumption of supermodularity, an MO-absorbing state does exist generically. These results established the equilibrium selection based on perfect foresight dynamics. Relations among perfect foresight dynamics, a global game, and a reactiondiffusion model also became clear. An open question is whether there is a set-valued extension of Theorem 1 that claims that every MO-absorbing curb set is globally accessible under appropriate conditions such as linear incentives. If this claim were correct, then perhaps we could prove that there exists a unique minimal (and hence a smallest) MO-absorbing curb set in every game with linear incentives. The minimal MO-absorbing curb set would be related to minimal p-best response sets, which is a set-valued extension of p-dominant equilibria recently proposed by Tercieux (2006). Appendix A.1 Proof of Theorem 1 Fix a static game with linear incentives, an MO-absorbing strict Nash equilibrium e i, and a state x e i. Assume zero rate of time preference. Lemma 3. For every τ [0, ) and every ε>0, there exist T [0, ) and a T -perfect foresight path φ T from e i such that φ T (t) e i <εfor every t [0,τ]. Proof. By Lemma 1 and the compactness of Φ, there exists a convergent sequence {φ T m } m N of T m -perfect foresight paths with T m as m. Since the limit of the sequence is a perfect foresight path from e i, and e i is MO-absorbing, the limit is a constant path at e i. Therefore, φ T m (t) e i as m uniformly in t 0 on every compact interval. Lemma 4. For every T [0, ) and every ε>0, there exists a T -perfect foresight path φ induced by α such that φ (0) = x, α (t) =e i for every t>t, and φ (T ) e i < 3ε. Proof. Choose φ T induced by α T as in Lemma 3. Let φ be the feasible path induced by α defined by α (t) =α T (T t) for all t. We have φ (0) = x and 20

21 α (t) =e i for every t>t. By Lemma 1, φ is a T -perfect foresight path. Here we have τ 0 2e t φ T (t)dt = = = = τ t=0 τ t=0 T r=t τ T r= 2e t t s= 2e t r=t t τ t=t r e s t α T (s)ds dt e T r t α (r)dr dt 2e T r 2t α (r)dt dr + e r T α (r)dr +(1 e 2τ )e i η r=t τ t=0 2e T r 2t e i dt dr = φ (T )+(1 e 2τ )e i η. (13) The first and the last equalities follow from (1). In the second equality, we change variables, writing r = T s. The third equality follows from Fubini s theorem and α (r) = e i for r > T. In the fourth equality, we set η = T τ er T α (r)dr + T T τ et r 2τ α (r)dr, which is written as (2e τ e 2τ )y for some y. Choose τ such that 2e τ e 2τ <ε. Then φ (T ) e i < (φ (T ) e i ) (2e τ e 2τ )(y e i ) + ε τ = 2e t φ T (t)dt 2(1 e τ )e i + ε 0 τ 0 2e t φ T (t) e i dt + ε<3ε by y e i 1, (13), and φ T (t) e i <εfor every t [0,τ]. Proof of Theorem 1. Since e i is a strict Nash equilibrium, we can choose ε>0 such that i br(z) for every z with z e i < 3ε. Take the T -perfect foresight path φ in Lemma 4. Then i BR(φ,t) for every t>t. Therefore, φ is a perfect foresight path from x to e i. A.2 A Game with Multiple MO-Absorbing States We will show that both e 1 and e 3 in game (6) are MO-absorbing under zero rate of time preference. Using the symmetry between pure actions 1 and 3, we consider e 1 only. Since game (6) is supermodular, we can apply Proposition 2. Thus it is sufficient to show that there exists no increasing subpath φ T from e 1 to e 2 with T = T 2 = 0 or from e 1 to e 3 with T =(T 2,T 3 )=(0,T). In the case T = T 2 = 0, we have V i (φ T, 0) = (u i (1, 1) + u i (1, 2) + u i (2, 2))/3. Since BR(φ T, 0) = {1}, φ T is not an increasing subpath. 21

22 Similarly, in the case T =(T 2,T 3 )=(0,T), we have V i (φ T, 0) = 1 3 u i(1, 1) + 1 λ2 u i (1, 2) + λ2 3 3 u i(1, 3) (1 λ)2 + u i (2, 2) + 3 V i (φ T,T)= λ2 3 u i(1, 1) + + (1 λ)2 3 λ(1 λ) u i (2, 3) + λ 3 3 u i(3, 3), (14) 2λ(1 λ) u i (1, 2) + λ 3 3 u i(1, 3) u i (2, 2) + 1 λ u i (2, 3) u i(3, 3), (15) where λ =e T (0, 1]. We have max BR(φ T, 0) 2 if and only if ( )/2 λ 1, whereas we have max BR(φ T,T) = 3 if and only if 0 <λ 1/2. Since 1/2 < ( 3+ 17)/2, there is no λ such that φ T is an increasing subpath. A.3 Proof of Proposition 3 Proof of Proposition 3. The only if part follows from Lemma 2. The if part is proved by contradiction. Suppose that e i is not MM-absorbing. For a sequence {ε m } m N of positive numbers with ε m 0asm, let φ m be a perfect foresight path induced by α m from an ε m -neighborhood of e i that does not converge to e i. Taking a subsequence if necessary, we can assume without loss of generality that, for every m N, there exists k>isuch that α m k (t) > 0 for some t 0. Set T m k = inf{t [0, ) :αm l (t) > 0 for some l k} for each k i +1 under the convention that inf =, and Tk m = T k m T i+1 m. Let ˆφ m be the path from (1 ε m )e i +ε m e Ā induced by ˆα m such that ˆα m (t) =e k for k i + 1 and Tk m t<tk+1 m. Since φm is a perfect foresight path and ˆφ m (t) φ m (t + T i+1), m we have max BR(ˆφ m,t m k ) max BR(φ m, T m k ) k for every k i + 1 such that Tk m <. Taking a subsequence if necessary, we can find the limit T k of {Tk m} m N for k i + 1. Since we have Ti+1 m =0 for every m, there is some k such that T k <. Letting j be the maximum of all such k, we define φ T as in (7) with T =(T i+1,...,t j ). For every k with i +1 k j, since Tk m T k and ˆφ m φ T as m, we have max BR(φ T,T k ) k. Hence φ T is a subpath from e i. 22

23 A.4 Proof of Theorem 2 For i < j, e j is strictly increasingly accessible from e i if there exists an increasing strict subpath from e i to e j. Lemma 5. Strict increasing accessibility is transitive in a supermodular game. Proof. For i<j<k, let T 1 =(Ti+1 1,...,T1 j ) and T2 =(Tj+1 2,...,T2 k )be such that increasing paths φ T1 and φ T2 are strict subpaths from e i to e j and from e j to e k, respectively. For t Tj 1, define T (t) =(T i+1,... (t),t (t) j,t j+1,... (t),t (t) by T (t) l = Tl 1 for i +1 l j, and T (t) l = Tl 2 + t for j +1 l k. Since φ T(t) φ T1 and φ T1 is a strict subpath, we have min BR(φ T(t),T (t) l ) min BR(φ T1,T l ) l for every l with i +1 l j. Let φ (t) (s) =φ T(t) (s + t) for each s 0. Since φ (t) φ T2, φ (t) φ T2 as t, and φ T2 is a strict subpath, we have min BR(φ T(t),T (t) l ) = min BR(φ (t),t 2 l ) min BR(φ T2,T 2 l ) l as t k ) for every l with j +1 l k. Therefore, φ T(t) e k for a sufficiently large t. is a strict subpath from e i to Lemma 6. Suppose that the rate of time preference is zero. In a supermodular game with linear incentives, if e i is strictly increasingly accessible from e 1, then there is no decreasing superpath from e i. Proof. For a path φ, let γ(φ) be the feasible path induced by α satisfying α(t) =e min BR(φ,t) for every t R. By the supermodularity, γ is nondecreasing. Let φ T be an increasing strict subpath from e 1 to e i. Define {φ m } m N by φ 0 = φ T and φ m+1 = γ(φ m ) for every m. Since φ T is a strict subpath, φ 0 φ 1. Since γ is nondecreasing, {φ m } is nondecreasing and converges to some φ by the compactness of Φ. For each s R, define the shifted path φ (s) of φ T by φ (s) (t) =φ T (t + s) for every t R. Since φ T is a strict subpath, min BR(φ T,T k ) k for every k with 2 k i. Since BR(φ T,t) is upper hemicontinuous in t, there exists ε>0 such that, for every k with 2 k i, min BR(φ T,T k ε) k. Thus γ(φ T ) φ (ε). Since perfect foresight dynamics is time-homogeneous, γ(φ (s) ) φ (s+ε) for every s R. Here we have φ m φ (mε) for every m N. This claim is proved by induction on m. The case m = 0 is trivial. For each m N, ifφ m φ (mε), 23

24 then, by the monotonicity of γ, φ m+1 = γ(φ m ) γ(φ (mε) ) φ (mε+ε) = φ ((m+1)ε). Taking m, we have φ (t) e i for every t R. Suppose that there exists a decreasing superpath φ from e i induced by α. Let φ be the reversed path of φ, i.e., φ is induced by α given by α (t) =α ( t) for every t. Since φ is a superpath, γ(φ ) φ. By Lemma 1, we also have γ(φ ) φ. Since α T α, we have φ T φ. Operating γ infinitely to the both sides of φ T φ, we have φ φ. Since φ (t) e i for every t R, we have φ (t) e i for every t R, which contradicts the fact that φ is a decreasing path. Proof of Theorem 2. Let e(u) be the largest state strictly increasingly accessible from e 1 in G =(N, A, u). (If there is no such state, let e(u) =e 1.) Let U i = {u R AN : u is supermodular, u j is permutation-invariant for every j A, u k+1 (j 2,...,j N ) u k (j 2,...,j N ) >u k+1 (j 2,...,j N ) u k (j 2,...,j N ) for every k< i, u k+1 (j 2,...,j N ) u k (j 2,...,j N ) <u k+1 (j 2,...,j N ) u k (j 2,...,j N ) for every k i for some supermodular game G =(N, A, u) with e(u) =e i }. Note that i A U i contains an open dense set in the set of all supermodular functions. For every u U i, the game G =(N, A, u ) has neither an increasing subpath from e i by the maximality of e i and Lemma 5, nor a decreasing superpath from e i by Lemma 6. Therefore, by Proposition 2, e i is MO-absorbing in G. A.5 Supermodular Games with No MO-Absorbing State We will show that there is no MO-absorbing state under zero rate of time preference in game (9) and in its small payoff perturbations. Since game (9) is strictly supermodular, for small enough payoff perturbations, every perturbed game is supermodular. Let φ T be the increasing path from e 1 to e 3 with T =(T 2,T 3 )=(0, log(3/2)). Since we have BR(φ T, 0) = {2} and BR(φ T, log(3/2)) = {3} by (14) and (15) in Appendix A.2, φ T is an increasing strict subpath in game (9), which implies that φ T is an increasing subpath in every game close enough to game (9). By Lemma 2, in the perturbed game, no state other than e 3 is MO-absorbing. Since pure actions 1 and 3 are symmetric in game (9), we can show similarly that no state other than e 1 is MO-absorbing in the perturbed game. A.6 Proof of Proposition 5 Let Φ T be the set of feasible paths induced by α satisfying α(t) =e 1 for t<0 and α(t) = e Ā for t>t. 24