Monotone Games with Positive Spillovers

Games and Economic Behavior 37, 295–320 (2001) doi:10.1006/game.2000.0842, available online at http://www.idealibrary.com on

Monotone Games with Positive Spillovers1 Douglas Gale Department of Economics, New York University, New York, NY 10003 E-mail: [email protected] Received November 3, 1998; published online September 14, 2001

A monotone game comprises the infinitely repeated play of an n-person stage game, subject to the constraint that players’ actions be monotonically nondecreasing over time. These games represent a variety of strategic situations in which players are able to make (partial) commitments. If the stage games have positive spillovers and satisfy certain other conditions, then the limit points of the subgame perfect equilibria are precisely the approachable action profiles. This characterization is applied to voluntary contribution games, market games, and coordination games. Journal of Economic Literature Classification Number: C7.  2001 Academic Press Key Words: monotone games; positive spillovers; efficiency; subgame perfect equilibrium; voluntary contribution games; market games; coordination games.

1. INTRODUCTION Coordination games represent an important class of examples showing how rational behavior can lead to suboptimal outcomes even when economic agents have common interests. From a game-theoretic perspective, this is a surprising departure from collective rationality (Aumann and Sorin, 1989). From a macroeconomic perspective, coordination games have been adopted as a paradigm of market failure (Cooper and John, 1988). A familiar example is the following 2 × 2 game: T B

L

R

2 2 0 0

0 0 1 1

1 Earlier versions of this paper were presented in seminars at New York University and the London School of Economics and at the Northwestern University Summer Workshop (Microeconomics Week), where participants made many useful comments. I am particularly indebted to Bob Rosenthal for reading the paper and to Steven Matthews for pointing out the connections with his own work. The financial support of the National Science Foundation, under grants SBR 9422194 and SBR 9810912, and the C. V. Starr Center for Applied Economics at NYU are gratefully acknowledged.

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Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

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The players’ interests are perfectly aligned in the sense that they receive identical payoffs for each strategy profile. Nonetheless, there are two Pareto-ranked equilibria (top, left) and (bottom, right). The fact that rational players can end up choosing (bottom, right) despite their common interest in choosing (top, left) suggests the surprisingly limited power of rationality to guarantee satisfactory outcomes. When the coordination game is cast in a dynamic framework, additional structure can be added to the game, and it is sometimes possible to show that agents will succeed in coordinating on the Pareto-optimal equilibrium. In earlier work (Gale 1995), I studied an n-player coordination game in which each player has an investment option that can be exercised at any date (waiting forever corresponds to no investment). There is a fixed cost of investment, and the investment generates a stream of revenues. The revenue in each period is an increasing function of the number of players who have invested so far. Investment is assumed to be socially desirable, and players discount the future, so any delay is inefficient. There are many subgame-perfect equilibria, most of which involve delay. However, as the period length becomes vanishingly short, all subgame-perfect equilibrium outcomes converge to the Pareto-efficient outcome in which all players invest at the first date. The fact that investment is irreversible is crucial to this result. Once a player has invested, he or she is committed to that decision. At any stage of the game, the subgame is characterized by the number of players who remain uncommitted (i.e., have not yet invested). The number of uncommitted players is monotonically nonincreasing over time, so it is possible to analyze the game by backward induction on the family of subgames, indexed by the number of uncommitted players. In this paper I extend and clarify the role of irreversibility (commitment) in my earlier analysis (Gale 1995). Other studies have analyzed Pareto-efficient equilibrium in a dynamic coordination game. Aumann and Sorin (1989) studied the repeated play of two-person coordination games when the players have finite memory and are uncertain about the rationality of their opponents. They showed that the Pareto-dominant outcome will be played in the long run with high probability. Lagunoff and Matsui (1997) studied the repeated play of n-person coordination games under the assumption that the players can change their strategies only asynchronously. Here too the Pareto-dominant outcome is observed with high probability in the long run. Admati and Perry (1991) analyzed a game in which two players make alternate contributions to a public project. The project is undertaken when the contributions exceed the cost of the project. Delay is costly, but under certain circumstances there is a unique equilibrium outcome, and as the period length converges to 0, the equilibrium outcome converges to the Pareto-efficient outcome.

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All of these games have the property that an increase in one player’s action increases the other players’ payoffs. In other respects, they are different. The Admati–Perry game has an irreversibility structure (i.e., contributions cannot be retrieved) and the Lagunoff–Matsui game has a high degree of inertia (i.e., actions are changed infrequently). The Aumann– Sorin result depends on building a reputation. To study the role of irreversibility in general environments, I introduce a class of monotone games in Section 2. A monotone game is defined by a normal-form game  together with an irreversibility structure. In this paper I focus on games with positive spillovers. A game  has positive spillovers if each player’s payoff is nondecreasing in the actions of his opponents. The game  is played repeatedly over time, subject to the constraint that the players’ actions be monotonically nondecreasing over time. (The players’ action sets are subsets of a finite-dimensional Euclidean space with the usual vector partial ordering.) Section 3 characterizes the subgame-perfect equilibrium (SPE) outcomes of monotone games with positive spillovers. The structure of these equilibria is different from that of repeated games. In particular, there is no analogue of the “folk theorem.” Instead, the SPE outcomes are precisely the approachable outcomes, where an outcome is defined to be approachable if it is the limit of a nondecreasing sequence along which no player can ever guarantee himself more than the payoff he receives at the desired outcome. Section 4 provides sufficient conditions for an outcome to be approachable. An outcome is strongly minimal if it is impossible for any coalition to guarantee each of its members at least as high a payoff by choosing smaller strategies, when every member of the countercoalition chooses the minimal action. This condition has a family resemblence to the definition of the core. The resemblence to the core makes clear that “extreme” payoff distributions cannot be SPE, even though they may be efficient. In particular, a player can typically be sure of doing better than his security level in any SPE, so the set of SPE payoffs of a monotone game is smaller than the set of SPE payoffs of a repeated game. Characterizing the set of other SPE outcomes is more difficult, but we can at least say that the set of pure Nash equilibrium outcomes of the game  is always contained in the set of SPE outcomes. The Nash property immediately implies strong minimality in games with positive spillovers. Section 5 applies these results to voluntary contribution games, market games, and coordination games. The theory developed in this paper is applied to games with extensive structure and this structure is used to obtain the results. This is not very surprising once we realize that monotonicity by itself does not greatly restrict the set of equilibrium outcomes. On the other hand, the more economic structure we are prepared to build into these games, the more interesting

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results we may hope to derive. This preliminary study suggests that future research on monotone games will be very fruitful indeed. 2. MONOTONE GAMES This section formally introduces the concept of a monotone game, in which a normal-form game is played repeatedly, subject to the constraint that the players’ actions be nondecreasing. Let  = N X u be an n-player  game, where N = 1
n  n is the set of players, Xi = R+ is the action set for player i, X = i=1 Xi is the set of action profiles, ui  X → R is the payoff function for player i, and u ≡ u1   un  is the payoff function for the game. The function u  X → R n is assumed to be continuous unless otherwise noted. I focus on stage games in which each player’s payoff is monotonically nondecreasing in the actions of other players. Following Cooper and John (1988), I describe such games as having positive spillovers. For any vectors r r  ∈ R H , write

r ≥ r ⇐⇒ rh ≥ rh  ∀h = 1  H

r > r   ⇐⇒ r ≥ r   r = r   and

r  r ⇐⇒ rh > rh  ∀h = 1  H

Then the payoff function u exhibits positive spillovers if for any player i, for any action xi of player i, and for any action profiles x−i and x−i of player i’s opponents,

x−i ≥ x−i  ⇒ ui xi  x−i  ≥ ui xi  x−i  Similarly, u is said to exhibit strictly positive spillovers if for any player i, for any action xi of player i, and for any action profiles x−i and x−i of player i’s opponents,

x−i > x−i  ⇒ ui xi  x−i  > ui xi  x−i  The assumption of positive spillovers is maintained throughout this paper. The stage game  is played at a countable set of dates t = 1 2 A monotone game differs from a repeated game because the players’ strategies are restricted to be nondecreasing. This restriction transforms what would have been a repeated game into a stochastic game. Let xt ∈ X denote the action profile observed at date t. The initial profile of the game is denoted by x0 ∈ X. Because the players’ strategies are nondecreasing, the only action profiles that will be observed in the course of the game

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must satisfy xt ≥ x0 , so without essential loss of generality we assume that n ). x0 = 0 (recall that X = R+ The order of play is a sequence π = πt ∞ t=1 that specifies the set of players πt ⊆ N who are allowed to move at each date t. The order of player allows for the possibility that some players cannot move at date t. In the standard infinitely repeated game, for example, every player can choose a new action at date t, so πt = N. Alternatively, if can be assumed that one player moves at each date. For example, player 1 moves first, then player 2 moves, and so on until all the players have chosen a new action, after which the same pattern is repeated indefinitely. Formally, πt = t mod n ∀t Many other specifications of π are possible. We impose two mild restrictions on the order of play π: Every player can move infinitely often and some player can move at each date; that is, t  i ∈ πt  is infinite for every i and πt =  for every t. The monotone game based on the stage game  and the fixed order of play π is denoted by M. 2.1. Equilibrium A history of the monotone game M is a monotonically nondecreasing sequence of action profiles h = x1   xt  that is consistent with the order of play, i∈ / πs ⇒ xis = xis−1  ∀i ∀s ≤ t Let H denote the set of feasible histories. A strategy for player i in the monotone game M is a function fi  H → Xi , where fi h denotes player i’s choice of action after the history h. A strategy fi must satisfy monotonicity and be consistent with the order of play; that is, for every player i and history h = x1   xt , i i ∈ πt+1 ⇒ fi h ≥ xit and

ii

i∈ / πt+1 ⇒ fi h = xit


Let Fi denote the set of strategies for player i, and let F = ni=1 Fi denote the set of strategy profiles. A feasible outcome for the monotone game M is a monotonically nondecreasing sequence of action profiles x = xt ∞ t=1 that is consistent with the order of play; that is, for every player i and history t, i∈ / πt ⇒ xit+1 = xit

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For any strategy profile f = f1   fn  ∈ F, there is a unique feasible outcome x = xt ∞ t=1 defined recursively by putting x1 = f  and

xt+1 = f x1   xt  ∀t > 1

A player’s objective in the monotone game M is to maximize the long-run payoff associated with the strategy profile. Players do not discount the future. The payoff to player i from the strategy profile f is denoted by Ui f  and defined by Ui f  = lim inf ui xt 

t→∞ ∞ A feasible outcome x = xt t=1 is monotonically tion, so the sequence xt ∞ t=1 is convergent if it

nondecreasing by definiis bounded above. Continuity of ui then implies that the sequence of period payoffs ui xt  is convergent and that Ui f  = lim ui xt  t→∞

Under these conditions, any reasonable definition of the long-run payoff will be equivalent to the one given earlier. The state of the game at any date t is defined by the ordered pair x t, where x is the action profile chosen in the previous period. The subgame beginning at date t is uniquely defined by the state x t. Let Mt  x denote the monotone game beginning at date t with initial profile x. The subgame Mt  x depends on the date, because the order of play πt  πt+1   depends on the initial date t. For any strategy f and any history h, let f h denote the continuation strategy defined by putting f hh  = f h h  for every continuation game history h . A monotone game is a game of complete information and perfect recall, so the appropriate solution is the subgame-perfect equilibrium (SPE). A SPE is a strategy profile f ∈ F such that for any feasible history h = x1   xt  ∈ H, the continuation strategies f h constitute a Nash equilibrium of Mt  xt ; that is, for every player i, Ui f h ≥ Ui fˆi h f−i h  ∀fˆi ∈ Fi 3. APPROACHABILITY In a game with positive spillovers, the worst threat the opponents of player i can make is to maintain their current actions. Player i’s security level in the subgame Mt  x is denoted by u∗i x and defined by u∗i x = supui xi  x−i xi ≥ xi 

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Note that the security level is a function of the current profile x but is independent of the date t. Player i can guarantee himself at least u∗i x in the subgame Mt  x, so u∗i x plays the role of the status quo. The following assumption plays an important role in the analysis. Boundedness Assumption. For each i and x ∈ X, there exists a number k such that for any xˆ ≥ x, ˆ ≤ ui x if xˆ i  > k and uj x ˆ ≥ uj x ∀j = i ui x The boundedness assumption says that for any profile x reached in the play of the game, if player i increases his action far enough, then he is sure to be worse off, no matter what individually rational actions the opponents choose. This property will be used to show that for any player i and any action profile x, there is a SPE of the continuation game Mt  x in which player i just achieves his security level u∗i x. The boundedness assumption ensures that u∗i is well defined; that is, the maximum exists. I note without proof that u∗i satisfies the same properties as ui . Lemma 1. u∗i  X → R is a well defined, continuous function. It is nonincreasing in xi and nondecreasing in xj for all j = i. If u exhibits strictly positive spillovers, then u∗i is increasing in xj for all j = i. The boundedness assumption also immediately implies the following useful fact. Lemma 2. For any profile x, the set of profiles satisfying x ≥ x and ux  ≥ u∗ x is bounded. A satiation point is an action profile x ∈ X such that u∗ x = ux. Satiation points play a role somewhat like Nash equilibria in the analysis of monotone games. Once a satiation point has been reached, no player can make himself better off by deviating unilaterally. Hence it is a Nash equilibrium to remain at a satiation point forever. The key concept in the characterization of SPE outcomes is approachability. For any subgame Mt  x and any profile x , the profile x is approachable in Mt  x if there exists an outcome x = xs ∞ s=t that is feasible in Mt  x and satisfies the conditions i and

ii

u∗i xis−1  x−is  ≤ ui x  for any date s ≥ t and player i lim xs = x

s→∞

Condition (ii) says that x is the limit of the feasible outcome x. Condition (i) says that no player can guarantee himself a payoff higher than he

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would receive at the limit point x . To see this, suppose that player i deviates from the prescribed play at date s and chooses an action xˆ i ≥ xis−1 . At the beginning of date s, the new action profile will be xˆ i  x−is , so player i can guarantee himself at least ui xˆ i  x−is . Condition (i) ensures that ui xˆ i  x−is  ≤ u∗i xis−1  x−is  ≤ ui x , so player i is no better off than he would have been if he had followed the play prescribed by outcome x. A profile x is strongly approachable in Mt  x if there exists an outcome x = xs ∞ s=t that is feasible in Mt  x and satisfies conditions (i) and (ii) such that at most one player moves at each date s. In other words, x is approachable in Mt  x, and one player at a time moves along the path leading to x . A strongly approachable profile x is obviously approachable. If x is strongly approachable in Mt  x, then condition (i) is equivalent to the following condition: i

u∗i xt−1  ≤ ui x  ∀t ∀i

It is clear that condition (i) implies condition i , but the converse is not generally true. Suppose that x is an outcome in which at most one player moves at any date and that condition i is satisfied. If player i moves at date t, then x−it = x−it−1 , and condition i implies condition (i). If player i does not move at date t, then let s be the first date after t when player i moves. Then xit−1 = xis−1 and u∗i xit−1  x−it  ≤ u∗i xs−1  ≤ ui x  so condition (i) is satisfied. If x is strongly approachable in Mt  x, then x is strongly approachable in Ms  x for any s > t. The only difference between the games Mt  x and Ms  x is the order of play, so that an outcome x that is consistent with the order of play in Mt  x may not be consistent with the order of play in Ms  x. But x can always be transformed into an essentially equivalent outcome that is feasible in Ms  x simply by changing the dates at which the players change their actions. To see this, let x = xt+k ∞ k=0 be a feasible outcome of Mt  x and define a feasible outcome of Ms  x as follows. Let τ  N → N be a nondecreasing function such that player i can move at s + τk if i moves moves at date t + k. Because every player gets to move infinitely often, this construction is always feasible. Then define the outcome z =xs+j ∞ j=0 recursively by putting xs+j = xt+k  ∀k ∀τk ≤ j < τk + 1 By construction, z involves the same moves as x in the same order, so x and z are equivalent except for the timing of moves, which is varied to accommodate the different orders of play. Thus the set of strongly approachable

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profiles is the same for Mt  x and Ms  x, and I am justified in calling x approachable from x without specifying the date at which the subgame starts. By a similar argument, I can show that the subgames Mt  x and Ms  x have the same set of approachable profiles, if I assume that each set πt occurs infinitely often. I do not need to make this assumption here, because in the analysis of subgames, it is possible to restrict attention to the strongly approachable profiles. I note in passing the following useful property of approachable profiles. Lemma 3.

Any approachable profile is a satiation point.

Proof. If x is approachable in Mt  x, then there is an outcome x that is feasible for Mt  x and satisfies conditions (i) and (ii). The continuity of u∗ implies that for every i, u∗i x  = lim u∗i xt  ≤ lim u∗i xit−1  x−it  ≤ ui x  t→∞

∗



t→∞



Because u x  ≥ ux  by definition, it follows that u∗ x  = ux . An equilibrium outcome of M is the unique outcome generated by a SPE of M. A limit point of the game is the asymptotic limit x∞ = limt→∞ xt of an equilibrium outcome x = xt ∞ t=1 . The limit is well defined, because Lemma 2 ensures that x is bounded. The main result of this section is to show that the approachable profiles are precisely the limit points of a monotone game with positive spillovers. Theorem 1. Under the maintained assumptions, for any profile x and any date t, a profile x is a limit point of the game Mt  x if and only if x is approachable in Mt  x. Proof.

The proof is given in Section 6.

In one direction, the proof is easy. If x is a limit point of Mt  x, then it is approachable in Mt  x because the equilibrium path satisfies the conditions for approachability. To prove the converse, suppose that x is approachable in Mt  x and let x be the feasible outcome of Mt  x that approaches x . Take x as the equilibrium path. Next, to show that x is a limit point, I have to construct the entire SPE strategy that supports x as an equilibrium path. The strategy of the proof is to show that for any action profile xˆ ≥ x and any player i there exists a profile xˆ  , strongly approachable from x, ˆ such that ui xˆ   = u∗i x. ˆ Suppose that the game has reached the action profile xˆ when player i deviates. Then there exists a feasible outcome xˆ leading to the profile xˆ  satisfying ui xˆ   = u∗i x, ˆ and I take xˆ to be the equilibrium path in the continuation game following the deviation. Thus the most that player i can

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gain from a deviation is u∗i x. ˆ By construction, this is no greater than the payoff that player i expects along the equilibrium path, so he or she has no incentive to deviate. 4. SUFFICIENT CONDITIONS Although approachability provides a characterization of the asymptotic outcomes of the monotone game, it is not obvious whether or not a profile is approachable. In this section I look at sufficient conditions for approachability that are easier to check. 4.1. Minimal Profiles A profile x is Pareto-efficient if there does not exist a profile xˆ such that ux ˆ > ux. Pareto-efficient profiles are interesting because of their welfare properties. It turns out that they are natural candidates for limit points as well. A profile x is minimal if there does not exist a profile 0 < xˆ < x such that u∗ x ˆ ≥ u∗ x. The following proposition explores the relationship between efficiency and minimality. Proposition 1. (i) Every Pareto-efficient profile x ∈ X is a satiation point; that is, u∗ x = ux. (ii) For any Pareto-efficient profile x ∈ X, there exists a minimal (Pareto-efficient) profile x ∈ X such that u∗ x  = u∗ x. Proof. (i) Let x ∈ X be a Pareto-efficient profile. If x is not a satiation point, then for some i and some xi > xi , ui xi  x−i  > ui x. Because u has positive spillovers, it follows that uxi  x−i  > ux, contradicting Pareto efficiency. Thus every Pareto-efficient profile is a satiation point. (ii) Let x ∈ X be a Pareto-efficient profile. Let A = xˆ ∈ Xxˆ ≤ x u∗ x ≤ u∗ x ˆ Because u∗ is continous, A is compact. There must exist a profile x ∈ A such that xˆ < x implies that xˆ ∈ / A. For example, choose x to minimize   x  subject to x ∈ A. By construction, ux  ≥ ux, and because x is Pareto-efficient, that ux  = ux must follow. Clearly, x is also Paretoefficient and hence a satiation point, so u∗ x  = u∗ x. This completes the proof that x is the required minimal satiation point. Proposition 1 shows that under mild regularity conditions (the existence of Pareto-efficient profiles), minimal satiation points will exist. The relationship between minimal profiles and approachable profiles is explored in the next two results.

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A profile x is positive if xi > 0 for all i. A profile x is said to be locally approachable if for any ε > 0, there exists a profile xˆ < x such that x − x ˆ < ε and ux ˆ < ux. Lemma 4. Let x be a positive satiation point. If x is minimal, then x is locally approachable. Proof.

See Section 6.

My strategy is to begin at a profile xˆ near the target profile x and, using backward induction, construct a finite sequence that “approaches” xˆ from 0. I then generate a sequence of these sequences by taking xˆ increasingly closer to x. In the limit, I get a feasible outcome approaching x. The next property ensures that I can construct the initial finite sequence without getting “stuck” along the way. A positive satiation point x is said to be strongly minimal if there does not exist a profile 0 < xˆ < x such that for every i, either xˆ i = 0 or u∗i x ˆ ≥ ui x. Strong minimality obviously implies minimality. Lemma 5. Let x be a positive satiation point. If x is strongly minimal, then x is strongly approachable from 0. Proof.

See Section 6.

Although the concept of strongly minimal profiles is just as abstract as the concept of approachability, it is closely related to familiar solution concepts in special cases. Some of these are explored in the remainder of this section. 4.2. Minimal Profiles and the Core Strongly minimal profiles have a core-like property. Let x be a fixed but arbitrary profile. Imagine a coalition S that wants to reduce its action profile without reducing its members’ payoffs. Suppose that the coalition members choose actions xˆ i ≤ xi for each i ∈ S. The worst threat that the countercoalition N − S can make is to choose actions xˆ i = 0 for all i ∈ N − S. Then the coalition succeeds in reducing its profile without reducing its members’ payoffs if and only if ui x ˆ ≥ ui x for every i ∈ S. This is precisely what the definition of a minimal profile rules out. Proposition 2. Let x be a minimal positive satiation point. Then there does not exist a coalition S and an allocation xˆ < x such that ui x ˆ ≥ ui x for every i ∈ S and xˆ i = 0 for every i ∈ N − S. Proof. The proof is by contradiction. Suppose, contrary to what I want to prove, that x is a minimal, positive satiation point and xˆ < x is a profile ˆ ≥ ui x for every i ∈ S and xˆ i = 0 for every i ∈ N − S. such that ui x Then u∗i x ˆ ≥ ui x ˆ ≥ ui x = u∗i x for every i ∈ S. Because xˆ i = 0 for

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all i ∈ N − S, u∗i x ˆ ≥ u∗i x for every i such that x∗i > 0, contradicting the assumption that x is minimal. The notion of a strongly minimal profile differs from the usual definition of the core in two ways. To see this, examine the definition of the core more closely. An action profile x belongs to the core of  if there does not exist a coalition S =  and a profile xˆ such that ui x ˆ > ui x for any i ∈ S and xˆ i = 0 for any i ∈ N − S. The first difference is that every member of the coalition is strictly better off, not just weakly better off. The second difference is that any profile xˆ can be considered, not just a profile xˆ < x. The first difference means that a core profile is not necessarily strongly minimal; the second difference means that a strongly minimal profile is not necessarily a core profile. To bring the two concepts closer together, consider a profile x to be a strict core profile of  if there does not exist a coalition S =  and a profile xˆ = x such that ui x ˆ ≥ ui x for any i ∈ S and xˆ i = 0 for any i ∈ N − S. All I have done is to replace the strict inequalities in the definition of the core with weak inequalities. I require that xˆ = x to avoid trivialities. A strict core profile obviously belongs to the core, but the converse is not necessarily true. More interesting is the fact that a strict core profile is strongly minimal. Proposition 3. satiation point.

If x is a strict core profile of , then x is a strongly minimal

Proof. A strict core profile is Pareto-efficient and hence a satiation point. If x is a satiation point, but not strongly minimal, then there exists a profile xˆ < x such that u∗i x ˆ ≥ ui x for every i such that xˆ i > 0. Let S = i ∈ N  xˆ i > 0. Construct an approachable profile x ≥ xˆ using the method of Lemma 6, but allowing only i ∈ S to change strategies. Then ui x  ≥ u∗i x ˆ ≥ ui x for every i ∈ S and xi = 0 for every i ∈ N − S, contradicting the assumption that x is a strict core profile. Hence any positive strict core profile is approachable and thus is a limit point of the monotone game M. 4.3. Minimal Profiles in Two-Person Stage Games The relationship between the core and strongly minimal profiles gives a clearer idea of which action profiles in  may be approachable from 0 and hence limit points of SPE in the monotone game M based on . In special cases, the core reduces to the set of Pareto-efficient individually rational profiles, which are easily identified. A profile x is strictly individually rational if ux  u∗ 0. I now show that in a two-person game, a minimal profile that is strictly individual rational is strongly minimal.

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Proposition 4. Let x be a positive, strictly individually rational satiation point of the two-person game . Then x is minimal if and only if it is strongly minimal. Proof. Let x be a positive, strictly individually rational satiation point. If x is not strongly minimal, then there exists a profile 0 < xˆ < x such that u∗i x ˆ ≥ ui x if xˆ i > 0. Because x is minimal, there must be at least ˆ ≥ uj x, because one i such that xˆ i = 0. Then for j = i, u∗j 0 ≥ u∗j x u∗j is nondecreasing in xˆ j . This contradicts the assumption that x is strictly individually rational. So in the case of two-person games, we have a clear indication of which profiles are approachable. A sufficient condition is that they be positive, minimal and strictly individually rational satiation points. Corollary 1. Let x be a positive, Pareto-efficient, and strictly individually rational profile of a two-person game . Then x is approachable and hence is a limit point of M. Conversely, because (weak) individual rationality is a necessary condition for x to be a limit point of M, these conditions are close to being necessary. 4.4. Minimal Profiles and Nash Equilibria Pareto-efficient profiles are not the only approachable profiles. An important class of profiles, typically not Pareto-efficient, is the set of strict Nash equilibrium points of . An action profile x is a strict Nash equilibrium of  if for every player i and every action xˆ i = xi , ui xˆ i  x−i  < ui x. Proposition 5. Suppose that u has strictly positive spillovers. Let x ∈ X be a positive strict Nash equilibrium of . Then x is a strongly minimal satiation point, and hence x is approachable. Proof. Let x be a positive strict Nash equilibrium. A Nash equilibrium is a satiation point by definition. To see that x is strongly minimal, suppose instead that 0 < xˆ < x and u∗i x ˆ ≥ ui x if xˆ i > 0. There must be at least one i such that xˆ i < xi and some j such that xˆ j > 0. Two cases must be considered. If, on the one hand, j = i, then, because u has positive spillovers and xˆ −j ≤ x−j , u∗j xˆ j  x−j  ≥ u∗j x ˆ ≥ uj x, contradicting the assumption that x is a strict Nash equilibrium. If, on the other hand, j = i, then, because u has strictly positive spillovers and xˆ −j < x−j , u∗j xˆ j  x−j  > u∗j x ˆ ≥ uj x, contradicting the assumption that x is a Nash equilibrium of . So x is strongly minimal, and approachability follows from Lemma 5.

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douglas gale 5. APPLICATIONS 5.1. Voluntary Contribution Games

An example of a game with strictly positive spillovers is the voluntary contribution game with public goods. Each of the players i = 1  n contributes an amount xi ≥ 0 of the numeraire good for the provision of  a public good. The total amount of the public good produced is y = G i xi , where G· is a continuous and increasing production function. Each player i has a utility function Wi y mi , where y is the consumption of the public good and mi = m ¯ i − xi is player i’s consumption of the numeraire. Wi · is assumed to be continuous and increasing in both arguments. Then the payoff function can be defined by putting      ¯ i − xi  xi  m ui x ≡ Wi G i

for every i and for any x = x1   xn . It is clear that the payoff function ui · defined in this way is continuous and exhibits strictly positive spillovers. A Lindahl equilibrium comprises an input level z ≥ 0 and a vector α = α1   αn  of nonnegative weights, summing to 1, such that z maximizes ¯ i − αi z for every i. If z α is a Lindahl equilibrium and x Wi Gz m is an action profile defined by xi = αi z, for every i, then x belongs to the core of the game . Under standard conditions, a Lindahl equilibrium exists, so the core is nonempty. To apply the results from the preceding section, however, I need to prove the existence of a strict core profile. This is implied by the existence of a strict Lindahl equilibrium; that is, a Lindahl equilibrium in which z is the unique maximizer of Wi Gz m ¯ i − αi z for every i. Uniqueness of the maximizer is guaranteed by strict concavity of Wi and G, for example. Given the existence of a strict Lindahl equilibrium, there exists at least one strict core profile, so we can conclude that some Pareto-efficient allocations are limit points of M. However, the (strict) Nash equilibrium points of the voluntary contribution game  are also limit points of M, so the monotone game does not “solve” the free-rider problem. Admati and Perry (1991) studied a similar game in which two players make successive contributions to the production of a public good. The public good is assumed to be socially desirable and to have a fixed cost K. The players make contributions alternately, beginning with player 1 in period 1. The contributions are sunk (nonrecoverable) and players incur the cost of each contribution at the date it is made. The public good is produced, and the players enjoy the benefit of it as soon as the contributions total K. Players are impatient and discount the future using a common

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discount factor 0 < δ < 1. Admati and Perry showed that under certain conditions there is an essentially unique equilibrium. The equilibrium may be inefficient; in the case of linear costs, there is a loss of welfare whenever the project is not completed at the earliest possible date. But as long as the project is completed, it can be shown that as the period length converges to 0 (the discount factor converges to 1), the unique subgame perfect equilibrium outcome converges to the Pareto-dominant outcome. In other words, when the period length is short, there is virtually no equilibrium delay. The assumption that the project be completed is crucial. For some parameter values, the project is not completed in any SPE, even though doing so is efficient. For example, with linear costs, the project is completed if and only if each player’s individual valuation of the project is higher than the total cost. This is another source of inefficiency in the Admati–Perry model. The indivisibility of the project is crucial here. As I have shown, if the public good is perfectly divisible, then it may be possible to support Pareto-efficient equilibria under weaker conditions. Marx and Matthews (2000) extended the model of Admati and Perry (1991) and provided a thorough investigation of a number of interesting points. In particular, they suggested that simultaneous moves may improve efficiency by admitting equilibria in which the public project can be provided even though neither of the players would be willing to provide it individually. In a simultaneous move game, each player is in effect willing to provide part of the cost based on the assumption (validated in equilibrium) that the other player is providing the rest. This raises the question of the robustness of equilibria; if players are unable to coordinate their moves perfectly (i.e., move simultaneously), then the efficient equilibrium may disappear, leaving only the equilibrium corresponding to the alternating move game.

5.2. Market Games Another example of a game with positive spillovers is the gift-giving game, in which players give bundles of goods to each other. In a monotone game, gifts are irrevocable, and the process of gift giving eventually comes to an end. Nonetheless, it is still possible to sustain “altruism.” It turns out that every individually rational payoff is a limit point of the monotone game based on the stage game. This result is analogous to the “folk theorem” of repeated games. Kurz (1978) showed that “altruism” can be supported as an equilibrium in an infinitely repeated gift-giving game. The difference is that in a repeated game, the opportunities for gift giving do not diminish over time.

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The boundedness assumption is not satisfied in this case. Thus the limit points must be characterized by a different method, based on the fact that gifts reduce the donor’s utility, ceteris paribus, so every profile is a satiation point. Consider an exchange economy with n agents i = 1  n. Each agent i H H is characterized by a consumption set R+ , an endowment ei ∈ R+ , and a H continuous, increasing utility function Ui  R+ → R. The process of exchange is modeled as a gift-giving game in which agents give nonnegative bundles of commodities to other players. Each agent i H chooses a nonnegative bundle of commodities xi j ∈ R+ to give to agent j. The sum of the commodity bundles given away cannot exceed the agent’s endowment,  xi j ≤ ei  ∀i j=i

This ensures that the action xi is feasible regardless of the strategies chosen by other agents. Agent i’s final consumption bundle is   ci = ei + xj i − xi j! j=i

j=i

that is, his initial endowment plus his gifts from agents j = i minus his gifts to agents j = i. The gift-giving game for the exchange economy is denoted by  and defined as follows. The set of players is N = 1  n; the set of strategy profiles X = X1 × · · · × Xn is defined by putting N−i ≡ N − i and     H Xi = xi  N−i → R+  xi j ≤ ei  ∀i! j=i

and the payoff function u = u1   un  is defined by putting     ui x = Ui ei + xj i − xi j  ∀x ∈ X ∀i j=i

j=i

 is a game with strictly positive spillovers if Ui is an increasing function for every i. The monotone game M is defined in the usual way for any order of play π = πt . If xit − xit−1 is the vector of gifts that i gives at date t, then the monotonicity condition xit − xit−1 ≥ 0 simply says that agent i can make additional gifts at date t, but cannot take back the gifts given at previous dates. Monotonicity is a natural restriction in a society in which gifts are irrevocable. Trade is voluntary in the sense that each agent i can choose the action xi = 0; that is, he does not give any commodities away. Thus, even if the

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agents j = i give nothing to agent i, agent i can guarantee that his final consumption is at least ei and his payoff is at least Ui ei . An action profile x is individually rational if ui x ≥ Ui ei  for every i. A payoff vector u¯ 1   u¯ n  is individually rational if there exists an individually rational profile x such that u¯ i = ui x for every i. Note that more than one individually rational profile x may correspond to an individually rational payoff vector u. ¯ Theorem 2. Suppose that u¯ is an individually rational payoff vector . Then there exists a (SPE) limit point x such that u¯ = ux. Proof. For any individually rational payoff vector u, ¯ let z = z1   zn  H be a vector of net trades such that e + z ∈ R and Ui ei + zi  = u¯ i , for i i +  every player i, and i zi = 0. Let x ∈ X be an action profile such that   xj i − xi j ∀ i zi = j=i

and

 j=i

j=i

xih j = − minzih  0 ∀i h

The first condition says that x achieves the net trades z. The second condition requires that there be no indirect trades or wash trades. In particular, if agent i has a net demand for commodity h, then he does not give away any of that good and, if he has an excess supply, he will give it only to those agents who have an excess demand. An action profile satisfying these condi i = xˆ i ∈ Xi xˆ i ≤ xi . tions exists if i zi = 0. For this fixed profile x, let X Construct a feasible outcome x = xt  as follows. For each date t, select a unique player it ∈ πt so that every player i is chosen infinitely often. We use the selection it as the order of play in the construction that follows. For each date t, choose xt so that xjt = xjt−1 for j = it and let xit ≥ xit−1 i and be maximal subject to the constraints xit ∈ X uj xit  x−it  ≤ uj x ∀j = it Because the sequence x is nondecreasing and bounded, it must converge to a limit x∞ , say, and ux∞  ≤ ux by construction. If ui x∞  < ui x for some i, then ui xt  < ui x for all t sufficiently large. Then there must be some agent j who could give more to i without violating the constraint ui xt  ≤ ui x. But this contradicts the way in which the feasible outcome x was constructed. This proves that xt → x∞ and ux∞  = u. ¯ To show that x is a limit point, it is sufficient to define SPE strategies f that will support this outcome. The construction is simple. Have the agents follow the equilibrium path until there is a deviation. Then for

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any history h = xˆ 1   xˆ t  that involves a deviation, have every agent i put fi h = xˆ it . Because the profile xˆ t is a satiation point, this is clearly a Nash equilibrium for any history h, and this proves that the strategy profile f is a SPE of M. The boundedness assumption is not satisfied in this game because of the possibility of wash trades. If two agents give each other equal quantities of some commodity, then their payoffs do not change. Wash trades are explicitly ruled out in the construction of the SPE strategy. 5.3. Coordination Games So far, I have studied games in which various profiles, efficient and otherwise, can be supported as limit points of SPE. In some games, the set of limit points is more restricted, at least to the extent that the outcome is Pareto-efficient under certain conditions. The following example is motivated by my earlier work (Gale 1995). This is a fairly easy result, but it makes the point that without the possibility of self-punishment, the set of limit points may be sharply restricted.  Theorem 3. Suppose that Xi = xi ∈ R+  xi ≤ x∗i  for some fixed vector ∗ If x strictly Pareto-dominates any other satiation point, then x∗ is the unique limit point of M.

x∗i .

Proof. I first show that x∗ is a limit point. For any history h, put fi h = if i ∈ πt . It is clear that the strategy profile f = f1   fn  is a SPE. (Note that positive spillovers are not required for this result.) To show that x∗ is the unique limit point, I use induction. Obviously, it is true for n = 1, so assume that it also is true for n = k. Consider the case n = k + 1 and suppose that at some point in the game, some player i chooses xi = x∗i . By our induction hypothesis, the remainder choose strategies leading to x∗−i . Thus each player can guarantee himself at least ui x∗  in any SPE. But any limit point must be a satiation point (assuming positive spillovers!), and any satiation point other than x∗ is strictly Paretodominated by x∗ . Thus x∗ is the only limit point. x∗i

In this example, the boundedness assumption is not satisfied. In particular, players cannot punish themselves by choosing large actions. If players are allowed to choose actions larger than x∗i , then the result in the theorem is no longer true (see the end of this section for a counterexample). In my earlier work (Gale 1995), investment is irreversible (i.e., the game is monotone), and the Pareto-dominant profile requires every player to invest the maximum (i.e., x∗ is Pareto-dominant). Lagunoff and Matsui (1997) studied repeated coordination games under the assumption that players can change their strategies only asynchronously.

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This assumption of inertia implies that whenever an individual player has the opportunity to change his action, he can assume that all of the other players will hold their strategies constant for some time. That player also anticipates the effect that his choice will have on the next player to have an opportunity to change his action. In this context, SPE implies that eventually they must all coordinate on the Pareto-dominant equilibrium. To illustrate the idea behind the Lagunoff–Matsui model, consider a game with common preferences. Formally, a stage game  has common preferences if ui = u for i = 1  n. This is a slight generalization of the familiar coordination game, in which players have common preferences and also prefer the diagonal outcomes to all others. Consider the monotone game M β in which the order of play π allows exactly one player to move at each date t and players discount the future using the common discount factor 0 < β < 1. If x is an outcome of the game, then player i’s payoff will be ∞  t=1

βt−1 ui xt 

Because exactly one player moves at each date and preferences are identical, a SPE of this game is essentially the same as the solution to a dynamic problem. So it is hardly surprising that an efficient outcome results. Theorem 4 (Lagunoff–Matsui). Let  = N X u be a stage game with common preferences and suppose that u has a maximum at x∗ . Then for any initial state x ≤ x∗ and any date t, lim vt x! β = ux∗ 

β→1

where vt x! β is the infimum of the payoffs from the SPE of the monotone game Mt  x β. Although the proof respects the monotonicity conditions of the game, these conditions are not used at any point to obtain the result. So the theorem holds for any dynamic game of this form, even without the constraint that the states must be nondecreasing over time. This shows that it is only sequential decision making, together with a positive but vanishing rate of discounting, that ensures optimality in the limit. An example shows that the result fails if there is no discounting. Suppose that there are two players, i = 1 2, with action sets Xi = 0 2 and payoff functions ux = −maxx1  x2  − 12 Clearly, the maximum value of ux is 0. Yet without discounting the profile x = 2 2 can be supported as a limit point of a SPE. To see this, consider

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the strategies that require each player to put xi = 2 whenever he has the chance to move. Each player, anticipating that his opponent will choose xi = 2 and ensure a payoff of −1, will find it optimal to follow the strategy described. Discounting makes a difference to this analysis, because even if the player anticipates a poor outcome in the future, he would still like to postpone it as long as possible. If both players do this, then they end up cooperating indefinitely. An important topic for future research is to characterize the limit points of monotone games with positive discounting as the discount factor converges to 1. 6. PROOFS 6.1. Proof of Theorem 1 Before attempting the proof of the theorem, I prove the following simpler result. Lemma 6. Under the maintained assumptions, for any initial profile x there exists a profile x ≥ x that is strongly approachable from x. Proof. The proof is constructive. Let I = It be a selection from the order of play π = πt such that each player gets to move an infinite number of times. That is, It ∈ πt for every t and I −1 i = ∞ for every i. At each date t, let player It control the play and have him or her choose a myopic best response to the current profile, subject to the usual condition that profiles be nondecreasing. Formally, x = xt ∞ t=1 is a feasible path satisfying the condition that, for every date t, xt ∈ arg maxuIt x  x ∈ SIt xt−1  where Si xt−1  = x ≥ xt−1 xj = xjt−1  ∀j = i is the set of profiles that can be reached by a unilateral move by player i. By construction, x is a feasible outcome (it is nondecreasing and consistent with the order of play). It is bounded by Lemma 2. Because x is bounded and nondecreasing, it has a limit point x = limt→∞ xt . Because u is continuous, it follows that limt→∞ uxt  = ux . The sequence of security levels u∗ xt ∞ t=1 is also nondecreasing. To see this, consider two cases. First, for any date t, if i = It, then xit = xit−1 and x−it ≥ x−it−1 . Then the result follows from the fact that u∗i is monotonically nondecreasing in x−it . Second, for any date t, if i = It,

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then xt is chosen so that ui xt  = u∗i xt−1 . Then the result follows from the fact that u∗i xt  ≥ ui xt . I have shown that the sequence u∗ xt  is nondecreasing and bounded and hence convergent. Because each player i gets to move infinitely often, ui xt  = u∗i xt−1  infinitely often for any given player i. It follows that limt→∞ u∗ xt  = limt→∞ uxt  = ux , and the fact that u∗ xt  is nondecreasing implies that u∗ xt  ≤ ux  for all t. This completes the proof that x is strongly approachable from x. The next lemma shows that certain profiles used as “punishments” are also strongly approachable. Lemma 7. Under the maintained assumptions, for any profile x and any player i there exists a profile x ≥ x that is strongly approachable from x and satisfies ui x  = u∗i x. Proof. The strategy of the proof is as follows. Let x and i be a fixed but arbitrary profile and player. Suppose that player i increases his or her action from xi to xˆ i . Then Lemma 6 shows that there exists a profile xˆ that is strongly approachable from xˆ i  x−i . If xˆ i = xi , then u∗i x ˆ ≥ u∗i x. If xˆ i is chosen large enough, then boundedness implies that u∗i x ˆ < u∗i x for any xˆ that is approachable from xˆ i  x−i . So “continuity” implies that ˆ = u∗i x. The rest of the proof for some appropriate choice of xˆ i , u∗i x involves making this argument precise. For any initial profile x and any xˆ = xˆ i  x−i  ∈ Si x, let ζx ˆ denote the set of profiles that are strongly approachable from x. ˆ I first establish a number of properties of the correspondence ζ. 1. For every profile xˆ the set ζx ˆ is nonempty. This follows directly from Lemma 6. 2. The correspondence ζ  X → X is upper hemicontinuous. Let xk  be a sequence of profiles converging to x0 and let y k  be a sequence of profiles such that y k ∈ ζxk  for every k. From the definition of strong approachability, for each k there is a feasible outcome xk = xrk ∞ r=1 such that x1k = xk  lim xrk = y k 

r→∞

and u∗ xrk  ≤ uy k  ∀r Lemma 2 implies that the sequence y k  is bounded and so has a convergent subsequence. Without loss of generality this can be taken to

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be the original sequence. For a fixed value of r, the sequence xrk  is also bounded, because xk ≤ xrk ≤ y k and xk  and y k  are convergent sequences and hence bounded. Thus, by the usual diagonalization procedure, a subsequence of r k can be found such that xrk converges to z r along this subsequence. The sequence z =z r  inherits the properties of xrk —nondecreasing, consistent, and satisfying u∗ z r  ≤ uy 0  for every r—so z is a feasible outcome. Also, z 1 = x0 and limr→∞ z r = y 0 . This shows that y 0 is strongly approachable from x0 as claimed. ˆ = supui x  x ∈ ζx ˆ is upper semicontin3. The function φi x uous. This follows from the upper hemicontinuity of ζ and the maximum theorem. ˆ ≥ u∗i x is nonempty. Any 4. The set Ai x = xˆ ∈ Si x  φx profile xˆ that is strongly approachable from x satisfies ui x ˆ ≥ u∗i x, so ∗ φi x ≥ ui x by construction. Furthermore, Ai x is bounded by Lemma 2 and closed because φi is upper semicontinuous. Thus Ai x is a compact set, and so it contains a maximal element x¯ with respect to the partial ordering ≥. ¯ = u∗i x. To see this, suppose to the 5. The profile x¯ satisfies φi x ∗ ¯ > ui x. Then by upper semi-continuity, there exists a contrary that φi x profile xˆ > x¯ such that φi x ˆ > u∗i x, contradicting the assumption that x¯ is maximal. I have now shown that there exists a profile x that is strongly approachable from x¯ and such that ui x  = u∗i x. The final step is to show that x is strongly approachable from x. Construct a feasible outcome x =xt ∞ t=1 as follows. Let k be the first date at which i ∈ πk and then put  if t = 1  k − 1 x if t = k xt = x¯  zt−k if t = k + 1 k + 2 ,  where zt ∞ t=1 is the feasible path by which x is strongly approachable from x. ¯ The path so constructed has all of the required properties, so x is strongly approachable from x.

The profile x whose existence is guaranteed by Lemma 7, can be used to hold player i’s payoff down to his security level in the subgame Mt  x. Now I am ready to prove the theorem in two steps. The next lemma shows that a limit point is approachable. Lemma 8. Suppose that x is a limit point of the game Mt  x. Then x is approachable in Mt  x. Proof. If x is a limit point of Mt  x, then the equilibrium path is a  feasible outcome x =xs ∞ s=t starting at x and converging to x . To show

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that x is approachable, I only need to show that u∗i xis−1  x−is  ≤ ui x  for every s and i. Suppose, contrary to what I want to prove, that u∗i xis−1  x−is  > ui x  for some date s ≥ t and some player i. Let s be the first date after s such that i ∈ πs ; that is, s = mins ≥ s  i ∈ πs  Then u∗i xis −1  x−is  > ui x  because x is nondecreasing, xis = xis −1 , and u∗i is nondecreasing in the actions of players j = i. At date s , player i can choose an action xi such that ui xi  x−is  > ui x  = Ui f  = lim inf t→∞ ui xt . In this way, player i guarantees himself a payoff greater than ui x , because whatever actions the other players choose at date s and after, his payoff cannot be lower than ui xi  x−is . This contradicts the equilibrium conditions and proves the result. To prove the converse, I assume that x is an approachable profile in Mt  x and construct a SPE that has x as a limit point. Lemma 9. Suppose that x is approachable in Mt  x. Then x is a limit point of the game Mt  x. Proof. Because the game Mt  x is fixed, there is no essential loss of generality in taking t = 1 and x = 0; that is, I prove the result for Mt  x = M. Let x = xt ∞ t=1 be a fixed but arbitrary feasible outcome and let H 1 x denote the set of finite histories that involve deviations from x in the last period; that is, 

H 1 x = h ∈ Hh = x1   xt−1  xˆ t  and xˆ t = xt Next I construct a set
1 x of feasible outcomes beginning with histories in H 1 x as follows. Let h = x1   xt−1  xˆ t  ∈ H 1 x be a fixed but arbitrary history. I consider two cases. In the first case, xˆ it = xit for some i and xˆ jt = xjt for all j = i. Then the single deviator, player i, must be punished. By Lemma 7 there exists a profile x that is strongly approachable from xˆ t and such that ui x  = u∗i xˆ t . In this case let x be the target profile in the subgame beginning with the finite history h. In the second case, xˆ it = xit for more than one value of i. Then x can be an arbitrary profile that is strongly approachable from xˆ t . In either case, let xˆ s ∞ s=t+1 be the feasible outcome strongly approaching x from xˆ t and let xˆ h = x1   xt−1  xˆ t  xˆ t+1  .
1 x is just the set of all infinite histories derived in this way; that is,
1 x = ˆxh  h ∈ H 1 x. Then xˆ h is the unique feasible outcome in
1 x that contains the initial segment h. Now I am ready to describe the equilibrium play of the game for every possible history h ∈ H. Because x is approachable, there exists a feasible outcome x =xt  converging to x . In other words, the feasible outcome x is the equilibrium path. I define SPE strategies f that support this equilibrium

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path recursively, for all possible subgames, as follows. Let
1 =
1 x, and for any k, let 


k = ∪
1 ˆx  xˆ ∈
k−1 Clearly, for any finite history h ∈ H there exists a feasible outcome xˆ ∈
k for some finite k such that h is an initial segment of xˆ . The equilibrium strategy f requires all players to follow the path xˆ once h is reached, f h  = xt+k  for any h = h xt   xt+k−1 , where xˆ = h xt  . (By construction, there is a unique equilibrium continuation of h.) To show that f is a SPE, I need to show that after any finite history h, no player can benefit by unilaterally deviating. Consider the subgame equilibrium path represented by the feasible outcome xˆ with limit point xˆ  . Suppose that at some date t, player i unilaterally deviates so that the profile at t is x˜ rather than xˆ t+1 . Because there is a single deviation from the prescribed play, the equilibrium strategy requires players to behave in a way that causes the game to end up at a profile x˜ i , say, which is strongly approachable from x˜ and gives player i a payoff ui x˜ i  = u∗i x ˜ ≤ u∗i xit−1  x−it  ≤ ui xˆ   because xˆ  is approachable. This completes the proof that f is a SPE. 6.2. Proof of Lemma 4 Fix some small number 0 < ε < 1 and for each i, define xi ∈ X by putting  if j = i 1 − εxi xij = if j = i. xj Let K denote the convex hull of the profiles xi ni=1 and let ˆ ≥ ui x Ai = xˆ ∈ K u∗i x Because u∗ is continuous and K is closed, Ai is closed for every i. For each i, there is some j such that u∗j xi  < uj x, because x is minimal and xi < x. Similarly, the fact that x is minimal implies that ∩ni=1 Ai = . For any xˆ ∈ K, let gi x ˆ denote the distance between xˆ and Ai ; that is, ˆ = min xˆ − x  gi x  x ∈Ai

Without loss of generality, K can be indentified with the unit simplex,    n K = t ∈ R+  ti = 1 i

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and then a mapping T  K → K can be defined by putting ˆ g x ˆ =i Ti x g ˆ j j x

 Clearly, T is well defined, because ∩ni=1 Ai =  implies that j gj x > 0 for every xˆ ∈ K. T is continuous because each gi is continuous. The conditions of Brouwer’s fixed point theorem are satisfied, so there exists a point xˆ ∈ K such that T x ˆ = x. ˆ Let I = i  xˆ i > 0 = i  gi x ˆ > 0. Recalling that gi x ˆ > 0 implies that xˆ ∈ / Ai , see that u∗i x ˆ < ui x for all i ∈ I. For all i ∈ / I, xˆ < x and xˆ i = xi , so positive spillovers imply that u∗i x ˆ ≤ ui x. Then xˆ is the required profile, and because the choice of ε is arbitrary, x is locally approachable. 6.3. Proof of Lemma 5 From the previous lemma, x is clearly locally approachable, so for any small ε > 0 there exists xˆ < x such that xˆ is ε-close to x and u∗ x ˆ < ux. Construct a sequence xk  as follows. Choose x1 = xˆ and for each k = 1 2 , choose xk+1 to maximize xk − xk+1  subject to the constraints a xk ∈ Si xk+1  ∃i and

b

u∗ xk+1  ≤ ux

In other words, we choose xk+1 to satisfy two conditions. First, xk is reached from xk+1 by exactly one player increasing his or her action. Second, no player can achieve more than his or her target payoff at x by unilaterally deviating at xk+1 . Condition (b) is the analogue of condition (i) . Condition (i) is the appropriate condition here, because exactly one player moves at a time in this construction. At each step, the constraints can be satisfied by putting xk = xk+1 so the sequence is well defined at each step. Furthermore, as long as xk = 0, we must have xk+1 < xk , because the fact that x is strongly minimal implies that u∗i xk  < ui x for some i such that xik > 0 for each k. I claim that the sequence xk must reach 0 in a finite number of steps. If not, then the sequence can be continued indefinitely and, by compactness, there exists a subsequence converging to a limit point x∞ , say. With a slight abuse of notation, let xk  denote the subsequence that converges to x∞ . Because xk − xk+1  → 0 along this subsequence, in the limit u∗ x∞  = ux, contradicting the minimality of x. This contradiction implies that xk = 0 for some finite k. The sequence xk K ˆ backward to the origin in a finite k=1 goes from x number of steps. I now reverse the order of the sequence. What results

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is a nondecreasing sequence xk K ˆ and k=1 with x1 = 0 and xK = x u∗ xk  ≤ ux for every k. By construction, exactly one player changes his action at each step. Now suppose that I take a sequence xˆ s ∞ s=1 of target profiles converging to x and denote the corresponding approaching Ks sequences by xsk k=1 ∞ s=1 . Using the standard diagonalization procedure, I can select a sequence x ∞ =1 such that x is a limit point of xs , for each fixed  and x → x as  → ∞. Then x  is the required sequence. Note that I have not worried about consistency. So to turn x  into a feasible outcome, I must construct another sequence in which players wait after x is reached until the right player has the chance to move and reach the next profile x+1 . This construction completes the proof that x is strongly approachable from 0. REFERENCES Admati, A., and Perry, M. (1991). “Joint Projects Without Commitment,” Rev. Econ. Stud. 58, 259–276. Aumann, R., and Sorin, S. (1989). “Cooperation and Bounded Recall,” Games and Econ. Behav. 1, 5–39. Cooper, R., and John, A. (1988). “Coordinating Coordination Failures in Keynesian Models,” Quart. J. Econ. 103, 441–463. Gale, D. (1995). “Dynamic Coordination Games,” Econ. Theory 5, 1–18. Kurz, M. (1978). “Altruism as an Outcome of Social Interaction,” Am. Econ. Rev. 68, 216–222. Lagunoff, R., and Matsui, A. (1997). “An ‘Anti-Folk Theorem’ for a Class of Asynchronously Repeated Games,” Econometrica 65, 1467–1477. Marx, L., and Matthews, S. (2000). “Dynamic Voluntary Contribution to a Public Project,” Rev. Econ. Stud. 67, 327–358.