RECURSIVE GLOBAL GAMES

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1 RECURSIVE GLOBAL GAMES C G F T May 27, 2012 A. We consider a finite horizon, continuum player game with strategic complementarities and incomplete information. In each period, there is aggregate uncertainty and players productivity types are also subject to idiosyncratic shocks. While players know their current type, they only receive noisy signals about the aggregate shock realization. We provide suffi cient conditions on the primitives of the game to allow us to analyze each period s continuation game as an independent static reduced game. Each such reduced game is amenable to techniques developed for static global games. We show that as signals become suffi ciently precise, the game has a unique Markov perfect equilibrium. Keywords: Global games, supermodular games, dynamic complementarities, equilibrium selection. JEL Classification: C73, D43, E I It has been widely recognized that complementarities may be at the heart of a satisfactory explanation of why there are large shifts and fluctuations in economic activity. A variety of specific channels, such as increasing returns to scale in specific activities or sectors, and search and thick market externalities, yield a tendency to cluster and agglomerate. Models with multiple equilibria have been offered as descriptions of how such shifts may come about, with their proponents arguing e.g. that when an economy slides into recession, it is nothing but a transition to a low activity equilibrium. While these types of explanations clearly have merit, We wish to thank Guilherme Carmona, David Frankel, Christopher Harris, Sergiu Hart, Frank Heinemann, Christian Hellwig, Jonathan Levin, Stephen Morris, Mario Pascoa, Alessandro Pavan, Michael Peters, Hamid Sabourian, Jakub Steiner, Mich Tvede for useful comments. We are also pleased to acknowledge suggestions and comments from participants at the 2003 Winter Meetings of the Econometric Society, Washington, D. C., the 2003 Summer Meetings of the Econometric Society, Evanston, IL., the 2003 Money, Macro and Finance meetings in Cambridge, the 2004 Econometric Society European Meetings, Madrid, the Coordination Failures Workshop in Paris, 2004, the 2005 ESSET meetings in Gerzensee, the 2007 Winter Meetings of the Econometric Society, Chicago, IL and seminar participants at the Hebrew University of Jerusalem, the University of Copenhagen and Universidade Nova de Lisboa. The present version of this paper contains a substantially different set of assumptions than those contained in previously circulated versions, although the main ideas of the analysis remain unchanged. Faculty of Economics, University of Cambridge and CEPR. cg349@cam.ac.uk. Faculty of Economics, University of Cambridge and CEPR. fmot2@cam.ac.uk. 1

2 R G G 2 they leave important issues unexplained. In particular, they are silent about the important question of how these equilibria are reached and why there are shifts between them. 1 Partly as a response to this type of reservations about multiple equilibrium explanations, recent years have seen an increasing interest in equilibrium selection in games with strategic complementarities. In this paper, we analyze a class of such fully dynamic games with complementarities. We consider an incomplete information model, in which a continuum of players choose actions simultaneously over a finite number of periods. We show that under certain assumptions on payoff functions and transition laws, expected current and future payoffs are characterized by strategic complementarities. Next, we provide suffi cient conditions on the primitives of the model such that results on static global games by Frankel, Morris and Pauzner 2003 can be applied at each stage of our dynamic framework in order to identify a unique Markov perfect equilibrium. In our setting, there are three types of intertemporal links. First, there is a purely exogenous economic fundamental which evolves according to a Markov process. Second, each player s private productivity or type evolves according to a partially endogenous transition law. Specifically, a player s current action and type determines the distribution of that player s type in the next period. Last, since each player conditions play on its type, the cross-sectional distribution of types across the population can be thought of as a state variable which influences players future payoffs. 2 Our model features two types of complementarities. First, each player s incentive to raise its action is increasing in other players actions, i.e. actions are contemporaneous strategic complements. Second, payoffs are monotone and supermodular in types, and in an economic fundamental variable, the distributions of which are stochastically increasing and supermodular. Therefore, each player s payoffs display intertemporal complementarities between actions, types and the economic fundamental. Our set of conditions ensure that these different complementarities reinforce each other. The proof of equilibrium uniqueness is by induction. We start by showing that in the last period, there is a unique equilibrium that survives iterative strict dominance this follows from our assumptions and the results of Frankel et al., Because the underlying complete information game is supermodular, the unique equilibrium has certain properties, such as continuity and monotonicity. With these results in hand, we then use the assumed properties of the transition laws to show that the game played in the penultimate period satisfies all the assumptions for a unique equilibrium to obtain, under the assumption that equilibrium play takes place in the last period. In a similar way, we can then show equilibrium uniqueness in an arbitrary period, under the assumption that the continuation equilibrium is well-defined, i.e. is uniquely determined given past and present play. Our paper relates closely to two strands of literature. First, the basic model of complete information in which complementarities are analyzed, shares features with an existing literature on supermodular games. Curtat 1996 considers stochastic games with strategic complementarities in which a finite set of players can influence the evolution of a state variable probabilistically. He shows the existence of a Markov perfect equilibrium. In contrast, we work with a continuum of players that are subject to both aggregate and idiosyncratic shocks and thus our results complement his analysis. Vives 2009 considers a finite horizon, finite player 1 An exponent of this critique is Durlauf 1991 who states that Most of these models describe multiple steady states in economies rather than multiple non-degenerate time-series paths, and consequently cannot address issues of aggregate fluctuations. Further, this literature has not shown how economies can shift across equilibria, inducing periods of boom and depression. 2 Although players can in principle condition play on this distribution, the distribution is common knowledge at the time of choosing actions.

3 R G G 3 game with strategic complementarities, with both aggregate and private state variables, but considers deterministic transitions. 3 Sleet 2001 considers a model of a continuum of pricesetting firms that are subjected to idiosyncratic shocks and price adjustment costs. Last, our analysis has similarities with the work of Milgrom, Qian and Roberts They consider an up-stream down-stream model in which knowledge in the two industries is accumulated over time and where actions in each period are complementary with the state of knowledge. They prove a momentum theorem which shows that when knowledge accumulation is increasing in actions and actions are complementary with the state of knowledge, the system may enter a self-enforcing dynamic path where actions and knowledge increase over time. Second, our paper contributes to the literature on dynamic global games. Building on insights from Carlsson and van Damme 1993, a fast-growing literature has evolved in which many coordination games have been analyzed within the so-called global games framework. 4 The powerful results of the global games literature has allowed applied modelers to pin down equilibria in games that notoriously have very rich equilibrium sets and extending the success to dynamic settings is now an active research field. While a lot or the literature has focused on static game settings, four distinct approaches to dynamic global equilibrium selection have previously been developed. The first, exemplified by the work of Burdzy, Frankel and Pauzner 2001 is to consider players that are randomly matched over a sequence of periods to play a 2 2 coordination game, but where the payoff matrix is parametrized by a stochastically evolving economic fundamental. The second approach, exemplified by Frankel and Pauzner 2000, is to consider continuous time games in which the stochastic economic fundamental is governed by a Brownian motion and where players receive revision opportunities according to a Poisson process. The third approach is that of Levin 2001 who considers players who only live for a single period, but whose payoffs may depend on actions of players living before or after them. The fourth approach is that of Toxvaerd 2008 who considers a timing game in which players decide when to move and where players at each point in time trade off current payoffs with expected future payoffs. Although formally not a supermodular game, the recursive payoff structure allows for static global games techniques to be employed at each stage of the game and thus the fully dynamic game can be analyzed as a sequence of interrelated static games. The present analysis is most closely related to this fourth approach. Recent work in the same spirit includes that by Steiner 2008a, 2008b and Ordoñez T G 2.1. Basics. Time is discrete and finite, indexed by t T = {1,..., T }. The recursive global game we analyze is a sequence of global games denoted by {G t ν t } t T, where ν t > 0, t = 1,..., T are scale factors that parametrize the game. 5 There is a continuum of players, all in a set I = [0, 1]. At a given point in time, each player has a type from the set K = {1, 2,..., K}, i.e. a player i I in period t is of type τ it K. The set of players of a given type from K can at a point in time contain a continuum of identical players of finite measure, be a singleton, or be empty. Denote by T the set of all possible distributions over the set K, with typical element τ t T, i.e. τ t is the cross-sectional distribution of types in the population at time t. In a given period t, a state θ t R is drawn from the real line. The realization of the economic fundamental variable θ t is not common knowledge at time t. Rather, in each period t 3 The assumption of deterministic transitions means that Vives 2009 cannot show existence of equilibrium, but he is able to characterize extremal Markov perfect equilibria under the assumption that these are continuous. 4 See Morris and Shin 2003 for a survey of the literature and its relation to other strands of research. Global games methodology in more general environments is found in Mathevet 2009 and in Oury The parameter ν t parametrizes the precision of agents signals in period t. While the precision parameters can be identical for all periods of the game, we choose to denote them with a subscript t in order to be able to take limits sequentially and not simultaneously when we prove uniqueness of equilibrium in Section 3.

4 R G G 4 each player i observes a signal x it = θ t +ν t η it. The noise variables η it [ 1, 1 ] are independent 2 2 across players and over time, and identically distributed. The action set of a player i at time t is A = [0, 1], that is we assume that the action sets are time and player invariant. A strategy for a player i at time t is a mapping from the crosssectional type distribution, his type and his signal to his action set s it : T K R A, while a strategy profile at time t is a collection of strategies s t = s it i I. Although we do allow the strategies of players to depend on the cross-sectional distribution of types, we suppress this dependence in the notation in order not to clutter the exposition. A strategy profile s t at time t is increasing, if s it is weakly increasing in all its arguments, for all i I. 6 We restrict attention to Markov strategies. 7 Throughout, we maintain the following informational assumptions. At any point in time t, all realizations up to t 1 of the economic fundamental are common knowledge, as are all past action profiles. Furthermore, each agent knows his own type. Last, all cross-sectional type distributions up to and including the one at time t are commonly known. In short, the players have common knowledge of all past data and the current period s cross-sectional type distribution and only have uncertainty about the current value of the economic fundamental Payoffs. For a player i, we denote by O it the set of types that are i s opponents at time t. Next, we define the one stage return for an arbitrary player i of type τ it at time t. All players of the same type at time t have the same one stage return function. The return will depend on i the action of player i, i.e. on a it A, ii the action profile of i s opponents, denoted by a it = aˆτ t ˆτ O it, where aˆτ t is the cumulative distribution function of actions chosen by players of type ˆτ at time t, iii the player s type τ it at time t and finally iv on the realization of the economic fundamental. Let a it A, where A is the set of possible cumulative distribution functions of actions across types of i s opponents. We then denote the one stage return by r it : A K R A R and r it = r a it, τ it, θ t ; a it. We assume that the return function r is bounded. Throughout, we assume that players seek to maximize the sum of discounted expected payoffs Dynamics. We assume that all agents discount the future with the same discount factor, 0 < δ < 1. For a player i, the dynamics are described by two stochastic state variables, summarized in a vector χ it τ it, θ t. The exogenous dynamic element is the aggregate state variable θ t, which is assumed to follow a first-order Markov process. The endogenous dynamic element is the individual state variable τ it, which is the player s type. We assume that the type of a player i at time t depends stochastically on his type and action at time t 1, in a Markovian way. We denote the transition probabilities of players types by p κk a i,t 1 Pr τ it = κ τ i,t 1 = k; a i,t 1. 1 We include a i,t 1 in the notation to be explicit about the fact that a player s future type not only depends stochastically on his current type, but also on his current action. We assume that for a given agent i, the two variables θ t and τ it a i,t 1 are independent. The economic fundamental θ t and the players types generate intertemporal links that make the game dynamic, in the sense that play at each point in time is influenced by past play and in turn influences the game to be played subsequently. We assume that players payoffs 6 The distributions of types are ordered in the sense of first-order stochastic dominance. 7 This restriction simplifies the analysis by allowing us to study the game as a sequence of appropriately defined strategic form games in which future play is summarized by expected equilibrium value functions from continuation play. As shown in Echenique 2004, without a Markov restriction on strategies, strategic complementarities in extensive form games is diffi cult to obtain.

5 R G G 5 depend on both the exogenous aggregate state variable the economic fundamental and on the private state variables the types. Specifically, we assume that for a given player, the two state variables increase payoffs directly. Furthermore, higher states make higher actions more desirable. These interactions will be made more precise in section 3.1. To understand the several sources of complementarities in our model, it is instructive to consider two extreme versions in which they feature in isolation. First, suppose that players types are fixed at the beginning of the game and do not change over time. In this case, our game is essentially a sequence of static global games, where the only intertemporal link is the exogenous economic fundamental. At each stage, there are strategic complementarities and the value of increasing one s strategy is increasing in others strategies. This type of setup was analyzed by Morris and Shin 1999 in a dynamic model of currency crises. Second, suppose that the players payoffs are independent of other players actions. In this case, each player faces an independent Markov decision process of the class studied by Jorgenson, McCall and Radner 1967 and Topkis In the individual s problem, the randomly evolving state variable is only partially controlled. Importantly, there are complementarities between the controlled part of the state variable and the decision maker s actions. In our model, these two sources of complementarities are combined. 8 Additionally, there is an interesting further intraand intertemporal link through the evolution of the cross-sectional type distribution, which players need to keep track of. Each agent conditions play only on its own type, but must keep track of the cross-sectional type distribution in forecasting continuation play. In what follows, we will be using two notations for the state variables interchangeably, namely the compact vector notation χ and the notation τ and θ, depending on whether we need to economize or refer explicitly to a particular element of the state variable The reduced game. Consider the complete information game, i.e. where θ t is directly observed. Then at a given time t and for a player i, the total payoff W it from playing action a it equals the one stage return from playing this action, r it, plus the expected discounted continuation value from playing optimally in period t + 1 and onwards, conditional on the current state θ t. This is summarized in the following recursion. In the final period T, the payoff from playing a particular action is simply the return r, i.e. For t = 1,..., T 1 we then define W it W it a it, τ it, θ t ; a it = r a it, τ it, θ t ; a it + δ V W it = r a it, τ it, θ T ; a it. 2 Q Vi,t+1 κ, ϑ dp κ, ϑ τ it, θ t ; a it 3, it τ it, θ t = V it τ it, θ t ; s t, 4 V it τ it, θ t = max it a it, τ it, θ t ; a it, a it A 5 where s t = { s it τ it, θ t } i I is an equilibrium strategy profile of the complete information game and s it : K R A are strategies mappings of the type and exogenous state to the action space. Also, P is a probability measure on Q = K R. 8 While this setup has similarities with that of Curtat 1996, there are also important differences. In his work, players types do not change over time, but aggregate play influences an aggregate state variable. We could easily extend our setting to include an additional endogenous aggregate state variable, whose distribution depends on its previous value and aggregate play. We have decided against this as we are working with a continuum of players in contrast to Curtat. When each agent is infinitesimal, the aggregate state variable does not add anything substantial to the analysis. This is because players ignore their impact on aggregate outcomes.

6 R G G 6 Next, we clarify some important aspects of the notation and definitions. First, W denotes the sum of discounted expected payoffs for an individual player when all play some arbitrary action profile in the current period but play according to some continuation equilibrium henceforth. Second, V denotes the value function for a player given a fixed contemporaneous action profile a it of the opponents and given some continuation equilibrium. Last, we use V to denote equilibrium value functions, i.e. the payoffs that prevail when all players choose their equilibrium strategies. In general, a star will denote equilibrium values of variables. It is implicit in 3 that the integral of Vi,t+1 is well-defined. Next we use results from Billingsley 1995, chapter 3, section 18, to express the integral in 3 in the two following useful ways. The first is K κ=1 p κk a it Vi,t+1 κ, ϑ dp ϑ θ t. 6 R The sum is well-defined, since a player s type is a simple random variable and therefore integrable. For the integral to be well defined, we need that Vi,t+1 is integrable with respect to the probability measure P ϑ θ t, which is true as long as V is bounded, given that the probability measure is finite. This is indeed true since we have assumed that the return function r is bounded. With these clarifications in place, we can also write the integral in 3 as Vi,t+1 χi,t+1 df χi,t+1 χ it ; a it, 7 Q where F χ i,t+1 χ it ; a it is the distribution of χi,t+1 conditional on the two state variables, for a given action a it. Both expressions will be useful in later derivations and proofs An application. There are many interesting examples of dynamic models with complementarities, e.g. those presented in Curtat 1996, Cooper 1999, Sleet 2001 and Vives To fix ideas, consider a modified version of Diamond s 1982 search model. 9 Economic agents search for trading partners. A player s action is its search intensity. Search intensities are strategic complements, i.e. the marginal value of increasing one s search intensity a it is increasing in other agents search intensities a it. Each player s search effort is characterized by the individual s search effi ciency or experience τ it, which is in turn a function of the agent s accumulated experience τ i,t 1 and past search intensity a i,t 1. In particular, we assume that the agent s search effi ciency is positively but stochastically related to last period s effi ciency and search intensity. The value of a successful match is influenced by an exogenous economic fundamental θ t, which may reflect overall economic conditions. This variable, which evolves according to a Markov process, is common to and imperfectly observed by all agents. Last, for suffi ciently favorable economic conditions, maximum search intensity is optimal while for suffi ciently unfavorable conditions, it is optimal to cease search at least temporarily E In this section, we set out a set of assumptions and conditions under which the uniqueness result of Frankel et al can be extended to our dynamic game. We will show that as the signals of the players become very precise, i.e. as ν t 0, the dynamic global game {G t ν t } t T has a unique Markov perfect equilibrium. We will work with the properties of the underlying complete information game throughout, but call on known correspondences between 9 This version of the model is inspired by that of Curtat This is relevant for the existence of dominance regions, to be introduced in section 3.1.

7 R G G 7 Nash equilibria of the complete information game and the unique equilibrium selected by the global games approach in the incomplete information version of the game. The structure of the proof is as follows. We first note that the game in the last period T is a static global game and therefore there is a unique strategy s T that survives iterative strict dominance in period T. We know from Frankel et al that as ν T 0, in period T the players play arbitrarily close to a pure strategy Nash equilibrium s T of the underlying complete information game. This allows us to conclude that the equilibrium value function ViT for the complete information game is i increasing in the state variables, for all i I, and ii continuous in θ T. We then show our main result by induction. Each induction step contains the following sequence of arguments: a given that the equilibrium continuation value of the subsequent period V is increasing in the state variables, we can show that the payoff function W is supermodular; given that it is continuous, we can show that the payoff function is continuous in the current fundamental state variable; we can also show that that the equilibrium continuation value of the subsequent period V is Lipschitz continuous in the opponents current play; b supermodularity, continuity and Lipschitz continuity of the payoff function implies that we can apply Theorem 5 of Frankel et al and pin down a unique equilibrium strategy profile in the current period, which is increasing in states; c from Theorem 6 of Frankel et al we know that the players will be playing arbitrarily close to a pure strategy Nash equilibrium for the current period s complete information game that shares the properties of the unique equilibrium of the incomplete information game, and finally d this implies that the equilibrium continuation value of the current period V is also increasing, continuous and Lipschitz continuous, and the loop starts again. To summarize our approach, we are interested in the properties of the incomplete information game as the noise in players information vanishes. We know from Frankel et al that there is a tight correspondence between the equilibrium of this limit game and an equilibrium of the underlying complete information game, namely that players in the unique equilibrium of the incomplete information game play arbitrarily close to some Nash equilibrium of the complete information game. Instead of working directly with the incomplete information game, we will work with the complete information version, but using properties of continuation equilibrium strategies that we know are compatible with the unique equilibrium of the incomplete information game. Thus we make no claims about what happens in the incomplete information game away from the limit, i.e. when players signals are imprecise. We should also emphasize that the way we show uniqueness requires the limits on signal precisions to be taken sequentially and backwards, i.e. we first take ν T 0 to establish uniqueness in period T, which is then used to take ν T 1 0 and establish uniqueness in period T 1, etc. We believe that this is the sensible approach to adopt, since it corresponds to the order in which we solve for finite horizon games i.e. backwards Assumptions. To simplify the exposition, we introduce the following notation: W it a it, a it, χ it ; a it = W it a it, χ it ; a it W it a it, χ it ; a it, 8 r a it, a it, χ it ; a it = r a it, χ it ; a it r a it, χ it ; a it, 9 i.e. W it and r denote differencing with respect to own action of the relevant functions. In what follows, we denote assumptions A1-A5 in Frankel et al by FMP1-FMP5. 11 We also note the fact that because we have a continuum of players with a finite number of types, assumptions FMP2 and FMP5 have to be modified according to footnote 23 in Frankel 11 For reference, we remind the reader that FMP1 is strategic complementarities, FMP2 is dominance regions, FMP3 is state monotonicity, FMP4 is payoff continuity and FMP5 is Lipschitz continuity.

8 R G G 8 et al The suffi cient assumptions for our main result are: A1. Complementarity: For all t, the return function ra it, χ it ; a it is supermodular in a it and in the pairs a it, a it and a it, χ it. The distribution function F χ i,t+1 χ it ; a it is stochastically supermodular in a it, χ it. A2. Dominance regions: For all t, there exist extreme values θ and θ with θ < θ such that for all i I and types τ it and opposing action profiles a it, if a it 0 and θ θ and W it 0, a it, χ it ; a it > 0, 10 W it 1, a it, χ it ; a it > 0, 11 if a it 1 and θ θ. Note that this has to hold for any possible continuation equilibrium. 12 A3. Monotonicity: For all t, the return function ra it, χ it ; a it is increasing in χ it and a it and F χ i,t+1 χ it ; a it is stochastically increasing in χit for a given fixed strategy profile a it, a it. A4. Regularity: For all t, the return function ra it, χ it ; a it is continuous in a it, θ t and a it. Also, we assume that the conditional distribution of θ t+1 θ t is continuously differentiable and that the corresponding density function is jointly continuous in θ t+1 and θ t. Finally, the transition probabilities p κk a i,t 1 are continuous in a i,t 1. A5. Lipschitz continuity: 13 For all t we assume the following: a For all θ t, τ it and a it, there exists a constant K 1 > 0 such that for all a it and a it, r a it, a it; a it, τ it, θ t K 1 a it a it. 12 b For all θ t and τ it, there exists a constant K 2 > 0 such that for all a it, a it, a it and a it, r a it, a it; a it, τ it, θ t r a it, a it; a it, τ it, θ t K 2 a it a it a it a it. 13 c There exists a constant K 3 > 0 such that for all a i,t 1 and a i,t 1, pκk a i,t 1 p κk a i,t 1 K3 ai,t 1 a i,t 1, 14 where the variation p κk a i,t 1 p κk a i,t 1 is defined with respect to the Kantorovich metric K on the set of probability distributions with finite first moment vector Since θ t is not iid across periods, the dominance regions from period to period may differ. Here, we implicitly assume that θ and θ are the maximum possible upper dominance cutoff and the minimum possible dominance cutoff respectively, across periods and states. 13 We define a metric for the space of strategy profiles as follows: The distance between two distributions of actions a and a is given by a t a t = sup {λ : for all a it A, either aa it + λ a a it or a a it + λ aa it }. Note that since the strategy spaces are not type dependent, there is no need to distinguish between differences in strategies across player types. 14 The Kantorovich metric for measures µ, κ is defined as } { κ µ, κ = sup hdµ hdκ : h Lip 1 ρ S where Lip 1 ρ S is the set of functions on S such that l ρs h 1 and l is the Lipschitz module for details, see Hinderer, 2005.

9 R G G Auxiliary results. With these assumptions in place, we first show that there exists a unique equilibrium strategy profile in the last period. This strategy profile is increasing in the states. The first lemma essentially shows that the game in the last period is a standard static global game for which we can apply Theorem 5 of Frankel et al Lemmas 2, 3 and 4 show that the equilibrium value function ViT is increasing in the states and continuous in the economic fundamental, for all i I. Lemma 1. In period T, the payoff function W it for player i is supermodular in a i, a i, a i, τ i, and a i, θ, and as ν T 0, there exists a unique increasing strategy profile s T surviving iterative strict dominance. Proof. Since W it = ra it, χ it ; a it, we have that by A1, ra it, χ it ; a it has increasing differences in a it, a it, i.e. FMP1 is true. Also, A2 implies FMP2. The third assumption of state monotonicity FMP3 holds because of A1, since ra it, χ it ; a it has increasing differences in a it, θ T and therefore FMP3 is also satisfied. FMP4 is satisfied by A4. Finally, FMP5 is true because we have directly assumed it in A5. Therefore, as ν T 0 and from Frankel et al. 2003, Theorem 5, there exists a unique increasing strategy profile s T surviving iterative strict dominance. Since the last period game is supermodular, there exist largest and smallest equilibrium strategy profiles that are increasing in the players states Topkis, 1998, Theorem These two coincide when ν T 0. Therefore the unique equilibrium strategy is increasing in the states χ it. Lemma 2. In G T ν T as ν T 0, players play arbitrarily close to some pure strategy Nash equilibrium of the underlying complete information game. The implied strategy profile is denoted by s T and is increasing, i.e. s it χ it is increasing in the states χ it. Proof. Follows directly from Theorems 5 and 6 of Frankel et al Lemma 3. The equilibrium payoff in the final period V it χ it is increasing in χ it for all i I. Proof. For any i, and V it χ it = max a it A ra it, χ it ; a it, 15 V it χ it = V it χ it ; s T max a it A ra it, χ it ; s T, 16 where s T is the Nash equilibrium strategy profile, which is increasing by Lemma 2. Then, from A3 i.e. that the return function is increasing in χ it and s T and the fact that monotonicity is preserved under maximization, it follows that ViT χ it is increasing in χ it. Lemma 4. The equilibrium payoff in the final period V it χ it corresponding to the pure strategy Nash equilibrium described in Lemma 2 is continuous in θ T for all i I. Proof. Recall that V it τ it, θ T = max a it A ra it, τ it, θ T ; s T. 17 First we have that because A is compact, then by Berge s theorem of the maximum, s T θ T is continuous in θ T. To see this, recall that each player s best response correspondence is upper hemi-continuous in θ T. We know that there is a unique strategy profile s T θ T that survives iterative strict dominance. This implies that the set of best responses for each player that survives this process is single valued. It follows that in equilibrium, each player s equilibrium strategy is in fact a continuous function of θ T. The fact that s T θ T is continuous in θ T and the regularity conditions from A4 ensure that ViT is continuous in θ T, since it is composed of continuous functions.

10 R G G Main result. We are now ready to state and prove our main result. Theorem. Under assumptions A1-A5, the game {G t ν t } t T has an essentially unique increasing strategy profile s t s it i I surviving iterative strict dominance, in each period t = 1,..., T as ν t 0. Proof. We prove the result by backward induction. For each period we analyze, there are four steps. The first step applies the relevant theorem of Frankel et al Steps two to four provide properties that are required for proving the result in the preceding period. Period T : The result is true as shown in Lemma 1. From Lemmas 2-4, we also have that a the equilibrium strategy is arbitrarily close to a pure strategy Nash equilibrium of the underlying complete information game played in period T ; b this implies that ViT χ it under this equilibrium strategy is increasing in χ it and continuous in θ T. Period T 1: Step 1: We check the validity of each of the assumptions in Frankel et al in turn. First we verify FMP1. For this, we note that the problem of a player i is to maximize W i,t 1 = r a i,t 1, χ i,t 1 ; a i,t 1 + δ ViT χ it df χ it χ i,t 1 ; a i,t From A1, we have that F χ it a i,t 1, χ i,t 1 is stochastically supermodular in ai,t 1, χ i,t 1. We also have that ViT χ it is increasing. Then, from Corollary of Topkis 1998 it follows that δ ViT χ it df χ it χ i,t 1 ; a i,t 1 19 is supermodular in a i,t 1, χ i,t 1. Since by A1, r is supermodular in ai,t 1, χ i,t 1, then by Lemma in Topkis 1998, W i,t 1 ai,t 1, χ i,t 1 ; a i,t 1 is supermodular in ai,t 1, χ i,t 1. Next, to show supermodularity in a i,t 1, a i,t 1, note first that by A1, r is supermodular in a i,t 1, a i,t 1. Also, the change in the second term of W i,t 1 can be written as K τ it =1 p τ it,τ i,t 1 ai,t 1, a i,t 1 V it χ it ; s T f θ T θ T 1 dθ T. 20 R where p τ it,τ i,t 1 ai,t 1, a i,t 1 pτ it,τ a i,t 1 i,t 1 p τ it,τ i,t 1 a i,t 1. We need to establish that expression 20 is increasing in a i,t 1. To do this, recall that an increase in a i,t 1 leads to an increase in the cross-sectional distribution of types in period T. 15 In turn, an arbitrary agent j s best response at time T is increasing in his type τ jt in the pure strategy Nash equilibrium that corresponds to the unique equilibrium in the global game. But this means 15 As in Bergin and Bernhardt 1992, we will work under the assumption that the evolution of the cross sectional distribution of types is deterministic, conditional on strategies and the exogenous shocks.

11 R G G 11 that aggregate play at time T is increasing in a i,t Last, since V it χ it ; s T = max a it A ra it, χ it ; s T 21 and r is increasing in s T, the result follows. Next, FMP2 and FMP3 are trivially true because of our assumptions A2 and A1 respectively. To check the validity of FMP4, and given that we have assumed A3, we only need to verify that ViT χ it df χ it χ i,t 1 ; a i,t 1 22 is continuous in a i,t 1 and θ T 1. As we explained in section 2.4, and using A3, the above integral can be written as K τ it =1 p τ it,τ i,t 1 a i,t 1 ViT χ it f θ T θ T 1 dθ T, 23 R where, with some abuse of notation, f θ T θ T 1 denotes the conditional density of θ T given θ T 1, which is jointly continuous in θ T, θ T 1. Given that from Lemma 4, ViT χ it is continuous in θ T, it follows that the integral is continuous in θ T 1. Continuity in a i,t 1 follows immediately from the fact that p τ it,τ i,t 1 a i,t 1 is continuous in a i,t 1 see A4. Finally, verifying FMP5 involves two parts. First, we show part a of FMP5, i.e. that W i,t 1 is Lipschitz continuous in a i,t 1. Lipschitz continuity of the return function in a i,t 1 is ensured directly by A5. Next, we have that the expected continuation equilibrium payoff is also Lipschitz continuous in a i,t 1. This is true because p τ it,τ i,t 1 a i,t 1 is Lipschitz continuous in a i,t 1 by A5, and therefore the integral is Lipschitz continuous in a i,t 1 by Theorem 4.1 in Hinderer Last, since ViT χ it is bounded and using Lemmas 2.1c and 3.2 of Hinderer 2005, it follows that W i,t 1 is Lipschitz continuous in a i,t 1. To show part b of FMP5, we need to establish that W i,t 1 is Lipschitz continuous in a i,t 1. Recall that W i,t 1 = r a i,t 1, χ i,t 1 ; a i,t 1 K + p τ it,τ i,t 1 a i,t 1 V it χ it ; s T f θ T θ T 1 dθ T. 24 R τ it =1 We first note that r a i,t 1, χ i,t 1 ; a i,t 1 satisfies this property by part b of A5. The actions of player i s opponents a i,t 1 enter the second term of the above expression indirectly through their influence on next period s distribution of player types. Since players at time T condition their play on their types, and their period T 1 actions influence these through 16 To be precise, for any period t = 1,..., T, although each player s payoff function r a it, τ it, θ t ; a it depends only on the player s own type τ it and has increasing differences in a it, τ it, one may always write the payoffs as r a it, τ it, θ t ; a it, τ t, i.e. as a function of the entire distribution of types τ t and then assume that these payoffs have weakly increasing differences in a it, τ jt for any player j i. We can now take two distributions τ t and τ t where the former first-order stochastically dominates the latter, i.e. has higher cross-sectional type distribution. In order to trace the effect on equilibrium play of this shift in type distribution, we start by changing the type of a single player i, from τ it to τ it, i.e. from his type under distribution τ t to his type under distribution τ t. Recall that we have a parametrized supermodular game, where the parameter is simply player i s type τ it. Since each player s payoffs have increasing differences in own action and this parameter perhaps weakly so and there are strategic complementarities, then extremal equilibria will be monotone in player i s type. Next, we take another player j i and change his type from τ jt to τ jt etc. Running through the entire population in this manner, it follows that when moving from distribution τ t to τ t, aggregate play increases.

12 R G G 12 the transition probabilities, there is an effect of period T 1 play on period T play. In other words, as a i,t 1 varies, so does the continuation equilibrium value V it χ it ; s T through the dependence of the cross-sectional distribution of types on s T. Because each player s transition probability p τ it,τ i,t 1 a i,t 1 is Lipschitz continuous in a i,t 1 by A5, Lipschitz continuity of the cross-sectional distribution in a i,t 1 follows. Since V it χ it ; s T = max a it A ra it, χ it ; s T 25 and r is Lipschitz continuous in s T, it follows from Lemma 2.1c of Hinderer 2005 that ViT is Lipschitz continuous in a i,t This shows that part b of FMP5 holds. We can therefore conclude that the global game in period T 1 satisfies all the assumptions that are required for Theorems 5-6 of Frankel et al to apply. Thus, once ν T 0 and uniqueness obtains in period T, we conclude that as ν T 1 0, there exists a unique strategy profile s T 1 surviving iterative strict dominance and this strategy is increasing. Furthermore, this equilibrium is arbitrarily close to a pure strategy Nash equilibrium s T 1 of the underlying complete information game, which must then be increasing too. Step 2 : From A3, F χ it χ i,t 1 ; a i,t 1 is stochastically increasing in χi,t 1, for a given a i,t 1, and thus from Corollary of Topkis 1998 it follows that δ ViT χ it df χ it χ i,t 1 ; a i,t 1 26 is increasing in χ i,t 1. Together with the assumption in A3 that r is increasing and the fact that monotonicity is preserved under maximization, it follows that V i,t 1 χi,t 1 is increasing in χ i,t 1. Last, since Vi,T 1 χi,t 1 = Vi,T 1 χi,t 1 ; s T 1 27 and since s T 1 is increasing, it follows that V i,t 1 χi,t 1 is increasing in the states. Step 3: Next, we show continuity of Vi,T 1 χi,t 1 in θt 1. We use the same argument as in the proof of Lemma 4. From the uniqueness of s T θ T, it follows from the upper hemi-continuity of the best response correspondence that the strategies surviving iterative strict dominance are continuous functions. Thus Vi,T 1 χi,t 1 is a composite function of continuous functions and the result follows. Step 4: Last we show that V i,t 1 is Lipschitz continuous in a i,t 2. We have already established that Vi,T 1 χi,t 1 = Vi,T 1 χi,t 1 ; s T 1 is Lipschitz continuous in s T 1 see footnote 12. Next, the actions of player i s opponents a i,t 2 influence next period s distribution of player types. Since players at time T 1 condition their play on their types, and their period T 2 actions influence these through the transition probabilities, there is an effect of period T 2 play on period T 1 play. Because each player s transition probability p τ i,t 1,τ i,t 2 a i,t 2 is Lipschitz continuous in a i,t 2 by A5, Lipschitz continuity of the cross-sectional distribution in a i,t 2 follows. Because W i,t 1 is bounded and types take discrete values, the response of an agent i to a change in his type at T 1 is 17 Note that this last sequence of arguments can be replicated exactly to show that W i,t 1 is Lipschitz continuous in a i,t 1, given that A5 ensures Lipschitz continuity of r in a i,t 1. Therefore, Vi,T 1 is Lipschitz continuous in s T 1 from Lemma 2.1c of Hinderer

13 R G G 13 bounded; in turn this implies that aggregate play at time T 1 varies Lipschitz continuously with a i,t 2. Period 1 t < T 1: The theorem is true for t = T 1. It then suffi ces to show that, if Vi,t+1 χi,t+1 is increasing and continuous in θt+1, Lipschitz continuous in a i,t and there exists a unique increasing strategy profile s t+1 s i,t+1, which is arbitrarily close to a pure i I strategy Nash equilibrium s t+1 of the complete information game, then there exists a unique equilibrium strategy profile s t s it i I surviving iterative strict dominance, which implies the existence of an arbitrarily close pure strategy Nash equilibrium s t of the complete information game, and Vit χ it is increasing and continuous in θ t, and Lipschitz continuous in a i,t 1. Step 1: We first confirm the validity of FMP1-FMP5 for this period. The problem of a player i at time t is to maximize W it = r a it, χ it ; a it + δ Vi,t+1 χi,t+1 df χi,t+1 χ it ; a it. 29 From A1, we have that F χ i,t+1 χ it ; a it is stochastically supermodular in ait, χ it. We have also assumed that Vi,t+1 χi,t+1 is increasing. Then, from Corollary of Topkis 1998 it follows that the integral in 29 is supermodular in a it, χ it. Since by A1, r is supermodular in a it, χ it, and by Lemma in Topkis 1998, W it a it, χ it ; a it is supermodular in a it, χ it. Next, to show supermodularity in a it, a it, note first that by A1, r is supermodular in a it, a it. Next, we establish that W it a it, χ it ; a it is increasing in a it. An increase in a it leads to an increase in cross-sectional distribution of types in period t + 1. In turn, agent j s best response at time t + 1 is increasing in his type τ j,t+1. Thus aggregate play at time t + 1 is increasing in a it. Last, since V i,t+1 χi,t+1 ; s t+1 and r are increasing in s t+1 under the pure strategy Nash equilibrium that corresponds to the unique equilibrium of the global game, the result follows. This confirms FMP1. Next, FMP2 and FMP3 are trivially true because of our assumptions A2 and A1 respectively. To confirm FMP4, and given A3, we only need to verify that the integral in 29 is continuous in a it and θ t. Using A3, the integral can be written as K τ i,t+1 =1 p τ i,t+1,τ it a it Vi,t+1 χi,t+1 f θt+1 θ t dθ t+1, 30 R where the conditional density f θ t+1 θ t is jointly continuous in θ t+1, θ t. By the induction step assumption, Vi,t+1 χi,t+1 is continuous in θt+1 and therefore the integral is continuous in θ t. Continuity in a it follows immediately from the fact that p τ i,t+1,τ it a it is continuous in a it, from A4. Finally, verifying FMP5 involves two parts. First, we show part a of FMP5, i.e. that W it is Lipschitz continuous in a it. First, Lipschitz continuity of the return function in a it is ensured directly by A5. Next, we have that the expected continuation equilibrium payoff is also Lipschitz continuous in a it. This is true because p τ i,t+1,τ it a it is Lipschitz continuous in a it by A5, and therefore the integral is Lipschitz continuous in a it by Theorem 4.1 in Hinderer Last, since Vi,t+1 χi,t+1 is bounded and using Lemmas 2.1c and 3.2 of Hinderer 2005, it follows that W it is Lipschitz continuous in a it. To show part b of FMP5, we need to establish that W it a it, χ it ; a it is Lipschitz continuous in a it. First, r a it, χ it ; a it satisfies this property by part b of A5. The actions of player i s opponents a it enter the second term of the above expression indirectly through their influence on next period s distribution of player types.

14 R G G 14 Players at time t + 1 condition their play on their types, and their period t actions influence these through the transition probabilities, therefore as a it varies, so does the continuation equilibrium value V i,t+1 χi,t+1 ; s t+1 through the dependence of the cross-sectional distribution of types on s i,t+1 χi,t+1. Because each player s transition probability pτ i,t+1,τ it a it is Lipschitz continuous in a it by A5, Lipschitz continuity of the cross-sectional distribution in a it follows. Since by the induction hypothesis it is assumed that V i,t+1 χi,t+1 ; s t+1 is Lipschitz continuous in a it, it follows from Lemma 2.1c of Hinderer 2005 that W it a it, χ it ; a it is Lipschitz continuous in a it. 18 This confirms part b of FMP5. With these results in place, we conclude that the global game in period t satisfies all the assumptions that are required for Theorems 5-6 of Frankel et al to hold. Therefore, once ν t+1 0 and uniqueness obtains in period t + 1, we conclude that as as ν t 0, there exists a unique increasing strategy profile s t surviving iterative strict dominance, which implies a unique pure strategy Nash equilibrium s t for the complete information game, which is increasing. Step 2: From A3, F χ i,t+1 χ it ; a it is stochastically increasing in χit, for a given a it. Then from Corollary of Topkis 1998 it follows that the integral in 29 is increasing in χ it. Together with the assumption that r is increasing A3 and the fact that monotonicity is preserved under maximization, it follows that V it χ it is increasing in χ it. Last, since Vit χ it = V it χ it ; s t and because we have shown in Step 1 that the unique Nash equilibrium strategy profile s t is increasing, it follows that Vit χ it is increasing. Step 3: Arguments that parallel those in Lemma 4 establish that V it χ it is a continuous function of θ t. Step 4 : Last, we show that Vit is Lipschitz continuous in a i,t 1. We have already established that Vit χ it is Lipschitz continuous in s t. Next, the actions of player i s opponents a i,t 1 influence next period s distribution of player types. Since players at time t condition their play on their types, and their period t 1 actions influence these through the transition probabilities, there is an effect of period t 1 play on period t play. Because each player s transition probability p τ it,τ i,t 1 a i,t 1 is Lipschitz continuous in a i,t 1 by A5, Lipschitz continuity of the cross-sectional distribution in a i,t 1 follows. Since W it is bounded and types take discrete values, the response of an agent i to a change in his type at time t is bounded. In turn, this implies that aggregate play at time t varies Lipschitz continuously with a i,t 1. This concludes the induction. QED. 4. C C We should briefly comment on some aspects of our model and results. First, because our assumptions are suffi cient rather than necessary and because we have two distinct sources of complementarities, our framework allows for the possibility that our uniqueness results carry over to settings in which there are not strategic complementarities but where the type-action complementarities are strong enough to outweigh this and vice versa. Second, because our game features strategic complementarities and positive spillovers, the unique equilibrium that survives iterative strict dominance has interesting welfare properties. In particular, higher economic fundamental or higher cross-sectional type profiles induce Pareto welfare improvements. Third, the strategic complementarities in our model provide a mechanism for the amplification of exogenous shocks, while the type-action complementarities provide a mechanism for the propagation of shocks across time. 18 As for period T 1, we can repeat the same arguments to establish that V it is Lipschitz continuous in s t.

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