Elsevier

Journal of Economic Theory

Volume 155, January 2015, Pages 95-130
Journal of Economic Theory

Extremal choice equilibrium with applications to large games, stochastic games, & endogenous institutions

https://doi.org/10.1016/j.jet.2014.11.010Get rights and content

Abstract

We prove existence and purification results for strategic environments possessing a product structure that includes classes of large games, stochastic games, and models of endogenous institutions. Applied to large games, the results yield existence of pure-strategy equilibria allowing for infinite-dimensional externalities. Applied to stochastic games, the results yield existence of stationary Markov perfect equilibria with extremal payoffs, which in turn yields existence of pure strategy stationary Markov perfect equilibria for games with sequential moves. Applied to the model of institutions, we obtain equilibrium existence with general group decision correspondences.

Introduction

We study the question of existence of solutions in a general class of strategic environments that includes large games, stochastic games, and endogenous institutions. In the context of large games, we obtain existence of Nash equilibria, and when underlying pure action sets are finite, our results deliver existence in pure strategies. Ours is the first such result that accommodates infinite-dimensional externalities but still uses standard measure spaces (e.g., [0,1] with Lebesgue measure), as opposed to saturated measure spaces. For stochastic games, we extend the existence theorem of Duggan [14] for noisy stochastic games to obtain stationary Markov perfect equilibria such that payoffs are in the closure of the extreme points of the set of Nash equilibrium payoffs of the associated auxiliary one-shot games. In the special case of sequential move games, our results imply existence of stationary Markov perfect equilibrium in pure strategies. Finally, we consider a framework of endogenous institutions such that a large number of individuals sort into groups and then take collective decisions within groups. Generalizing a result due to Caplin and Nalebuff [10], we establish existence of an equilibrium when group decisions can be multi-valued, as is the case for common voting rules.

We obtain these results as applications of an abstract existence theorem in a general setting characterized by a product structure of the form T×U, where the interpretation of the spaces T and U depends on the application considered. In the large game application, we view T×U as the set of players, so we identify a player with a pair (t,u), where t is a general characteristic and u is a personal characteristic. We assume payoffs depend on own actions and the profile of average actions across general characteristics. That is, we “integrate out” personal characteristics, and we let average actions vary arbitrarily across general characteristics to accommodate infinite-dimensional externalities. In the stochastic games application, we view T×U as the set of states, where t is a general component of the state and u is a noise component that is payoff relevant in the current period but is not directly affected by last period's state and actions. Applied to endogenous institutions, we view T×U as a society consisting of individuals who must select into groups, where t is a public characteristic of an individual that affects others and u is a private characteristic.1 In particular, our arguments permit the space T to be infinite by exploiting this product structure, integrating out the idiosyncratic variable u and using known results on parameterized integrals of correspondences to verify key continuity conditions. Adding an assumption of non-atomicity of the idiosyncratic variable u, we sharpen our results to deliver solutions with extremal properties, which translate into pure strategy existence results in applications.

Our main existence theorem may be viewed as an abstract fixed point result, with no immediate interpretation in terms of a game—there are, for example, no players and no payoff functions—but it exploits structure that appears in a number of economic environments, as evidenced by the applications we provide. To illustrate the usefulness of the result, consider the following example of a static model of competition among firms.

Consider a market composed of many firms, where each firm is characterized by its location t and technological characteristic u. Assume that there is a continuum of locations and of technologies, each represented by the unit interval [0,1]. Each firm i=(t,u) produces a vector q(i)d of commodities belonging to a production set Q(i). Let α(t)=uq(t,u)du denote the aggregate production vector at location t, averaging over technologies u. Assume that prices are determined by product and factor market clearing in each location, where consumers and workers may (at some cost) travel to transact in markets at different locations, so that prices and profits depend on the aggregate production function α. We can then write the profit of firm i producing vector q given aggregate production α as πi(q;α). Thus, we have a large game with N=T×U being the player set (the firms), where T=U=[0,1] are the spaces of location and technologies (endowed with the standard Lebesgue measure), Q(i) being the action set, and πi being the payoff function of player iN. The externality caused by other firms on a given firm i is summarized by the infinite-dimensional aggregate production function α. This is to be contrasted with similar models in the literature that either only allow for finite-dimensional externalities (see, e.g., Yu and Zhu [38]) or manage to allow for infinite-dimensional externalities by imposing a much richer measurable structure on T and U (i.e., by using a super-atomless measure instead of the Lebesgue measure, as in Carmona and Podczeck [12]). Let M(i;α) denote the best response correspondence of firm i: that is, given the aggregate production α determined by the choices of all other firms, we compute the set of production vectors qQ(i) that maximize πi(q;α). An equilibrium of the game is thus a function q() such that q(i)M(i;α) for all firms i, where α is the aggregate production associated with q(), i.e., α(t)uq(t,u)du. As we argue next, our general fixed point result allows us to establish fairly general conditions ensuring existence (and purification) of equilibria in such a general Cournot game.

Our general framework is formulated abstractly, and our main theorem can be viewed as a fixed point theorem that exploits a product structure on its domain.2 To convey the idea, we define a choice function γ as assigning to each pair (t,u) a choice in d, where we view t as a systematic variable and u as an idiosyncratic variable. We then calculate the corresponding average choice function, α, by taking the marginal, α(t)=uγ(t,u)du, of γ across the idiosyncratic variable pointwise for each t. We then assign a choice set M(t,u;α) to each pair (t,u), where by construction these sets are parameterized by average choices, and we define a choice equilibrium as a mapping γ such that for all (t,u) pairs, γ(t,u) belongs to the choice set M(t,u;α) determined by the corresponding average choices. The firm competition example above is immediately seen as a special case of the general framework upon setting γ=q. Beyond existence, assuming u is non-atomically distributed, we provide an “extremization” result: for every choice equilibrium, there is an extremal choice equilibrium γˆ that chooses from the (closure of) extreme points of choice sets M(t,u;αˆ) such that γˆ is equivalent to γ, in the sense that it determines the same average choices and, therefore, the same choice sets for all (t,u) pairs.

The existence argument takes place in the space of average choice functions. We define S(α) as the set of selections of the correspondence tuM(t,u;α)du, and we prove existence of a fixed point αS(α) that is generated by an equilibrium choice function γ. The fixed point argument surmounts a number of technical challenges. To ensure sequential upper hemicontinuity of S, we apply a result of Artstein [3] on weak limits of sequences of integrable functions, and as the space of average choice functions is not necessarily (weakly) compact or metrizable, we apply a recent result of Agarwal and O'Regan [1] to obtain a fixed point, α. Finally, we employ the theorem of Artstein [4] to back out an equilibrium choice function γ consistent with α. Our purification argument relies on an application of a version of Lyapunov's theorem pointwise for each t, using non-atomicity of u; we then apply Artstein's theorem again to back out an extremal choice function. The latter step relies on a result establishing lower measurability of the extreme points of a lower measurable correspondence with nonempty, compact values in d.

We apply the results above to large games by indexing players as (t,u), where t is a general characteristic of the player and u is a personal characteristic, and we interpret choice set M(t,u;α) as the set of best response actions of player (t,u), given the average action α as a function of the general characteristic. Thus, given a strategy profile σ, the average action α(t)uσ(t,u)du is an infinite-dimensional statistic of the actions of the players. We then consider a large game such that this statistic (the “externality”) captures the influence of the rest of the players on a given player's outcome, and a choice equilibrium corresponds to a Nash equilibrium of the large game. We provide general sufficient conditions on the players' preferences to generate well-behaved choice sets M(t,u;α) and therefore existence of an extremal equilibrium. We illustrate the approach with the above-mentioned example of a Cournot game with many firms characterized by their location and production technology: our results deliver existence of equilibrium while allowing market clearing conditions in infinitely many locations to affect the price received by a given firm, generalizing results in the literature. When feasible action sets have a simplicial structure and payoffs are multilinear, our result delivers existence in pure strategies, the first such result using standard measure spaces; the cost is that underlying pure action sets must be finite, whereas with saturated measure spaces they can be more general.

The application to noisy stochastic games interprets a pair (t,u) as a decomposition of a state in a stochastic game, with the first component t satisfying the usual assumptions in the literature and the second component u being conditionally independent of the previous state and actions. That is, t is a general component of the state, and u is the “noise” component, as in Duggan [14]. Choice sets M(t,u;α) are the sets of Nash equilibrium payoffs of auxiliary one-shot games indexed by the vector of (interim) continuation values, given by the average choice α, so a choice equilibrium selects Nash equilibria from each such one-shot game. Dynamic programming ideas then show that a choice equilibrium corresponds to a stationary Markov perfect equilibrium of the noisy stochastic game. We therefore establish existence of a particular kind of stationary equilibrium, namely one that selects extreme points from the set of Nash equilibrium payoffs of auxiliary one-shot games. This refinement implies existence of a pure strategy stationary Markov perfect equilibrium for noisy stochastic games with sequential moves. We illustrate with a dynamic sequential oligopoly game with random movers, where our results guarantee existence of a pure strategy stationary Markov perfect equilibrium.

In the application to endogenous institutions, we identify an individual with a pair (t,u), where t is a public characteristic and u is a private characteristic that does not affect other individuals. We establish existence of a pure strategy equilibrium in a general framework for endogenous sorting into groups and collective decisions within groups, allowing for general spaces of individual characteristics and preferences (including crowding effects within groups), for constraints on group membership that depend on individual characteristics, and for group decision correspondences, which arise naturally for common voting rules; the latter assumption generalizes Caplin and Nalebuff [10], who restrict attention to single-valued mappings. The existence argument proceeds by transforming the model into our abstract framework by adding artificial agents to represent collective decisions within groups. Here, M(t,u;α) consists of mixtures over the optimal groups for the original individuals (t,u), given the distribution of general characteristics across groups summarized by α, and for the artificial agent representing group j, M(t,u;α) consists of the set of group decisions for the group given membership profile α. We apply our general existence and extremization results to obtain the desired equilibrium in the institutional model. We illustrate with an example of super majority voting over local public goods, in which group decisions are made by voting (according to a quota rule) and individuals select into groups, where our results guarantee existence of an endogenous institutional equilibrium.

Our existence result for choice equilibria in the abstract framework is comparable but non-nested with Theorem 2.2.1 of Balder [8]. The latter paper establishes existence of pure strategy equilibria in pseudogames that are more general than our framework in that his action sets may be infinite-dimensional, but less general in that it assumes externalities are finite-dimensional. At a finer level of detail, the papers also differ in how convexity conditions are formulated; see Remark 3.6 for an explanation of how the latter distinction arises from our general treatment of externalities. Note that when the idiosyncratic variable u is non-atomically distributed for all t, as would be the case in most applications, convexity is not needed in either framework, so the distinction is moot in this case.

In large games, Schmeidler [34] first provides conditions for existence of Nash equilibrium in pure strategies with finite sets of pure actions using the integral of the strategy as the “societal response” (also called the “externality”). Since action sets are finite, such externalities are finite-dimensional; our Corollary 3.2 generalizes that result by allowing richer, infinite-dimensional, externalities. Mas-Colell [28] uses the distribution of the strategy as the externality, and under such alternative formulation is able to establish existence of mixed strategy equilibria for general action sets.3 As indicated by Khan, Rath, and Sun [24], once infinite action sets are considered, it makes a difference how one considers the effect of the other players' actions on a given player's payoffs. We follow their paper and others in capturing externalities as an integral (an average) rather than a distribution, and we refer to their arguments in favor of the integral approach over the distribution approach. In this class of games, our existence results for Nash equilibria in large games is non-nested with respect to the results in Martins-da-Rocha and Topuzu [27] and Balder [8], as we allow for infinite-dimensional externalities at the cost of finite-dimensional action sets. With respect to Khan, Rath, and Sun [24], we provide a modeling approach that allows us to handle infinite-dimensional externalities (as integrals) without relying on an infinite-dimensional version of Lyapunov's theorem. In particular, letting σ denote a strategy profile and σ(t,u) denote the action of player (t,u), the approach in Khan, Rath, and Sun [24] would be to condense externalities to the finite-dimensional statistic β=(t,u)σ(t,u)d(t,u), with the implication that two strategy profiles σ and σˆ with β=βˆ would be considered equivalent by all players. In contrast, in our model, it is the infinite-dimensional statistic α()=uσ(,u)du to which players best respond; it is obviously possible to have ααˆ while β=βˆ, so players react to a richer set of externalities in our formulation.

This modeling strategy circumvents the failure of Lyapunov's theorem in infinite dimensions, without resorting to extremely diffuse environments (i.e., saturated measure spaces with super-atomless measures), as in Podczeck [31], Carmona and Podczeck [11], [12], and Khan, Rath, Sun, and Yu [25].4 It is worth noting that, under Mas-Colell's distribution approach for a large game, Noguchi [29] has recently proved existence of a pure strategy equilibrium allowing for general action sets, under the assumption that there are “many agents of every type.” This assumption allows him to circumvent the failure of Lyapunov's theorem in infinite dimensions without resorting to extremely diffuse environments, but in our framework, the assumption means that u is payoff irrelevant; as such, it reduces to a randomization device for player t, limiting the usefulness of the results for applications.

The existence of an extremal stationary Markov perfect equilibrium in a noisy stochastic game is new, as is the extension to pure strategy equilibria in sequential move games. These results build on Duggan [14], which in turn generalizes the result of Nowak and Raghavan [30]. We note that the product structure on states is crucial in obtaining existence of pure strategy stationary equilibria in sequential move games: restricting to sequential move games in the framework of Nowak and Raghavan [30] only allows one to remove correlation from their equilibrium; with the product structure, we can remove mixing altogether. The pure strategy existence result generalizes Theorem 5 in Duggan and Kalandrakis [16], which holds for a special class of sequential move stochastic games such that states are finite-dimensional, the transition on the general component of the state is smooth, and the noise component of the state enters into payoffs in a non-degenerate way. On the other hand, they show that all equilibria are essentially pure and possess desirable continuity properties, results we do not replicate in our more general framework. Recently, Jaśkiewicz and Nowak [23] show existence of a pure strategy Markov perfect equilibrium in stochastic overlapping generations models with sequential moves allowing for more general payoff functions (including hyperbolic discounting). Rather than assuming an atomless noise component of the state, those authors assume that the state transition is a convex combination of a fixed, finite set of atomless transition probabilities.

Models of local public goods with mobility trace back to early work of Tiebout [35], and existence of equilibrium has been a focus since Westhoff [36], Epple et al. [18], and Konishi [26]. We adopt the general framework of Caplin and Nalebuff [10], who show that equilibria may fail to exist, due to an incentive of some types of individuals to “run toward” others, and an incentive for the latter types to “run from” the former. Those authors propose three routes to equilibrium existence, the second of which endows the set of individuals with a product structure, so that an individual is identified as (t,u), where t is a general characteristic that may affect the well-being of others, and un is a vector of additive preference shocks on the utility of belonging to any given institution. Our results maintain the decomposition of individuals into public and private characteristics and generalize this route to existence in several ways: we let the set of individuals be the product of Polish spaces, rather than Euclidean spaces, and we weaken continuity of utility to measurability in the individuals' types; we allow the private component u to enter into individual preferences in a general way, instead of the additive form; we allow for arbitrary membership constraints that restrict entry into the groups; we allow an individual to care not only about the vector of group decisions but also about the composition of the group to which she belongs; and we allow for group decision correspondences, which associate sets of possible decisions to each group given the selection of individuals into groups.

In Section 2, we present the abstract framework and our general existence and purification result. Section 3 provides an application of our general results to large games, Section 4 takes up the case of noisy stochastic games, and Section 5 presents the application to endogenous institutions. The abstract theorem is proved in Appendix A; Appendix B contains the proofs of the results in Sections 3 Large games, 4 Stochastic games, 5 Endogenous institutions; and Appendix C contains a technical lemma on lower measurability of extreme points of a correspondence.

Section snippets

Informal discussion

We construct an abstract framework that takes as given a set N=T×U with a product structure and a correspondence A:Nd such that for each iN, A(i) represents a set of alternatives that are feasible at i. In the large game setting, A(i) is the set of actions available to player i; in the stochastic game setting, an element iN represents a state, and the set A(i) is a fixed set that bounds the payoff vectors of the players; in the endogenous institution setting, A(i) is a face of the unit

Large games

In this section, we formulate a class of large games as a special case of the abstract framework. We endow the set N of players with a product structure, so that a player is described by a general characteristic t and a personal characteristic u, where the latter are distributed independently conditional on t in the space of players. Letting T and U be complete, separable metric spaces with Borel sigma-algebras T and U, a product large game is described by a tuple (T,U,A,P,κ,ν) such that

  • N=T×U

Stochastic games

In this section, we consider the class of discounted stochastic games studied in Duggan [14] and map it into the framework of the current paper. Letting T and U be complete, separable metric spaces with Borel sigma-algebras T and U, a noisy stochastic game is a tuple (T,U,κ,νT,νU,(Xi,Bi,πi,δi)i=1m) with {1,,m} being the set of players such that:

  • T×U, with sigma-algebra TU, is the measurable space of states,

  • κ is a Borel probability measure on T,

  • νT:T×U×X×T[0,1] is a transition probability, with

Endogenous institutions

In this section, we consider a general framework for endogenous selection of individuals into groups (e.g., institutions, jurisdictions, or clubs) and collective decision within groups. We seek equilibria such that each of a continuum of individuals selects into her optimal group, given the distribution of other individuals across groups and collective decisions within the groups, and such that the collective decision of each group is consistent with the membership of that group. Letting T and U

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We thank the editor and an anonymous referee for helpful comments and suggestions.

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