Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Convergence to competitive equilibria and elimination of no-trade (in a strategic market game with limit prices)

The strategic market games literature contains many results that predict Walrasian equilibria in the competitive limit. However, they usually come at the expense of ad hoc assumptions that rule out “pathological” no trade equilibria. This paper studies a strategic market game with limit prices. The set of Nash equilibrium allocations of this game converges to the set containing all competitive equilibria and no-trade, when players are replicated. Moreover, two rounds of iterated deletion of weakly dominated strategies eliminate the no-trade equilibria. Hence, replication paired with two rounds of iterated dominance gives a clean prediction of competitive equilibrium.     

Determinacy of equilibrium in outcome game forms

We show the generic finiteness of the number of probability distributions on outcomes induced by Nash equilibria for two-person game forms such that either (i) one of the players has no more than two strategies or (ii) both of the players have three strategies, and (iii) for outcome game forms with three players, each with at most two strategies. Finally, we exhibit an example of a game form with three outcomes and three players for which the Nash equilibria of the associated game induce a continuum of payoffs for an open non-empty set of utility profiles.     

Equilibria in a class of aggregative location games

Consider a multimarket oligopoly, where firms have a single license that allows them to supply exactly one market out of a given set of markets. How does the restriction to supply only one market influence the existence of equilibria in the game? To answer this question, we study a general class of aggregative location games where a strategy of a player is to choose simultaneously both a location out of a finite set and a non-negative quantity out of a compact interval. The utility of each player is assumed to depend solely on the chosen location, the chosen quantity, and the aggregated quantity of all other players on the chosen location. We show that each game in this class possesses a pure Nash equilibrium whenever the players’ utility functions satisfy the assumptions negative externality, decreasing marginal utility, continuity, and Location–Symmetry. We also provide examples exhibiting that, if one of the assumptions is violated, a pure Nash equilibrium may fail to exist.     

Essential stability in games with endogenous sharing rules

We prove that essential games with endogenous sharing rules form a dense residual set and that every game with endogenous sharing rules has at least one minimal essential set of solutions. Furthermore, we establish that essential continuous games form a dense residual set and that every continuous game has at least one minimal essential set of Nash equilibria.     

Endogenous mechanisms and Nash equilibrium in competitive contracting games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each firm competes for the business of a buyer of unknown type by offering the buyer a catalog of products and prices. The timing in our model is Stackelberg: in the first stage, given the distribution of buyer types known to all firms and the deducible, type-dependent best responses of the agent, firms simultaneously and noncooperatively choose their catalog offers. In the second stage the buyer, knowing his type, chooses a single firm and product-price pair from that firm’s catalog. By backward induction, this Stackelberg game with asymmetric information reduces to a game over catalogs with payoff indeterminacies. In particular, due to ties within catalogs and/or across catalogs, corresponding to any catalog profile offered by firms there may be multiple possible expected firm payoffs, all consistent with the rational optimizing behavior of the agent for each of his types. The resolution of these indeterminacies depends on the tie-breaking mechanism which emerges in the market. Because each tie-breaking mechanism induces a particular game over catalogs, a reasonable candidate would be a tie-breaking mechanism which supports a Nash equilibrium in the corresponding catalog game. We call such a mechanism an endogenous Nash mechanism. The fundamental question we address in this paper is, does there exist an endogenous Nash mechanism—and therefore, does there exist a Nash equilibrium for the catalog game? We show under fairly mild conditions on primitives that catalog games naturally possess tie-breaking mechanisms which support Nash equilibria.     

An indifference trap of voting

In this paper strategic situations of voting and abstentions are analysed in a three-candidate voting model where voters are indifferent to various alternatives and candidates are interested in winning the election and seeing their most preferred alternative being realized. A specific decision problem is analysed, described by an Indifference Trap Game, with respect to, e.g., the corresponding Nash equilibria, perfect equilibria, and maximin solution. A second-best outcome is contained in the choice set of all three solutions while the Nash equilibrium concept is compatible with the Pareto efficient outcomes of the game.An alternative scenario where candidates suffer from incumbency but voters still are indifferent to some alternatives also supports the second-best outcome. Again, various solutions concepts are applied. We conclude that indifferent voters imply an eminent coordination problem for the candidates in the given voting game which, in general, ends up in inefficient outcomes. The inherent complexity of the decision situation cannot be sufficiently reduced to single out one and only one outcome.     

Strong Nash equilibrium in games with common and complementary local utilities

A rather general class of strategic games is described where the coalitional improvements are acyclic and hence strong Nash equilibria exist: The players derive their utilities from the use of certain facilities; all players using a facility extract the same amount of local utility therefrom, which amount depends both on the set of users and on their actions, and is decreasing in the set of users; the ultimate utility of each player is the minimum of the local utilities at all relevant facilities. Two important subclasses are “games with structured utilities,” basic properties of which were discovered in 1970s and 1980s, and “bottleneck congestion games,” which attracted researchers’ attention quite recently. The former games are representative in the sense that every game from the whole class is isomorphic to one of them. The necessity of the minimum aggregation for the existence of strong Nash equilibria, actually, just Pareto optimal Nash equilibria, in all games of this type is established.     

Implementing the nash extension bargaining solution for non-convex problems

Conley and Wilkie (1993) introduced an axiomatization at the Nash extension bargaining solution defined on a domain of comprehensive but not necessarily convex problems. In this paper we present a non-cooperative game which implements the Nash extension solution in subgame perfect equilibria in the limit as the discount rate applied between rounds of play vanishes.     

Double implementation of the ratio correspondence by a market mechanism

To overcome deficits of the Lindahl solution concept when the economy does not exhibit constant returns to scale, Kaneko (1977a) introduced the concept of aratio equilibrium. Theratio correspondence selects for each economy its set of ratio equilibrium allocations. In this paper we provide a simple market game thatdouble implements the ratio correspondence in Nash and strong equilibria.     

Equilibrium selection in games: the mollifier method

This paper introduces an embedding of a Nash equilibrium into a sequence of perturbed games, which achieves continuous differentiability of best responses by mollifying them over a continuously differentiable density with compact support (window size). Along any sequence with shrinking window size, the equilibria are single-valued whenever the function has a regular Jacobian and the set of equilibria where it is singular has measure zero. We achieve a further reduction of the equilibrium set by inserting within the embedding a procedure that eliminates the strict interior of equilibrium sets. The mollifier can be approximated consistently using kernel density regression, and we sketch a non-stationary stochastic optimization algorithm that uses this approximation and converges with probability one to an equilibrium of the original game.     

A note on the symmetry of all Nash equilibria in games with increasing best replies

It is well known that a symmetric game has only symmetric (pure strategy) Nash equilibria if its best-reply correspondences admit only increasing selections and its strategy sets are totally ordered. Several nonexamples of the literature show that this result is generally false when the totality condition of the relation that orders the strategy sets is simply dropped. Making use of the structure of interaction functions, this note provides sufficient conditions for the symmetry of all (pure strategy) Nash equilibria in symmetric games where best-reply correspondences admit only increasing selections, but strategy sets are not necessarily totally ordered.     

Pre-play communication in procurement auctions: Silence is not golden

I study the effect of cheap talk between bidders on the outcome of a first-price procurement auction in which participation is costly. Although no side payments or commitments are allowed, there exists a family of equilibria in which sellers use communication to collude on a subset of participants and/or reveal information about their cost. Cheap talk matters in the sense that it strictly enlarges the set of Nash equilibria (symmetric and asymmetric) and the set of public correlated equilibria of the game. I show that the buyer may benefit from cheap talk between sellers and that the surplus increases in the amount of information revealed in equilibrium under one fairly general condition. This is because when communication is cheap, sellers cannot directly collude on higher prices. Rather, communication leads to competition between fewer, but more aggressive bidders, which entails greater allocative efficiency and a decrease in the total wasteful entry cost.     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

Debt contracts and cooperative improvements

In this paper, we consider a dynamic game with imperfect information between a borrower and lender who must write a contract to produce a consumption good. In order to analyze the game, we introduce the concept of a coalitional perfect Bayesian Nash equilibrium (cPBNE). We prove that equilibria exist and are efficient in a precise sense, and that deterministic contracts that resemble debt are optimal for a general class of economies. The cPBNE solution concept captures both the non-cooperative aspect of firm liquidation and the cooperative aspect of renegotiation.     

Large economies and two-player games

We characterize the core and the competitive allocations of a continuum economy as strong Nash equilibria of an associated game with only two players.     

Full Nash implementation of neutral social functions

This paper characterizes neutral social functions that are fully implementable. A neutral social function must be monotonic and simple to be fully implementable under either the Nash equilibrium concept or the strong Nash equilibrium concept. If a neutral social function is monotonic and simple, then the social function is fully implementable by a set of Nash equilibria. For finite alternative sets a neutral social function will be fully implementable by a set of strong Nash equilibria if and only if it is monotonic, simple and dictatorial.     

Bargaining solutions with non-standard objectives

I examine the pure-strategy solutions of the sealed-bid bargaining game with incomplete information, when the buyer's and seller's objectives are other than the standard objective, namely maximization of expected profit. The motivation for this exploration lies in three problems of the standard formulation: the necessity of assuming common priors, the existence of uncountably many Nash equilibria, with no means for the players to coordinate on any one of them, and the uncertain relationship between these equilibria and observed behavior in bargaining experiments. Specifically, I consider two alternative objectives: minimization of maximum regret, and maximization of maximum profit. The solution concept here is not Nash equilibrium, but rather -individually rational strategy bundle. For that reason, I shall, where appropriate, use the word “solution” in place of “equilibrium.” Yet we find that the notion of Nash Equilibrium reappears, in a sense to be explained. In the minimax-regret case I find (in contrast to the case of expected profit) a unique solution; this solution reduces, for priors with coincident support, to the linear equilibrium of Chatterjee-Samuelson. In the maximum-profit case there are many solutions; they turn out to be slight generalizations of the one-step equilibria of Leininger-Linhart-Radner.     

Nash equilibrium reconsidered and an option for maximin

In this paper we will discuss some peculiarities of the Nash equilibrium which are at odds with its standard applications: (a) the underlying dynamic interpretation, (b) the incentive independency if equilibria are mixed, and (c) the unprofitability if equilibrium and maximin are mixed and the game is 2-by-2. Maximin is proposed as an alternative solution concept in “relevant situations”.     

The stochastic lake game: A numerical solution

In this paper, we numerically solve a stochastic dynamic programming problem for the solution of a stochastic dynamic game for which there is a potential function. The players select a mean level of control. The state transition dynamics is a function of the current state of the system and a multiplicative noise factor on the control variables of the players. The particular application is for lake water usage. The control variables are the levels of phosphorus discharged (typically by farmers) into the watershed of the lake, and the random shock is the rainfall that washes the phosphorus into the lake. The state of the system is the accumulated level of phosphorus in the lake. The system dynamics are sufficiently nonlinear so that there can be two Nash equilibria. A Skiba-like point can be present in the optimal control solution.We analyze (numerically) how the dynamics and the Skiba-like point change as the variance of the noise (the rain) increases. The numerical analysis uses a result of Dechert (1978. Optimal control problems from second order difference equations. Journal of Economic Theory 19, 50–63) to construct a potential function for the dynamic game. This greatly reduces the computational burden in finding Nash equilibria solutions for the dynamic game.