Variational convergence: Approximation and existence of equilibria in discontinuous games

We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.     

On the existence of pure-strategy perfect equilibrium in discontinuous games

We provide sufficient conditions for a (possibly) discontinuous normal-form game to possess a pure-strategy trembling-hand perfect equilibrium. We first show that compactness, continuity, and quasiconcavity of a game are too weak to warrant the existence of a pure-strategy perfect equilibrium. We then identify two classes of games for which the existence of a pure-strategy perfect equilibrium can be established: (1) the class of compact, metric, concave games satisfying upper semicontinuity of the sum of payoffs and a strengthening of payoff security; and (2) the class of compact, metric games satisfying upper semicontinuity of the sum of payoffs, strengthenings of payoff security and quasiconcavity, and a notion of local concavity and boundedness of payoff differences on certain subdomains of a player's payoff function. Various economic games illustrate our results.     

Stationary equilibria in bargaining with externalities

This paper studies infinite-horizon bargaining between a seller and multiple buyers when externalities are present. We extend the analysis in Jehiel and Moldovanu by allowing for both pure and mixed equilibria [Jehiel, P., Moldovanu, B., 1995a. Cyclical delay in bargaining with externalities. Rev. Econ. Stud. 62, 619–637]. A characterization of the stationary subgame perfect equilibria in generic games is presented. Equilibria with delay exist only for strong positive externalities. Since each buyer receives a positive payoff when the seller makes an agreement with some other buyer, positive externalities induce a war of attrition between buyers.     

Understanding some recent existence results for discontinuous games

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players’ action spaces are metric and locally convex, it implies the existence results of Reny (Econometrica 67:1029–1056, 1999) and Carmona (J Econ Theory 144:1333–1340, 2009) and that it is equivalent to a recent result of Barelli and Soza (On the Existence of Nash Equilibria in Discontinuous and Qualitative Games, University of Rochester, Rochester, 2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property.     

Maskin's Theorem with limited veto power

We weaken the no-veto power condition of Maskin [Maskin, E.,1999. Nash equilibrium and welfare optimality. Rev. Econ. Stud. 66, 23–38] to limited veto power, and prove that any monotonic social choice rule is Nash implementable if it satisfies this weaker condition. The result is obtained by using the canonical Maskin mechanism without modification. An immediate corollary is that the weak core is Nash implementable in any coalitional game environment. An example is given to show that the strong core need not be implementable, even when it is monotonic.     

Essential equilibria of discontinuous games

We introduce spaces of discontinuous games in which games having essential Nash equilibria are the generic case. In order to prove the existence of essential Nash equilibria in such spaces, we provide new results on the Ky Fan minimax inequality. In the setting of potential games, we show that games with essential Nash equilibria are the generic case when their potentials satisfy a condition called weak upper pseudocontinuity that is weaker than upper semicontinuity.     

A Simple Extension of the Dasgupta–Maskin Theorem,with an Application

This paper generalizes the classical existence results for games with discontinuous payoffs, developed by Dasgupta and Maskin in 1986. This new existence result is then applied to a simple duopoly model with quantity precommitment and Bertrand competition.     

Subgame perfect reaction function equilibria in discounted duopoly supergames are trivial

This paper is concerned with infinitely repeated duopoly games with discounting. A question which has been open since Friedman's (Rev. Econ. Stud. 35 (1968), 257–272) reaction function article is settled for a general class of games. The question is wheter nontrivial reaction function equilibria can be subgame perfect. This question is answered in the negative. Such equilibria must be trivial in the sense of prescribing the stage game noncooperative equilibrium actions in every period, independent of prior history.     

Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players’ payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of e{\epsilon}-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.      

A simple sufficient condition for strong implementation

In an important step forward Maskin [E. Maskin, Nash equilibrium and welfare optimality, Rev. Econ. Stud. 66 (1999) 23–38] showed that two properties – monotonicity and no veto power – are together sufficient for Nash implementation. In contrast to the vast literature that followed, this characterization has two major advantages: First, it is often easy to verify, and second, it has an elegant and simple interpretation. However, there does not exist a similar condition for social choice correspondences that are implementable in strong equilibrium. All existing characterizations are either hard to verify or apply only to comprehensive preference domains. In this paper we improve the situation by giving one such condition. Moreover, using well-known examples we show that this is a practical tool.     

Rapid evolution under inertia

This paper demonstrates that inertia driven by switching costs leads to more rapid evolution in a class of games that includes m×m pure coordination games. Under the best-response dynamic and a fixed rate of mutation, the expected waiting time to reach long-run equilibrium is of lower order in the presence of switching costs, due to the creation of new absorbing states that allow Ellison's [Ellison, G., 2000. Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev. Econ. Stud. 67, 17–45] “step-by-step” evolution to occur.     

Bargaining, binding contracts, and competitive wages

In a model where many workers bargain with one firm and sign binding contracts, we show existence of a stationary subgame perfect equilibrium. If the production function satisfies decreasing returns, each worker receives a share of his marginal product (treating all other workers as employed) in equilibrium. Thus, wages are competitive. This is in contrast to Stole and Zweibel (1996, Rev. Econ. Stud. 63, 375–410), who assume that contracts are non-binding and find that the payoff of a worker is a weighted average of the inframarginal contributions. Hence, binding contracts imply lower wages than non-binding contracts.     

Markov equilibria in dynamic matching and bargaining games

Rubinstein and Wolinsky [Rev. Econ. Stud. 57 (1990) 63] show that a simple homogeneous market with exogenous matching has a continuum of (non-competitive) perfect equilibria; however, the unique Markov-perfect equilibrium of this model is competitive. By contrast, in the more general case of heterogeneous markets, even the Markov property is not enough to guarantee the perfectly competitive outcome. We define a market game that allows for heterogeneous values on both sides of the market and exhibit a number of examples of (non-competitive) Markov-perfect equilibria, with and without discounting. Unlike the homogeneous case, these equilibria allow for inefficient trades and for trade at non-uniform prices. The non-competitive equilibrium may be unique.     

On equilibrium existence in payoff secure games

We propose a single framework for studying the existence of approximate and exact pure strategy equilibria in payoff secure games. Central to the framework is the notion of a multivalued mapping with the local intersection property. By means of the Fan-Browder collective fixed point theorem, we first show an approximate equilibrium existence theorem that covers a number of known games. Then a short proof of Reny’s (Econometrica 67:1029–1056, 1999) equilibrium existence theorem is provided for payoff secure games with metrizable strategy spaces. We also give a simple proof of Reny’s theorem in its general form for metric games in an appendix for the sake of completeness.     

Competition between asymmetrically informed principals

Summary. This paper derives the set of equilibria for common agency games in which the principals compete in piece rates and lump sum payments and one principal has incomplete information about the agent's preferences. We show that the uninformed principal's expected payoff function is discontinuous with respect to the identity of the marginal agent type. This discontinuity is shown to support an open set of equilibria. For games in which the first-best equilibrium strategies are measurable with respect to the uninformed principal's information partition, this result implies the existence of an open set of Pareto inefficient equilibria. Received: December 5, 1995; revised version August 18, 1996     

Nash equilibrium and party polarization in an electoral model with mixed motivations

We study the existence problem of Nash equilibrium as well as the patterns of equilibrium policy outcomes in an electoral competition model with mixed motivations. Each party maximizes a sum of party members’ expected utility and office rent. The inclusion of office rent renders the payoff of each party discontinuous. This makes it difficult to apply usually fixed point arguments to prove the existence of Nash equilibria. By using a recently developed concept, multiple restrictional security (MR‐security) we provide conditions under which a pure‐strategy Nash equilibrium exists within fairly general settings, and further the analysis by presenting conditions under which various patterns of policy choices, including polarization, arise in equilibrium.     

Coordination and transfer

We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.