Markov perfect equilibria in industries with complementarities

Summary.   This paper considers the existence and computation of Markov perfect equilibria in games with a “monotone” structure. Specifically, it provides a constructive proof of the existence of Markov perfect equilibria for a class of games in which a) there is a continuum of players, b) each player has the same per period payoff function and c) these per period payoff functions are supermodular in the player's current and past action and have increasing differences in the player's current action and the entire distribution of actions chosen by other players. The Markov perfect equilibria that are analyzed are symmetric, not in the sense that each player adopts the same action in any period, but rather in the sense that each player uses the same policy function. Since agents are typically distributed across many states they will typically take different actions. The formal environment considered has particular application to models of industries (or economies) in which firms face costs of price adjustment. It is in this context that the results are developed. Received: November 9, 1999; revised version: February 10, 2000     

Subgame perfect implementation: A necessary and almost sufficient condition

We present a necessary and almost sufficient condition for subgame perfect implementation of social choice correspondences. In societies with at least three individuals, any social choice correspondence which satisfies no veto power and our necessary Condition α is subgame perfect implementable. Thus Condition α is analogous to monotonicity which, by Maskin's celebrated characterization, is necessary and, in a similar way, almost sufficient for Nash implementation.     

Cooperative Outcomes through Noncooperative Games

This paper shows the equivalence between the stable solution set of any cooperative game in characteristic form (G1) and the subgame perfect Nash equilibria in pure strategies of a certain noncooperative game (G2). Players of G1 are named "agents." G2 is played by different players ("principals") who compete in wages to attract agents. The equivalence result holds when there are enough principals (if the game is superadditive, two principals suffice). Finally, another related cooperative game (G3) is constructed with both principals and agents as players. For G2 and G3 the same result is then proven, for any number of principals. Journal of Economic Literature Classification Numbers: C71 and C72.     

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately . Letting M=N→∞, the expected number of Nash equilibria is , where is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.     

Infinite Histories and Steady Orbits in Repeated Games

We study a model of repeated games with the following features: (a) Infinite histories. The game has been played since days of yore, or is so perceived by the players: (b) Turing machines with memory. Since regular Turing machines coincide with bounded recall strategies (in the presence of infinite histories), we endow them with "external" memory; (c) Nonstrategic players. The players ignore complicated strategic considerations and speculations about them. Instead, each player uses his/her machine to update some statistics regarding the others′ behaviour, and chooses a best response to observed behaviour. Relying on these assumptions, we define a solution concept for the one shot game, called steady orbit. The (closure of the) set of steady orbit payoffs strictly includes the convex hull of the Nash equilibria payoffs and is strictly included in the correlated equilibria payoffs. Assumptions (a)–(c) above are independent to a large extent. In particular, one may define steady orbits without explicitly dealing with histories or machines.     

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Evolution and Mixed Strategies

Selten (1980, J. Theoret. Biol., 84, 93–101) showed that no mixed equilibria are evolutionarily stable when players can condition their strategies on their roles in a game. Alternatively, Harsanyi's (1973, Int. J. Game Theory, 2, 1–23) purification argument implies that all mixed equilibria are approximations of strict, and hence evolutionarily stable, equilibria of games with slightly perturbed payoffs. This paper reconciles these results: Approximations of mixed equilibria have high invasion barriers, and hence are likely to persist, when payoff perturbations are relatively important and role identification is relatively noisy. When payoff perturbations are unimportant and role identification is precise, approximations of mixed equilibria will have small invasion barriers and are unlikely to persist. Journal of Economic Literature Classification Numbers: C70, C78.     

The local best response criterion: An epistemic approach to equilibrium refinement

The standard refinement criteria for extensive form games, including subgame perfect, perfect, perfect Bayesian, sequential, and proper, reject important classes of reasonable Nash equilibria and accept many unreasonable Nash equilibria. This paper develops a new refinement criterion, based on epistemic game theory, that captures the concept of a Nash equilibrium that is plausible when players are rational. I call this the local best response (LBR) criterion. This criterion is conceptually simpler than the standard refinement criteria because it does not depend on out-of-equilibrium, counterfactual, or passage to the limit arguments. The LBR is also informationally richer because it clarifies the epistemic conditions that render a Nash equilibrium reasonable. The LBR criterion appears to render the traditional refinement criteria superfluous.     

Monetary equilibrium in an overlapping generations model with productive capital

Summary We study perfect foresight competitive equilibrium in an overlapping generations model with productive capital and a fixed nominal stock of money. We obtain almost-complete characterizations of (a) the existence of a monetary equilibrium from an arbitrary initial capital stock, and (b) the existence of anefficient monetary equilibrium from an arbitrary initial capital stock. When the initial capital stock is no larger than the golden rule stock, the necessary and sufficient condition for both (a) and (b) is the dynamic inefficiency (in the sense of Malinvaud) of the autarkic (or nonmonetary) equilibrium from the same initial stock. However, this condition, though necessary, isnot sufficient for the existence of a monetary equilibrium when the initial stock exceeds the golden rule stock (and still more conditions are needed for anefficient monetary equilibrium to exist). We provide characterizations for these cases, and as corollaries obtain examples in which (a) the nonmonetary equilibrium is inefficient but no monetary equilibrium exists, and (b) monetary equilibria exist but no efficient monetary equilibrium does.We are grateful to a co-editor and an anonymous referee for comments that greatly improved the exposition in the paper.     

Congestion games with malicious players

We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counterintuitive phenomenon which we demonstrate is the “windfall of malice”: paradoxically, when a myopically malicious player gains control of a fraction of the flow, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

Commitment Robust Equilibria and Endogenous Timing

This paper investigates which equilibria of a game are still viable when players have the opportunity to commit themselves. To that end we study a model of endogenous timing in which players face the trade-off between committing early and moving late. It is shown that mixed (resp. pure) equilibria of the original game are subgame perfect (resp. persistent) in the timing game only when no player has an incentive to move first. Consequently, mixed equilibria are viable only if no player has an incentive to move first. One needs strong evolutionary solution concepts to draw that conclusion for pure equilibria.Journal of Economic LiteratureClassification Number: C72.     

Undominated Nash Implementation with Collusion and Renegotiation

This article looks at implementation in economic environments when agents have perfect information about the state of the world, but cannot commit not to renegotiate bad outcomes or to collude against each other. If renegotiation satisfies a weak condition of disagreement point monotonicity, then any Pareto-efficient social choice function can be implemented if there are at least three agents who play undominated Nash equilibria. The mechanism does not use modulo or integer games, has no bad mixed strategy equilibria, and is “bounded.”Journal of Economic LiteratureClassification Number: D71.     

Strong equilibrium in cost sharing connection games

We study network games in which each player wishes to connect his source and sink, and the cost of each edge is shared among its users either equally (in Fair Connection Games—FCG's) or arbitrarily (in General Connection Games—GCG's). We study the existence and quality of strong equilibria (SE)—strategy profiles from which no coalition can improve the cost of each of its members—in these settings. We show that SE always exist in the following games: (1) Single source and sink FCG's and GCG's. (2) Single source multiple sinks FCG's and GCG's on series parallel graphs. (3) Multi source and sink FCG's on extension parallel graphs. As for the quality of the SE, in any FCG with n players, the cost of any SE is bounded by H(n) (i.e., the harmonic sum), contrasted with the Θ(n) price of anarchy. For any GCG, any SE is optimal.     

Convergence to perfect competition of a dynamic matching and bargaining market with two-sided incomplete information and exogenous exit rate

Consider a decentralized, dynamic market with an infinite horizon and incomplete information in which buyers and sellers' values for the traded good are private and independently drawn. Time is discrete, each period has length δ, and each unit of time a large number of new buyers and sellers enter the market. Within a period each buyer is matched with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, the seller and winning buyer exit with their realized utility. Traders who fail to trade either continue in the market to be rematched or exit at an exogenous rate. We show that in all steady state, perfect Bayesian equilibria, as δ approaches zero, equilibrium prices converge to the Walrasian price and realized allocations converge to the competitive allocation.     

Meaning and Credibility in Cheap-Talk Games

I define neologism-proofness, a refinement of perfect Bayesian equilibrium in cheap-talk games. It applies when players have a preexisting common language, so that an unexpected message′s literal meaning is clear, and only credibility restricts communication. I show that certain implausible equilibria are not neologism-proof; in some games, no equilibrium is. Journal of Economic Literature classification numbers: D83 D82 C73.     

Dynamic legislative policy making

We prove existence of stationary Markov perfect equilibria in an infinite-horizon model of legislative policy making in which the policy outcome in one period determines the status quo for the next. We allow for a multidimensional policy space and arbitrary smooth stage utilities, and we assume preferences and the status quo are subject to arbitrarily small shocks. We prove that equilibrium continuation values are differentiable and that proposal strategies are continuous almost everywhere. We establish upper hemicontinuity of the equilibrium correspondence, and we provide weak conditions under which each equilibrium of our model determines an aperiodic transition probability over policies. We establish a convergence theorem giving conditions under which the invariant distributions generated by stationary equilibria must be close to the core in a canonical spatial model. Finally, we extend the analysis to sequential move stochastic games and to a version of the model in which the proposer and voting rule are determined by play of a finite, perfect information game.     

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].     

Consistency of beliefs and epistemic conditions for Nash and correlated equilibria

A consistency condition (action-consistency) on the interim beliefs of players in a game is introduced. Action-consistency is weaker than common priors and, unlike common priors, is characterized by a “no-bets” condition on verifiable events. Using action-consistency, we provide epistemic conditions to Nash and correlated equilibria weakening the common knowledge restrictions in Aumann and Brandenburger [Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180] and Aumann [Aumann, R., 1987. Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55, 1–18].