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.     

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games.     

Cournot-Walras and Cournot equilibria in mixed markets: A comparison

Summary In this paper, we show that, in markets with a continuum of traders and atoms, the set of Cournot-Walras equilibria and the set of Cournot equilibria may be disjoint. We show also that, when the preferences of the traders are represented by Cobb-Douglas utility functions, the set of Cournot-Walras equilibria and the set of Cournot equilibria have a nonempty intersection.I would like to thank R. Amir, B. de Meyer, J. J. Gabszewicz, J.-F. Mertens, H. Polemarchakis and L. Ventura for some helpful discussions about the questions examined in the present work.     

Adaptation and complexity in repeated games

The paper presents a learning model for two-player infinitely repeated games. In an inference step players construct minimally complex inferences of strategies based on observed play, and in an adaptation step players choose minimally complex best responses to an inference. When players randomly select an inference from a probability distribution with full support the set of steady states is a subset of the set of Nash equilibria in which only stage game Nash equilibria are played. When players make ‘cautious’ inferences the set of steady states is the subset of self-confirming equilibria with Nash outcome paths. When players use different inference rules, the set of steady states can lie between the previous two cases.     

Pareto optima and equilibria when preferences are incompletely known

An exchange economy in which agents have convex incomplete preferences defined by families of concave utility functions is considered. Sufficient conditions for the set of efficient allocations and equilibria to coincide with the set of efficient allocations and equilibria that result when each agent has a utility in her family are provided. Welfare theorems in an incomplete preferences framework therefore hold under these conditions and efficient allocations and equilibria are characterized by first order conditions.     

Regularity and Pareto improving on financial equilibria with price-dependent borrowing restrictions

We consider a pure exchange, general equilibrium model, with two periods and a finite number of states, commodities, numeraire assets, and households. Participation in the asset markets is restricted in a household specific manner, imposing upper bounds on the amounts of borrowing which can be obtained using assets. Those bounds are assumed to depend on asset and commodity prices. After establishing existence of equilibria, we show that, generically in the set of the economies, equilibria are finite and regular. Then, we restrict our attention to the significant set of economies in which some associated equilibria exhibit a sufficiently high number of strictly binding participation constraints. We prove that, generically in that set, those equilibria are Pareto improvable through a local change of the participation constraints.     

On the (non-)lattice structure of the equilibrium set in games with strategic substitutes

This paper studies models where the optimal response functions under consideration are not increasing in endogenous variables, and weakly increasing in exogenous parameters. Such models include games with strategic substitutes, and include cases where additionally, some variables may be strategic complements. The main result here is that the equilibrium set in such models is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Indeed, for a given parameter value, a pair of distinct equilibria are never comparable. Therefore, with multiple equilibria, some of the established techniques for exhibiting increasing equilibria or computing equilibria that use the largest or smallest equilibrium, or that use the lattice structure of the equilibrium set do not apply to such models. Moreover, there are no ranked equilibria in such models. Additionally, the analysis here implies a new proof and a slight generalization of some existing results. It is shown that when a parameter increases, no new equilibrium is smaller than any old equilibrium. (In particular, in n-player games of strategic substitutes with real-valued action spaces, symmetric equilibria increase with the parameter.)      

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     

Menu theorems for bilateral contracting

This paper studies bilateral contracting where multiple principals negotiate contracts with multiple agents independently. It is shown that pure-strategy equilibrium allocations relative to any ad hoc set of feasible mechanisms are supported by pure-strategy perfect Bayesian equilibria relative to the set of menus. This paper also shows that all equilibrium allocations to any ad hoc set of feasible mechanisms are supported by correlated equilibria relative to the set of menus, where a state is a probability distribution function over payoff-relevant variables. Furthermore, all equilibrium allocations relative to the set of menus persist even if principals use more complex mechanisms.     

The Structure of Sunspot Equilibria: The Role of Multiplicity

This paper examines the structure of sunspot equilibria in a standard two period exchange economy with real assets. We show that for a generic choice of utility functions and endowments, there exists an open set of real asset structures whose payoffs are independent of sunspots such that the economy with this asset structure has a regular sunspot equilibrium. An important implication of our result is that the multiplicity of non-sunspot equilibria is not necessary for the existence of sunspot equilibria. Our technique is general and can be applied to show the existence of sunspot equilibria in other frameworks.     

Endogenous public policy and multiple equilibria

One of the well known roles of public policy in models with indeterminacy is to reduce the set of equilibria. However, agents' expectations regarding future policy may be self fulfilling when public policy is endogenized. We show that a simple overlapping generations economy with public education may yield multiple equilibria. Under a laissez-faire system, our model has a unique equilibrium. Thus, the presence of public policy may generate, rather than eliminate, multiple equilibria.     

Citizen-candidates,lobbies, and strategic campaigning

We study a spatial model of political competition in which potential candidates need a fixed amount of money from lobbies to enter an election. We show that the set of pure strategy Nash equilibria in which lobbies finance candidates whose policies they prefer among the set of entrants coincides with the set of Nash equilibria with weakly less than two entering candidates. Fixing lobbies’ preferences, if the total amount of money held by lobbies is finite, there exists some minimal distance between the two candidates’ positions. This minimal distance is a bound for all such Nash equilibria and is independent of the distribution of voters’ preferences. I would like to thank John Duggan, Al Slivinski, and William Thomson for useful comments and suggestions. Dan Kovenock and two anonymous referees also provided detailed comments and pointed out several errors. All errors are my own.     

Time-inconsistent preferences in a general equilibrium model

This paper introduces time-inconsistent preferences in a multicommodity general equilibrium framework with incomplete markets. The standard concept of competitive equilibrium is extended in order to allow for changes in intertemporal preferences. Depending on whether or not agents recognize that their intertemporal preferences change, agents are called sophisticated or naïve. This paper presents competitive equilibrium notions for economies with naïve agents and economies with sophisticated agents and provides assumptions under which both types of equilibria exist. Surprisingly, the set of naïve equilibria in societies populated by time-consistent households is not allocationally equivalent to the set of competitive equilibria. For sophisticated equilibria, the equivalence holds. Time-inconsistency also raises conceptual issues about the appropriate concept of efficiency. Choices have to be made concerning the incorporation of future preferences and the appropriate instruments to create Pareto improvements. For both naïve and sophisticated societies, we present four possible efficiency concepts. Suitable conditions are specified for which both naïve and sophisticated equilibria satisfy appropriate efficiency concepts.     

Secure implementation experiments: Do strategy-proof mechanisms really work?

Strategy-proofness, requiring that truth-telling is a dominant strategy, is a standard concept used in social choice theory. Saijo, Sjöström and Yamato [Saijo, T., Sjöström, T., Yamato, T., 2003. Secure implementation: Strategy-proof mechanisms reconsidered. Working paper 4-03-1. Department of Economics, Pennsylvania State University] argue that this concept has serious drawbacks. In particular, many strategy-proof mechanisms have a continuum of Nash equilibria, including equilibria other than dominant strategy equilibria. For only a subset of strategy-proof mechanisms do the set of Nash equilibria and the set of dominant strategy equilibria coincide. For example, this double coincidence occurs in the Groves mechanism when preferences are single-peaked. We report experiments using two strategy-proof mechanisms. One of them has a large number of Nash equilibria, but the other has a unique Nash equilibrium. We found clear differences in the rate of dominant strategy play between the two.     

Secure implementation experiments: Do strategy-proof mechanisms really work?

Strategy-proofness, requiring that truth-telling is a dominant strategy, is a standard concept used in social choice theory. Saijo, Sjöström and Yamato [Saijo, T., Sjöström, T., Yamato, T., 2003. Secure implementation: Strategy-proof mechanisms reconsidered. Working paper 4-03-1. Department of Economics, Pennsylvania State University] argue that this concept has serious drawbacks. In particular, many strategy-proof mechanisms have a continuum of Nash equilibria, including equilibria other than dominant strategy equilibria. For only a subset of strategy-proof mechanisms do the set of Nash equilibria and the set of dominant strategy equilibria coincide. For example, this double coincidence occurs in the Groves mechanism when preferences are single-peaked. We report experiments using two strategy-proof mechanisms. One of them has a large number of Nash equilibria, but the other has a unique Nash equilibrium. We found clear differences in the rate of dominant strategy play between the two.     

EQUILIBRIA IN NON-COOPERATIVE GAMES I: PERTURBATIONS BASED REFINEMENTS OF NASH EQUILIBRIUM

The concept of Nash equilibrium is widely used to analyse non-cooperative games. However, one of the problems with that concept is that many games have multiple equilibria. Recent work has concentrated on reducing or refining the set of Nash equilibria in some games. In this paper, we survey some equilibrium concepts based on perturbations of strategies that refine the set of Nash equilibria. We discuss the pros and cons of each concept and its relationship to the others by the use of numerous examples and intuition. It is hoped that this survey will enable the economist to consider the relevance of a particular equilibrium concept to a given economic model of interest. Journal of Economic Literature Classification Number: C72.     

Efficient Equilibria in the Voluntary Contributions Mechanism with Private Information

We characterize generally the Bayesian Nash equilibria of a voluntary contributions public goods game for two consumers with private information.The two consumers simultaneously make voluntary contributions to the public good, and the contributions are refunded if the total falls short of the cost of the public good. Several families of equilibria (step‐function, regular and semi‐regular) are studied. Necessary and sufficient conditions for regular and semi‐regular equilibrium allocations to be interim incentive efficient are derived. In the uniform distribution case we prove (i) the existence of an open set of incentive efficient regular equilibria when the cost of production is large enough and (ii) the existence of an open set of incentive efficient semi‐regular equilibria when the cost of production is low enough. Step‐function equilibra are proved to be interim incentive inefficient.     

Real Indeterminacy of Stationary Monetary Equilibria in Centralized Economies

We show that real indeterminacy of stationary equilibria, by which the set of stationary equilibria is a continuum and the real allocation varies among equilibria, may arise in some general equilibrium models with fiat money. The conditions under which such equilibria arise are: (i) each household optimally saves a constant amount of money; and (ii) at least two households face different budget constraints. We present various models, including a decentralized money search model and a centralized model with a monopoly firm, to explain how these conditions lead to real indeterminacy. Finally, we present a policy that uniquely implements any desirable outcome.