Equilibrium Refinement Research Articles

Rationality and Coherent Theories of Strategic Behavior

A non-equilibrium model of rational strategic behavior that can be viewed as a refinement of (normal form) rationalizability is developed for both normal form and extensive form games. This solution concept is called aτ-theory and is used to analyze the main concerns of the Nash equilibrium refinements literature such as dominance, iterative dominance, extensive form rationality, invariance, and backward induction. The relationship betweenτ-theories and dynamic learning is investigated.Journal of Economic LiteratureClassification Number: C72.

Equilibria and Pareto optima of markets with adverse selection

This paper examines the efficiency properties of competitive equilibrium in an economy with adverse selection. The agents (firms and households) in this economy exchange contracts, which specify all the relevant aspects of their interaction. Markets are assumed to be complete, in the sense that all possible contracts can, in principle, be traded. Since prices are specified as part of the contract, they cannot be used as free parameters to equate supply and demand in the market for the contract. Instead, equilibrium is achieved by adjusting the probability of trade. If the contract space is sufficiently rich, it can be shown that rationing will not be observed in equilibrium. A further refinement of equilibrium is proposed, restricting agents' beliefs about contracts that are not traded in equilibrium. Incentive-efficient and constrained incentive-efficient allocations are defined to be solutions to appropriately specified mechanism design problems. Constrained incentive efficiency is an artificial construction, obtained by adding the constraint that all contracts yield the same rate of return to firms. Using this notion, analogues of the fundamental theorems of welfare economics can be proved: all refined equilibria are constrained incentive-efficient and all constrained incentive-efficient allocations satisfying some additional conditions can be decentralized as refined equilibria. A constrained incentive-efficient equilibrium is typically not incentive-efficient, however. The source of the inefficiency is the equilibrium condition that forces all firms to earn the same rate of return on each contract.

Fully Revealing Equilibria with Suboptimal Investment

This paper examines investment and financing policy in equilibria--equilibria in which information asymmetries are resolved. Since all securities are priced correctly in a fully revealing equilibrium, it seems plausible that such equilibria would be free of the well known Myers-Majluf (1984) problem of inefficient investment. I show to the contrary that, for a large class of problems, whenever there is an equilibrium with efficient investment, there are also infinitely many equilibria in which almost all firms invest inefficiently. These inefficient outcomes survive the standard signaling-game equilibrium refinements. There are also examples that have fully revealing equilibria with inefficient investment but none with efficient investment. These findings contradict the claim of Constantinides and Grundy (1989) that firms invest the socially optimal amount in any fully revealing equilibrium.

Deviations, Dynamics, and Equilibrium Refinements

Existing equilibrium refinements rule out Nash equilibria susceptible to deviations. We propose a framework for considering not only equilibria impervious to deviations, but also equilibria likely to recur in the long run because they are repeatedly deviated to. We explore which equilibria are recurrent with respect to the deviations underlying some existing signaling refinements. We show that the set of recurrent equilibria based on Cho and Kreps's (1987) intuitive criterion is equivalent to their solution concept, but that applying our framework to existing cheap-talk refinements make those solution concepts more realistic and guarantee existence where their current formulations do not.Journal of Economic LiteratureClassification Numbers: B49, C72, C73.

Equilibrium Refinement for Infinite Normal-Form Games

We present three distinct approaches to perfect and proper equilibria for infinite normal form games. In the first two approaches, players tremble in the infinite games playing full support approximate best responses to others' strategies. In the strong approach, a tremble assigns high probability to the set of pure best responses ; in the weak approach, it assigns high probability to a neighborhood of this set. The third, limit-of-finite approach applies traditional refinements to sequences of successively larger finite games. Overall, the strong approach to equilibrium refinement most fully respects the structure of infinite games.

Pareto efficiency, simple game stability, and social structure in finitely repeated games

Simple game (sensu Brown and Vincent, 1987) evolutionary theory, when coupled with social structure measured as non‐random encounter of strategy “clones”, often permits equilibrium refinement leading to Pareto superior outcomes (e.g., Axelrod, 1981; Myerson et al., 1991), a foundational goal of economic game theory (Myerson, 1991: 370–375). This conclusion, derived from analyses of one‐shot and infinitely repeated games, fails for finitely repeated games. While mutant cluster invasion enhances Pareto efficiency of equilibria in the former, it can depress Pareto efficiency in the latter. Cooperative equilibria of finitely repeated games (under economic analysis) can be susceptible to cluster‐invasion by even more Pareto efficient strategies which are not themselves evolutionarily stable. Evolutionary (simple) game theory's ability to eliminate Pareto inferior Nash equilibrium strategies induces vulnerabilities foreign to economic analysis. Simple game analysis of finitely repeated games suggests that socia...

An early paper on the refinement of Nash equilibrium

An early paper on the refinement of Nash equilibrium

Game theory and applications

Contributed Papers: Refinement of Nash Equilibrium: The Main Ideas. Supergames. Infinitely Repeated Games with Incomplete Information. Repeated Games. Bounded Rationality and Strategic Complexity in Repeated Games. The Shapley Value. Advances in Value Theory. Axiomatizations of the Core, The Nucleolus, And the Prekernel. Consistency Principle. Discrete Concepts in N-Person Game Theory: Nondegeneracy and Homogeneity. Two-Sided Matching Markets: An Overview of Some Theory and Empirical Evidence. Strategic Market Game Models of Exchange Economies. Information Transmission. Monotonic Surplus Sharing and the Utilization of Common Property Resources. Developments in Stable Set Theory. Game Theoretic Models of Voting in Multidimensional Issue Spaces. Israel and the PLO: A Game with Differential Information . A Survey of Some Results Closely Related to the Knaster-Kuratowski-Mazurkiewz Theorem. Selected Abstracts: On the Core of the Assignment Game. The Second Welfare Theorem in Nonconvex Economies. An Evolutionary Game Theory Model for Risk-Taking. Values of Nonatomic Vector Measure Games: Are They Linear Combinations of the Measures? Subgame-Perfect Equilibria in Discrete and Continous Games-Does Discretization Matter? To Vote or Not to Vote: What Is the Quota? Some Bounds for the Banzhaf Index and Other Semivalues. Comparative Cooperative Games Theory. Escalation and Cooperation in International Conflicts-The Dollar-Auction Revisited. An Axiomatization of the Nonsymmetric Nontransferable Utility Value. A Milnor Condition for Nonatomic Lipschitz Games and Its Applications. Zero-Sum Nonstationary Stochastic Games with General State Space. Perfect Equilibrium Points and Lexicographic Domination. A Two-Person Repeated Bargaining Game with Long-Term Contracts. Independence of Irrelevant Alternative and Revealed Group Preferences. The Sealed-Bid Mechanism: An Experimental Study. Rate of Convergence to Full Efficiency in the Buyers' Bid Double Auction as the Market Becomes Large Stationary Strategies in Deterministic Games. Big Boss Games, Clan Games, And Information Market Games. Large Games and Economics with Near-Exhaustion of Gains to Coalition Formation. Values of Large Finite Games. Index.

The Predictive Utility of Generalized Expected Utility Theories

Many alternative theories have been proposed to explain violations of expected utility (EU) theory observed in experiments. Several recent studies test some of these alternative theories against each other. Formal tests used to judge the theories usually count the number of responses consistent with the theory, ignoring systematic variation in responses that are inconsistent. We develop a maximum-likelihood estimation method which uses all the information in the data, creates test statistics that can be aggregated across studies, and enables one to judge the predictive utility-the fit and parsimony-of utility theories. Analyses of 23 data sets, using several thousand choices, suggest a menu of theories which sacrifice the least parsimony for the biggest improvement in fit. The menu is: mixed fanning, prospect theory, EU, and expected value. Which theories are best is highly sensitive to whether gambles in a pair have the same support (EU fits better) or not (EU fits poorly). Our method may have application to other domains in which various theories predict different subsets of choices (e.g., refinements of Nash equilibrium in noncooperative games).

Nash refinement of equilibria

A method for choosing equilibria in strategic form games is proposed and axiomatically characterized. The method as well as the axioms are inspired by the Nash bargaining theory. The method can be applied to existing refinements of Nash equilibrium (e.g., perfect equi- librium) and also to other equilibrium concepts, like correlated equi- librium.

Equilibrium Refinements in Sender-Receiver Games

This paper examines the effectiveness of perturbation refinements in sender-receiver games. It is shown that babbling equilibria are always perfect and even proper. However, they need not be strategically stable. An example is given where the only strategically stable pooling equilibria are pure strategy equilibria. Furthermore, there exist examples in which none of the pooling equilibria is strategically stable. Persistence is effective in games with small message spaces. It rules out pooling equilibria in games which have strict separating equilibria but its effectiveness is not confined to these games. Journal of Economic Literature Classification Number: C72.

The Algebraic Geometry of Perfect and Sequential Equilibrium

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Selten's (1975) perfect equilibrium and Kreps and Wilson's (1982) more inclusive sequential equilibrium. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are semi-algebraic sets; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential equilibrium correspondences have this semi-algebraic structure follows from a deep result from mathematical logic, the Tarski-Seidenberg Theorem; that this structure has important game-theoretic consequences follows from deep properties of semi-algebraic sets.

Drift

Which equilibria should command our attention in a game with multiple equilibria? Attempts to solve this equilibrium selection problem have recently turned from the invention of equilibrium refinements to evolutionary models of games. Unfortunately, the basic equilibrium concept for evolutionary games, an evolutionarily stable strategy, frequently fails to exist. In this paper, we examine some of the modified and alternative stability concepts that have been applied when evolutionarily stable strategies do not exist.

Information, rational beliefs and equilibrium refinements

Given an extensive game G, three subsets of the normal-form equivalence class of G are defined: the subset of simultaneous games [denoted by Sim( G)] the subset of subgame-proserving quasi-simultaneous games [denoted by SubSim( G)] and, finally, the subset consisting of the game G itself. We show that by applying the notion of rational profile of beliefs (which is formulated independently of the notion of strategy and therefore of Nash equilibrium) to the games in Sim( G) one obtains exactly the Nash equilibria of G, by applying it to the games in SubSim( G) one obtains exactly the subgame-perfect equilibria of G and, finally, by applying it to G itself one obtains a (strict) refinement of subgame-perfect equilibrium.

An Experimental Analysis of Nash Refinements in Signaling Games

Refinements of Nash equilibrium are investigated in two-person signaling game experiments. The experiments cover several nested refinements: Bayes–Nash, sequential, intuitive, divine, universally divine, NWBR, and stable. The experimental data suggest that subjects select the more refined equilibria up to divinity. However, an anomaly occurs in one game in which the stable equilibrium is preferred to an NWBR equilibrium. Since the refinements are nested this anomaly implies that outcomes are game specific. Deviations from Nash behavior do not seem to follow any specific decision rule (e.g., Nash, minimax, principle of insufficient reason, etc.). Most choices by both players are part of some Nash equilibrium, but deviations from equilibrium behavior occur when the choices are part of different equilibria. Journal of Economic Literature Classification Numbers: 026, 215.

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.

A path-following procedure to find a proper equilibrium of finite games

We propose a procedure to find a proper equilibrium of finiten-person games, which was introduced by Myerson as a refinement of perfect equilibrium. The procedure is a new variable dimension fixed point algorithm having Π =1/ m i ! directions in which it may leave the starting point, wherem i is the number of thei-th player's pure strategies.

Self-Confirming Equilibrium

rium path of play. Thus, if a equilibrium occurs repeatedly, no player ever observes play that contradicts his beliefs, even though beliefs about play at off-path information sets need not be correct. We characterize the ways in which equilibria and Nash equilibria can differ, and provide conditions under which self-con- firming equilibria correspond to standard solution concepts. NASH EQUILIBRIUM AND ITS REFINEMENTS describe a situation in which (i) each player's strategy is a best response to his beliefs about the play of his opponents, and (ii) each player's beliefs about the opponents' play are exactly correct. We propose a new equilibrium concept, equilibrium, that weakens condition (ii) by requiring only that players' beliefs are correct along the equilibrium path of play. Thus, each player may have incorrect beliefs about how his opponents would play in contingencies that do not arise when play follows the equilibrium, and moreover the beliefs of different players may be wrong in different ways. The concept of equilibrium is motivated by the idea that noncooperative equilibria should be interpreted as the outcome of a learning process, in which players revise their beliefs using their observations of previous play. Suppose that each time the game is played, the players observe the actions chosen by their opponents, but that players do not observe the actions their opponents would have played at the information sets that were not reached along the path of play. Then, if a equilibrium occurs repeatedly, no player ever observes play that contradicts his beliefs, so the equilibrium is self-confirming in the weak sense of not being inconsistent with the evidence. By analogy with the literature on the bandit problem (e.g., Rothschild (1974)) one might expect that a non-Nash equilibrium can be the outcome of plausible learning processes. This point was made by Fudenberg and

Are game theoretic concepts suitable negotiation support tools? From Nash equilibrium refinements toward a cognitive concept of rationality

If game theory is to be used as a negotiation support tool, it should be able to provide unambiguous recommendations for a target to aim at and for actions to reach this target. This need cannot be satisfied with the Nash equilibrium concept, based on the standard instrumental concept of rationality. These equilibria, as is well known, are generally multiple in a game. The concept of substantive or instrumental rationality has proved to be so pregnant, however, that researchers, instead of re-evaluating its use in game theory, have simply tried to design concepts related to the Nash equilibrium, but with the property of being unique in a game — i.e., they have devised ways ofselecting among Nash equilibria. These concepts have been labeledrefined Nash equilibria.

Two-sided uncertainty in the monopoly agenda setter model

We extend the Romer-Rosenthal model of representative democracy to a signaling environment, in which (i) only the representatives knows the ‘status quo’ outcome resulting if her take-it-or-leave-it policy proposal is rejected by the voters, while (ii) only the voters know their true preferences over policies. A separating sequential equilibrium is shown to exist, and to uniquely satisfy a common equilibrium refinement. Furthermore, this equilibrium has the property that, relative to the environment where the status quo is known to the voter, there is a downward bias in the setter's proposal, and an associated upward bias in the probability of the proposal's acceptance by the voter.