Iterated regret minimization: A new solution concept

For some well-known games, such as the Traveler?s Dilemma or the Centipede Game, traditional game-theoretic solution concepts—most notably Nash equilibrium—predict outcomes that are not consistent with empirical observations. We introduce a new solution concept, iterated regret minimization, that exhibits the same qualitative behavior as that observed in experiments in many games of interest, including Traveler?s Dilemma, the Centipede Game, Nash bargaining, and Bertrand competition. As the name suggests, iterated regret minimization involves the iterated deletion of strategies that do not minimize regret.     

On general periodic time-varying bilinear processes

We study in this note the class of bilinear processes with periodic time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of strict and second order stationary solutions (in periodic sense) and for the existence of higher order moments. The given conditions can be applied to periodic ARMA or periodic GARCH models. The central limit theorem and the law of iterated logarithm (LIL) for higher order sample moments are showed.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

An experimental study of bidding in contests of incomplete information

In many contest situations, such as R&D competition and rent seeking, participants’ costs are private information. We report the results of an experimental study of bidding in contests under different information and symmetry conditions about players’ costs of effort. The theory predicts qualitatively different comparative statics between bids under complete and incomplete information in contests of two and more than two players. We use a 2×3 experimental design, (n=2, n=4)×(symmetric complete information, asymmetric complete information, incomplete information), to test the theoretical predictions. We find the comparative statics of bids across the information and symmetry conditions, and the qualitative differences in comparative statics across group sizes, in partial agreement with the theory.     

On derivatives and information costs

Information plays a central role in capital markets and in the process of asset pricing. The specific features of over-the-counter (OTC) markets require often an investment in information acquisition. Information costs can be defined in the context of Merton's [Merton, R. (1987). A simple model of capital market equilibrium with incomplete information. Journal of Finance, 42, 483–510] model of capital market equilibrium with incomplete information (CAPMI). In this context, hedging portfolios can be constructed and analytic formulas can be derived using the Black and Scholes technology or the martingale method. This paper presents a simple framework for the valuation of exotic derivatives and OTC traded securities in this context. We incorporate information costs into a model, and then use this new model to price a variety of exotic options using the general context in Bellalah [Bellalah, M. (2001). Market imperfections, information costs and the valuation of derivatives: Some general results. International Journal of Finance, 13, 1895–1928]. In each case, simple analytic formulae are derived.From a pedagogical viewpoint, we illustrate the methodology and propose simple analytic formulas for pay-on-exercise options, power derivatives, outperformance options, guaranteed exchange-rate contracts in foreign stock investments, equity-linked foreign exchange options and quantos in the same context. These formulae are simple and have the potential to explain some deviations with respect to the standard Black–Scholes model. We can use also stochastic volatilities and information costs to explain the smiles and skews found in options price data as in Bellalah, Prigent, and Villa [Bellalah, M., Prigent, J. L., & Villa, C. (2001). Skew without skewness: Asymmetric smiles, information costs and stochastic volatilitiy, International Journal of Finance, 2001, 1826, 1837] or Bellalah and Mahfoudh (2004) [Bellalah M. and Mahfoudh S. (2004). Option pricing under stochastic volatility with incomplete Information, Wilmott Magazine]. Our methodology can be applied for the valuation of several OTC and real options in the presence of incomplete information.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Generalized risk-dominance and asymmetric dynamics

This paper proposes two (ordinal and cardinal) generalizations of [J.C. Harsanyi, R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA and London, 1988] risk-dominance to multi-player, multi-action games. There are three reasons why generalized risk-dominance (GR-dominance) is interesting. Extending the logic of risk-dominance, GR-dominant actions can be interpreted as best responses to conjectures that satisfy a certain type of symmetry. Second, in a local interaction game of [G. Ellison, Learning, local interaction, and coordination, Econometrica 61 (5) (1993) 1047], if an action is risk-dominant in individual binary interactions with neighbors, it is also GR-dominant in the large game on a network. Finally, we show that GR-dominant actions are stochastically stable under a class of evolutionary dynamics. The last observation is a corollary to new abstract selection results that applies to a wide class of so-called asymmetric dynamics. In particular, I show that a (strictly) ordinal GR-dominant profile is (uniquely) stochastically stable under the approximate best-response dynamics of [M. Kandori, G.J. Mailath, R. Rob, Learning, mutation, and long run equilibria in games, Econometrica 61 (1) (1993) 29]. A (strictly) cardinal GR-dominant equilibrium is (uniquely) stochastically stable under a class of payoff-based dynamics that includes [L.E. Blume, The statistical-mechanics of strategic interaction, Games Econ. Behav. 5 (3) (1993) 387-424]. Among others, this leads to a generalization of a result from [G. Ellison, Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution, Rev. Econ. Stud. 67 (230) (2000) 17] on the -dominant evolutionary selection to all networks and the unique selection to all networks that satisfy a simple, sufficient condition.     

The swing voter's curse with adversarial preferences

We analyze voting behavior in a large electorate in which voters have adversarial state-contingent preferences with incomplete information about the state of the world. We show that one type of voter can suffer from the swing voter's curse à la Feddersen and Pesendorfer [The swing voter's curse, Amer. Econ. Rev. 86 (1996) 408-424], and go on to characterize the symmetric Nash equilibria of this model under different parameter values. We prove that unlike settings with nonadversarial preferences, there are equilibria in which in one state of the world, a minority-preferred candidate almost surely wins the election and thus the election may fail to correctly aggregate information. Indeed, we show that the fraction of the electorate dissatisfied with the result can be as large as .     

Correlated Nash equilibrium

We modify the epistemic conditions for Nash equilibrium only to accommodate Gilboa and Schmeidler's [I. Gilboa, D. Schmeidler, Maxmin expected utility with nonunique prior, J. Math. Econ. 18 (1989) 141-153] maxmin expected utility preferences, and identify the equilibrium concept in n-player strategic games that characterizes the modified epistemic conditions. The epistemic characterization supports the equilibrium concept as a minimal generalization of Nash equilibrium, in the sense that it deviates from Nash equilibrium only in terms of players' attitude towards ambiguity. Consequently, comparing it with Nash equilibrium constitutes a ceteris paribus study of the effects of ambiguity on how a game is played. For example, with ambiguity, (beliefs about) action choices are in general correlated.     

Some extensions of a claim of Aumann in an axiomatic model of knowledge

The paper begins by presenting an axiomatic model of simple and iterated knowledge. A formal definition of the intuitive notion of common knowledge is given and shown equivalent to previous characterizations. It is shown that agents have information partitions. The second part generalizes Aumann's (Ann. Statist.4 (1976), 1236–1239) well-known propositions about common knowledge between two rational agents of each other's probability assignments. It is shown that: common knowledge of decisions—if these are rational—implies a common decision for like-minded agents; and that a “dialogue” in decisions leads to a common decision. A “no-trade” theorem is given which includes trade under complete uncertainty.     

Sequential equilibrium in monotone games: A theory-based analysis of experimental data

A monotone game is an extensive-form game with complete information, simultaneous moves and an irreversibility structure on strategies. It captures a variety of situations in which players make partial commitments and allows us to characterize conditions under which equilibria result in socially desirable outcomes. However, since the game has many equilibrium outcomes, the theory lacks predictive power. To produce stronger predictions, one can restrict attention to the set of sequential equilibria, or Markov equilibria, or symmetric equilibria, or pure-strategy equilibria. This paper explores the relationship between equilibrium behavior in a class of monotone games, namely voluntary contribution games, and the behavior of human subjects in an experimental setting. Several key features of the symmetric Markov perfect equilibrium (SMPE) are consistent with the data. To judge how well the SMPE fits the data, we estimate a model of Quantal Response Equilibrium (QRE) [R. McKelvey, T. Palfrey, Quantal response equilibria for normal form games, Games Econ. Behav. 10 (1995) 6-38; R. McKelvey, T. Palfrey, Quantal response equilibria for extensive form games, Exp. Econ. 1 (1998) 9-41] and find that the decision rules of the QRE model are qualitatively very similar to the empirical choice probabilities.     

Information transmission in coalitional voting games

A core allocation of a complete information economy can be characterized as one that would not be unanimously rejected in favor of another feasible alternative by any coalition. We use this test of coalitional voting in an incomplete information environment to formalize a notion of resilience. Since information transmission is implicit in the Bayesian equilibria of such voting games, this approach makes it possible to derive core concepts in which the transmission of information among members of a coalition is endogenous. Our results lend support to the credible core of Dutta and Vohra [Incomplete information, credibility and the core, Math. Soc. Sci. 50 (2005) 148-165] and the core proposed by Myerson [Virtual utility and the core for games with incomplete information, Mimeo, University of Chicago, 2005] as two that can be justified in terms of coalitional voting.     

Optimal ownership of public goods reconsidered

Consider a non-governmental organization (NGO) that can invest in a public good. Should the government or the NGO own the public project? In an incomplete contracting framework with split-the-difference bargaining, Besley and Ghatak (2001) argue that the party who values the public good most should be the owner. We demonstrate the robustness of their insight when the split-the-difference rule is replaced by the deal-me-out solution. Our finding is in contrast to the private good results of Chiu (1998) and De Meza and Lockwood (1998), who show that the optimal ownership structure crucially depends on whether the split-the-difference rule or the deal-me-out solution is used.     

Computationally restricted unmediated talk under incomplete information

Summary. We show the role of unmediated talk with computational complexity bounds as both an information transmission and a coordination device for the class of two-player games with incomplete information and rational parameters. We prove that any communication equilibrium payoff of such games can be reached as a Bayesian-Nash equilibrium payoff of the game extended by a two phase universal mechanism of interim computationally restricted pre-play communication. The communication protocols are designed with the help of modern cryptographic tools. A familiar context in which our results could be applied is bilateral trading with incomplete information.Received: 9 September 2002, Revised: 14 March 2003, JEL Classification Numbers: C72. Correspondence to: Amparo UrbanoWe wish to thank financial aid from the Valencian Institute of Economic Research (IVIE) and partial support by DIGCYT under project PB95 - 1074. A previous version of this work appears as IVIE Working Paper WP-AD 99-07, under the title: "Unmediated talk under incomplete information".     

Information structures with unawareness

I construct a state space model with unawareness following [R.J. Aumann, Agreeing to disagree, Ann. Stat. 76 (1976) 1236-1239]. Dekel et al. [E. Dekel, B.L. Lipman, A. Rustichini, Standard state-space models preclude unawareness, Econometrica 66 (1998) 159-173] show that standard state space models are incapable of representing unawareness. The model circumvents the impossibility result by endowing the agent with a subjective state space that differs from the full state space when he has the unawareness problem. Information is modeled as a pair, consisting of both factual information and awareness information. The model preserves the central properties of the standard information partition model.     

The roommates problem revisited

One of the oldest matching problems is Gale and Shapley's (1962) [8] “roommates problem”: is there a stable way to assign 2N students into N roommate pairs? Unlike the classic marriage problem or college admissions problem, there need not exist a stable solution to the roommates problem. However, stability ignores the key physical constraint that roommates require a room and is therefore too restrictive. This motivates a new matching problem: matching agents subject to an initial assignment. A particularly important example is kidney exchange where after an assignment has been made, subsequent tests may determine that a patient and donor are incompatible. This paper introduces an efficient algorithm for finding a Pareto improvement starting from any status quo roommates assignment.     

Agreeable bets with multiple priors

This paper considers a two-agent model of trade with multiple priors. Firstly, we characterize the existence of an agreeable bet on some event in terms of the set of priors. It is then shown that the existence of an agreeable bet on some event is a strictly stronger condition than the existence of an agreeable trade, whereas the two conditions are equivalent in the standard Bayesian framework. Secondly, we show that the two conditions are equivalent when the set of priors is the core of a convex capacity. These results are also related to the no trade theorems under asymmetric information.     

Signaling and mediation in games with common interests

Players who have a common interest are engaged in a game with incomplete information. Before playing they get differential stochastic signals that depend on the actual state of nature. These signals provide the players with partial information about the state of nature and may also serve as a means of correlation.Different information structures induce different outcomes. An information structure is better than another, with respect to a certain solution concept, if the highest solution payoff it induces is at least that induced by the other structure. This paper characterizes the situation where one information structure is better than another with respect to various solution concepts: Nash equilibrium, strategic-normal-form correlated equilibrium, agent-normal-form correlated equilibrium and belief-invariant Bayesian solution. These solution concepts differ from one another in the scope of communication allowed between the players. The characterizations use maps that stochastically translate signals of one structure to signals of another.     

Market structure and matching with contracts

Ostrovsky (2008) [9] develops a theory of stability for a model of matching in exogenously given networks. For this model a generalization of pairwise stability, chain stability, can always be satisfied as long as agents' preferences satisfy same side substitutability and cross side complementarity. Given this preference domain I analyze the interplay between properties of the network structure and (cooperative) solution concepts. The main structural condition is an acyclicity notion that rules out the implementation of trading cycles. It is shown that this condition and the restriction that no pair of agents can sign more than one contract with each other are jointly necessary and sufficient for (i) the equivalence of group and chain stability, (ii) the core stability of chain stable networks, (iii) the efficiency of chain stable networks, (iv) the existence of a group stable network, and (v) the existence of an efficient and individually stable network. These equivalences also provide a rationale for chain stability in the unrestricted model. The (more restrictive) conditions under which chain stability coincides with the core are also characterized.