Market Games in Large Economies with a Finite Number of Types

We study market games derived from an exchange economy with a continuum of agents, each having one of finitely many possible types. The type of agent determines his initial endowment and utility function. It is shown that, unlike the well-known Shapley–Shubik theorem on market games (Shapley and Shubik in J Econ Theory 1:9–25, 1969), there might be a (fuzzy) game in which each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a game needs also to be homogeneous. We also study investment games – which are fuzzy games obtained from an economy with a finite number of agents cooperating in one or more joint projects. It is argued that the usual definition of the core is inappropriate for such a model. We therefore introduce and analyze the new notion of comprehensive core. This solution concept seems to be more suitable for such a scenario. We finally refer to the notion of feasibility of an allocation in games with a large number of players. Some of the results in this paper appear in a previous draft distributed by the name “Cooperative investment games or Population games”. An anonymous referee of Economic Theory is acknowledged for his/her comments     

Conjectural variations and evolutionary stability in finite populations

Recently it has been shown that consistent conjectures are evolutionarily stable. In this note we show that this finding depends on the use of the infinite population ESS (Maynard-Smith, Evolution and the theory of games, Cambridge University Press, Cambridge, UK, 1982). When applying the finite-population ESS (Schaffer, J Theor Biol 132:469–478, 1988) we show that the conjectures surviving in the long run are not consistent.      

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

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.     

Evolutionary Stability in Games of Communication

This paper identifies evolutionarily stable outcomes in games in which one player has private information and the other takes a payoff-relevant action. The informed player can communicate at little cost. Outcomes satisfying a set-valued evolutionary stability condition must exist and be efficient in common-interest games. When there is a small cost associated with using each message the outcome preferred by the informed player is stable. The paper introduces a nonequilibrium, set-valued stability notion of entry resistant sets. For games with partial common interest, the no-communication outcome is never an element of an entry resistant set. Journal of Economic Literature Classification Numbers: C72,D82.     

Does Neutral Stability Imply Lyapunov Stability?

A central concept in evolutionary game theory is that of an evolutionarily stable strategy. However, many games of interest for economics have only strategies which meet the weaker requirement of neutral stability. While neutral stability implies Lyapunov stability under the replicator dynamics in pairwise interactions, we give here an example in which an evolutionarily stable strategy is not Lyapunov stable in a more general context. We propose to use instead the notions of (strong) uninvadability and (strong) unbeatability, refinements of evolutionary and neutral stability, respectively. Strong uninvadability implies asymptotic stability, and strong unbeatability implies Lyapunov stability under the replicator dynamics. Journal of Economic Literature Classification Number: C72.     

An equilibrium closure result for discontinuous games

For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.     

Homotopy methods to compute equilibria in game theory

This paper presents a survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present the relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke–Howson algorithm and the van den Elzen–Talman algorithm to compute Nash equilibria in 2-person games, and the Herings–van den Elzen, Herings–Peeters, and McKelvey–Palfrey algorithms to compute Nash equilibria in general n-person games. We explain how the main ideas can be extended to compute equilibria in extensive form and dynamic games, and how homotopies can be used to compute all Nash equilibria.     

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.     

Evolution of Interdependent Preferences in Aggregative Games

We study the evolution of preference interdependence in aggregative games which are symmetric with respect to material payoffs but asymmetric with respect to player objective functions. We identify a class of aggregative games whose equilibria have the property that the players with interdependent preferences (who care not only about their own material payoffs but also about their payoffs relative to others) earn strictly higher material payoffs than do the material payoff maximizers. Implications of this finding for the theory of preference evolution are discussed. Journal of Economic Literature Classification Numbers: C72, D62.     

Corrigendum to “On randomized matching mechanisms” [Economic Theory 8(1996)377–381]

Ma (in Econ. Theory 8, 377–381, 1996) studied the random order mechanism, a matching mechanism suggested by Roth and Vande Vate (Econometrica 58, 1475–1480, 1990) for marriage markets. By means of an example he showed that the random order mechanism does not always reach all stable matchings. Although Ma's (1996) result is true, we show that the probability distribution he presented – and therefore the proof of his Claim 2 – is not correct. The mistake in the calculations by Ma (1996) is due to the fact that even though the example looks very symmetric, some of the calculations are not as “symmetric.” We thank two anonymous referees for their helpful comments. B. Klaus’s and F. Klijn’s research was supported by Ramón y Cajal contracts of the Spanish Ministerio de Ciencia y Tecnología. The work of the authors was also partially supported through the Spanish Plan Nacional I+D+I (BEC2002-02130 and SEJ2005-01690) and the Generalitat de Catalunya (SGR2005-00626 and the Barcelona Economics Program of CREA).     

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.     

Computing uniformly optimal strategies in two-player stochastic games

We provide a computable algorithm to calculate uniform ε-optimal strategies in two-player zero-sum stochastic games. Our approach can be used to construct algorithms that calculate uniform ε-equilibria and uniform correlated ε-equilibria in various classes of multi-player non-zero-sum stochastic games.     

Stability and Largeness of the Core

In general, there are examples of TU games where the core is stable but is not large. In this paper, we show that the extendability condition introduced by Kikuta and Shapley (1986, “Core Stability in n-Person Games,” Mimeo) is sufficient for the core to be stable as well as large, for TU games with five or fewer players. We provide a counter example when the number of players is six. We then introduce a stronger extendability condition and show that it is necessary and sufficient for the core to be large. Our proof makes use of a well-known result from the theory of convex sets. Journal of Economic Literature Classification Number: C71.     

Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

Financial liberalization and short-run housing price dynamics

We develop a continuous time, rational expectations, multi-cohort model of an exchange economy with housing, the purchase of which is subject to a down payment (DP) constraint. The timing of the house purchase decision is a crucial endogenous variable, and four determinants of it are identified – the housing services effect, the interest discounting effect, the consumption smoothing effect, and the rate of price increase effect. Cohort effects, and supply constraints, play crucial roles at the aggregative level. We explore in detail the effects of a discrete financial liberalization, and show that if the liberalization is not announced sufficiently far in advance, housing prices will initially overshoot the new stationary equilibrium, and vice versa. Particular attention is paid to the possibility that for a subset of cohorts along the transition path the DP constraint will not bind. An interesting ‘Prisoners’ Dilemma’ is also identified, and policy implications discussedJEL Classification Numbers: E3, R21Valuable comments and suggestions from Phillip Brock, Ho Kong-Weng, Liu Haoming, David McKenzie, David Miles, Jacques Olivier, Phang Sock-Yong, J. Thampapillai, Ping Wang, Wong Wing-Keung, and Zeng Jinli are gratefully acknowledged. I am also immensely indebted to an anonymous referee, whose incisive, deep and patient comments, on successive drafts, helped greatly to sharpen and improve the paper, as well as to the Editor and the Co-Editor, Professor Mordecai Kurz, for their invaluable advice and encouragement. An earlier version was presented at a Conference in Honour of Ronald McKinnon, held at Stanford University in June 2002     

Cooperation,stability and social welfare under majority rule

We resolve collective irrationality in the stability set, a solution concept for voting games proposed by Rubinstein (J Econ Theory 23:150–159, 1980) to resolve the well-known paradox of voting in the core. A lack of cooperation within winning coalitions prevents their members from eliminating an alternative globally less preferred to another one when they observe the rational behavior defining the stability set. Allowing for the possibility of accompanying a coalitional vote with a binding solidarity agreement is a natural response to this cooperation failure. This leads to a new standard of behavior and a new solution concept called stability set with binding solidarity agreement or S _c -stability set. It is shown that the S _c -stability set and a newly defined version of the Mas-Colell bargaining set for simple games are not comparable with respect to set inclusion. The S _c -stability set includes the core, is included in the stability set and contains only Pareto-optimal alternatives. When individual preferences are complete linear orders, the S _c -stability set is always non-empty.      

The role of ethics and product personality in the intention to purchase organic food products: a structural equation modeling approach

The aim of the present study is to examine the role of ethical dimensions and product personality in the purchasing intention of organic food products. The Prospect method (Caprara et al. in Test Psicomet Metodol 7(3–4):113–128, 2000), which integrates the Five factors model of personality (cf. Digman in Annu Rev Psychol 41(1):417–440, 1990) and the Theory of planned behavior (Ajzen in Organ Behav Hum Decis Process, 50(2):179–211, 1991) extended to an ethical dimension, was employed, by using a Structural Equation Modeling approach. Results showed that moral norms—i.e., personal beliefs regarding what is right or wrong (Parker et al. in Br J Soc Psychol, 34(2):127–137, 1995)—can be considered the main motivator of purchasing intention, and they are, in turn, affected by subjective norms and product personality traits of Naturalness and Authenticity. Marketing implications for firms operating in the organic food industry are discussed, in their intent to shift from a “niche” market to a broader diffusion of these products.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

Robust rational turnout

We show that equilibria of a class of participation games (Palfrey and Rosenthal in Public Choice 41(1):7–53, 1983; Journal of Public Economics 24(2):171–193, 1984) exhibit minimal heterogeneity of behavior so that players’ mixed strategies are summarized by at most two probabilities. We then establish that, except for a finite set of common costs of participation, these games are regular. Thus, equilibria of these voting games are robust to general payoff perturbations and survive in nearby games of incomplete information. Thanks to participants of the 2006 MPSA conference for comments on an early version.