Belief-free equilibria in games with incomplete information: Characterization and existence

We generalize the results of Hörner and Lovo (2009) [15] to N-player games with arbitrary information structure. First, we characterize the set of belief-free equilibrium payoffs under low discounting as the set of feasible payoffs that are individually rational, jointly rational, and incentive compatible. Second, we provide necessary and sufficient conditions on the information structure for this set to be non-empty.     

The target projection dynamic

We study the target projection dynamic, a model of learning in normal form games. The dynamic is given a microeconomic foundation in terms of myopic optimization under control costs due to a certain status-quo bias. We establish a number of desirable properties of the dynamic: existence, uniqueness and continuity of solution trajectories, Nash stationarity, positive correlation with payoffs, and innovation. Sufficient conditions are provided under which strictly dominated strategies are wiped out. Finally, some stability results are provided for special classes of games.     

Quantal response equilibria with heterogeneous agents

This paper introduces a model of quantal response equilibrium with heterogeneous agents and demonstrates the existence of a representative agent for such populations. Except in rare cases, the representative agent?s noise terms cannot be independently and identically distributed across the set of actions, even if that is assumed for the individual agents. This result demonstrates a fundamental difference between a representative agent and truly heterogeneous quantal responders and suggests that when fitting quantal response specifications to aggregate data from a population of subjects, the noise terms should be allowed to be jointly dependent across actions. Even though this introduces additional degrees of freedom, it makes the model well specified. The representative agent inherits a regular quantal response function from the actual agents, so this model does impose falsifiable restrictions.     

On the folk theorem with one-dimensional payoffs and different discount factors

Proving the folk theorem in a game with three or more players usually requires imposing restrictions on the dimensionality of the stage-game payoffs. Fudenberg and Maskin (1986) assume full dimensionality of payoffs, while Abreu et al. (1994) assume the weaker NEU condition (“nonequivalent utilities”). In this note, we consider a class of n-player games where each player receives the same stage-game payoff, either zero or one. The stage-game payoffs therefore constitute a one-dimensional set, violating NEU. We show that if all players have different discount factors, then for discount factors sufficiently close to one, any strictly individually rational payoff profile can be obtained as the outcome of a subgame-perfect equilibrium with public correlation.     

Learning from private information in noisy repeated games

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games where players observe imperfect public signals of the actions played, and both the payoff functions and the map from actions to signal distributions depend on an unknown state. The PTXE payoffs when players are patient are determined by the solutions to a family of linear programming problems. Using this characterization, we develop conditions under which play can be as if the players have learned the state. We provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.     

The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition

Abstract. We analyze the Nash equilibria of a standard Bertrand model. We show that in addition to the marginal-cost pricing equilibrium there is a possibility for mixed-strategy equilibria yielding positive profit levels. We characterize these equilibria and find that having unbounded revenues is the necessary and sufficient condition for their existence. Hence, we demonstrate that under realistic assumptions the only equilibrium is marginal-cost pricing.     

Nonminimal product differentiation as a bargaining outcome

It is well known that in a duopoly model of product choice with quadratic transportation cost, the firms locate at the extreme endpoints of the market. Jehiel (1992, Int. J. Ind. Organ, 10, 633–641) has examined this model in an infinite horizon setting where in the initial period the firms choose location and in subsequent periods charge the Nash bargaining solution prices. Then, in the unique equilibrium both firms locate at the center of the market. This paper examines the case when the firms instead charge the prices determined by either the egalitarian bargaining solution or the Kalai–Smorodinski bargaining solution. It is shown that central agglomeration is an equilibrium. Furthermore, there is a continuum of symmetric equilibria in addition where the firms locate apart from each other. So the products are not necessarily minimally differentiated. Thus different bargaining solutions provide quite different outcomes.     

Stochastic evolutionary selection in finite populations

We primarily focus on a wide range of stochastic evolutionary game dynamics between two strategies which are characterized by a condition we call monotonicity: the sign of the difference between the probabilities of increasing and decreasing an A-individual completely depends on the difference of payoffs based on different strategies. When mutations are excluded, we provide sufficient conditions for selection to favor one strategy over the other and necessary conditions for selection to favor or oppose change, respectively. Moreover, we discuss which strategy will be favored in case of rare mutations and give a simple rule to determine evolutionary selection of strategies for large populations under some specific stochastic mutation–selection dynamics.     

Non-cooperative solutions for estate division problems

In an estate division problem an estate has to be divided among several players whose total entitlement to the estate exceeds its size. This paper extends the non-cooperative approach through a claim game, as initiated by O?Neill (1982), by allowing players to put multiple claims on the same part of the estate, and by considering the case where individual entitlements may exceed the estate. A full characterization of the set of Nash equilibria of the claim game is obtained both for restricted estate division problems, where individual entitlements do not exceed the estate, and for the general case. Variations on the claim game are considered, which result in proportional division in equilibrium.     

Uniqueness of stationary equilibria in bargaining one-dimensional policies under (super) majority rules

We consider negotiations selecting one-dimensional policies. Individuals have instantaneous preferences represented by continuous, concave and single-peaked utility functions, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. We provide sufficient conditions that guarantee the existence of a unique stationary subgame perfect equilibrium, and we provide its explicit characterization. The uniqueness of the equilibrium permits an analysis of the set of Pareto optimal voting rules. For symmetric distributions of peaks and uniform recognition probabilities unanimity is the unanimously preferred majority rule.     

Transferable utility games with uncertainty

We introduce the concept of a TUU-game, a transferable utility game with uncertainty. In a TUU-game there is uncertainty regarding the payoffs of coalitions. One out of a finite number of states of nature materializes and conditional on the state, the players are involved in a particular transferable utility game. We consider the case without ex ante commitment possibilities and propose the Weak Sequential Core as a solution concept. We characterize the Weak Sequential Core and show that it is non-empty if all ex post TU-games are convex.     

Large deviations and multinomial probit choice

We consider a discrete choice model in which the payoffs to each of an agent?s n actions are subjected to the average of m i.i.d. shocks, and use tools from large deviations theory to characterize the rate of decay of the probability of choosing a given suboptimal action as m approaches infinity. Our model includes the multinomial probit model of Myatt and Wallace (2003) [5] as a special case. We show that their formula describing the rates of decay of choice probabilities is incorrect, provide the correct formula, and use our large deviations analysis to provide intuition for the difference between the two.     

A characterization of strategic complementarities

I characterize games for which there is an order on strategies such that the game has strategic complementarities. I prove that, with some qualifications, games with a unique equilibrium have complementarities if and only if Cournot best-response dynamics has no cycles; and that all games with multiple equilibria have complementarities. As applications of my results, I show that: (1) generic 2×2 games either have no pure-strategy equilibria, or have complementarities; (2) generic two-player finite ordinal potential games have complementarities.     

Exploiting old customers and attracting new ones: The case of bank deposit pricing

Recent theoretical contributions suggest that deposit interest rates should be higher in geographic areas characterized by greater in-migration and lower for depositors at banks with greater shares of existing (or so-called “locked-in”) depositors. These hypotheses are tested using a rich data set obtained for the Spanish banking industry. Results confirm that, all else equal, banks offer higher deposit rates in territories characterized by greater in-migration, and also that they tend to offer lower rates, the larger the number of their locked-in depositors. These findings confirm the existence of the trade-off between exploiting old customers and attracting new ones.     

Advice and behavior in intergenerational ultimatum games: An experimental approach

In the real world, when people play a game, they often receive advice from those that have played it before them. Such advice can facilitate the creation of a convention of behavior. This paper studies the impact of advice on the behavior subjects who engage in a non-overlapping generational Ultimatum game where after a subject plays he is replaced by another subject to whom he can offer advice.Our results document the fact that allowing advice fosters the creation of a convention of behavior in Ultimatum games. In addition, by reading the advice offered we conclude that arguments of fairness are rarely used to justify the offers of Senders but are relied upon to justify rejections by Receivers.