Pure strategy equilibria in games with private and public information

We introduce a new game form which allows the players’ strategies to depend on their strategy-relevant private information as well as on some publicly announced information. The players’ payoffs depend on their own payoff-relevant private information and some payoff-relevant common information. Under the assumption that the players’ strategy-relevant private information is diffuse and their private information is conditionally independent given the public and payoff-relevant common information, we prove the existence of a pure strategy equilibrium for such a game by developing a distribution theory of correspondences via vector measures.     

Markov Perfect equilibria in repeated asynchronous choice games

This paper examines the issue of multiplicity of Markov Perfect equilibria in alternating move repeated games. Such games are canonical models of environments with repeated, asynchronous choices due to inertia or replacement. Our main result is that the number of Markov Perfect equilibria is generically finite with respect to stage game payoffs. This holds despite the fact that the stochastic game representation of the alternating move repeated game is “non-generic” in the larger space of state dependent payoffs. We further obtain that the set of completely mixed Markov Perfect equilibria is generically empty with respect to stage game payoffs.     

Analysis of information feedback and selfconfirming equilibrium

Recent research emphasizes the importance of information feedback in situations of recurrent decisions and strategic interaction, showing how it affects the uncertainty that underlies selfconfirming equilibrium (e.g., Battigalli et al., 2015, Fudenberg and Kamada, 2015). Here, we discuss in detail several properties of this key feature of recurrent interaction and derive relationships. This allows us to elucidate different notions of selfconfirming equilibrium, showing how they are related to each other given the properties of information feedback. In particular, we focus on Maxmin selfconfirming equilibrium, which assumes extreme ambiguity aversion, and we compare it with the partially-specified-probabilities (PSP) equilibrium of Lehrer (2012). Assuming that players can implement any randomization, symmetric Maxmin selfconfirming equilibrium exists under either “observable payoffs,” or “separable feedback.” The latter assumption makes this equilibrium concept essentially equivalent to PSP-equilibrium. If observability of payoffs holds as well, then these equilibrium concepts collapse to mixed Nash equilibrium.     

Rational expectations equilibrium and the strategic choice of costly information

This paper studies costly information acquisition in one-good production economies when agents acquire private information and prices transmit information. Before asset markets open, agents choose the quality of their private information. After this information stage, agents trade assets in sequentially complete markets taking into account their private information and the information revealed by equilibrium prices (rational expectations equilibrium, (Radner, R., 1979. Rational expectations equilibrium: generic existence and the information revealed by prices, Econometrica 47, 655–678.)). An overall equilibrium in asset and information market is defined as a Nash equilibrium of the information game in which agents’ actions are information choices and their utility payoffs are the ex-ante expected utilities of the corresponding rationale expectations equilibrium. This paper shows that for a generic set of economies parameterized by endowments and productivity shocks, an overall equilibrium in information and asset market (a Nash equilibrium of the induced information game) with costly information acquisition and fully-revealing prices exists. In other words, informational efficiency is in general consistent with costly information acquisition.     

Inefficient entry order in preemption games

In a preemption game, players decide when to take an irreversible action. Delaying the action exogenously increases payoffs, but there is an early mover advantage. Riordan (1992) shows that in a preemption game with two asymmetric players, players act in decreasing order of efficiency. This provides a microfoundation to the assumption that entry in a market occurs in the order of profitability, commonly used in the empirical analysis of market entry. We provide a counterexample showing that with more than two players this intuitive result can be reversed. We present a preemption game of entry into a new market. The potential entrants are three asymmetric firms: one “efficient” firm with high post-entry profits, and two “inefficient firms”. We show that the set of parameters such that the equilibrium entry order does not reflect the efficiency ranking is nonempty, and analyse which changes in post-entry profits preserve this entry order.     

Anonymous price taking equilibrium in Tiebout economies with a continuum of agents: Existence and characterization

We introduce a model of a local public goods economy with a continuum of agents and jurisdictions with finite but unbounded populations, where the set of possible projects for each jurisdiction/club is unrestricted in size. Under boundedness of per capita payoffs, which simply ensures that equal treatment payoffs are bounded above, we apply results of Kaneko and Wooders (1986) to obtain nonemptiness of the core of the economy. We then demonstrate, under the stronger condition of strict small group effectiveness, that the equal treatment core coincides with the set of price-taking equilibrium outcomes with anonymous prices—that is, prices for public goods depend only on observable characteristics of agents. Existence of equilibrium follows from nonemptiness of the core and equivalence of the core to the set of equilibrium outcomes. Our approach provides a new technique for showing existence of equilibrium in economies with a continuum of agents.     

Coordinating under incomplete information

We show that, in a minimum effort game with incomplete information where player types are independently drawn, there is a largest and smallest Bayesian equilibrium, leading to the set of equilibrium payoffs (as evaluated at the interim stage) having a lattice structure. Furthermore, the range of equilibrium payoffs converges to those of the deterministic complete information version of the game, in the limit as the incomplete information vanishes. This entails that such incomplete information alone cannot explain the equilibrium selection suggested by experimental evidence. We thank Atila Abdulkadiroglu, Hans Carlsson, Ani Guerdjikova and an anonymous referee for helpful comments. Part of this work was done while Asheim was visiting Cornell University, which hospitality is gratefully acknowledged.     

Piecewise closed-loop equilibria in differential games with regime switching strategies

We propose a new methodology exploring piecewise closed-loop equilibrium strategies in differential games with regime switching actions. We develop a general game with two players. Players choose an action that influences the evolution of a state variable, and decide on the switching time from one regime to another. Compared to the optimal control problem with regime switching, necessary optimality conditions are modified for the first player to switch. When choosing her optimal switching strategy, this player considers the impact of her choice on the other player’s actions and consequently on her own payoffs. In order to determine the equilibrium timing of regime changes, we derive conditions that help eliminate candidate equilibrium strategies that do not survive deviations in switching strategies. We then apply this new methodology to an exhaustible resource extraction game.     

Bounds for best response functions in binary games

This paper studies the identification of best response functions in binary games without making strong parametric assumptions about the payoffs. The best response function gives the utility maximizing response to a decision of the other players. This is analogous to the response function in the treatment-response literature, taking the decision of the other players as the treatment, except that the best response function has additional structure implied by the associated utility maximization problem. Further, the relationship between the data and the best response function is not the same as the relationship between the data and the response function in the treatment-response literature. We focus especially on the case of a complete information entry game with two firms. We also discuss the case of an entry game with many firms, non-entry games, and incomplete information. Our analysis of the entry game is based on the observation of realized entry decisions, which we then link to the best response functions under various assumptions including those concerning the level of rationality of the firms, including the assumption of Nash equilibrium play, the symmetry of the payoffs between firms, and whether mixed strategies are admitted.     

Stopping with congestion and private payoffs

This paper analyses a two-player stopping game with multiarmed bandits in which each player chooses between learning about the quality of her private risky arm and competing for the use of a single shared safe arm. The qualities of the players’ risky arms are independent. A player whose risky arm produces a success no longer competes for the safe arm. We assume that a player observes her opponent’s actions but not his realised payoffs. She is therefore never certain whether her opponent is still competing for the safe arm. When the players’ prior probabilities of success are sufficiently close, there exists no pure strategy equilibrium, and we characterise the unique mixed strategy equilibrium. Otherwise, the unique equilibrium is in pure strategies. The amount of experimentation performed in equilibrium is inefficiently low but, for many priors, higher than if successes are publicly observed.     

A noncooperative approach to the Mas-Colell bargaining set

We propose an extensive form game to give a noncooperative interpretation of the Mas-Colell bargaining set for transferable utility games. Based on the objection and counterobjection in the sense of the Mas-Colell bargaining set, a bilateral negotiation is designed in our model. We show that the equilibrium outcomes of our extensive form game coincide with the Mas-Colell bargaining set in the supperadditive environments.     

Manipulation of the Walrasian mechanism in production economies with unbounded short-selling

Hurwicz (1979) and Otani and Sicilian (1982, 1990) characterized the Nash equilibrium allocations of the Walrasian demand manipulation game in successively more general exchange environments. In this paper, I extend the analysis to production economies with short-selling. First, I generalize Hurwicz’s and Otani and Sicilian’s theorem that any allocation at which each agent’s consumption bundle lies above her true offer curve can be supported in Nash equilibrium. I then show that for finite economies of any size the set of such allocations is often topologically large.Received: 17 January 2003, Accepted: 4 April 2005, JEL Classification: D51, D82For comments on this and earlier versions of the paper, I wish to thank Rick Bond, Bhaskar Chakravorti, Tom Gresik, Costas Syropoulos and William Thomson. I would especially like to thank Mike Jerison for helping to overcome a difficulty with a previous version. Also, the comments of the anonymous referees are gratefully acknowledged.     

On the non-existence of oligopolistic equilibria in differentiated products spaces

In a large class of product differentiation models á la Hotelling, the firms' payoffs in the game involving location in the differentiated products space exclusively depends on the distance to their neighbouring firms, rather than on the firms' locations proper. This leads to a degeneracy on the firms' strategy space. Using alternative equilibrium concepts, this is shown to be the reason for the non-existence of non-cooperative equilibria, especially if the firms' technologies are not identical.     

Convergence to Walrasian equilibrium with minimal information

We consider convergence to Walrasian equilibrium in a situation where firms know only market price and their own cost function. We term this a situation of minimal information. We model the problem as a large population game of Cournot competition. The Nash equilibrium of this model is identical to the Walrasian equilibrium. We apply the best response (BR) dynamic as our main evolutionary model. This dynamic can be applied under minimal information as firms need to know only the market price and the their own cost to compute payoffs. We show that the BR dynamic converges globally to Nash equilibrium in an aggregative game like the Cournot model. Hence, it converges globally to the Walrasian equilibrium under minimal information. We extend the result to some other evolutionary dynamics using the method of potential games.

     

Set-valued solution concepts using interval-type payoffs for interval games

Uncertainty is a daily presence in the real world. It affects our decision making and may have influence on cooperation. Often uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e., payoffs lie in some intervals. A suitable game theoretic model to support decision making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players’ payoffs. In this paper, some set-valued solution concepts using interval payoffs, namely the interval core, the interval dominance core and the interval stable sets for cooperative interval games, are introduced and studied. The main results contained in the paper are a necessary and sufficient condition for the non-emptiness of the interval core of a cooperative interval game and the relations between the interval core, the interval dominance core and the interval stable sets of such a game.     

The commuting time paradox

We examine the conditions for the ‘commuting time paradox’ which states that the average commuting time does not vary between different periods. We develop therefore an equilibrium job search model with endogenously chosen commuting costs. Presuming wage bargaining between workers and firms, the optimally chosen maximum commuting costs jointly maximise the worker's and firm's payoffs. We demonstrate that when productivity levels increase over time, average commuting costs and average wages both increase, which affects the optimally chosen commuting time. We establish the conditions under which the paradox holds.     

Linear learning in changing environments

The decision maker receives signals imperfectly correlated with an unobservable state variable and must take actions whose payoffs depend on the state. The state randomly changes over time. In this environment, we examine the performance of simple linear updating rules relative to Bayesian learning. We show that a range of parameters exists for which linear learning results in exactly the same decisions as Bayesian learning, although not in the same beliefs. Outside this parameter range, we use simulations to demonstrate that the consumption level attainable under the optimal linear rule is virtually indistinguishable from the one attainable under Bayes’ rule, although the respective decisions will not always be identical. These results suggest that simple rules of thumb can have an advantage over Bayesian updating when more complex calculations are more costly to perform than less complex ones. We demonstrate the implications of such an advantage in an evolutionary model where agents “learn to learn.”     

Sovereign debt: Reputation, seizure and reputation

The interaction between a creditor and a sovereign debtor is described as a ‘one-shot’ game with discrete actions—total or no debt-repudiation and seizure of asset holding abroad. Possible Nash equilibria where each player chooses an action as to maximize his expected payoff given his beliefs about the other player’s action and the implications of those actions on the players’ trustworthy reputation are identified. However, if reputation losses rise convexly with the players’ relative hostility, partial repudiation and seizure can be the preferred strategies. The preferred repudiation and seizure rates are analyzed under asymmetric and symmetric information about the state of the world. (JEL classification F34)