How common are common priors?

To answer the question in the title we vary agents? beliefs against the background of a fixed knowledge space, that is, a state space with a partition for each agent. Beliefs are the posterior probabilities of agents, which we call type profiles. We then ask what is the topological size of the set of consistent type profiles, those that are derived from a common prior (or a common improper prior in the case of an infinite state space). The answer depends on what we term the tightness of the partition profile. A partition profile is tight if in some state it is common knowledge that any increase of any single agent?s knowledge results in an increase in common knowledge. We show that for partition profiles that are tight the set of consistent type profiles is topologically large, while for partition profiles that are not tight this set is topologically small.     

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.     

The veil of public ignorance

A theory of cooperative choice under incomplete information is developed in which agents possess private information at the time of contracting and have agreed on a utilitarian “standard of evaluation” governing choices under complete information. The task is to extend this standard to situations of incomplete information. Our first main result generalizes Harsanyi's (J. Polit. Econ. 63 (1955) 309) classical result to situations of incomplete information, assuming that group preferences satisfy Bayesian Coherence and Interim Pareto Dominance. These axioms are mutually compatible if and only if a common prior exists. We argue that this result partly resolves the impossibility of Bayesian preference aggregation under complete information.     

Information in conflicts

We consider two-player contests for a prize of common but uncertain value. For settings where one player knows the value of the prize, while the other only knows its prior distribution, we give conditions for when the uninformed agent is ex ante strictly more likely to win the prize than is the informed agent. In the special case of a lottery contest, equilibrium expenditures are lower under asymmetric information than if either both agents are informed or neither agent is informed.     

Iterated Expectations and Common Priors

A necessary and sufficient condition for the existence of a common prior for several players is given in terms of the players' present beliefs only. A common prior exists iff for each random variable it is common knowledge that all its iterated expectations converge to the same value; this value is its expectation with respect to the common prior. The proof is based on the presentation of type functions as Markov matrices.Journal of Economic LiteratureClassification Numbers: C70, D82.     

Monotonic Knowledge Models,Cycles, Linear Versions and Auctions with Differential,Finite Information

In games of incomplete information where each player's information is represented by a partition of the state space, this paper presents a characterisation of monotonic models in terms of connected partitions and another in terms of versions . A model is monotonic if and only if there is a linear version, and this is true if and only if there is a real‐valued function on the state space such that every partition of the model is connected with respect to this function. These results help us understand the strength of the connectedness assumption on common‐value, second‐price auctions with differential, finite information. We offer a simple sufficient condition for non‐monotonicity to check if models are monotonic.     

On Non-Nash Equilibria

I consider generalisations of the Nash equilibrium concept based on the idea that in equilibrium the players' beliefs should not be contradicted, even if they could possibly be incorrect. This possibility depends on the information about opponents' behaviour available to the players in equilibrium. Therefore the players' information is crucial for this notion of equilibrium, called Conjectural Equilibrium in general and Rationalizable Conjectural Equilibrium (Rubinstein-Wolinsky 1994) when the game and the players' Bayesian rationality are common knowledge. In this paper I argue for a refinement of Rationalizable Conjectural Equilibrium showing by propositions and by examples how this equilibrium notion works and how the suitable equilibrium concept depends on the players' information.Journal of Economic LiteratureClassification Numbers: C72, D83, D82.     

Representation and aggregation of preferences under uncertainty

We axiomatize in the Anscombe-Aumann setting a wide class of preferences called rank-dependent additive preferences that includes most known models of decision under uncertainty as well as state dependent versions of these models. We prove that aggregation is possible and necessarily linear if and only if (society's) preferences are uncertainty neutral. The latter means that society cannot have a non-neutral attitude toward uncertainty on a subclass of acts. A corollary to our theorem is that it is not possible to aggregate multiple prior agents, even when they all have the same set of priors. A number of ways to restore the possibility of aggregation are then discussed.     

Bayesian consistent belief selection

A subjective expected utility agent is given information about the state of the world in the form of a set of possible priors. She is assumed to form her beliefs given this information. A set of priors may be updated according to Bayes' rule, prior-by-prior, upon learning that some state of the world has not obtained. In a model in which information is completely summarized by this set of priors, we show that there exists no decision maker who obeys Bayes' rule, conditions her prior only on the available information (by selecting a belief in the announced set), and who updates the information prior-by-prior using Bayes' rule.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

On the impossibility of achieving no regrets in repeated games

Regret-minimizing strategies for repeated games have been receiving increasing attention in the literature. These are simple adaptive behavior rules that lead to no regrets and, if followed by all players, exhibit nice convergence properties: the average play converges to correlated equilibrium, or even to Nash equilibrium in certain classes of games. However, the no-regret property relies on a strong assumption that each player treats her opponents as unresponsive and fully ignores the opponents’ possible reactions to her actions. We show that if at least one player is slightly responsive, it is impossible to achieve no regrets, and convergence results for regret minimization with responsive opponents are unknown.     

Bargaining power and enforcement in credit markets

In a credit market with enforcement constraints, we study the effects of a change in the outside options of a potential defaulter on the terms of the credit contract, as well as on borrower payoffs. The results crucially depend on the allocation of “bargaining power” between the borrower and the lender. We prove that there is a crucial threshold of relative weights such that if the borrower has power that exceeds this threshold, her expected utility must go up whenever her outside options come down. But if the borrower has less power than this threshold, her expected payoff must come down with her outside options. In the former case a deterioration in outside options brought about, say, by better enforcement, must create a Lorenz improvement in state-contingent consumption. In particular, borrower consumption rises in all “bad” states in which loans are taken. In the latter case, in contrast, the borrower's consumption must decline, at least for all the bad states. These disparate findings within a single model permit us to interpret existing literature on credit markets in a unified way.     

Strategic step-by-step negotiation

We analyze a specific type of negotiation process where parties proceed in stages taking into consideration that negotiation may end prematurely with an inefficient agreement. Parties negotiate only one increasing pie, thereby avoiding inefficiencies which are typical for issue-by-issue negotiations. For ann-stage game, we prove the existence of a unique subgame-perfect equilibrium. We then show that step-by-step negotiation can only improve players' expected payoffs if negotiation in stages reduces the difference between their equilibrium offers. For this to occur, however, the risk of negotiation must be affected by the agenda of the bargaining problem.     

Does Transparency of Central Banks Produce Multiple Equilibria on Currency Markets?

A recent strand of literature shows that multiple equilibria in models of markets for pegged currencies vanish if there is slightly diverse information among traders; see Morris and Shin (2001). It is known that this approach works only if the common knowledge in the market is not too precise. This has led to the conclusion that central banks should try to avoid making their information common knowledge. We develop a model in which more transparency of the central bank implies better private information, because each trader utilises public information according to her own private information. Thus, transparency makes multiple equilibria less likely.     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

Impartial division of a dollar

For impartial division, each participant reports only her opinion about the fair relative shares of the other participants, and this report has no effect on her own share. If a specific division is compatible with all reports, it is implemented.We propose a family of natural methods meeting these requirements, for a division among four or more participants. No such method exists for a division among three participants.     

“Timing Is Everything” and Marital Bliss

We first show that in a marriage market, when the stability of a matching is disturbed when a new agent joins the game, natural greedy behavior defines an equilibration procedure that converges to a stable matching for the extended problem. We then consider the iterative procedure under which agents join the game sequentially, and the natural greedy procedure is applied after the entrance of each agent. It is shown that this procedure converges to a stable matching for the original (global) problem and that for each agent, if the order of all other agents is given, he/she weakly improves his/her final outcome by deferring his/her arrival time. The agent that arrives last gets his/her optimal outcome under stable matchings. Journal of Economic Literature Classification Numbers: C78, C62.     

Contest success functions

Summary Tournaments, conflict, and rent-seeking have been modelled as contests in which participants exert effort to increase their probability of winning a prize. A Contest Success Function (CSF) provides each player's probability of winning as a function of all players' efforts. In this paper the additive CSF employed in most contests is axiomatized, with an independence from irrelevant alternatives property as the key axiom. Two frequently used functional forms are also axiomatized: one in which winning probabilities depend on the ratio of players' efforts and the other in which winning probabilities depend on the difference in efforts.I am grateful to three anonymous referees, Jack Hirshleifer, Guillermo Owen, and especially Duncan Luce for their helpful comments, and to the UC Irvine Academic Senate and the NSF for financial support.     

Core equivalence with congested public goods

Summary This paper examines a model of an infinite production economy with a finite number of types of agents andsemi- public goods, which are subjected to crowding and exclusion. The utility of an agent depends not only on the vector of public commodities produced by the coalition to which she belongs, but also on the mass of agents of her type who are the members of this coalition. The main purpose of the paper is to derive necessary and sufficient conditions on the local degrees of congestion which would guarantee the equivalence between the core and the set of equal treatment Lindahl equilibria. We prove that this equivalence holdsif and only if there are constant returns to group size for each type of agents. It implies that linearity of each agent's congestion function with respect to the mass of the agents of her own type is necessary for the core equivalence to hold.The final version of this paper was written while Shlomo Weber was visiting the Technical University of Dresden as the Fellow of the Alexander von Humboldt Foundation, whose support is gratefully acknowledged. The authors are grateful to Peter Meyer, Nicholas Yannelis and the anonymous referees for useful remarks and suggestions.     

Intertemporal equilibria with Knightian uncertainty

We study a dynamic and infinite-dimensional model with incomplete multiple prior preferences. In interior efficient allocations, agents share a common risk-adjusted prior and subjective interest rate. Interior efficient allocations and equilibria coincide with those of economies with subjective expected utility and priors from the agents? multiple prior sets. A specific model with neither risk nor uncertainty at the aggregate level is considered. Risk is always fully insured. For small levels of ambiguity, there exists an equilibrium with inertia where agents also insure fully against Knightian uncertainty. When the level of ambiguity exceeds a critical threshold, full insurance no longer prevails and there exist equilibria with inertia where agents do not insure against uncertainty at all. We also show that equilibria with inertia are indeterminate.