Is it Possible to Define Subjective Probabilities in Purely Behavioral Terms? A Comment on Epstein-Zhang (2001)

It is shown that well-behaved preference orderings may exhibit the Ellsberg paradox on the set of unambiguous events as defined by Epstein and Zhang (2001). Moreover, since such counterexamples can be constructed even when the set of unambiguous events is rich, EZ’s main representation result does not clarify satisfactorily when the proposed definition delivers probabilistic sophistication on unambiguous events. We conclude by conjecturing that these problems indicate the existence of inherent limitations of a strictly behavioral approach to identifying probabilistic beliefs in the presence of ambiguity, rather than deficiencies in EZ’s implementation of that approach.

Decision-Making in the Context of Imprecise Probabilistic Beliefs

Coherent imprecise probabilistic beliefs are modelled as incomplete comparative likelihood relations admitting a multiple-prior representation. Under a structural assumption of Equidivisibility, we provide an axiomatization of such relations and show uniqueness of the representation. In the second part of the paper, we formulate a behaviorally general axiom relating preferences and probabilistic beliefs which implies that preferences over unambiguous acts are probabilistically sophisticated and which entails representability of preferences over Savage acts in an Anscombe-Aumann-style framework. The motivation for an explicit and separate axiomatization of beliefs for the study of decision-making under ambiguity is discussed in some detail.

MONOTONICITY IMPLIES STRATEGY-PROOFNESS FOR CORRESPONDENCES

We show that Maskin monotone social choice correspondences on sufficiently rich domains satisfy a generalized strategy-proofness property, thus generalizing Muller and Satterthwaite''s (1977) theorem to correspondences. From the point of view of Nash implementation theory, the result yields a partial characterization of the restrictions entailed by Nash implementability. Alternatively, the result can be viewed as a possibility theorem on the dominant-strategy-implementability of monotone SCCs via set-valued mechanisms for agents who are completely ignorant about the finally selected outcome. It is shown by examples that stronger strategy-proofness properties fail easily.

INCENTIVE-COMPATIBLE AND EFFICIENT RESOURCE ALLOCATION IN LARGE ECONOMIES: AN EXACT AND LOCAL APPROACH

The main result of this paper characterizes possibly non-symmetric strategy-proof and efficienct choice functions as Perfectly Competitive. Efficiency is defined as impossibility of improvement by reallocation of commodity among finite sets of agents, and largeness of the economy is captured by a weak aggregation-condition called ""local separability."" Individual rationality constraints with respect to an assignment of endowments imply that the resulting allocations must be Walrasian relative to the assignment of endowments. The exact, local approach combined with a normality assumption on the domain of preferences allows the proofs to remain elementary throughout.

INTRODUCTION TO THE SEMANTICS OF BELIEF AND COMMON BELIEF

We provide an introduction to interactive belief systems from a qualitative and semantic point of view. Properties of belief hierarchies are formulated locally. Among the properties considered are ""Common belief in no error"" (which has been shown to have important game theoretic applications), ""Negative introspection of common belief"" and ""Truth about common belief."" The relationship between these properties is studied.

A THEORY OF RATIONAL CHOICE UNDER COMPLETE IGNORANCE

This paper contributes to a theory of rational choice under uncertainty for decision-makers whose preferences are exhaustively described by partial orders representing ""limited information."" Specifically, we consider the limiting case of ""Complete Ignorance"" decision problems characterized by maximally incomplete preferences and important primarily as reduced forms of general decision problems under uncertainty. ""Rationality"" is conceptualized in terms of a ""Principle of Preference-Basedness,"" according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called ""Simultaneous Expected Utility Maximization"" which in particular satisfies a choice-functional independence and a context-dependent choice-consistency condition; it can be interpreted as the fair agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the different possible states (respectively extermal priors in the general case).

EPISTEMIC FOUNDATIONS OF SOLUTION CONCEPTS IN GAME THEORY: AN INTRODUCTION

We give an introduction to the literature on the epistemic foundations of solution concepts in game theory. Only normal-form games are considered. The solution concepts analyzed are rationalizability, strong rationalizability, correlated equilibrium and Nash equilibrium. The analysis is carried out locally in terms of properties of the belief hierarchies. Several examples are used throughout to illustrate definitions and concepts.

AGREEING TO DISAGREE: A SURVEY

Aumann (1976) put forward a formal definition of common knowledge and used it to prove that two ""like minded"" individuals cannot ""agree to disagree"" in the following sense. If they start from a common prior and update the probability of an event E (using Bayes'' rule) on the basis of private information, then it cannot be common knowledge between them that individual 1 assigns probability p to E and individual 2 assigns probability q to E with p ¹ q. In other words, if their posteriors of event E are common knowledge then they must coincide. Aumann''s Agreement Theorem has given rise to a large literature which we review in this paper. The results are classified according to whether they are probabilistic (Bayesian) or qualitative. Particular attention is paid to the issue of how to interpret the notion of Harsanyi consistency as a (local) property of belief hierarchies.

Global Identifiability Under Uncorrelated Residuals

Suppose in each equation, not counting covariance restrictions, we need one more restriction to meet the order condition. If we now add to each equation a restriction that its structural residual is uncorrelated with the residual of some other equation, is the parameter of the new model identifiable globally? That is the question. In general the answer is no. The parameter could remain either not identifiable or is locally identifiable, possibly globally under additional inequality restrictions. In this paper we find families of models for which the answer to the question is yes without the help of inequalities. The families share common characteristics. First, the sufficient condition for local identifiability must hold. Secondly, the string of zero correlations between residuals contains a closed cycle of length at least four. Thirdly, with the variables, equations and residuals all numbered as they are in the cycle, the odd numbered variables must satisfy a kinship relationship and lastly, the structural residuals can not all be uncorrelated. There are also differences in families, but these come from the difference in the required kinship relationship. When there are four or more equations containing external variables, the variety of models with uniquely identifiable parameter under a string of uncorrelated residuals is considerable. In particular, when correlated inverse demand shocks are uncorrelated with correlated supply shocks, our results show that many flexible inverse demand and supply equations reproducing exactly the observed price and quantity moments are members of the above families.

Condorcet admissibility: Indeterminacy and path-dependence under majority voting on interconnected decisions

Judgement aggregation is a model of social choice where the space of social alternatives is the set of consistent evaluations (`views') on a family of logically interconnected propositions, or yes/no-issues. Unfortunately, simply complying with the majority opinion in each issue often yields a logically inconsistent collection of judgements. Thus, we consider the Condorcet set: the set of logically consistent views which agree with the majority in as many issues as possible. Any element of this set can be obtained through a process of diachronic judgement aggregation, where the evaluations of the individual issues are decided through a sequence of majority votes unfolding over time, with earlier decisions possibly imposing logical constraints on later decisions. Thus, for a fixed profile of votes, the ultimate social choice can depend on the order in which the issues are decided; this is called path dependence. We investigate the size and structure of the Condorcet set ---and hence the scope and severity of path-dependence ---for several important classes of judgement aggregation problems.judgement aggregation; diachronic; path-dependence; indeterminacy; Condorcet; median rule; majoritarian