Rationality and Solutions to Nonconvex Bargaining Problems: Rationalizable, Asymmetric and Nash Solutions

Conditions α and β are two well-known rationality conditions in the theory of rational choice. This paper examines the implications of weaker versions of these two rationality conditions in the context of solutions to nonconvex bargaining problems. It is shown that, together with the standard axioms of efficiency and strict individual rationality, they imply rationalizability of solutions to nonconvex bargaining problems. We then characterize asymmetric Nash solutions by imposing a continuity and the scale invariance requirements. We also give a characterization of the Nash solution by using the two rationality conditions. These results make a further connection between solutions to non-convex bargaining problems and rationalizability of choice function in the theory of rational choice.

Capabilities and Equality of Health II: Capabilities as Options

The concept of capabilities, introduced originally by Sen, has inspired many researchers but has not found any simple formal representation which might be instrumental in the construction of a comprehensive theory of equality. In a previous paper (Keiding, 2005), we investigated whether preferences over capabilities as sets of functionings can be rationalized by maximization of a suitable utility function over the set of functionings. Such a rationalization turned out to be possible only in cases which must be considered exceptional and which do not allowfor interesting applications of the capability approach to questions of health or equality. In the present paper we extend the notion of rationalizing orderings of capabilities to a dynamical context, in the sense that the utility function is not yet revealed to the individual at the time when the capabilities are ordered. It turns out that orderings which are in accordance with such probabilistic utility assignments can be characterized by a smaller set of the axioms previously considered.Capabilities; characteristics; equality of health

Rationality and the Nash Solution to Non-convex Bargaining Problems

Conditions α and β are two well-known rationality conditions in the theory of rational choice. This paper examines the implication of weaker versions of these two rationality conditions in the context of solutions to non-convex bargaining problems. It is shown that, together with the standard axioms of efficiency, anonymity and scale invariance, they characterize the Nash solution. This result makes a further connection between solutions to non-convex bargaining problems and rationalizability of choice functions in the theory of rational choice.

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.

Essential Data, Budget Sets and Rationalization

According to a minimalist version of Afriat’s theorem, a consumer behaves as a utility maximizer if and only if a feasibility matrix associated with his choices is cyclically consistent. An ”essential experiment” consists of observed consumption bundles (x1,xn) and a feasibility matrix α. Starting with a standard experiment, in which the economist has specific budget sets in mind, we show that the necessary and sufficient condition for the existence of a utility function rationalizing the experiment, namely, the cyclical consistency of the associated feasibility matrix, is equivalent to the existence, for any budget sets compatible with the deduced essential experiment, of a utility function rationalizing them (and typically depending on them). In other words, the conclusion of the standard rationalizability test, in which the economist takes budget sets for granted, does not depend on the full specification of the underlying budget sets but only on the essential data that these budget sets generate. Starting with an essential experiment (x1,...,xn;α), we show that the cyclical consistency of α, together with a further consistency condition involving both (x1,...,xn) and α, guarantees that the essential experiment is rationalizable almost robustly, in the sense that there exists a single utility function which rationalizes at once almost all budget sets which are compatible with (x1,...,xn;α). The conditions are also trivially necessary.Afriat’s theorem, budget sets, cyclical consistency, rational choice, revealed preference

Sentiments and rationalizability

Sentiments are characteristics of players' beliefs. I propose two notions of sentiments, confidence and optimism, and I study their role in shaping the set of rationalizable strategy profiles in (incomplete information) games with complementarities. Confidence is related to a player's perceived precision of information; optimism is the sentiment that the outcome of the game will be ``favorable.'' I prove two main results on how sentiments and payoffs interact to determine the size and location of the set of rationalizable profiles. The first result provides an explicit upper bound on the size of the set of rationalizable strategy profiles, relating complementarities and confidence; the second gives an explicit lower bound on the change of location, relating complementarities and optimism. I apply these results to four areas. In models of currency crisis, the results suggest that the most confident investors may drive financial markets. In models of empirical industrial organization, the paper provides a classification of the parameter values for which the model is identified. In non-Bayesian updating, the results clarify the strategic implications of certain biases. Finally, the results generalize and clarify the uniqueness result of global games.Rationalizable strategy profiles; complementarities; sentiments; confidence; optimism

Rational Choice on Arbitrary Domains: A Comprehensive Treatment

The rationalizability of a choice function on arbitrary domains by means of a transitive relation has been analyzed thoroughly in the literature. Moreover, characterizations of various versions of consistent rationalizability have appeared in recent contributions. However, not much seems to be known when the coherence property of quasi-transitivity or that of P-acyclicity is imposed on a rationalization. The purpose of this paper is to fill this significant gap. We provide characterizations of all forms of rationalizability involving quasi-transitive or P-acyclical rationalizations on arbitrary domains