Itzhak Gilboa

Expected utility with purely subjective non-additive probabilities

Acts are functions from the set of states of the world into the set of consequences. Savage proposed axioms regarding a binary relation on the set of acts which are necessary and sufficient for it to be representable by the functional ʃu(*)dP for some real-valued (utility) function u on the set of consequences and a (probability) measure P on the set of states of the world. The Ellsberg paradox leads us to reject one of Savages main axioms - the Sure Thing Principle - and develop a more general theory, in which the probability measure need not be additive.

Nash and correlated equilibria: Some complexity considerations

This paper deals with the complexity of computing Nash and correlated equilibria for a finite game in normal form. We examine the problems of checking the existence of equilibria satisfying a certain condition, such as “Given a game G and a number r, is there a Nash (correlated) equilibrium of G in which all players obtain an expected payoff of at least r?” or “Is there a unique Nash (correlated) equilibrium in G?” etc. We show that such problems are typically “hard” (NP-hard) for Nash equilibria but “easy” (polynomial) for correlated equilibria.

Social Stability and Equilibrium

This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and the temporary effects on growth (TEG) specification, which generates semi-endogenous long-run economic growth. In the present model, the direction of the effect of the size of innovations on the pattern of trade and Homes relative wage depends on the way in which the scale effects property is removed. Under the PEG specification, changes in the size of innovations increase Homes comparative advantage and its relative wage, while under the TEG specification, an increase in the size of innovations increases Homes relative wage but with an ambiguous effect on its comparative advantage.

Additive representations of non-additive measures and the choquet integral

AbstractThis paper studies some new properties of set functions (and, in particular, “non-additive probabilities” or “capacities”) and the Choquet integral with respect to such functions, in the case of a finite domain.We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results:the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand;the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures;under fairly general conditions one may define a “Radon-Nikodym derivative” of one capacity with respect to another;the “optimistic” pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space. We also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.

Ambiguity and the Bayesian Paradigm

This is a survey of some of the recent decision-theoretic literature involving beliefs that cannot be quantified by a Bayesian prior. We discuss historical, philosophical, and axiomatic foundations of the Bayesian model, as well as of several alternative models recently proposed. The definition and comparison of ambiguity aversion and the updating of non-Bayesian beliefs are briefly discussed. Finally, several applications are mentioned to illustrate the way that ambiguity (or “Knightian uncertainty”) can change the way we think about economic problems.

A Theory of Case-Based Decisions: Acknowledgments

Gilboa and Schmeidler provide a new paradigm for modeling decision making under uncertainty. Case-based decision theory suggests that people make decisions by analogies to past cases: they tend to choose acts that performed well in the past in similar situations, and to avoid acts that performed poorly. The authors describe the general theory and its relationship to planning, repeated choice problems, inductive inference, and learning. They highlight its mathematical and philosophical foundations and compare it to expected utility theory as well as to rule-based systems.

A Theory of Case-Based Decisions: Frontmatter

Gilboa and Schmeidler provide a new paradigm for modeling decision making under uncertainty. Case-based decision theory suggests that people make decisions by analogies to past cases: they tend to choose acts that performed well in the past in similar situations, and to avoid acts that performed poorly. The authors describe the general theory and its relationship to planning, repeated choice problems, inductive inference, and learning. They highlight its mathematical and philosophical foundations and compare it to expected utility theory as well as to rule-based systems.

Utilitarian Aggregation of Beliefs and Tastes.

Harsanyi’s utilitarianism is extended here to Savage’s framework. We formulate a Pareto condition that implies that both society’s utility function and its probability measure are linear combinations of those of the individuals. An indiscriminate Pareto condition has been shown to contradict linear aggregation of beliefs and tastes. We argue that such a condition is not compelling: Society should not necessarily endorse a unanimous choice when it is based on contradictory beliefs. Our Pareto condition is restricted to choices that involve identical beliefs only.

IS IT ALWAYS RATIONAL TO SATISFY SAVAGE'S AXIOMS?

This note argues that, under some circumstances, it is more rational not to behave in accordance with a Bayesian prior than to do so. The starting point is that in the absence of information, choosing a prior is arbitrary. If the prior is to have meaningful implications, it is more rational to admit that one does not have sufficient information to generate a prior than to pretend that one does. This suggests a view of rationality that requires a compromise between internal coherence and justification, similarly to compromises that appear in moral dilemmas. Finally, it is argued that Savages axioms are more compelling when applied to a naturally given state space than to an analytically constructed one, in the latter case, it may be more rational to violate the axioms than to be Bayesian.

Expectation and Variation in Multi-period Decisions

Multi-period decisions are decisions which determine an individuals payoffs in several periods in the future. This paper examines the theoretical foundations of the prevalent weighted average assumption. More specifically, we use a multi-period interpretation of the famous Ellsberg paradox in decision under uncertainty to show that in many cases of interest additively-separable functionals (in general) and weighted average ones (in particular) do not seem appropriate for the representation of the decision makers preferences. We then suggest replacing the sure-thing principle, which may be used to axiomatize a weighted average functional, by a weaker version of it. Using the weakened axiom in Schmeidlers nonadditive measure model (reinterpreted for the multi-period context) yields an axiomatization of a larger class of decision rules which are representable by a weighted average of the utility in each period und the utility variation between each two consecutive periods. The weighted average assumption is a special case of the generalized model, a case in which the decision maker is variation neutral. Similarly, we define and characterize variation aversion and variation liking, and show an example of the economic implications of these properties.