David Schmeidler

Subjective Probability and Expected Utility Without Additivity

An act maps states of nature to outcomes: deterministic outcomes, as well as random outcomes, are included. Two acts f and g are comonotonic, by definition, if it never happens that f(s) > f(t) and g(t) > g(s) for some states of nature s and t. An axiom of comonotonic independence is introduced here. It weakens the von Neumann-Morgenstern axiom of independence as follows: If f > g and if f, g and h are comonotonic then

The Nucleolus of a Characteristic Function Game

f + (1 -

Integral representation without additivity

)h >

Equilibrium points of nonatomic games

g + (1 -

Egalitarian Equivalent Allocations: A New Concept of Economic Equity

)h. If a nondegenerate, continuous, and monotonic (state independent) weak order over acts satisfies comonotonic independence, then it induces a unique non-(necessarily-) additive probability and a von Neumann-Morgenstern utility. Furthermore, one can compute the expected utility of an act with respect to the nonadditive probability, using the Choquet integral. This extension of the expected utility theory covers situations, such as the Ellsberg paradox, which are inconsistent with additive expected utility. The concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included. Copyright 1989 by The Econometric Society.

Implementation in differential information economies

Abstract : In RM 23, a proof was given that the nucleolus is continuous as a function of the characteristic function. This proof is not correct; the author, at least, does not know how to complete it. In the paper a correct proof for this fact is given. The proof is based on an alternative definition of the nucleolus, which is of some interest in its own right. (Author)

Additive representations of non-additive measures and the choquet integral

Let I be a norm-continuous functional on the space B of bounded Y-measurable real valued functions on a set S, where E is an algebra of subsets of S. Define a set function v on E by: v(E) equals the value of I at the indicator function of E. For each a in B let J(a) = J (v(a > a) -v(S)) da + v(a > a) da. Then I = J on B if and only if I(b + c) = I(b) + I(c) whenever (b(s) b(t))(c(s) c(t)) > 0 for all s and t in S.

Existence of Correlated Equilibria

The Nash theorem on the existence of equilibrium points inTV-person non-cooperative games in normal form is generalized to the case when there is a continuum of players endowed with a nonatomic measure. The mathematical tools are those used in mathematical economics, in particular, markets with a continuum of traders. The main result shows that under a restriction on the payoff functions there exists an equilibrium in pure strategies.

Utility theory with uncertainty

Foreword, 671. — I. Introduction, 671. — II. The concept of Pareto-efficient-egalitarian-equivalent-allocations (PEEEA), 674. — III. PEEEA as a fair arbitration scheme for allocations, 676. — IV. Maximin properties of PEEEA, 678. — V. PEEEA in economies with production, 680. — Mathematical appendices, 682.

A Difficulty in the Concept of Fairness

Abstract We consider the problem of implementation of social choice correspondences in differential information economies. We provide necessary conditions for implementation and slightly stronger conditions which are sufficient to guarantee implementation