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.

Although their goal is to separate a decision maker's underlying beliefs (their subjective probabilities of events) from their preferences (their attitudes toward risk), classic choice-theoretic derivations of subjective probability all rely upon some form of the Marschak-Samuelson "Independence Axiom" or the Savage "Sure-Thing Principle, " which is equivalent to requiring that the decision maker's preferences over lotteries conform to the expected utility hypothesis. This paper presents a choice-theoretic derivation of subjective probability which satisfies the axioms of classical probability theory, but which neither assumes nor implies that the decision maker's preferences over lotteries necessarily conform to the expected utility hypothesis.

[This paper presents an expected utility theory for state-dependent preferences. It proposes axioms that permit the joint derivation of subjective probabilities and utilities when the decision maker's preferences are not independent of the prevailing state of nature. In addition to the usual von Neumann-Morgenstern axioms, these axioms also include the requirement that the decision-maker's actual preferences are consistent with his preferences contingent on an hypothetical probability distribution over the states of nature. Two versions of the consistency axiom are introduced and their significance in the context of Bayesian decision theory is discussed.]

IT IS WELL KNOWN that for a finite exchange economy, where preferences are not assumed to be convex, there may be no price or even the core may be empty. For this reason it was proposed to enlarge the set of price and the core by introducing the concepts of equilibrium and core. The existence of approximate for exchange economies, where preferences are not assumed to be convex, has been investigated by R. Starr [6]. He showed that there exists a quasi-equilibrium, provided the number of participants is large enough and there is a bound on the degree of non-convexity [6, p. 30, Assumption D]. In this note we shall show the existence of equilibria (a stronger concept than the one considered by Starr [6, p. 31]) for large economies where the preferences are neither assumed to be convex nor complete. To obtain our result we shall assume that the preferences and the endowments of all participating agents belong to a compact set. In [5] Shapley and Shubik proved that, for a large replica of a given economy with transferable utility, the e-core is nonempty. We shall generalize this result to large economies without transferable utility by using the concept of ?-core as introduced by Kannai [2]. The nonemptiness of the e-core follows easily from the existence of approximate and a relationship between the set of approximate and e-core. The existence of c-core for large economies (with a fixed number of types) can also be deduced from Kannai's Theorem C' [2] in its stronger form (Theorem C in [3]).

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of repetitions of past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventualitycase pairs, as follows. A memory is identified with a vector, counting the number of repetitions of each case. Multiplication of the matrix by a memory vector yields a numerical representation of the ordinal plausibility ranking given that memory. Interpreting this result for the ranking of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with their likelihood functions. 1 Introduction It is well known that inductive inference is not logically valid. As David Hume (1748) put it, "... The co...

SAMUELSON MADE THE CONJECTURE stated above in his 1967 paper [7]. He also formalized there the axiom of independence of irrelevant alternatives for cardinal preferences, used here. Preference are cardinal if their representation by a numerical function is invariant under, and only under, positive linear transformations. One may think that the disregard for intensity of preferences, embedded in Arrow's treatment of profiles of ordinal rankings of alternatives, leads to the impossibility result. Samuelson's conjecture points out that this is not the way to refute the conclusions of Arrow's theorem. There is also interest per se in aggregation of cardinal preferences. Such preferences are usually considered as von Neumann-Morgenstern utility, i.e., numerical representation of preferences over lotteries [11]. Since uncertainty is the rule and not the exception whenever decisions are involved, it is of some importance to obtain a social N-M utility over risky outcomes. Given such a utility, the society will be able to choose a best alternative among the several feasible risky actions (i.e., lotteries). However it is not necessary to restrict the interpretation of cardinal preferences to those induced by ordinal ranking over lotteries. One can think of cardinal preferences derived from comparisons between pairs of alternatives (as in an axiomatization of a regret relation). See Alt [1] for an early work of this kind. When working with cardinal preferences a continuity assumption is needed, in addition to unanimity and independence (see the example at the end of the next section). A standard reference for Arrow's theorem is the last chapter of his book [2]. For a general discussion of aggregation of cardinal preferences, see ShapleyShubik [10]. Some other impossibility results involving different notions of cardinal preferences appear in the works of Sen [9], DeMeyer-Plott [4], Schwartz [8], and Fishburn [5]. A model dealing with aggregation of cardinal preferences into social cardinal prefereiLces, as here, is that of Harsanyi [6]. However he is interested in 1 The work of the first author was done at Northwestern University and the work of the second author began at the University of Illinois in Urbana-Champaign and it was completed at the University of Minnesota in Minneapolis. Both authors are on leave from Tel-Aviv University. The authors wish to express their thanks to E. A. Pazner, M. A. Satterthwaite, J. Kelly and the referees for helpful comments. This research was partly supported by NSF Grant # SOC-75-05317.

We argue that the notion of Pareto dominance is not as compelling in the presence of uncertainty as it is under certainty. In particular, voluntary trade based on differences in tastes is commonly accepted as desirable, because tastes cannot be wrong. By contrast, voluntary trade based on incompatible beliefs may indicate that at least one agent entertains mistaken beliefs. We propose and characterize a weaker, No-Betting, notion of Pareto domination which requires, on top of unanimity of preference, the existence of shared beliefs that can rationalize such preference for each agent.

An agent is asked to assess a real-valued variable Yp based on certain characteristics Xp = (Xp1, …, Xpm), and on a database consisting of Xi1, … Xim, Yi) for i = 1, …, n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Ȳps, be the weighted average of all previously observed values Yi, where the weight of Yi for every i = 1, …, n, is the similarity between the vector Xp1, …, Xpm, associated with Yp, and the previously observed vector, Xi1, …, Xim. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations.We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction.

Fifty years ago, Harsanyi published the first of his seminal two papers on utilitarianism. His results were derived within the von Neumann Morgenstern expected utility theory. A year later, Savage incorporated subjective probability into expected utility theory in his famous book. In this note we extend Harsanyi’s utilitarianism to Savage’s framework. We show that a Pareto condition implies utilitarian aggregation: both society’s utility function and its probability measure are linear combinations of those of the individuals. This conclusion contrasts the impossibility of reconciling a Pareto condition and linear aggregation of beliefs and tastes, that was noted by several authors. We argue that the indiscriminate Pareto condition considered by these authors is not compelling. Society should not necessarily endorse a unanimous choice when it is based on contradictory beliefs. Restricting the Pareto condition to choices that only involve identical beliefs allows the extension of Harsanyi’s result to Savage’s framework.