Yaari dual theory without the completeness axiom.

This note shows how Yaari’s dual theory of choice under risk naturally extends to the case of incomplete preferences. This also provides an axiomatic characterization of a large and widely studied class of stochastic orders used to rank the riskiness of random variables or the dispersion of income distributions (including, e.g., second order stochastic dominance, dispersion, location independent riskiness).Yaari’s dual theory; incomplete preferences; stochastic orders

A strong law of large numbers for capacities

We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.Comment: Published at http://dx.doi.org/10.1214/009117904000001062 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Coherence without Additivity.

The Dutch book argument is a coherence condition for the existence of subjective probabilities. This note gives a general framework of analysis for this argument in a nonadditive probability setting. Particular cases are given by comonotonic and affinely related Dutch books that lead to Choquet expectations and Min expectations.Coherence; Dutch Book; Constant Linearity; Choquet Expectation; Multiple Priors

A strong law of large numbers for capacities.

We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.Capacities; Choquet integral; Strong law of large numbers

Disputed Lands

In this paper we consider the classical problem of dividing a land among many agents so that everybody is satisfied with the parcel she receives. In the literature, it is usually assumed that all the agents are endowed with cardinally comparable, additive, and monotone utility functions. In many economic and political situations violations of these assumptions may arise. We show how a family of cardinally comparable utility functions can be obtained starting directly from the agents’ preferences, and how a fair division of the land is feasible, without additivity or monotonicity requirements. Moreover, if the land to be divided can be modelled as a finite dimensional simplex, it is possible to obtain envy-free (and a fortiori fair) divisions of it into subsimplexes. The main tool is an extension of a representation theorem of Gilboa and Schmeidler (1989).Gender Fair Division; Envy-freeness; Preference Representation.

How to cut a pizza fairly: fair division with descreasing marginal evaluations.

Existential and constructive solutions to the classic problems of fair division are known for individuals with constant marginal evaluations. By considering nonatomic concave capacities instead of nonatomic probability measures, we extend some of these results to the case of individuals with decreasing marginal evaluations.

Ergodic Theorems for Lower Probabilities

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers

BV as a dual space.

Let C be a field of subsets of a set I. It is well known that the space FA of all the finitely additive games of bounded variation on C is the norm dual of the space of all simple functions on C. In this paper we prove that the space BV of all the games of bounded variation on C is the norm dual of the space of all simple games on C. This result is equivalent to the compactness of the unit ball in BV with respect to the vague topology.Set functions; duality; compactness; coalitional games