We establish an equivalence between two seemingly different theories: one is the traditional axiomatisation of incomplete preferences on horse lotteries based on the mixture independence axiom; the other is the theory of desirable gambles developed in the context of imprecise probability. The equivalence allows us to revisit incomplete preferences from the viewpoint of desirability and through the derived notion of coherent lower previsions. On this basis, we obtain new results and insights: in particular, we show that the theory of incomplete preferences can be developed assuming only the existence of a worst act---no best act is needed---, and that a weakened Archimedean axiom suffices too; this axiom allows us also to address some controversy about the regularity assumption (that probabilities should be positive---they need not), which enables us also to deal with uncountable possibility spaces; we show that it is always possible to extend in a minimal way a preference relation to one with a worst act, and yet the resulting relation is never Archimedean, except in a trivial case; we show that the traditional notion of state independence coincides with the notion called strong independence in imprecise probability---this leads us to give much a weaker definition of state independence than the traditional one; we rework and uniform the notions of complete preferences, beliefs, values; we argue that Archimedeanity does not capture all the problems that can be modelled with sets of expected utilities and we provide a new notion that does precisely that. Perhaps most importantly, we argue throughout that desirability is a powerful and natural setting to model, and work with, incomplete preferences, even in case of non-Archimedean problems. This leads us to suggest that desirability, rather than preference, should be the primitive notion at the basis of decision-theoretic axiomatisations.
The link between desirability and preference has been surfacing in the literature in a number of cases, but has apparently gone unnoticed. That it has surfaced is not surprising, because the theoretical study of preferences based on the mixture-independence axiom results in, and is worked out using, cones; cones are also the fundamental tool in desirability. That it has not been remarked and exploited explicitly, as we do in this paper, is. Perhaps the most evident case where the two theories have nearly touched each other is in Galaabaatar and Karni's work @cite_10 . In the next sections, we discuss this and two other main approaches in the literature that have dealt with the axiomatisation of incomplete preferences, and compare our approach with them.
@cite_10
2091277030 2081359028 2154367906 2099790472
We study the behavior of random walks along the edges of the stable marriage lattice for various restricted families of allowable preference sets. In the "k-attribute model," each man is valued in each of k attributes, and each woman's ranking of the men is determined by a linear function, representing her relative ranking of those attributes; men's rankings of the women are determined similarly. We show that sampling with a random walk on the marriage lattice can take exponential time, even when k = 2. Moreover, we show that the marriage lattices arising in the k-attribute model are more restrictive than in the general setting; previously such a restriction had only been shown for the sets of preference lists. The second model we consider is the "k-range model," where each person lies in a position in [i, i + k - 1], for some i, on every preference list of the opposite sex. When k = 1 there is a unique stable marriage. When k = 2 there already can be an exponential number of stable marriages, but we show that a random walk on the stable marriage lattice always converges quickly to equilibrium. However, when k ≥ 5, there are preference sets such that the random walk on the lattice will require exponential time to converge. Lastly, we show that in the extreme case where each gender's rankings of the other are restricted to one of just a constant k possible preference lists, there are still instances for which the Markov chain mixes exponentially slowly, even when k = 4. This oversimplification of the general model helps elucidate why Markov chains based on spouse-swapping are not good approaches to sampling, even in specialized scenarios. Much of the dissatisfaction with Professor Arrow's formulation (1963) of the problem of social choice rests on Arrow's imposition of the controversial "independence of irrelevant alternatives" condition (A3). Arrow himself has ceased to insist upon this condition in its strong original form (Arrow 1967). The chief objection to A3 is that it prevents the collectivechoice rule from responding to intensity of individual preference, making the choice rule unacceptable on a priori grounds (Kemp and Asimakopulos 1952; Hildreth 1953; Coleman 1966; Quirk and Saposnik 1968; Sen 1970). This view is not correct if intensity of preference is measured by an ability to make side payments in order to influence collective decision making. I show that a modified version of A3 which takes explicit account of intensity of preference actually implies the original independence condition for realistic environment sets. This will be true no matter how side payments of private goods and services are evaluated. A modest amount of notation is required before the independence condition can be given formal expression. The set X, the nonnegative orthant of m-dimensional Euclidean space, is the space of hypothetical states of the economy. There are n individuals whose preferences are to be considered by any collective decision-making process, and to each individual i there corresponds a binary relation Ri which is complete on X. The term This paper extends the subjective expected utility model of decision making under uncertainty to include incomplete beliefs and tastes. The main results are two axiomatizations of the multiprior expected multiutility representations of preference relations under uncertainty. The paper also introduces new axiomatizations of Knightian uncertainty and the expected multiutility model with complete beliefs. Couples looking for jobs in the same labor market may cause instabilities. We deter- mine a natural preference domain, the domain of weakly responsive preferences, that guarantees stability. Under a restricted unemployment aversion condition we show that this domain is maximal for the existence of stable matchings. We illustrate how small deviations from (weak) responsiveness, that model the wish of couples to be closer together, cause instability, even when we use a weaker stability notion that excludes myopic blocking. Our remaining results deal with various properties of the set of stable matchings for "responsive couples markets," viz., optimal- ity, filled positions, and manipulation. Journal of Economic Literature Classification Numbers: C78; J41.