Stochastic dominance with imprecise information

Abstract

Stochastic dominance, which is based on the comparison of distribution functions, is one of the most popular preference measures. However, its use is limited to the case where the goal is to compare pairs of distribution functions, whereas in many cases it is interesting to compare sets of distribution functions: this may be the case for instance when the available information does not allow to fully elicitate the probability distributions of the random variables. To deal with these situations, a number of generalisations of the notion of stochastic dominance are proposed; their connection with an equivalent p-box representation of the sets of distribution functions is studied; a number of particular cases, such as sets of distributions associated to possibility measures, are investigated; and an application to the comparison of the Lorenz curves of countries within the same region is presented.

Ranking of fuzzy intervals seen through the imprecise probabilistic lens

“…If a p-box is {0, 1} valued, that is corresponds to Dirac measures between a lower and an upper bounds, those extensions coincide with the interval comparisons bearing the same name (already in [32]). Additionally, implications (9) directly extend to the above extensions of first-order stochastic dominance.…”

“…When the cumulative distributions of X and Y are bounded by a lower and an upper one, respectively [F X , F X ] and [F Y , F Y ], the notion of stochastic dominance can again be extended in multiple ways [19,32]. Such extensions are the result of applying Equation (21) to the family of mappings { f c } c∈R , where f c : R 2 → R is defined as follows, for each c ∈ R:…”

“…Extending stochastic dominance to convex sets of probabilities comes down to check stochastic dominance on specific pairs of cumulative functions [32], and we can associate them to ideas used in interval ordering:…”

“…While some recent works have explored different extensions of classical probabilistic comparison methods to sets of probabilities (see, for instance [10,19,31]), only some of them ( [10,16,32]) have marginally considered the case of fuzzy sets/possibility distributions in general and of fuzzy intervals in particular. Studying the applications of the different extensions to the fuzzy interval case and relating them to previously proposed ranking in the fuzzy literature is the main topic of this paper.…”

A New Stochastic Dominance Degree Based on Almost Stochastic Dominance and Its Application in Decision Making

“…Construct relative dominant degree matrix with Equations (6)- (8). The relative dominant degree matrix is shown as follows: Normalize the relative dominant degree matrix with Equations (3) and (4): …”