Extension of stochastic dominance theory to random variables

SSRN Electronic Journal, 2000

This paper extends some well-known univariate stochastic dominance results to multivariate stochastic dominances for risk averters and risk seekers, respectively, when the attributes are assumed to be independent and the utility is assumed to be additively separable. Under these assumptions, we develop some properties of multivariate stochastic dominances for risk averters and risk seekers, respectively. For example, we prove that multivariate stochastic dominances are equivalent to the expected-utility maximization for risk averters and risk seekers, respectively. We show that the hierarchical relationship exists for multivariate stochastic dominances. We develop some properties for non-negative combinations and convex combinations of random variables of multivariate stochastic dominance.

Journal of Applied Mathematics and Decision …, 2006

extended the theory of mean-variance criterion to include the comparison among distributions that differ only by location and scale parameters and to include general utility functions with only convexity or concavity restrictions. In this paper, we make some comments on Meyer's paper and extend the results from Tobin (1958) that the indifference curve is convex upwards for risk averters, concave downwards for risk lovers, and horizontal for risk neutral investors to include the general conditions stated by Meyer (1987). We also provide an alternative proof for the theorem. Levy (1989) extended Meyer's results by introducing some inequality relationships between the stochasticdominance and the mean-variance efficient sets. In this paper, we comment on Levy's findings and show that these relationships do not hold in certain situations. We further develop some properties among the first-and second-degree stochastic dominance efficient sets and the mean-variance efficient set.

SSRN Electronic Journal, 2010

Stochastic dominance permits a partial ordering of alternatives (probability distributions on consequences) based only on partial information about a decision maker's utility function. Univariate stochastic dominance has been widely studied and applied, with general agreement on classes of utility functions for dominance of different degrees. Extensions to the multivariate case have received less attention and have used different classes of utility functions, some of which require strong assumptions about utility. We investigate multivariate stochastic dominance using a class of utility functions that is consistent with a basic preference assumption, can be related to well-known characteristics of utility, and is a natural extension of the stochastic order typically used in the univariate case. These utility functions are multivariate risk averse, and reversing the preference assumption allows us to investigate stochastic dominance for utility functions that are multivariate risk seeking. We provide insight into these two contrasting forms of stochastic dominance, develop some criteria to compare probability distributions (hence alternatives) via multivariate stochastic dominance, and illustrate how this dominance could be used in practice to identify inferior alternatives. Connections between our approach and dominance using different stochastic orders are discussed.

SSRN Electronic Journal, 2007

This paper characterizes higher order risk e¤ects, such as prudence and temperance, via preferences that partially order a set of simple 50-50 lotteries. In particular, consider the random variables e X N ; e Y N ; e X M and e Y M , and assume that e X i dominates e Y i via i th-order stochastic dominance for i = M; N. We show that the 50-50 lottery [ e X N + e Y M ; e Y N + e X M ] dominates the lottery [ e X N + e X M ; e Y N + e Y M ] via (N +M) th-order stochastic dominance. A preference ranking over these lotteries is shown to generalize the concept of risk apportionment, as introduced by Eeckhoudt and Schlesinger (2006). We apply our results in several examples of decision making under risk.

Wiley-Blackwell eBooks, 2011

The goals of this chapter are the following: • To explore the relationship between preference relations and quasi-semidistances. • To introduce a universal description of probability quasisemidistances in terms of a Hausdorff structure. • To provide examples with first-, second-, and higher-order stochastic dominance and to introduce primary, simple, and compound stochastic orders. • To explore new stochastic dominance rules based on a popular risk measure. • To provide a utility-type representation of probability quasisemidistances and to describe the degree of violation utilized in almost stochastic orders in terms of quasi-semidistances.

SIAM Journal on Optimization, 2002

We consider the problem of constructing mean{risk models which are consistent with the second degree stochastic dominance relation. By exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. This allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation. We also provide stochastic linear programming formulations of these models.

2017

This paper extends the theory between Kappa ratio and stochastic dominance (SD) and risk-seeking SD (RSD) by establishing several relationships between first- and higher-order risk measures and (higher-order) SD and RSD. We first show the sufficient relationship between the (n+1)-order SD and the n-order Kappa ratio. We then find that, in general, the necessary relationship between SD/RSD and the Kappa ratio cannot be established. Thereafter, we find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, we can get the necessary relationships between the (n+1)-order SD with the n-order Kappa ratio when we impose some conditions on the means. Our findings enable academics and practitioners to draw better decision in their analysis.

1998

Two methods are frequently used for modeling the choice among uncertain prospects: stochastic dominance relation and mean-risk approaches. The former is based on an axiomatic model of riskaverse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. The seminal Markowitz model uses the variance as the risk measure in the mean-risk analysis which results in a formulation of a quadratic programming model. Following the pioneering work of Sharpe, many attempts have been made to linearize the mean-risk approach. There were introduced risk measures which lead to linear programming mean-risk models. This paper focuses on two such risk measures: the Gini's mean (absolute) difference and the mean absolute deviation. Consistency of the corresponding mean-risk models with the second degree stochastic dominance (SSD) is reexamined. Both the models are in some manner consistent with the SSD rules, provided that the trade-off coefficient is bounded by a certain constant. However, for the Gini's mean difference the consistency turns out to be much stronger than that for the mean absolute deviation. The analysis is graphically illustrated within the framework of the absolute Lorenz curves.

Management Science, 2017

2010

Available characterizations of the various notions of stochastic dominance concern continuous random variables. Yet, discrete random variables are often used either in pedagogical presentations of stochastic dominance or in experimental tests of this no- tion. This note provides complete characterizations of the various notions of stochastic dominance for discrete random variables. J.E.L Classification Numbers: D80, D81, G11.