Stochastic Dominance Revisited

RAIRO - Operations Research, 1999

In this paper, we develop some stochastic dominance theorems for the location and scale family and linear combinations of random variables and for risk lovers as well as risk averters that extend results in Hadar and Russell (1971) and Tesfatsion (1976). The results are discussed and applied to decision-making.

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 Encyclopedia of Operations Research and Management Science, 2011

Management Science, 2013

L eshno and Levy Preferred by "all" and preferred by "most" decision makers:

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.

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.

1998

We analyse relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The concept of α-consistency of these approaches is defined as the consistency within a bounded range of mean-risk trade-offs. We show that mean-risk models using central semideviations as risk measures are α-consistent with stochastic dominance relations of the corresponding degree if the trade-off coefficient for the semideviation is bounded by one.

European Journal of Operational Research, 1999

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible tradeoff analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean-variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome-Risk diagram, which appears to be particularly useful for comparing uncertain outcomes.

Probability in the Engineering and Informational Sciences, 2015

The concept ofstochastic precedencebetween two real-valued random variables has often emerged in different applied frameworks. In this paper, we analyze several aspects of a more general, and completely natural, concept of stochastic precedence that also had appeared in the literature. In particular, we study the relations with the notions of stochastic ordering. Such a study leads us to introducing some special classes of bivariate copulas. Motivations for our study can arise from different fields. In particular, we consider the frame of Target-Based Approach in decisions under risk. This approach has been mainly developed under the assumption of stochastic independence between “Prospects” and “Targets”. Our analysis concerns the case of stochastic dependence.

Mathematical Programming, 2001

We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically.

Applied Stochastic Models in Business and Industry

Actuarial risks and nancial asset returns are tipically heavy tailed. In this paper, we introduce two criteria, called the right tail order and the left tail order, to compare stochastically these variables. The criteria are based on comparisons of expected utilities, for two classes of utility functions that give more weight to the right or the left tail (depending on the context) of the distributions. We study their properties, applications and connections with other classical orders, including the increasing convex and increasing concave orders. Finally, we provide empirical evidence of these orders with an example using real data. I This is a previous version of the article published in Applied Stochastic Models in

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.

Journal of Economic Theory, 2009

Consider a simple two-state risk with equal probabilities for the two states. In particular, assume that the random wealth variable X i dominates Y i via i th-order stochastic dominance for i = M,N. We show that the 50-50 lottery [X N + Y M , Y N + X M ] dominates the lottery [X N + X M , Y N + Y M ] via (N + M) th-order stochastic dominance. The basic idea is that a decision maker exhibiting (N + M) th-order stochastic dominance preference will allocate the statecontingent lotteries in such a way as not to group the two "bad" lotteries in the same state, where "bad" is defined via i th-order stochastic dominance. In this way, we can extend and generalize existing results about risk attitudes. This lottery preference includes behavior exhibiting higher order risk effects, such as precautionary effects and tempering effects. JEL Code: D81.

The concept of stochastic precedence between two real-valued random variables has often emerged in different applied frameworks, where it has been compared with the usual stochastic ordering. However it cannot be seen as a notion of stochastic ordering. In this paper we consider a slightly more general, and completely natural, concept of stochastic precedence and analyze its relations with the notions of stochastic ordering. Such a study leads us to introducing some special classes of bivariate copulas that reveal to be strictly related with the analysis of non-exchangeability of copulas. Some motivations for our study arise in the frame of the Target-Based Approach to the field of decisions under risk.

2000

Hanoch and Levy [6J presented a general first degree stochastic dominance theorem for the case in which the Von Neumann-Morgenstern utility function u(x) is non-decreasing in x. In contrast to other papers on this subject, Hanoch and Levy did not assume that utility is differentiable or even continuous. 1 Recently, Tesfatsion [7J has shown that the proof of the Hanoch and Levy first degree stochastic dominance theorem was incorrect. Tesfatsivll then stated and proved a Himilar theorem for utility functions ,h~H are non-decreaHing and continuous.