Multivariate Almost Stochastic Dominance

• Published date: 01 June 2018
• Date: 01 June 2018
• DOI: http://doi.org/10.1111/jori.12222

©2017 The Journal of Risk and Insurance. Vol.85, No. 2, 431–445 (2018).

DOI: 10.1111/jori.12222

Multivariate Almost Stochastic Dominance

Ilia Tsetlin

Robert L. Winkler

Abstract

Almost stochastic dominance allows small violations of stochastic domi-

nance rules to avoid situations where most decision makers prefer one al-

ternative to another but stochastic dominance cannot rank them. Wepresent

the concepts of multivariate almost stochastic dominance and multivariate

almost nth-degree risk and their connections with a preference for combin-

ing good with bad. Then, we show how a preference for combining good

with bad can be applied to obtain various comparative statics results, and

we extend our approach to risk-prone (convex) stochastic dominance, which

relates to the opposite preference, for combining good with good and bad

with bad.

Introduction

Often only partial information is available about a decision maker’s utility function.

Stochastic dominance (SD) was developed to provide a partial ranking of distributions

of outcomes associated with potential actions in such circumstances. For example, if

we know only that a utility function is increasing, we can use first-order stochastic

dominance (FSD) rules to eliminate some actions that are dominated by others. If

we know that utility is increasing and concave (risk averse), a common assumption

in financial decision making, then we can use second-order stochastic dominance

(SSD) rules to further reduce the set of actions under consideration. This can simplify

the decision-making process. Early articles in the financial literature on SD include

Hanoch and Levy (1969) and Hadar and Russell (1969).

The set of all risk-averse utility functions includes some “pathological” functions that

are not likely to represent very many (or any) decision maker’s preferences but can

cause SSD to be too restrictive in practice. To address this issue, Leshno and Levy

(2002) introduce the notion of almost stochastic dominance (ASD), which limits ratios

of marginal utilities and higher-order derivatives of utility functions in an attempt

to exclude some of the pathological functions from consideration. For example, in

Leshno and Levy’s approach, first-order ASD (FASD) limits ratios of marginal utili-

ties, whereas second-order ASD (SASD) limits ratios of the second derivatives of the

Ilia Tsetlin is at the INSEAD, 1 Ayer Rajah Avenue, Singapore 138676. Tsetlin can be contacted

via e-mail: ilia.tsetlin@insead.edu. Robert L. Winkler is at Fuqua School of Business, Duke

University,Durham, NC 27708-0120. Winkler can be contacted via e-mail: rwinkler@duke.edu.

431