A Characterization of the n-th Degree Bounded Stochastic Dominance

TL;DR

This paper characterizes the n-th degree bounded stochastic dominance order, linking it to decision-makers' risk preferences and utility functions with bounded risk aversion, revealing limitations and proposing related risk measures.

Contribution

It introduces a novel characterization of BSD linked to risk tolerance, clarifies its limitations, and proposes a related lower-partial-moment order for better risk assessment.

Findings

BSD reflects specific risk preferences through utility functions.

Limitations of BSD depend on support interval and risk aversion behavior.

A related lower-partial-moment order offers a trade-off between expected value and downside risk.

Abstract

We provide a novel characterization of the $n$-th degree bounded stochastic dominance (BSD) order, linking it to the risk tolerance of decision-makers and providing a decision-theoretic foundation for these stochastic orders. Our results reveal that BSD reflects specific risk preferences through the choice of the interval $[a,b]$, by characterizing it in terms of utility functions with globally bounded Arrow--Pratt risk aversion or that satisfy an $n$-convexity condition. They also highlight limitations of BSD, including its dependence on the chosen support interval and the resulting peculiar risk aversion behavior of decision-makers included in the generator of BSD. To partially address this issue, we use our characterization to separate two roles that are combined in BSD: the largest payoff in the lotteries and the upper endpoint of the interval that determines the Arrow--Pratt lower bound. We then introduce a related lower-partial-moment order that provides a clean trade-off between expected value and downside-risk protection. Using our characterization, we present comparative statics results for decision-making under uncertainty with globally bounded risk aversion measures and savings decisions under globally bounded prudence measures, and derive inequalities for $n$-convex functions.