Asymptotic Fractional Degree Stochastic Dominance for Lognormal Prospects

Abstract

In this paper, we propose a continuum of asymptotic stochastic dominance criteria as a novel rule for comparing and ranking prospects in long term decision-making contexts. The new criteria encompass the established asymptotic first-degree and second-degree stochastic dominance criteria, and between the two integer degrees, they effectively characterize the preferences of decision makers who are predominantly risk averse but do not categorically dislike all forms of risk, that is, those whose utility functions exhibit local convexities. To this end, we first introduce the concept of asymptotic fractional degree stochastic dominance, and then derive its equivalent conditions under the assumption of lognormal distributions. Furthermore, to enhance the tractability of asymptotic fractional degree stochastic dominance, under an additional condition on decision makers’ utility functions that marginal utilities are finite, we introduce a variant of asymptotic fractional degree stochastic dominance, referred to as operational asymptotic fractional degree stochastic dominance, and derive its corresponding equivalent distributional characterizations. The (operational) asymptotic fractional degree stochastic dominance offers a more comprehensive criterion for ranking prospects in long term decision-making contexts. This study elucidates how various constraints on marginal utilities of decision makers shape the equivalent distributional conditions of asymptotic stochastic dominance criteria. Empirical examples also illustrate that the newly proposed (operational) asymptotic fractional degree can be effectively utilized in practice. Funding: This work was supported by the National Natural Science Foundation of China [Grant 72271113] and the Natural Science Foundation of Jiangxi Province [Grants 20232ACB201003 and 20232BAB201022].