Quantile Maximization in Decision Theory*

Abstract

This paper introduces a model of preferences, in which, given beliefs about uncertain outcomes, an individual evaluates an action by a quantile of the induced distribution. The choice rule of Quantile Maximization unifies maxmin and maxmax as maximizing the lowest and the highest quantiles of beliefs distributions, respectively, and offers a family of less extreme preferences. Taking preferences over acts as a primitive, we axiomatize Quantile Maximization in a Savage setting. Our axiomatization also provides a novel derivation of subjective beliefs, which demonstrates that neither the monotonicity nor the continuity conditions assumed in the literature are essential for probabilistic sophistication. We characterize preferences of quantile maximizers towards downside risk. We discuss how the distinct properties of the model, robustness and ordinality, can be useful in studying choice behaviour for categorical variables and in economic policy design. We also offer applications to poll design and insurance problems. Copyright © 2009 The Review of Economic Studies Limited.

Stock market’s reaction to money supply: a nonparametric analysis

“…The investigation of the quantile-based Euler equation is the topic of on-going research. Our preliminary theoretical results indicate that obtaining a quantile version of the Euler equation is very possible by using quantile-type utility functions as in Manski (1988) and Rostek (2010) .…”

“…One direction that we discuss in the last section of this paper is how to investigate the theoretical justification behind the established stylized facts. We point out the possibility of extending the Money-based Capital Asset Pricing Model (M-CAPM) [see Chan, Foresi, and Lang (1996) and Balvers and Huang (2009) ], which is a statement about the conditional mean of asset returns, to a quantile-based M-CAPM using the quantile utility functions developed in Manski (1988) and Rostek (2010) . Another direction would be to shed light on the channels through which the monetary policy money supply affects stock returns.…”

An Axiomatization of Quantiles on the Domain of Distribution Functions

“…Manski [8] suggests using quantiles in a decision-theoretic framework similar to that of Savage [15]. Rostek [14] provides a decision-theoretic analysis of quantiles in terms of order structures. She uses Savage-style axioms to uncover the complete behavioral implications of a decision maker who behaves as if she forms a unique probability measure over the measurable space, as well as a state-independent utility index.…”

Setting Environmental Policy When Experts Disagree

“…A particularly challenging one would be to extend the α-maxmin model to take into account higher moments or quantiles (see, e.g., the quantile-based decision-theoretic framework of Rostek [39]) of the considered mixture distributions of Eq. (4).…”