Comparing Risks by Acceptance and Rejection

Stochastic dominance is a partial order on risky assets (“gambles”) that is based on the uniform preference, of all decision-makers (in an appropriate class), for one gamble over another. We modify this, first, by taking into account the status quo (given by the current wealth) and the possibility of rejecting gambles, and second, by comparing rejections that are substantive (that is, uniform over wealth levels or over utilities). This yields two new stochastic orders: wealth-uniform dominance and utility-uniform dominance. Unlike stochastic dominance, these two orders are complete: any two gambles can be compared. Moreover, they are equivalent to the orders induced by, respectively, the Aumann–Serrano (2008) index of riskiness and the Foster–Hart (2009a) measure of riskiness. This paper supersedes “A Simple Riskiness Order Leading to the Aumann–Serrano Index of Riskiness,” May 2008 (draft), August 2009 (Discussion Paper # 517, Center for the Study of Rationality). Research partially supported by a grant of the Israel Science Foundation. The author thanks Bob Aumann, Elchanan Ben-Porath, Dean Foster, Werner Hildenbrand, Gil Kalai, Haim Levy, Moti Michaeli, Phil Reny, Frank Riedel, Yossi Rinott, Benjy Weiss, and Shlomo Yitzhaki, for useful discussions and suggestions. Department of Economics, Institute of Mathematics, and Center for the Study of Rationality, The Hebrew University of Jerusalem. E-mail : [email protected] Web site: http://www.ma.huji.ac.il/hart