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 requirement, 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 (i.e., 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 index of riskiness and the Foster–Hart measure of riskiness.