A note on almost stochastic dominance

•We compare the almost stochastic dominance (ASD) developed by Leshno and Levy (LL, 2002) and Tzeng et al. (THS, 2012).•We define the third-degree ASD in the same way that LL's second-degree ASD.•We first find that LL's ASD has the hierarchy property but not expected-utility maximization.•We then find that THS's ASD possesses the property of expected-utility but not the hierarchy property.•The above phenomenon also holds for higher-degree ASD dominance for these two concepts.

Both the expected-utility maximization and the hierarchy property are very important properties in stochastic dominance. For almost stochastic dominance, Leshno and Levy (2002) propose a definition and Tzeng et al. (2013) modified it to give another definition. This note provides more information on the two definitions. The former has the hierarchy property but not the expected-utility maximization, whereas the latter has the expected-utility maximization but not the hierarchy property.