Fraction-degree reference dependent stochastic dominance

For addressing the Allis-type anomalies, a fractional degree reference dependent stochastic dominance rule is developed which is a generalization of the integer degree reference dependent stochastic dominance rules. This new rule can effectively explain why the risk comparison does not satisfy translational invariance and scaling invariance in some cases. The rule also has a good property that it is compatible with the endowment effect of risk. This rule can help risk-averse but not absolute risk-averse decision makers to compare risks relative to reference points. We present some tractable equivalent integral conditions for the fractional degree reference dependent stochastic dominance rule, as well as some practical applications for the rule in economics and finance.