Refined expected value decision rules

We consider the difficult problem of choosing between alternatives where the payoffs for the alternatives are uncertain and modeled via discrete probability distributions. One popular approach for making a choice in this complex environment is to use the idea of expected value; we prefer alternatives with larger expected value. Here we suggest an approach for refining the calculation of the expected value to allow for the inclusion of a requirement that we prefer an alternative with payoff probability distribution P1 to an alternative with payoff probability distribution P2 by assuring that expected value of P1 is larger then the expected value of P2.