I forgot to add that there are of course also approaches based on regret.
Let’s call each possible solution of ambiguity a scenario.
For each scenario Z and each possible strategy S, one can estimate the expected value of S in scenario Z, let’s denote that by v(S|Z).
Let’s call the difference in expected value between the chosen strategy S and the optimal one in Z the regret in Z, denoted r(S|Z) = max{v(S’|Z): strategies S’} – v(S|Z).
Let’s denote the minimal and maximal regret when choosing S by minr(S) = min{r(S|Z): all scenarios Z} and maxr(S) = max{r(S|Z): all scenarios Z}
Then Savage’s minimax regret criterion demands one should choose that S which minimizes maxr(S). The advantage over the Hurwicz criterion is that the latter only looks at the two most extreme scenarios, which might not be representative at all of what will actually happen, while Savage’s criterion takes into account the available information about all possible scenarios more comprehensively.
Obviously, one might combine the Hurwicz and Savage approaches into what one might call the regret-based Hurwicz or Savage–Hurwicz criterion that would demand choosing that S which minimizes h maxr(S) + (1–h) minr(S), where h is again some parameter aiming to represent one’s degree of ambiguity aversion. (I haven’t found this criterion in the literature but think it must be known since it is such an obvious combination.)
I forgot to add that there are of course also approaches based on regret.
Let’s call each possible solution of ambiguity a scenario.
For each scenario Z and each possible strategy S, one can estimate the expected value of S in scenario Z, let’s denote that by v(S|Z).
Let’s call the difference in expected value between the chosen strategy S and the optimal one in Z the regret in Z, denoted r(S|Z) = max{v(S’|Z): strategies S’} – v(S|Z).
Let’s denote the minimal and maximal regret when choosing S by minr(S) = min{r(S|Z): all scenarios Z} and maxr(S) = max{r(S|Z): all scenarios Z}
Then Savage’s minimax regret criterion demands one should choose that S which minimizes maxr(S). The advantage over the Hurwicz criterion is that the latter only looks at the two most extreme scenarios, which might not be representative at all of what will actually happen, while Savage’s criterion takes into account the available information about all possible scenarios more comprehensively.
Obviously, one might combine the Hurwicz and Savage approaches into what one might call the regret-based Hurwicz or Savage–Hurwicz criterion that would demand choosing that S which minimizes h maxr(S) + (1–h) minr(S), where h is again some parameter aiming to represent one’s degree of ambiguity aversion. (I haven’t found this criterion in the literature but think it must be known since it is such an obvious combination.)