Finding an Optimal Decisions’ Subset by Minimaximax Regret Criterion Regarding Instability of the Decision Function

Abstract

Background. A generalization of the minimax regret criterion is represented as even the best-assurance minimax regret criterion comes inconsistent under instable evaluations of decision situations. Objective. The goal is to formulate the minimaximax regret criterion. Methods. Unlike the classic one, the generalized regret criterion is minimaximax which operates over generalized regrets. These regrets are found from a generalized decision function which is defined on a Cartesian product of a decisions’ set, a set of states, and a set of metastates. Metastate describes instability of the decision function whose values change through a set of metastates. The instability destroys assurance of minimaxed regrets found classically, so regrets are found over a generalized decision function. For this, utility evaluations are subtracted from the utility maximized across a decision set, or the loss/risk minimized across a decision set is subtracted from loss/risk evaluations. Then regrets are minimized under uncertainty across two dimensions of states and metastates, that is they are minimaximaxed. Results. The suggested minimaximax regret criterion allows finding an optimal decisions’ subset with not only regarding instability of the decision function, but also with reducing the initial decisions’ set more, unlike the ultimate pes-simism criterion without regrets (minimaximax/maximinimin). This especially concerns nonnegative utility matrices with many zeros. Conclusions. A ratio of a number of optimal decisions by the without-regret maximinimin/minimaximax to a number of optimal decisions by the minimaximax regret criterion decreased by 1 can be interpreted as a gain of applying the represented minimax regret criterion generalization. This gain fundamentally depends on whether sets of decisions, states, and metastates are finite or not. If they all are finite, then the gain depends on values in a three-dimensional regret matrix and its dimensions. It is surprising but the gain may be negative, that is finding regrets may come non-effective.

Keywords

• Decision function
• Minimax regret criterion
• Optimal decisions’ subset
• Metastate
• Minimaximax regret criterion