New decision rules under strict uncertainty and a general distance-based approach

Abstract

<abstract><p>Strict uncertainty implies a complete lack of knowledge about the probabilities of possible future states of the world. However, there is complete information about the set of alternatives under consideration, the set of future states, and the scalar evaluation of choosing every alternative if a given state occurs. The principle of insufficient reason by Laplace, the maximin rule by Wald, the Hurwicz criterion, or the minimax regret criterion by Savage are examples of decision rules under strict uncertainty. Within the context of strict uncertainty, moderate pessimism implies the existence of a decision-maker who cautiously assumes that the most favorable state will not occur when the action has been taken with no conjecture being made about the other states. The criterion of moderate pessimism proposed by Ballestero implies the use of the inverse of the range of evaluation for each state as a weight system. In this paper, we extend the notion of moderate pessimism under strict uncertainty to solve some of its limitations. First, we propose a new domination analysis that avoids removing dominated alternatives that are still relevant in the final ranking of alternatives. Second, we propose additional score functions using the inverse of the standard deviation and the mean absolute deviation instead of the range of evaluations for each future state to reduce the impact of the possible existence of outliers in the decision table. This partial result is later generalized through the concept of average deviation of a given order. Finally, we show that all the mentioned decision rules are special cases of a general ranking method based on the Minkowski distance function. We illustrate the use of distance-based decision rules through an application in the context of portfolio selection.</p></abstract>