Weighted Sets of Probabilities and Minimax Weighted Expected Regret: New Approaches for Representing Uncertainty and Making Decisions

TL;DR

This paper introduces weighted sets of probabilities to better represent uncertainty and proposes minimax weighted expected regret (MWER) as a decision-making method, addressing issues with traditional probability updating.

Contribution

It develops a novel weighted probability framework and a new decision rule, MWER, with axiomatic foundations for static and dynamic decision-making.

Findings

Weighted sets of probabilities improve uncertainty representation.

MWER effectively guides decisions under uncertainty.

Axiomatization characterizes preferences induced by MWER.

Abstract

We consider a setting where an agent's uncertainty is represented by a set of probability measures, rather than a single measure. Measure-by-measure updating of such a set of measures upon acquiring new information is well-known to suffer from problems; agents are not always able to learn appropriately. To deal with these problems, we propose using weighted sets of probabilities: a representation where each measure is associated with a weight, which denotes its significance. We describe a natural approach to updating in such a situation and a natural approach to determining the weights. We then show how this representation can be used in decision-making, by modifying a standard approach to decision making -- minimizing expected regret -- to obtain minimax weighted expected regret (MWER). We provide an axiomatization that characterizes preferences induced by MWER both in the static and dynamic case.