A geometric and game-theoretic study of the conjunction of possibility measures

TL;DR

This paper investigates the conditions under which the conjunction of possibility measures remains a possibility measure or a coherent upper probability, using geometric and game-theoretic methods to analyze and adjust these measures.

Contribution

It introduces a graphical, game-theoretic approach to determine when the minimum of two possibility measures preserves their properties and how to modify measures to ensure this.

Findings

Conditions for the minimum of two possibility measures to be a possibility measure.

Graphical zero-sum game formulation for checking measure conjunction properties.

Methods to adjust possibility measures to guarantee their conjunction remains a possibility measure.

Abstract

In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical way to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures.