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

• We study the conjunction of possibility measures within imprecise probability theory.
• We identify conditions under which the conjunction remains a possibility measure.
• We provide a graphical way to check these conditions, using zero-sum games.
• We provide a graphical way to adjust possibility measures so their conjunction is a possibility measure.
• We identify conditions under which the conjunction is coherent.

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.