On the convexity of preferences in decisions and games under (quasi-)convex/concave imprecise probability correspondences

The Shafer and Sonnenshein convexity of preferences is a key property in game theory. Previous research has shown that, in case of decisions under uncertainty, the compliance with this property (jointly) depends on the concavity/convexity of the imprecise probabi- listic model with respect to the decision variable and on the attitudes towards imprecision of the decision maker. The present paper deepens the analysis by looking at set-valued imprecise probabilistic models that encompass sets of probability distributions and sets of almost desirable gambles. Moreover, it is shown that the required Shafer and Sonnenshein convexity property is obtained also in case the imprecise probability correspondences satisfy quasi-concavity/convexity with respect to the decision variable so that the set of admissible probabilistic models is significantly broadened. It is well known that sets of probability distributions and sets of almost desirable gambles are general models of representation of uncertainty that are connected to each other; moreover, they are both related to another model known as lower expectation. Therefore, the second part of this work explores the links between the (quasi-)concavity/convexity properties accross the three different models so as to understand to what extent the Shafer and Sonnenshein convexity results hold.