The imprecise total variation model and its connections with game theory

Abstract

A common approach used in robust statistics to robustify a probabilistic model is to distort a probability measure or to create a neighbourhood around it with a given radius and with respect to an appropriate distorting function. This approach establishes a clear connection with lower probabilities, also referred to as non-additive measures or capacities, which serve as tools to model uncertainty in a probability measure and are formally equivalent to normalised coalitional games. In this contribution, we take this idea a step further by analysing the problem of directly distorting a lower probability. For this purpose, we introduce in first place a formal definition of a generic distortion procedure and examine some desirable properties such a procedure may satisfy. Afterwards, we focus particularly on the distortion procedure based on the total variation distance and investigate the properties it satisfies. Finally, we demonstrate that the distortion of lower probabilities has a clear interpretation from the perspective of coalitional games, showing that the distortion based on the total variation distance aligns with a procedure commonly known as strong-δ-core, used to relax the constraints imposed by coalitions in order to ensure the non-emptiness of the core.