Distribution-Valued Games -- Overview, Analysis, and a Segmentation-Based Approach

TL;DR

This paper formalizes distribution-valued games using stochastic orders, analyzes the properties of the tail order, identifies limitations of previous claims, and proposes a segmentation-based solution approach.

Contribution

It extends the formal framework of distribution-valued games, corrects prior misconceptions about the tail order, and introduces a novel segmentation-based solution concept.

Findings

The tail order is not total on certain distribution sets.

Not all tail-ordered games have mixed-strategy Nash equilibria.

Almost all finite-support tail-ordered games only have pure-strategy Nash equilibria.

Abstract

The paper [Ras15a] introduced distribution-valued games. This game-theoretic model uses probability distributions as payoffs for games in order to express uncertainty about the payoffs. The player's preferences for different payoffs are expressed by a stochastic order which we call the tail order. This thesis formalizes distribution-valued games with preferences expressed by general stochastic orders, and specifically analyzes properties of the tail order. It identifies sufficient conditions for tail-order preference to hold, but also finds that some claims in [Ras15a] about the tail order are incorrect, for which counter-examples are constructed. In particular, it is demonstrated that a proof for the totality of the order on a certain set of distributions contains an error; the thesis proceeds to show that the ordering is not total on the slightly less restricted set of distributions with non-negative bounded support. It is also shown that not all tail-ordered games have mixed-strategy Nash equilibria, and in fact almost all tail-ordered games with finitely-supported payoff distributions can only have a Nash equilibrium if they have a pure-strategy Nash equilibrium. The thesis subsequently extends an idea from [AM19] and proposes a new solution concept for distribution-valued games. This concept is based on constructing multi-objective real-valued games from distribution-valued games by segmenting their payoff distributions.