Game-Theoretic Upper Expectations for Discrete-Time Finite-State Uncertain Processes

TL;DR

This paper explores game-theoretic upper expectations for finite-state uncertain processes, establishing their properties, relationships with coherence, and continuity behavior, extending the framework of Shafer and Vovk.

Contribution

It provides new insights into the continuity properties and characterizations of game-theoretic upper expectations in discrete-time finite-state models.

Findings

Proves continuity of upper expectations with respect to certain sequences.

Shows the unique determination of upper expectations by values on bounded limits.

Establishes the relationship between upper expectations and Walley's coherence.

Abstract

Game-theoretic upper expectations are joint (global) probability models that mathematically describe the behaviour of uncertain processes in terms of supermartingales; capital processes corresponding to available betting strategies. Compared to (the more common) measure-theoretic expectation functionals, they are not bounded to restrictive assumptions such as measurability or precision, yet succeed in preserving, or even generalising many of their fundamental properties. We focus on a discrete-time setting where local state spaces are finite and, in this specific context, build on the existing work of Shafer and Vovk; the main developers of the framework of game-theoretic upper expectations. In a first part, we study Shafer and Vovk's characterisation of a local upper expectation and show how it is related to Walley's behavioural notion of coherence. The second part consists in a study of game-theoretic upper expectations on a more global level, where several alternative definitions, as well as a broad range of properties are derived, e.g. the law of iterated upper expectations, compatibility with local models, coherence properties,... Our main contribution, however, concerns the continuity behaviour of these operators. We prove continuity with respect to non-increasing sequences of so-called lower cuts and continuity with respect to non-increasing sequences of finitary functions. We moreover show that the game-theoretic upper expectation is uniquely determined by its values on the domain of bounded below limits of finitary functions, and additionally show that, for any such limit, the limiting sequence can be constructed in such a way that the game-theoretic upper expectation is continuous with respect to this particular sequence.