2020/08/06 by Natan T’Joens, T'Joens, Natan, Jasper De Bock +3
Decision Sciences · #Decision-Making and Behavioral Economics #FOS: Mathematics #Probability (math.PR) #Risk and Portfolio Optimization
paper · pdf · doi:10.48550/arxiv.2008.03133
openalex publication_date 2020/08/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Game-theoretic upper expectations are joint (global) probability models that\nmathematically describe the behaviour of uncertain processes in terms of\nsupermartingales; capital processes corresponding to available betting\nstrategies. Compared to (the more common) measure-theoretic expectation\nfunctionals, they are not bounded to restrictive assumptions such as\nmeasurability or precision, yet succeed in preserving, or even generalising\nmany of their fundamental properties. We focus on a discrete-time setting where\nlocal state spaces are finite and, in this specific context, build on the\nexisting work of Shafer and Vovk; the main developers of the framework of\ngame-theoretic upper expectations. In a first part, we study Shafer and Vovk's\ncharacterisation of a local upper expectation and show how it is related to\nWalley's behavioural notion of coherence. The second part consists in a study\nof game-theoretic upper expectations on a more global level, where several\nalternative definitions, as well as a broad range of properties are derived,≠.g. the law of iterated upper expectations, compatibility with local models,\ncoherence properties,... Our main contribution, however, concerns the\ncontinuity behaviour of these operators. We prove continuity with respect to\nnon-increasing sequences of so-called lower cuts and continuity with respect to\nnon-increasing sequences of finitary functions. We moreover show that the\ngame-theoretic upper expectation is uniquely determined by its values on the\ndomain of bounded below limits of finitary functions, and additionally show\nthat, for any such limit, the limiting sequence can be constructed in such a\nway that the game-theoretic upper expectation is continuous with respect to\nthis particular sequence.\n