A link between Game-Theoretic Probability and Imprecise Probabilities

by gertekoo

Lecture at the Fourth Workshop on Game-Theoretic Probability and Related Topics (GTP2012), held in Tokyo, Japan, 12-14 November 2012.

Based on research I’ve done during the past few years (and for the last part in particular during the last few months) with Filip Hermans, Jasper De Bock and Enrique Miranda.

Here’s the abstract: In game-theoretic probability (GTP) there is a fundamental formula (which we will call the Shafer-Vovk-Ville, or SVV, formula) for expressing lower and upper prices for a variable defined on the terminal situations of an event tree associated with a game. In GTP it is used as a given: a starting point for much of the development of the theory. In earlier work, we have shown how, for event trees that are bounded, this formula can be derived on a behavioral approach and with a different interpretation, in the context of the theory of imprecise probabilities (IP), from two rationality requirements: coherence and cut conglomerability. In the present talk, we discuss how something similar, but more involved, can also be done for unbounded event trees: besides coherence, we impose two additional rationality axioms, bounded cut conglomerability and bounded cut continuity. Interestingly, our approach shows that in deriving the SVV formula, two types of infinity have a part, and can be treated separately: bounded cut conglomerability tries to cope with infinity in the width of the event tree, and bounded cut continuity with infinity in its depth. These additional requirements only need to be invoked when going from local to global modes: it concerns the global uncertainty models in the tree—the uncertainty about paths—, whereas for the local models—the uncertainty about the next move in a situation—we only need to impose coherence, nothing more. We explore a number of aspects and consequences of this connection between GTP and IP.