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# Chapter 6Game-theoretic probability

Vladimir Vovk1 and Glenn Shafer2

1Department of Computer Science, Royal Holloway, University of London, UK

2Department of Accounting and Information Systems, Rutgers Business School—Newark and New Brunswick, USA

## 6.1 Introduction

The standard theory of probability is based on Kolmogorov's  measure-theoretic axioms. A less known alternative is the game-theoretic approach to probability, which is perhaps as old as the measure-theoretic approach (see the end of Section 6.9). There has been a revival of interest in game-theoretic probability lately, and the purpose of this chapter is to give a brief introduction into its current state.

Measure-theoretic probability is, by its nature, precise: according to Kolmogorov's axioms, the probability of every event is a number (not, say, an interval). The usual approach to imprecise probability is also axiomatic: Kolmogorov's axioms are weakened to allow interval probabilities and expectations (cf. the first two sections of Chapter 2). Game-theoretic probability can be precise or imprecise; however, precise game-theoretic probability tends to be equivalent to measure-theoretic probability (we will see an example of this at the end of Section 6.3), and so is not particularly interesting from the point of view of this chapter.

An important difference of game-theoretic probability from measure-theoretic probability, and from the standard approaches to imprecise probability, is its constructive character: ...

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