## Game-theoretic probability

What is game-theoretic probability? Like the better known measure-theoretic framework, the game-theoretic framework for probability can be traced back to the 1654 correspondence between Blaise Pascal and Pierre Fermat, often said to be the origin of mathematical probability. In their correspondence, Pascal and Fermat explained their different methods for solving probability problems. Fermat's combinatorial method is a precursor of the measure-theoretic framework, now almost universally accepted by mathematicians as the result of work by Borel, Kolmogorov, Doob, Martin-Lof, and others. Pascal's method of backward recursion, using prices at each step in a game to derive global prices, can be seen as the precursor of the game-theoretic framework, to which von Mises, Ville, Kolmogorov, Schnorr, and Dawid contributed further ingredients.

The game-theoretic framework was presented in a comprehensive way, as an alternative to the measure-theoretic framework, by Shafer and Vovk in 2001 and Takeuchi in 2004 (see www.probabilityandfinance.com). As these authors explain, a classical probability theorem tells us that some event is very likely or even certain to happen. In the measure-theoretic framework, such a theorem becomes a statement about the measure of a set: the set has measure near or equal to one. In the game-theoretic framework, it is instead a statement about a game: a player has a strategy that multiplies the capital it risks by a large or infinite factor if the event fails to happen. The mere fact that we can translate theorems from measure theory into game theory in this way is of limited interest, but the translation opens up new ways of using probability theory and provides new insights into the meaning of probability and into many existing applications and related fields.

Some of the new insights come from the greater generality of the game-theoretic picture. Many classical theorems hold in generalized form even in games where relatively few payoffs are priced. In classical probability, all payoffs are priced (we call them random variables, and we call their prices expected values). This is not necessary in the game-theoretic picture. An investor in a financial market, for example, plays a game in which he can buy some payoffs (corresponding to various securities traded in the market) but not others. Because probabilities (or upper and lower probabilities) for global events are defined even in these situations, we see these probabilities as features that emerge from the structure of the game, not as features of objective reality or subjective belief external to the game.

One of the most active recent topics in game-theoretic probability, not adequately treated in the monographs by Shafer, Vovk, and Takeuchi, is continuous-time processes. The mathematical idea underlying recent work has been described by Takeuchi as high-frequency limit order trading. A player divides his capital among many different strategies, all of which rebalance a portfolio of bets when a continuous function reaches various discrete levels, but some of which operate at a much higher frequency than others. Various classical properties of stochastic processes emerge merely from the basic assumption that a strategy will not multiply the capital it risks by a very large factor.