Preface

Probability theory has always been closely associated with gambling. In the 1650s, Blaise Pascal and Christian Huygens based probability's concept of expectation on reasoning about gambles. Countless mathematicians since have looked to gambling for their intuition about probability. But the formal mathematics of probability has long leaned in a different direction. In his correspondence with Pascal, often cited as the origin of probability theory, Pierre Fermat favored combinatorial reasoning over Pascal's reasoning about gambles, and such combinatorial reasoning became dominant in Jacob Bernoulli's monumental Ars Conjectandi and its aftermath. In the twentieth century, the combinatorial foundation for probability evolved into a rigorous and sophisticated measure-theoretic foundation, put in durable form by Andrei Kolmogorov and Joseph Doob.

The twentieth century also saw the emergence of a mathematical theory of games, just as rigorous as measure theory, albeit less austere. In the 1930s, Jean Ville gave a game-theoretic interpretation of the key concept of probability 0. In the 1970s, Claus Peter Schnorr and Leonid Levin developed Ville's fundamental insight, introducing universal game-theoretic strategies for testing randomness. But no attempt was made in the twentieth century to use game theory as a foundation for the modern mathematics of probability.

Probability and Finance: It's Only a Game, published in 2001, started to fill this gap. It gave game-theoretic proofs of probability's most classical limit theorems (the laws of large numbers, the law of the iterated logarithm, and the central limit theorem), and it extended this game-theoretic analysis to continuous-time diffusion processes using nonstandard analysis. It applied the methods thus developed to finance, discussing how the availability of a variance swap in a securities market might allow other options to be priced without probabilistic assumptions and studying a purely game-theoretic hypothesis of market efficiency.

The present book was originally conceived of as a second edition of Probability and Finance, but as the new title suggests, it is a very different book, reflecting the healthy growth of game-theoretic probability since 2001. As in the earlier book, we show that game-theoretic and measure-theoretic probability provide equivalent descriptions of coin tossing, the archetype of probability theory, while generalizing this archetype in different directions. Now we show that the two descriptions are equivalent on a larger central core, including all discrete-time stochastic processes that have only finitely many outcomes on each round, and we present an even broader array of new ideas.

We can identify seven important new ideas that have come out of game-theoretic probability. Some of these already appeared, at least in part, in Probability and Finance, but most are developed further here or are entirely new.

1. Strategies for testing. Theorems showing that certain events have small or zero probability are made constructive; they are proven by constructing gambling strategies that multiply the capital they risk by a large or infinite factor if the events happen. In Probability and Finance, we constructed such strategies for the law of large numbers and several other limit theorems. Now we add to the list the most fundamental limit theorem of probability - Lévy's zero-one law. The topic of strategies for testing remains our most prominent theme, dominating Part I and Chapters 7 and 8 in Part II.
2. Limited betting opportunities. The betting rates suggested by a scientific theory or the investment opportunities in a financial market may fall short of defining a probability distribution for future developments or even for what will happen next. Sometimes a scientist or statistician tests a theory that asserts expected values for some variables but not for every function of those variables. Sometimes an investor in a market can buy a particular payoff but cannot sell it at the same price and cannot buy arbitrary options on it. Limited betting opportunities were emphasized by a number of twentieth-century authors, including Peter Williams and Peter Walley. As we explained in Probability and Finance, we can combine Williams and Walley's picture of limited betting opportunities in individual situations with Pascal and Ville's insights into strategies for combining bets across situations to obtain interesting and powerful generalizations of classical results. These include theorems that are one-sided in some sense (see Sections 2.4 and 5.1).
3. Strategies for reality. Most of our theorems concern what can be accomplished by a bettor playing against an opponent who determines outcomes. Our games are determined; one of the players has a winning strategy. In Probability and Finance, we exploited this determinacy and an argument of Kolmogorov's to show that in the game for Kolmogorov's law of large numbers, the opponent has a strategy that wins when Kolmogorov's hypotheses are not satisfied. In this book we construct such a strategy explicitly and discuss other interesting strategies for the opponent (see Sections 4.4, and 4.7).
4. Open protocols for science. Scientific models are usually open to influences that are not themselves predicted by the models in any way. These influences are variously represented; they may be treated as human decisions, as signals, or even as constants. Because our theorems concern what one player can accomplish regardless of how the other players move, the fact that these signals or "independent variables" can be used by the players as they appear in the course of play does not impair the theorems' validity and actually enhances their applicability to scientific problems (see Chapter 10 ).
5. Insuring against loss of evidence. The bettor can modify his own strategy or adapt bets made by another bettor so as to avoid a total loss of apparently strong evidence as play proceeds further. The same methods provide a way of calibrating the p-values from classical hypothesis testing so as to correct for the failure to set an initial fixed significance level. These ideas have been developed since the publication of Probability and Finance (see Chapter 11 ).
6. Defensive forecasting. In addition to the player who bets and the player who determines outcomes, our games can involve a third player who forecasts the outcomes. The problem of forecasting is the problem of devising strategies for this player, and we can tackle it in interesting ways once we learn what strategies for the bettor win when the match between forecasts and outcomes is too poor. This idea, which came to our attention only after the publication of Probability and Finance, is developed in Chapter 12.
7. Continuous-time game-theoretic finance. Measure-theoretic finance assumes that prices of securities in a financial market follow some probabilistic model such as geometric Brownian motion. We obtain many insights, some already provided by measure-theoretic finance and some not, without any probabilistic model, using only the actual prices in the market. This is now much clearer than in Probability and Finance, as we use tools from standard analysis that are more familiar than the nonstandard methods we used there. We have abandoned our hypothesis concerning the effectiveness of variance swaps in stabilizing markets, now fearing that the trading of such instruments could soon make them nearly as liquid and consequently treacherous as the underlying securities. But we provide game-theoretic accounts of a wider class of financial phenomena and models, including the capital asset pricing model (), the equity premium puzzle, and portfolio theory (see Part IV).

The book has four parts.

• Part I, Examples in Discrete Time, uses concrete protocols to explain how game-theoretic probability generalizes classical discrete-time limit theorems. Most of these results were already reported in Probability and Finance in 2001, but our exposition has changed substantially. We seldom repeat word for word what we wrote in the earlier book, and we occasionally refer the reader to the earlier book for detailed arguments that are not central to our theme.
• Part II, Abstract Theory in Discrete Time, treats game-theoretic probability in an abstract way, mostly developed since 2001. It is relatively self-contained, and readers familiar with measure-theoretic probability will find it accessible without the introduction provided by Part I.
• Part III, Applications in Discrete Time, uses Part II's theory to treat important applications of game-theoretic probability, including two promising applications that have developed since 2001: calibration of lookbacks and p-values, and defensive forecasting.
• Part IV, Game-Theoretic Finance, studies continuous-time game-theoretic probability and its application to finance. It requires different definitions from the discrete-time theory and hence is also relatively self-contained. Its first chapter uses a simple concrete protocol to derive game-theoretic versions of the Dubins-Schwarz theorem and related results, while the remaining chapters use an abstract and more powerful protocol to develop a...