November 12 - 16, 2014, CIMAT (Centro de Investigación en Matemáticas, or Mathematics Research Center), Guanajuato, Mexico. The arrival date is November 12 and the departure date is November 16; the actual talks will be spread over three days, November 13 - 15.

General information — Objectives — Game-theoretic probability — Related topics — Links

This time the workshop is organized under the auspices of CIMAT. For local information, please click here. For a list of attendees (most of whom will give talks), please click here. Registration is now closed.

For the program please click here. Some abstracts are here.

This workshop will bring together researchers studying game-theoretic probability with others who have also been studying probability-free frameworks and frameworks depending on weak probabilistic assumptions. This year's workshop will have "Probability-free finance" as its area of special emphasis. Previous workshops with the same title, Game-Theoretic Probability and Related Topics, were held in 2006, 2008, and 2012 in Tokyo and in 2010 in the London area.

Traditionally, this series of workshops have included the following topics related to game-theoretic probability: imprecise probabilities, prequential statistics, on-line prediction, and algorithmic randomness. Another related topic is conformal prediction. However, this year, due to the emphasis on probability-free finance, not all of them will be represented.

The objective of the workshop will be to make researchers using different approaches to probability-free finance (both mathematical and philosophical) aware of each others' work and to explore commonalities between their frameworks and points of view.

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.

These related topics some of which might be represented in this workshop: (1) imprecise probabilities, (2) prequential statistics, (3) on-line prediction, (4) algorithmic randomness, and (5) conformal prediction.

- Imprecise probabilities (see www.sipta.org) is now a fairly broad field, which includes Walley's upper and lower probabilities, Dempster-Shafer theory, and other approaches to loosening the classical axioms of probability. The imprecise-probabilities community has accumulated expertise in studying various classes of set functions, including several important classes of Choquet capacities. Inasmuch as game-theoretic probability leads to upper and lower probabilities for events, it can be considered a topic within imprecise probabilities, and the extent to which other work on imprecise probabilities can be understood game-theoretically is an interesting and sometimes open question.
- Philip Dawid introduced prequential ("predictive sequential") statistics in the 1980s. It helped inspire the development of game-theoretic probability in the 1990s, because it re-conceptualized the notion of a probability distribution for a sequence of events as a strategy for a forecaster in a sequential forecasting game, which the forecaster can also play without thinking through a complete strategy. It has recently attracted renewed attention, because it brings statistics closer to the spirit of machine learning, shifting emphasis from parameter estimation and model selection to predictive performance.
- On-line prediction is a computer-science counterpart of prequential statistics. Performance of prediction strategies is measured either by evaluating a loss function or by measuring calibration and resolution. There are three important recent threads in on-line prediction: (i) prediction with expert advice, (ii) well-calibrated prediction, and (iii) defensive forecasting. All three are related to game-theoretic probability, and an elucidation of the relation may help the three learn from each other.
- The algorithmic theory of randomness, which continues to develop, shares with game-theoretic probability roots in the pioneering work by Ville in the 1930s and Schnorr in the 1970s. It still provides a theoretical underpinning and a convenient testbed for game-theoretic probability and its applications. It also contributes to prequential statistics, on-line prediction, and conformal prediction. This workshop will help the number of connections, already significant, to grow further.
- Conformal prediction is a method of producing prediction sets that can be applied on top of a wide range of prediction algorithms. The method has a guaranteed coverage probability under the standard IID assumption regardless of whether the assumptions (often considerably more restrictive) of the underlying algorithm are satisfied. The method shares the origins and techniques with game-theoretic probability and algorithmic randomness. There are annual workshops on conformal prediction (third planned for October 2014) for discussing applied aspects of conformal prediction. Recent theoretical work has included the analysis of the method of "conformalizing" for Bayesian algorithms. For the method to be really useful it is desirable that in the case where the assumptions of the underlying algorithm are satisfied, the conformal predictor loses little in efficiency as compared with the underlying algorithm (whereas being a conformal predictor, it has the stronger guarantee of validity). Asymptotic results have been obtained for Bayesian ridge regression, and work is underway for other classes of prediction algorithms. This workshop might serve as a forum for discussing new results and approaches in theoretical conformal prediction.

The two previous workshops:

- The Third Workshop (Egham, UK, June 21 - 23, 2010)
- The Fourth Workshop (Tokyo, November 12 - 14, 2012)

Some further links:

- Description of CIMAT on Wikipedia
- The web site for "Probability and Finance: It's only a Game!", with links to many subsequent papers on game-theoretic probability
- Description of imprecise probability on Wikipedia
- Description of Dempster - Shafer theory on Wikipedia
- The web site for "Algorithmic Learning in a Random World", with links to many subsequent papers on conformal prediction
- The web site for "Conformal Prediction for Reliable Machine Learning"
- The on-line prediction wiki