Using only the game-theoretic framework and an efficient-market hypothesis, this paper derives predictions that are similar to those of the standard CAPM, but are clearer and more precise. International Journal of Approximate Reasoning 49 175–197 (2008).
This article describes a continuous-time version of the game-theoretic capital asset pricing model described in Working Paper 1.
Instead of asking whether a person is willing to pay given prices for given risky payoffs, we ask whether the person believes he can make a lot of money at those prices. International Journal of Approximate Reasoning 31 1–49 (2003).
The Grundbegriffe appeared in 1933. We examine the work of the earlier scholars whose ideas Kolmogorov synthesized and the developments in the decades immediately following. A shorter version, which does not cover the later period, appeared as "The sources of Kolmogorov's Grundbegriffe" in Statistical Science, 21, 70–98 (2006).
In the game-theoretic framework, market volatility is a consequence of the absence of riskless opportunities for making money.
We review three stages of Kolmogorov's work on the foundations of probability: (1) his formulation of measure-theoretic probability, 1933, (2) his frequentist theory of probability, 1963, and (3) his algorithmic theory of randomness, 1965–1987. A version of this paper appeared in Problems of Information Transmission 39 21–31 (2003).
It is possible, using randomization, to make sequential probability forecasts that will pass any given battery of statistical tests. A version of this paper appeared in the Journal of the Royal Statistical Society, Series B 67 747–763 (2005).
For any continuous gambling strategy used for detecting disagreement between forecasts and actual labels, there exists a forecasting strategy whose forecasts are ideal as far as this gambling strategy is concerned. A version of this paper appeared in the AI & Statistics 2005 proceedings.
The K29 algorithm for probability forecasting (proposed in Working Paper 8) is studied empirically on a popular benchmark data set.
The K29 algorithm is generalized from binary to arbitrary linear forecasting protocols. A version of this paper appeared in the ALT 2005 proceedings.
For a wide range of infinite-dimensional benchmark classes one can construct a prediction algorithm whose cumulative quadratic loss over the first N examples does not exceed the cumulative loss of any prediction rule in the class plus O(sqrt(N)). The proof technique is based on defensive forecasting. A version of this paper appeared in the TAMC 2006 proceedings.
Even if the price of a security is not governed by a probability measure, a European option in the security can be hedged in discrete time by trading in the security and an instrument that pays its variance. A non-probabilistic bound on the error of the hedging is given.
We analyze a new algorithm (K29*, a modification of the K29 algorithm) for probability forecasting of binary observations. A version of this paper appeared in Theoretical Computer Science (ALT 2005 Special Issue) 387, 77–89 (2007).
Standard on-line learning algorithms can only deal with finite-dimensional (often countable) benchmark classes. In this paper we give results for decision rules ranging over an arbitrary reproducing kernel Hilbert space. Our proof technique is based on defensive forecasting. A version of this paper appeared in the ALT 2005 proceedings.
A revival of Cournot's principle can help us distinguish clearly among different aspects of market efficiency.
The regularity of a prediction rule D is measured by its "Holder exponent" h, informally defined by the condition that |D(x+dx)-D(x)| scales as |dx|h for small |dx|. The usual Hilbert-space methods cease to work for h<1/2. In this paper we develop Banach-space methods to construct, for each p in [2,infinity), a prediction algorithm whose average loss over the first N examples does not exceed the average loss of any prediction rule of Holder exponent h > 1/p + epsilon plus O(N-1/p). A version of this paper appeared in Machine Learning (COLT 2006 Special Issue) 69, 193–212 (2007).
The theory of competitive on-line learning can benefit from kinds of prediction that are now foreign to it, first of all from the kinds studied in game-theoretic probability. An abstract of this paper appeared in the COLT 2006 proceedings.
For any class of prediction strategies constituting a reproducing kernel Hilbert space we construct a leading strategy: the loss of any prediction strategy whose norm is not too large is determined by how closely it imitates the leading strategy. The loss function is assumed to be given by a Bregman divergence or by a strictly proper scoring rule. Theoretical Computer Science (ALT 2006 Special Issue) 405 285–296 (2008).
This paper gives constructive, point-wise, and non-asymptotic game-theoretic versions of several results on "merging of opinions" previously obtained in measure-theoretic probability and algorithmic randomness theory. To appear in the Annals of the Institute of Statistical Mathematics.
Defensive forecasting is competitive with the Aggregating Algorithm and handles "second-guessing" experts, whose advice depends on the learner's prediction.
There are two varieties of defensive forecasting: continuous and randomized. This note shows that the randomized variety can be obtained from the continuous variety by smearing Sceptic's moves to make them continuous.
This expository article reviews the game-theoretic framework for probability and the method of defensive forecasting that derives from it.
Game-theoretic efficient market hypotheses identify the same lead-lag anomalies as the conventional approach: statistical significance for the autocorrelations of small-cap portfolios and equal-weighted indices, as well as for the ability of other portfolios to lead them. Because the game-theoretic approach bases statistical significance directly on trading strategies, it allows us to measure the degree of market friction needed to account for this statistical significance. We find that market frictions provide adequate explanation.
A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or order topology) rather than in terms of the precise values taken by the price processes (assumed continuous). To appear in Stochastics.
This paper shows that the strong variation exponent of non-constant continuous price processes has to be 2, as in the case of Brownian motion. Electronic Communications in Probability 13 319–324 (2008).
This paper suggests a perfect-information game, along the lines of Lévy's characterization of Brownian motion, that formalizes the process of Brownian motion in game-theoretic probability.
This note shows that in Philip Dawid's prequential framework game-theoretic probability can be given a natural measure-theoretic definition. In particular, it makes game-theoretic laws of probability in the prequential framework with a finite outcome space corollaries of the corresponding measure-theoretic laws. However, the resulting strategies for Sceptic are very complex, in contrast with the strategies designed in game-theoretic probability.
This paper establishes a non-stochastic analogue of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. It contains the main results of Working Papers 24 and 25 as special cases.
We prove a game-theoretic version of Lévy's zero-one law, and deduce several corollaries from it, including Kolmogorov's zero-one law, the ergodicity of Bernoulli shifts, and a zero-one law for dependent trials.
Finance: 1, 2, 5, 12, 23; general: 3, 15, 19, 27, 29; history: 4, 6; defensive forecasting: 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 22; continuous time: 24, 25, 26, 28.
We construct an explicit strategy that weakly forces the strong law of large numbers in the bounded forecasting game with rate of convergence O(\sqrt{\log n / n}). Annals of the Institute of Statistical Mathematics 60, 801–812 (2008).
We illustrate the generality of discrete finite-horizon game-theoretic probability protocols. The game-theoretic framework is advantageous because no a priori probabilistic assumption is needed. Journal of the Japan Statistical Society 37, 87–104 (2007).
We study capital process behavior in the fair-coin and biased-coin games. A Bayesian strategy for Sceptic with a beta prior weakly forces the strong law of large numbers with rate of convergence O(sqrt(log n / n)). If Reality violates the law, then the exponential growth rate of the capital process is very accurately described in terms of Kullback divergence. We also investigate optimality properties of Bayesian strategies. Stochastic Analysis and Applications 26, 1161–1180 (2008).
We prove several versions of the game-theoretic strong law of large numbers in the case where Reality's moves are unbounded. Stochastics 79, 449–468 (2007).
We derive results on contrarian and one-sided strategies for Skeptic in the fair-coin game. For the strong law of large numbers, we prove that Skeptic can prevent the convergence from being faster than n-1/2. We also derive a corresponding one-sided result. Stochastic Processes and their Applications 118, 2125–2142, 2008.
We introduce a new formulation of continuous-time asset trading in the game-theoretic framework for probability. The market moves continuously but an investor trades at discrete times which can depend on the past path of the market. To appear in Bernoulli.
We study multistep Bayesian betting strategies in coin-tossing games in the framework of game-theoretic probability. By a countable mixture of these strategies, a gambler or an investor can exploit arbitrary patterns of deviations of nature's moves from independent Bernoulli trials. We apply our scheme to asset trading games in continuous time and derive the exponential growth rate of the investor's capital when the variation exponent of the asset price path deviates from two.
We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit. The ideas of this paper are further developed in Working Paper 29.