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The problem of optimizing a number of simultaneous bets is considered, using primarily log-utility. Stochastic gradient-based algorithms for solving this problem are developed and compared with the simplex method. The solutions may be regarded as a generalization of 'Kelly staking' to the case of many simultaneous bets. Properties of the solutions are examined in two example cases using real odds from sports bookmakers. The algorithms that are developed also have wide applicability beyond sports betting and may be extended to general portfolio optimization problems, with any reasonable utility function. Copyright 2007 Royal Statistical Society.

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... Portfolio optimization The approach of splitting the trader's workflow into the two steps of predictive modeling and investment optimization has a long tradition, and has been exploited in absolute majority of works [49,30,56,51,71,18], with some notable exceptions [24,38]. Extracting the parameter estimation out of the portfolio optimization problem then enabled the respective economic research to thrive in an isolated mathematical environment, giving rise to the frameworks of Markowitz [47] and Kelly [32], and their many successors [10,77,71,36,51]. While widely adopted, the optimality of the resulting portfolios is based on rather unrealistic assumptions, which has been progressively criticized by many [27,61,49,57,62,43]. ...

It is a common misconception that in order to make consistent profits as a trader, one needs to possess some extra information leading to an asset value estimation that is more accurate than that reflected by the current market price. While the idea makes intuitive sense and is also well substantiated by the widely popular Kelly criterion, we prove that it is generally possible to make systematic profits with a completely inferior price-predicting model. The key idea is to alter the training objective of the predictive models to explicitly decorrelate them from the market. By doing so, we can exploit inconspicuous biases in the market maker’s pricing, and profit from the inherent advantage of the market taker. We introduce the problem setting throughout the diverse domains of stock trading and sports betting to provide insights into the common underlying properties of profitable predictive models, their connections to standard portfolio optimization strategies, and the commonly overlooked advantage of the market taker. Consequently, we prove the desirability of the decorrelation objective across common market distributions, translate the concept into a practical machine learning setting, and demonstrate its viability with real-world market data.

... In (Smoczynski and Tomkins, 2010), a closed form solution for the use of the Kelly strategy when betting on horse racing was explored. Another practical extension for betting on multiple simultaneous games was discussed in a number of works (Whitrow, 2007;Grant et al., 2008;Buchen and Grant, 2012), where various approximations for large bet aggregations were proposed. ...

We investigate the most popular approaches to the problem of sports betting investment based on modern portfolio theory and the Kelly criterion. We define the problem setting, the formal investment strategies, and review their common modifications used in practice. The underlying purpose of the reviewed modifications is to mitigate the additional risk stemming from the unrealistic mathematical assumptions of the formal strategies. We test the resulting methods using a unified evaluation protocol for three sports: horse racing, basketball and soccer. The results show the practical necessity of the additional risk-control methods and demonstrate their individual benefits. Particularly, we show that an adaptive variant of the popular ``fractional Kelly'' method is a very suitable choice across a wide range of settings.

... Portfolio optimization The approach of splitting the trader's workflow into the two steps of predictive modeling and investment optimization has a long tradition, and has been exploited in absolute majority of works [49,30,56,51,71,18], with some notable exceptions [24,38]. Extracting the parameter estimation out of the portfolio optimization problem then enabled the respective economic research to thrive in an isolated mathematical environment, giving rise to the frameworks of Markowitz [47] and Kelly [32], and their many successors [10,76,71,36,51]. While widely adopted, the optimality of the resulting portfolios is based on rather unrealistic assumptions, which has been progressively criticized by many [27,61,49,57,62,43]. ...

It is a common misconception that in order to make consistent profits as a trader, one needs to posses some extra information leading to an asset value estimation more accurate than that reflected by the current market price. While the idea makes intuitive sense and is also well substantiated by the widely popular Kelly criterion, we prove that it is generally possible to make systematic profits with a completely inferior price-predicting model. The key idea is to alter the training objective of the predictive models to explicitly decorrelate them from the market, enabling to exploit inconspicuous biases in market maker's pricing, and profit on the inherent advantage of the market taker. We introduce the problem setting throughout the diverse domains of stock trading and sports betting to provide insights into the common underlying properties of profitable predictive models, their connections to standard portfolio optimization strategies, and the, commonly overlooked, advantage of the market taker. Consequently, we prove desirability of the decorrelation objective across common market distributions, translate the concept into a practical machine learning setting, and demonstrate its viability with real world market data.

... spread betting (Haigh, 2000, andChapman, 2007). Other variations on the simple scheme are multiple outcomes (Barnett, 2010) and simultaneous betting (Whitrow, 2007). ...

The Kelly betting criterion ignores uncertainty in the probability of winning the bet and uses an estimated probability. In general, such replacement of population parameters by sample estimates gives poorer out-of-sample than in-sample performance. We show that to improve out-of-sample performance the size of the bet should be shrunk in the presence of this parameter uncertainty, and compare some estimates of the shrinkage factor. From a simulation study and from an analysis of some tennis betting data we show that the shrunken Kelly approaches developed here offer an improvement over the “raw” Kelly criterion. One approximate estimate of the shrinkage factor gives a “back of envelope” correction to the Kelly criterion that could easily be used by bettors. We also study bet shrinkage and swelling for general risk-averse utility functions and discuss the general implications of such results for decision theory.

... Consequently, it can be seen the exponential rate is: the second one is used: the boundary value p c is determined by the intersection of these two results. The numerical solution can be obtained by using the algorithms of [25], while a comparison of the derived approximate results with numerical solutions of Eq.(9) is shown in [21] and displayed a good agreement. Fig. 2 (a) shows the total investment fraction is not simply M times of a single game, and Fig. 2 (b) shows that diversification significantly improves the return. ...

The growth-optimal portfolio optimization strategy has been investigated in many ways since firstly pioneered by Kelly. This paper firstly introduces the research progress of this so-called Kelly game. Based on the original Kelly game the optimality is shortly proofed. Especially generalized research is introduced such as the relation between M–V approach and the Kelly approach, the question of diversification, the influence of transaction fees and limited information, etc. Then the application of Kelly strategy is discussed with some conclusions.

... The maximization of G then yields the optimal fraction f * = 2p − 1 (13) which is the celebrated Kelly criterion (if short positions are not allowed, f * = 0 for p < 1/2: it is optimal to abstain from the game). Eq. (12), allowing no analytical solution for M ≥ 5, can be maximized by numerical techniques (Whitrow, 2007) or by analytical approximations (Medo et al., 2008) as we do below. ...

When assets are correlated, benefits of investment diversification are reduced. To measure the influence of correlations on investment performance, a new quantity--the effective portfolio size--is proposed and investigated in both artificial and real situations. We show that in most cases, the effective portfolio size is much smaller than the actual number of assets in the portfolio and that it lowers even further during financial crises.

... For M ≥ 5, Eq. (9) has no closed solution and thus in the following sections we seek for approximations. In complicated cases where such approximations perform badly, numerical algorithms are still applicable [13]. ...

Financial markets, with their vast range of different investment opportunities, can be seen as a system of many different simultaneous games with diverse and often unknown levels of risk and reward. We introduce generalizations to the classic Kelly investment game [Kelly (1956)] that incorporates these features, and use them to investigate the influence of diversification and limited information on Kelly-optimal portfolios. In particular we present approximate formulas for optimizing diversified portfolios and exact results for optimal investment in unknown games where the only available information is past outcomes.

We investigate the most popular approaches to the problem of sports betting investment based on modern portfolio theory and the Kelly criterion. We define the problem setting, the formal investment strategies and review their common modifications used in practice. The underlying purpose of the reviewed modifications is to mitigate the additional risk stemming from the unrealistic mathematical assumptions of the formal strategies. We test the resulting methods using a unified evaluation protocol for three sports: horse racing, basketball and soccer. The results show the practical necessity of the additional risk-control methods and demonstrate their individual benefits. Particularly, an adaptive variant of the popular ‘fractional Kelly’ method is a very suitable choice across a wide range of settings.

In this paper we present a smart portfolio management methodology, which advances existing portfolio management techniques at two distinct levels. First, we develop a set of investment models that target regimes found in the data over different time horizons. We then build a meta-model which uses the Kelly criterion to determine an optimal allocation over these investment strategies, thus simultaneously capturing regimes operating in the data over different time horizons. Finally, in order to detect changes in the relevant data regime, and hence investment allocations, we use a forecasting algorithm which relies on a Kalman filter. We call our combined method, that uses both the Kelly criterion and the Kalman filter, the K2 algorithm. Using a large-scale historical dataset of both stocks and indices, we show that our K2 algorithm gives better risk adjusted returns in terms of the Sharpe ratio, better average gain to average loss ratio and higher probability of success compared to existing benchmarks, when measured in out-of-sample tests.

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: The optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the di usion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then nd conditions under which the discrete-time process converges to a di usion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limiting di usion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuoustime gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the nancial value

This paper develops a sequential model of the individual's economic decision problem under risk. On the basis of this model, optimal consumption, investment, and borrowing-lending strategies are obtained in closed form for a class of utility functions. For a subset of this class the optimal consumption strategy satisfies the permanent income hypothesis precisely. The optimal investment strategies have the property that the optimal mix of risky investments is independent of wealth, noncapital income, age, and impatience to consume. Necessary and sufficient conditions for long-run capital growth are also given.

The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet. In the stock market (more inclusively, the securities markets), the problem is similar but more complex. The gambler, who is now an "investor", looks for "excess risk adjusted return". In both these settings, this chapter explores the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth ("maximize expected logarithmic utility"). The criterion is known to economists and financial theorists by names such as the "geometric mean maximizing portfolio strategy", maximizing logarithmic utility, the growth-optimal strategy, and the capital growth criterion. It initiates the practical application of the Kelly criterion by using it for card counting in blackjack. It presents some useful formulas and methods to answer various natural questions about it that arise in blackjack and other gambling games. It illustrates its recent use in a successful casino sports betting system. It discusses its application to the securities markets where it has helped the author to make a 30-year total of 80 billion dollars worth of "bets".

The Kelly strategy, risking a fixed fraction of one's gambling capital each time when faced with a series of comparable favourable bets, is known to be optimal under several criteria. We review this work, interpret it in the context of spread betting and describe its operation with a performance index. Interlocking spread bets on the same sporting event are frequently offered. We suggest ways of investigating which of these bets may be most favourable, and how a gambler might make an overall comparison of the bets from different firms. Examples illustrate these notions.

We focus on modelling the 92 soccer teams in the English Football Association League over the years 1992–1997 using refinements of the independent Poisson model of Dixon and Coles. Our framework assumes that each team has attack and defence strengths that evolve through time (rather than remaining constant) according to some unobserved bivariate stochastic process. Estimation of the teams' attack and defence capabilities is undertaken via a novel approach involving an approximation that is computationally convenient and fast. The results of this approximation compare very favourably with results obtained through the Dixon and Coles approach. We note that the full model (i.e. the model before the above approximation is made) may be implemented using Markov chain Monte Carlo procedures, and that this approach is vastly more computationally expensive. We focus on the probabilities of home win, draw or away win because these outcomes constitute the primary betting market. These probabilities are estimated for games played between any two of the 92 teams and the predictions are compared with the actual results.

A parametric model is developed and fitted to English league and cup football data from 1992 to 1995. The model is motivated by an aim to exploit potential inefficiencies in the association football betting market, and this is examined using bookmakers' odds from 1995 to 1996. The technique is based on a Poisson regression model but is complicated by the data structure and the dynamic nature of teams' performances. Maximum likelihood estimates are shown to be computationally obtainable, and the model is shown to have a positive return when used as the basis of a betting strategy.

This paper concerns the problem of optimal dynamic choice in discrete time for an investor. In each period the investor is faced with one or more risky investments. The maximization of the expected logarithm of the period by period wealth, referred to as the Kelly criterion, is a very desirable investment procedure. It has many attractive properties, such as maximizing the asymptotic rate of growth of the investor's fortune. On the other hand, instead of focusing on maximal growth, one can develop strategies based on maximum security. For example, one can minimize the ruin probability subject to making a positive return or compute a confidence level of increasing the investor's initial fortune to a given final wealth goal. This paper is concerned with methods to combine these two approaches. We derive computational formulas for a variety of growth and security measures. Utilizing fractional Kelly strategies, we can develop a complete tradeoff of growth versus security. The theory is applicable to favorable investment situations such as blackjack, horseracing, lotto games, index and commodity futures and options trading. The results provide insight into how one should properly invest in these situations.

Results of Kelly [5] and Breiman [2] relating optimal growth rates for gambling and investing to information distances are generalised to include return distributions for virtually any type of game or asset. These results are achieved by first introducing the notion of the optimal financial derivative instrument for a given gamble or investment and then solving the related optimisation problem. For assets varying continuously over time, a formula for optimal dynamic portfolio adjustment follows for commonly occurring models, assuming no transaction costs. The latter results are applied to assets with normal and lognormal returns. The results for these are demonstrated using simulation.

Information theory answers two fundamental questions in communication theory: what is the ultimate data compression (answer: the entropy H), and what is the ultimate transmission rate of communication (answer: the channel capacity C). For this reason some consider information theory to be a subset of communication theory. We will argue that it is much more. Indeed, it has fundamental contributions to make in statistical physics (thermodynamics), computer science (Kolmogorov complexity or algorithmic complexity), statistical inference (Occam's Razor: “The simplest explanation is best”) and to probability and statistics (error rates for optimal hypothesis testing and estimation). The relationship of information theory to other fields is discussed. Information theory intersects physics (statistical mechanics), mathematics (probability theory), electrical engineering (communication theory) and computer science (algorithmic complexity). We describe these areas of intersection in detail.

In the previous literature, two approaches have been used to model match outcomes in association football (soccer): first, modelling the goals scored and conceded by each team; and second, modelling win–draw–lose match results directly. There have been no previous attempts to compare the forecasting performance of these two types of model. This paper aims to fill this gap. Bivariate Poisson regression is used to estimate forecasting models for goals scored and conceded. Ordered probit regression is used to estimate forecasting models for match results. Both types of models are estimated using the same 25-year data set on English league football match outcomes. The best forecasting performance is achieved using a ‘hybrid’ specification, in which goals-based team performance covariates are used to forecast win–draw–lose match results. However, the differences between the forecasting performance of models based on goals data and models based on results data appear to be relatively small.

A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n + 1) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The
simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and
computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum,
needed in statistical estimation problems.

If the input symbols to a communication channel represent the outcomes of a chance event on which bets are available at odds consistent with their probabilities (i.e., “fair” odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially. The maximum exponential rate of growth of the gambler's capital is equal to the rate of transmission of information over the channel. This result is generalized to include the case of arbitrary odds.
Thus we find a situation in which the transmission rate is significant even though no coding is contemplated. Previously this quantity was given significance only by a theorem of Shannon's which asserted that, with suitable encoding, binary digits could be transmitted over the channel at this rate with an arbitrarily small probability of error.

The Mathematics of Gambling

- E O Thorp

Thorp, E. O. (1984) The Mathematics of Gambling. Secaucus: Lyle Stuart.

Fortune's Formula: the Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street The Sharpe ratio

- W Poundstone

Poundstone, W. (2005) Fortune's Formula: the Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street. New York: Hill and Wang. Sharpe, W. F. (1994) The Sharpe ratio. J. Prtfol. Mangmnt, 21, 49–58.

- Kelly J. L.