๐Ÿ– Thorp () Optimal gambling systems for favorable games

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By L. Breiman; Abstract: The following sections are included:IntroductionThe nature of ฮ›*The asymptotic time minimization problemAsymptotic.


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It is an easy problem to find, by backward induction, the betting system that maximizes. E(X20|X0), your () Optimal gambling systems for favorable games.


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A look at optimal betting and the Kelly criterion digging into some of the math. Optimal Gambling Systems for Favorable Games, E. O. Thorp.


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We determine Kelly criterion for a game with variable pay-off. The [1] BREIMANโ€‹, L.; Optimal gambling systems for favourable games, Fourth.


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Breiman, L. Optimal Gambling Systems for Favorable Games. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability.


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We determine Kelly criterion for a game with variable pay-off. The [1] BREIMANโ€‹, L.; Optimal gambling systems for favourable games, Fourth.


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L. Breiman โ€œOptimal Gambling Systems for Favorable Gamesโ€. () criterion in blackjack, sports betting, and the stock market. Elsevier.


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We say that the game is favorable if there is a gambling strategy such that almost. Only quite recently has the question of gambling systems optimal with respect.


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A look at optimal betting and the Kelly criterion digging into some of the math. Optimal Gambling Systems for Favorable Games, E. O. Thorp.


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There are many types of wagers that can be placed on sporting events. In this paper, we take a statistical approach to the problem where we account for the uncertainty in p. Therefore, an objective of the bookie is to set a competitive point spread that encourages a balance of the bets on both sides of the point spread. Then the gambling system is profitable if the expected return is positive. In Section 3 , we develop modified Kelly criteria by gradually increasing our assumptions. The second approach is based on Bayes estimation which requires the introduction of a prior distribution on p. A short discussion is provided in Section 6. Whereas the profitability condition 2 may seem remarkably low and plausible, gambling has been around for a very long time. In determining the Kelly criterion, a gambler needs to specify the probability p of placing a correct i. In its application to sports gambling, the Kelly criterion Kelly provides a gambler with the optimal fraction of a bankroll that should be wagered on a given bet. This natural loss function is given by. Then the subsequent bankroll is the random quantity. With respect to mathematical and probabilistic treatments related to finance and investing, MacLean, Thorp, and Ziemba provide a comprehensive edited volume with contributions that focus on various financial problems involving the Kelly criterion. One sees immediately from the point spread that the Warriors are the favorite chalk whereas the Spurs are the dog. This formulation is helpful since we are now comparing scalar quantities. In the examples which we consider, the resulting fractions tend to be less than the Kelly criterion. However, loss functions are complex in the sense that they depend on parameters, data and decisions. In the context of the Bayes estimators, we discuss the selection of prior distributions for p with particular emphasis given to a default prior which we hope is appealing to a wide audience. However, there is only a scattered and limited scientific literature on the mathematical properties of sports gambling. The Kelly criterion has received widespread attention, and some of the attention has been negative Samuelson Experienced gamblers claim that the Kelly fraction is too high and often leads to financial loss Murphy The perplexing aspect of these negative experiences is that the Kelly criterion is based on mathematical proof. Therefore, the development of a gambling system that ensures 2 is difficult to attain. The first approach requiring the fewest assumptions is based on minimax estimation. Moreover, we provide explicit expressions and R code to evaluate optimal betting fractions. If the point spread is an integer, then the betting outcome can be a push where the gambler neither wins nor loses, and the wager is returned. Estimators are proposed that are developed from a decision theoretic framework. In Baker and McHale , a decision theory framework is developed where fractional Kelly systems are investigated. To assess the quality of an estimator, it is necessary to introduce a loss function. European decimal odds provide an alternative representation of American odds. In some cases, numerical methods are required for the resultant optimization problem. Clearly, bookies cannot systematically be exploited by profitable gamblers and continue to exist. Although the language in Baker and McHale differs from ours e. From the framework described in Section 2 , we know that the Kelly criterion k p is the optimal value of f. At this stage, it is useful to review both the statement and the proof leading to the Kelly criterion. Given the above framework, decision theoretic approaches typically begin with the introduction of the risk function of an estimator f given by. In Section 4 , we introduce alternative loss functions and investigate the corresponding Bayes estimators. Using a Beta prior distribution, an analytical expression for the optimal betting fraction is obtained. The simple but often overlooked explanation is that the input p used in determining the Kelly fraction is an unknown quantity. They also study alternative utility functions. In other words, we attempt to find an estimator f which minimizes. We observe that the minimax approach is too conservative and does not provide useful betting fractions. Concavity is appealing to the sports gambler who has an apriori belief concerning the most likely value p 0 with decreasing probability as we move away from p 0.{/INSERTKEYS}{/PARAGRAPH} As shown in Breiman , the Kelly desiderata of maximizing expected log growth has various appealing properties. Begin with an initial bankroll B 0 where we bet a fraction f of the bankroll. The approach is flexible since it accommodates different prior beliefs. These papers also focus on the unknown aspect of p when considering the use of the Kelly criterion. But we stress that this is only applicable under a gambling system that satisfies the profitability condition 2. Our problem therefore reduces to minimizing the loss l 0 f , p given in 6. Insley, Mok, and Swartz extended the Kelly criterion to situations where simultaneous wagers are placed. By recognizing that the probability p of placing a correct wager is unknown, modified Kelly criteria are obtained that take the uncertainty into account. Unfortunately, the minimax approach is not fruitful because it is overly conservative. The Kelly criterion is optimal from several points of view; for example, it maximizes the exponential rate of growth and it provides the minimal expected time to reach an assigned balance Breiman Therefore, how can it be that gamblers often experience losses when using the Kelly approach? What is apparent is that R f p is an average over the sample space, and that the preference of an estimator f 1 over f 2 involves the comparison of risk functions. This constitutes a wagering system i. Often, gamblers are overly optimistic concerning their gambling systems and the true p is less than the specified p. We observe that the resultant betting fractions can differ markedly based on the choice of loss function. We also review the derivation of the Kelly criterion. In other words, we should never wager even when the system is profitable according to 2. In Section 5 , we investigate a number of examples where various estimators of the betting fraction f are compared. Suppose further that the gambler is correct i. For example, in greyhound racing, Schumaker used data mining techniques to identify profitable wagers. When we look at the worst thing that can happen for a given f [i. We take the view that the determination of the optimal wagering fraction f is a statistical problem where the probability p of placing a winning wager is an unknown parameter. In Section 2 , we review the necessary terminology and foundations of sports gambling. A main takeaway from their paper is that shrunken Kelly i. We use the well-known result that a Bayes estimator minimizes expected posterior loss. Before proceeding, there are two papers that deserve special mention. The most common type of wager is known as a point spread wager. Utility is analyzed for specified values of p. However, more structure is required in the minimization problem since f is a function of x and the parameter p is unknown. There are many papers that propose systems and insights with respect to gambling. The data x arises in the context of historical data. What is often overlooked in this standard approach is that the true parameter p is unknown. {PARAGRAPH}{INSERTKEYS}This paper considers an extension of the Kelly criterion used in sports wagering. Therefore, it seems reasonable to consider a loss function which is the ratio of the optimal Kelly expected log growth to the expected log growth under an alternative fraction f. We therefore consider standard decision theoretic approaches with respect to the minimization [see chapter 12 of Wasserman ]. We therefore introduce a statistical model based on a proposed wagering system:. Under this restriction,. The resulting fractions of the bankroll which we derive tend to be less than the Kelly criterion. In our development, the consideration of Bayes estimators leads to optimal betting fractions without imposing a specified form on the betting fraction. Therefore, the minimization of 8 with respect to f requires the minimization of the term within the square parentheses for each value of x. We note that a major difference between the two papers and ours is that we make use of prior distributions whereas their approach is frequentist. Therefore, to make R f p in 7 as small as possible [i. To simplify comparisons, minimax estimators are those which minimize the maximum risk. Consequently, treatment of the underlying estimation problem can differ, and at this stage, Baker and McHale take a distinctly alternative approach. Thorp provided probabilistic results concerning optimal systems for favourable games. Under simultaneous wagering, the simple Kelly criterion is no longer applicable and computation is required to obtain optimal fractions. When 2 does not hold, gambling is a losing proposition. To move sentiment, bookies can modify the point spread and they can also offer different odds on the two teams e. Then the optimal fraction of the bankroll for wagering explained later is known as the Kelly criterion and is given by. To see this, note that l 0 f , p in 6 is non-negative for all f and p. In the cases that we study, the modified Kelly fractions are smaller than original Kelly. This is accomplished in a decision theoretic framework with a loss function that is natural in the Kelly context. For example, a sports gambler may consider placing bets on road teams in the NBA when the home team is playing its first game back from a road trip. For example, a number of papers in MacLean et al. One particular line took the form. For example, the estimator f 1 x is preferred to f 2 x if r f 1 x r f 2 x.