How to Gamble If Youre In a Hurry

📝 Abstract

The beautiful theory of statistical gambling, started by Dubins and Savage (for subfair games) and continued by Kelly and Breiman (for superfair games) has mostly been studied under the unrealistic assumption that we live in a continuous world, that money is indefinitely divisible, and that our life is indefinitely long. Here we study these fascinating problems from a purely discrete, finitistic, and computational, viewpoint, using Both Symbol-Crunching and Number-Crunching (and simulation just for checking purposes).

💡 Analysis

The beautiful theory of statistical gambling, started by Dubins and Savage (for subfair games) and continued by Kelly and Breiman (for superfair games) has mostly been studied under the unrealistic assumption that we live in a continuous world, that money is indefinitely divisible, and that our life is indefinitely long. Here we study these fascinating problems from a purely discrete, finitistic, and computational, viewpoint, using Both Symbol-Crunching and Number-Crunching (and simulation just for checking purposes).

📄 Content

arXiv:1112.1645v2 [math.PR] 26 Nov 2014 HOW TO GAMBLE IF YOU’RE IN A HURRY SHALOSH B. EKHAD AND DORON ZEILBERGER Preamble This article is a brief description of the Maple package HIMURIM downloadable from http://www.math.rutgers.edu/~zeilberg/tokhniot/HIMURIM . Sample input and output files can be obtained from the webpage: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/himurim.html . The Maple package HIMURIM is to be considered as the main output of this project, and the present article is to be considered as a short user’s manual. We also briefly describe another Maple package downloadable from http://www.math.rutgers.edu/~zeilberg/tokhniot/PURIM . How To Gamble If You Must Suppose that you currently have x dollars, and you enter a casino with the hope of getting out with N dollars, (with, x and N, being positive integral values). The probability of winning one round is p (0 < p < 1). You can stake any integral amount of dollars s(x) (that must satisfy 1 ≤s(x) ≤min(x, N −x)), until you either exit the casino with the hoped-for N dollars, or you become broke. Deciding the value of the stake s(x), for each 1 ≤x < N, constitutes your strategy. Naturally, the question of whether a strategy is optimal arises; the three main optimality criteria in gambling theory can be summarized as follows: 1 Maximizing the probability of reaching a specified goal (i.e., amount N), with no time limit. 2 Maximizing the probability of reaching a specified goal by a fixed time T . 3 Minimizing the expected time to reach a specified goal, subject to a pre-specified level of risk-aversion. In their celebrated masterpiece, Dubins and Savage [4] proved that the optimal strategy (using the first criterion), if p ≤1 2, is the bold one taking s(x) = min(x, N −x), always betting the maximum, and if p ≥1/2, then an optimal strategy is the timid one, with s(x) = 1, always betting the minimum. A beautiful, lucid, and accessible account of these results can be found in Kyle Siegrist’s [8] on-line article. Alas, if you play timidly, i.e. according to the classical “gambler’s ruin” problem ( [5], p. 348, Eq. (3.4)) your expected time until exiting is (let q := 1 −p) ( x q−p − N q−p 1−(q/p)x 1−(q/p)N if p ̸= 1 2; x(N −x), if p = 1 2, Date: November 27, 2014. D.Z. was supported in part by the USA National Science Foundation. 1 and this may take a very long time. If p > 1 2, but you’re in a hurry, then you may decide to take a slightly higher chance of exiting as a loser if that will enable you to expect to leave the casino much sooner. It turns out that the bold strategy is way too risky. For example, if p = 3/5 and right now you have 100 dollars and the exit amount is 200 dollars, with the bold strategy, sure enough, you are guaranteed to exit the casino after just one round, but your chance of leaving as a winner is only 3/5. As a compromise, we can employ a deterministic fixed fractional betting strategy, namely, the Kelly strategy1, with factor f denoting a fixed fraction of our money. This is inspired by [6], however, in that paper, the underlying assumptions are: money is infinitely divisible, the game continues indefinitely, the game has even payoff, and the opponent is infinitely wealthy. Under those circumstances, Kelly recommends to take f = 2p −1 for his agenda. Based on our set of assumptions – using integral values – we obtain, K(f)(x) := min(⌊xf⌋+ 1, N −x) . For example, the Kelly strategy with f = 1/10 (and still p = 3/5, x = 100, N = 200) enables you to exit as a winner with probability %99.98784517, but the expected duration is only 44.94509484 rounds, much sooner than the expected duration of 500 rounds (with a fat tail!) promised by the timid strategy. Inspired by Breiman [1] we can generalize the Kelly-type strategy, and “morph” it with the bold strategy, and play boldly once our capital is ≥Nc, in other words B(f, c)(x) :=  min(⌊xf⌋+ 1, N −x), if x < cN; min(x, N −x), if x ≥cN. For example, taking f = 1/10, c = 4/5 (and still p = 3/5, x = 100, N = 200), your probability of exiting as a winner is %99.98721302, only slightly less than Kelly with f = 1/10, but your expected stay at the casino is about one round less (43.81842784). Paradoxically, lowering the c to 2/5 is not advisable, since your probability of winning is lower (%99.93836900) and you should expect to stay longer! (52.61769977 rounds). We observed, empirically, that for any f, lowering the c from 1 until a certain place c0(f) reduces the expected duration-until-winning (with a slightly higher risk of ultimate loss), but setting c below c0 (i.e., playing boldly starting at cN) will not only lower your chance of ultimately winning, but would also prolong your agony of staying in the casino (unless you want to maximize your stay there, in which case you should play timidly). Our question is: what is the optimal strategy according to Criterion 2 (i.e. maximizing the probability of reaching a specified goal by a fixed time T )? Borrowing the colorful yet gruesome lang

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