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 ≤ \frac{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)

\{ \begin{matrix}\frac{x}{q−p} − \frac{N}{q−p} \frac{1−(q/p)^{x}}{1−(q/p)^{N}} & if p ≠ \frac{1}{2}; \\ x(N − x), & if p = \frac{1}{2}, \end{matrix}

Date: November 24, 2014.

D.Z. was supported in part by the USA National Science Foundation.

and this may take a very long time. If p > \frac{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 strategy^{1}, 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) := \{ \begin{matrix}min(⌊xf⌋ + 1, N − x), & if x < cN; \\ min(x, N − x), & if x ≥ cN. \end{matrix}

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 c_{0}(f) reduces the expected duration-until-winning (with a slightly higher risk of ultimate loss), but setting c below c_{0} (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 language of [3], you owe N dollars to a loan shark who would kill you if you don’t return the debt in ≤ T units time (rounds of gambling). Luckily, you are at a superfair casino, (i.e. p ≥ \frac{1}{2}). If your current capital is i dollars (so you need to make N − i additional dollars in ≤ T rounds to stay alive), if you want to maximize your chances of staying alive, how many dollars should you stake ?

Finding the Best Strategy If You’re in a Hurry

So suppose that you currently have i dollars, and you need to make N − i additional dollars, so that you can exit the casino with N dollars in ≤ T rounds of gambling, where at each round you can stake any amount between 1 and min(i, N − i). You want to maximize your chance of success. How much should you stake, and what is the resulting probability, let’s call it f(i, T). The probability of winning a single round is p.

Obviously f(i, T) satisfies the dynamical programming recurrence

f(i, T) = max{(1 − p)f(i − x, T − 1) + pf(i + x, T − 1) : 1 ≤ x ≤ min(i, N − i)},

with the obvious boundary conditions f(0, T) = 0, f(N, T) = 1 and initial conditions f(N, 0) = 1, and f(x, 0) = 0 if x < N.

1The Kelly strategy is also known as the Kelly system or the Kelly criterion; terms first coined by Ed Thorp in [9] and [10].

The theoretical underpinnings of this strategy were provided by Breiman in [1].

The set of x’s that attain this max constitutes your optimal strategy. It is most convenient to take the largest x (in case there are ties). By repeatedly computing f(j, T) and the stake-amount x that realizes it, where j is the current capital and T is the steadily decreasing time left, the gambler can always know how much to stake in order to maximize his chance of staying alive, and also know the actual value of that probability. The Maple Package HIMURIM HIMURIM is downloadable, free of charge, from

http://www.math.rutgers.edu/~zeilberg/tokhniot/HIMURIM .

We will only briefly describe some of the more important procedures, leaving it to the readers to explore and experiment with the many features on their own, using the on-line help. The most Important Procedures of HIMURIM The most important procedure is BestStake(p,i,N,T), that implements f(i, T) with the given p and N. If you want so see the full optimal strategy, a list of length N − 1 whose i-th entry tells you how much to stake if you have i dollars, use procedure BestStrat(p,N,T). See, for example,

 http://www.math.rutgers.edu/~zeilberg/tokhniot/oHIMURIMk1 .
for the output of BestStrat(11/20,1000,30); .

Procedure SimulateBSv(p,i,N,T) simulates one game that follows the optimal strategy. Finally, Procedure BestStratStory(m0,N0,T0,K) collects optimal strategies for various p’s (all superfair), N’s (exit capitals) and T (deadlines). Other Procedures of HIMURIM Procedure ezraLA() lists the procedures that use Linear Algebra to find the exact probabilities, expected duration, and probability generating functions for the random variables “duration” and “duration conditioned on ultimately winning” for a casino with exit capital N, probability of winning a round p (that may be either numeric or symbolic), and all possible initial incomes. For example, PrW(p,S) inputs a probability p (a number between 0 and 1 or left as a symbol p) and a list S, of length N − 1, say, where S[i] tells you how much to stake if you have i dollars. It outputs the list, let’s call it L, such that L[i] is the probability of ultimately winning (exiting with N dollars) if you currently have i dollars and always play according to strategy S. It works by solving the system of N − 1 equations for the N − 1 unknowns L[1], …, L[N − 1]

L[i] = (1 − p)L[i − S[i]] + pL[i + S[i]] , 1 ≤ i ≤ N − 1 ,

together with the boundary conditions L[0] = 0, L[N] = 1. For example, if N = 3, and the strategy S being [1, 1], (the only possible strategy when N = 3), then

PrW(1/3,[1,1]);

would yield

[1/7, 3/7] ,

that means that if the probability of winning a round is \frac{1}{3} and you exit the casino when you either reach 0 or 3 dollars, then your probability of exiting as a winner, if you currently have one dollar is \frac{1}{7}, and if you currently have 2 dollars, is \frac{3}{7}. Procedure ED(p,S) inputs p and S as above and outputs the list, let’s call it L, such that L[i] is the expected duration until getting out (either as a winner or loser) if you currently have i dollars and always play according to strategy S. It works by solving the system of N − 1 equations for the N − 1 unknowns L[1], …, L[N − 1]

L[i] = (1 − p)L[i − S[i]] + pL[i + S[i]] + 1 , 1 ≤ i ≤ N − 1 ,

together with the boundary conditions L[0] = 0, L[N] = 0. For example, still with N = 3 and S = [1, 1],

ED(1/3,[1,1]);

would yield

[12/7, 15/7] ,

that means that if the probability of winning a single round is \frac{1}{3} and you exit the casino when you either reach 0 or 3 dollars, and you follow strategy [1, 1], then the expected remaining duration, if you currently have one dollar, is \frac{12}{7} , and if you currently have 2 dollars, it is \frac{15}{7}. Procedure EDw(p,S) inputs p and S as above and outputs the list, let’s call it L, such that L[i] is the expected duration until getting out, conditioned on being an ultimate winner! if you currently have i dollars and always play according to strategy S. For example, with the above (trivial) input

EDw(1/3,[1,1]);

would yield

[18/7, 11/7] ,

that means that if the probability of winning a round is \frac{1}{3} and you exit the casino when you either reach 0 or 3 dollars, then the expected remaining duration until winning (assuming that you do win), if you currently have one dollar is \frac{18}{7}, and if you currently have 2 dollars is \frac{11}{7}. Procedure Dpgf(p,S,t) inputs p and S as above, as well as a symbol (variable name) t, and outputs the list, let’s call it L, such that L[i] is the probability generating function, in t, for the random variable “remaining duration” if you currently have i dollars, i.e. if you take the Maclaurin expansion of L[i] and extract the coefficient of t^{j} you would get the exact value of the probability that the game would last exactly j more rounds. For example, still with the same N and S,

lprint(Dpgf(1/3,[1,1],t));

would yield

[-t*(6+t)/(-9+2*t**2), -t*(3+4*t)/(-9+2*t**2)] .
Typing “taylor(-t*(6+t)/(-9+2*t**2),t=0,5);” would yield

2/3*t+1/9*t**2+4/27*t**3+2/81*t**4+O(t**5)

meaning that if you play the above game with p = 1/3, N = 3 and you currently have one dollar, you would have probability 2/3 of exiting after one round, probability 1/9 of exiting after two rounds, probability 4/27 of exiting after three rounds, and probability 2/81 of exiting after four rounds. Procedure DpgfW(p,S,t) is analogous to Dpgf(p,S,t) but the duration is conditioned on the fortunate event of exiting as an ultimate winner. For example, for the timid strategy, and N = 3,

lprint(DpgfW(1/3,[1,1],t));

would yield

[-t**2/(-9+2*t**2), -3*t/(-9+2*t**2)] .
Typing taylor(-t**2/(-9+2*t**2),t=0,5); would yield
1/9*t**2+2/81*t**4+O(t**6)

meaning that if you play the above game with p = 1/3, N = 3 and you currently have one dollar, and you are destined to leave as a winner, you would have probability of 0 of exiting after one round, probability of 1/9 of exiting after two rounds, probability of 0 of exiting after three rounds, probability of 2/81 of exiting after four rounds. The Best Breiman-Kelly Strategies If You are in A Hurry Procedure BestBKdd(p,N,i,T,h) tells you the best Breiman-Kelly Strategy if the probability of winning a round is p, you have i dollars and you must exit the casino with N dollars in <= T rounds, and you’re using resolution h. It also returns the expected duration until exit (either as a winner or loser). To get the story for various initial capitals, and various deadlines, try out procedure BestBKddStory(p,N,h,t0,MaxF,M0). See the on-line help, and the sample input and output files in

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/himurim.html .

The Best Kelly Factor With a Given Level of Risk-Aversion Try out procedure KellyContestx(p,N,x,h,conf). For example see:

http://www.math.rutgers.edu/~zeilberg/tokhniot/oHIMURIM8a ,
http://www.math.rutgers.edu/~zeilberg/tokhniot/oHIMURIM8b .

The Maple package PURIM For an “umbral” approach, see our Maple package PURIM that tells you much more. It explores the whole “tree” of possibilities. See the package itself and the input and output files

http://www.math.rutgers.edu/~zeilberg/tokhniot/inPURIM2 and
http://www.math.rutgers.edu/~zeilberg/tokhniot/oPURIM2 for an example.

Further Work There are many possible generalizations and extensions. See for example the interesting article [3]. Conclusions The three authors completely agree on the mathematics, but they have somewhat different views about the significance of this project. Here they are.

Doron Zeilberger’s Conclusion

Traditional mathematicians like Dubins and Savage use traditional proof-based mathematics, and also work in the framework of continuous probability theory using the pernicious Kolmogorov, measure-theoretic, paradigm. This approach was fine when we didn’t have computers, but we can do so much more with both symbol-crunching and number-crunching, in addition to naive simulation, and develop algorithms and write software, that ultimately is a much more useful (and rewarding) activity than “proving” yet-another-theorem in an artificial and fictional continuous, measure-theoretic, world, that is furthermore utterly boring.

Shalosh B. Ekhad’s Conclusion

These humans, they are so emotional! That’s why they never went very far.

References

[1] L. Breiman, Optimal gambling systems for favorable games, Fourth Berkley Symposium on Math. Stat. and Prob. 1 (1961), 65-78.

[2] D. A. Berry and D. C. Heath and W. D. Sudderth, Red-and-black with unknown win probability, Ann. Statist. 2 (1974), 602608.

[3] R.W. Chen, L.A. Shepp, Y.-C. Yao, and C.-H. Zhang, On optimality of bold play for primitive casinos in the presence of inflation, J. Appl. Prob. 42 (2005), 121-137.

[4] Lester E. Dubins and Leonard J. Savage, Inequalities for Stochastic Processes (How to Gamble If You Must), Dover, 1976.

[5] William Feller, “An Introduction to Probability Theory and Its Application”, Wiley,volume 1, third edition, 1968.

[6] J.L. Kelly, Jr, A new interpretation of information rate, Bell System Tech. J. 35 (1956), 917-926.

[7] A.P. Maitra and W.D. Sudderth, “Discrete Gambling and Stochastic Games”, Springer, 1996.

[8] Kyle Siegrist, How to Gamble if you must, JOMA v. 8, http://www.maa.org/joma/Volume8/Siegrist/RedBlack.pdf

[9] E.O. Thorp, “Beat the Dealer”, 2nd Ed., Vintage, New York 1966.

[10] E.O. Thorp and W. Walden, “A winning bet in Nevada baccarat.”, J. Amer. Statist. Assoc., 61 Part I, (1966) 313-328.

Email addresses: SBE: c/o zeilberg@math.rutgers.edu ; DZ: zeilberg@math.rutgers.edu .