How to Gamble If You're In a Hurry

By
Shalosh B. EKHAD and Doron ZEILBERGER

.pdf   .ps   LaTex source
Written: Dec. 7, 2011.
[Appeared in Journal of Difference Equations and Application, 19(2013) , 520-526.]
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).

## Maple Packages

• HIMURIM (the main package)
• PURIM (a small package that uses the "umbral" approach)

# The MOST IMPORTANT Sample Input and Output for HIMURIM

• Assume that you currently have i dollars (1 ≤ i ≤ 999), and you owe 1000 dollars to a loan shark who would kill you if you can't pay him 1000 dollars in ≤ 30 rounds of gambling. Suppose that the probability of winning a single round is 11/20. Naturally you want to maximize the probability of staying alive. So you want a list of length 999 such that its i-th entry would tell you how much to stake if you currently have i dollars. Then
the input gives the output.

• Suppose you have 30 dollars, and you need to make 50 extra dollars in at most 20 rounds, and the probability of winning any given round is 6/11, and you want to see a simulated game that follows the optimal strategy, then
the input gives the output.

• Suppose you want to see the optimal strategies for various winning probabilities (all superfair, of course, for subfair ones, the optimal strategy is always bold), various exit capitals, and various deadlines, then
the input gives the output.

• Suppose you want to check procedure BestStake by simulating 10000 games with p=3/5, initial capital of 30 dollars, exit capital of 60 dollars, and a deadline of 20 rounds,
the input gives the output.

# Other Sample Input and Output for HIMURIM

• Suppose that the probability of winning one round of gambling is p=3/5. If you want to get advice on what Kelly factor to choose if the exit capital is 1200 dollars, for various initial capitals, and various levels of risk aversions, the
the input gives the output.

• Suppose that the probability of winning one round of gambling is p=3/5. If you want to get advice on what Kelly factor to choose if the exit capital is 200 dollars, and how a Breiman modification can improve it (not much!, as it turns out), for various initial capitals, and various levels of risk aversions, the
the input gives the output.

• Suppose that the probability of winning one round of gambling is p=3/4. If you want to get advice on what Kelly factor to choose if the exit capital is 200 dollars, and how a Breiman modification can improve it (not much!, as it turns out), for various initial capitals, and various levels of risk aversions, the
the input gives the output.

• Suppose that the probability of winning one round of gambling is p=5/9. If you want to get advice on what Kelly factor to choose if the exit capital is 360 dollars, and how a Breiman modification can improve it (not much!, as it turns out), for various initial capitals, and various levels of risk aversions, the
the input gives the output.

• Suppose that the probability of winning one round of gambling is p=3/4. If you want to get advice on what Kelly factor to choose if the exit capital is 800 dollars, and how a Breiman modification can improve it (not much!, as it turns out), for various initial capitals, and various levels of risk aversions, the
the input gives the output.

• If you want to see trade-off between finishing fast and taking a chance of losing given by the Breiman profile for exit capital N=800, prob. of a single round being a win p=3/5, and with the Kelly-recommended factor f=2p-1=1/5 for various initial capitals
the input gives the output.

• If you want to see trade-off between finishing fast and taking a chance of losing given by the Kelly profile for exit capital N=800, prob. of a single round being a win p=3/5, with various factors (not necessarily f=2p-1) for various initial capitals
the input gives the output.

If you want to see the same as above but with exit capital N=1200
the input gives the output.

If you want to see the same as above but with exit capital N=2000
the input gives the output.

If you want to see the same as above but with exit capital N=3000
the input gives the output.

• If you want to find the Best Kelly factor when the prob. of winning a single round is p=5/9, you need to exit the casino with 100 dollars, your current capital is (1/2)*100=50 dollars, and your risk-aversion level is 95/100 (i.e. you are willing to take up to a 5 percent chance of losing your capital), then
the input gives the output

If as above, you want to find the Best Kelly factor when the prob. of winning a single round is p=5/9, you need to exit the casino with 100 dollars, your current capital is (1/2)*100=50 dollars, but now your risk-aversion level is 99/100 (i.e. you are only willing to take up to a 1 percent chance of losing your capital), then
the input gives the output

If as above, you want to find the Best Kelly factor when the prob. of winning a single round is p=5/9, you need to exit the casino with 100 dollars, your current capital is (1/2)*100=50 dollars, but now your risk-aversion level is 999/100 (i.e. you are only willing to take up to a .1 percent chance of losing your capital), then
the input gives the output

# Sample Input and Output for PURIM

• If you want to find out about your prospects if your exit capital is 500, the deadline for delivering is 100 rounds, the prob. of winning a single round is 3/5, if you pursue the Kelly strategy with factor 1/5 (recommended by Kelly in his version, since 1/5=(2)(3/5)-1), and your current capital is only 50 dollars, then
the input gives the output
• If you want to find out about your prospects if your exit capital is 500, the deadline for delivering is 100 rounds, the prob. of winning a single round is 3/5, if you pusue toe Kelly strategy with factor 1/5 (recommended by Kelly in his version, since 1/5=(2)(3/5)-1), and your current capital is only 200 dollars, then
the input gives the output

Doron Zeilberger's List of Papers