Explorations of the Discrete Kelly-Breiman Strategies for Leaving the Casino\ (With High Probability as a Winner) as Soon as Possible By Shalosh B. Ekhad Suppose that your goal in life is to leave the casino with a fortune of , 200, dollars and for each gamble you have a prob. of, 3/5, of winning. According to your initial capital, we have the following options ----------------------------------------------------------------------------\ ----------- If the initial capital is, 100, dollars, then of course the safest is to play timidly, and then your prob. of exiting a winner is 1.000000000 but the expected time until you exit is 500.0000000, rounds of gambling. If you don't mind taking a small chance of NOT winning, then you may be able\ to leave the casino much sooner as follows. 999 99 For example, with the prob. of winning being at most, ----, ---, 9/10, 1000 100 we can pursue the following strategies. If you are willing to take a chance of completely losing your current capit\ al of, 100, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/10, that will let you expect to leave the casino in , 44.94509484, rounds of gambling. The probability of winning is, 0.9998784517 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [1/10, 4/5], then you can expect to leave the casino (winning!) in 43.81842784, rounds of gambling, with prob., 0.9998721302 even though this is a little riskier than the Kelly prob. 0.9998784517 999 it is still in your safey net below, ---- 1000 If you are willing to take a chance of completely losing your current capit\ al of, 100, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 3/20, that will let you expect to leave the casino in , 35.79289566, rounds of gambling. The probability of winning is, 0.9975458702 99 which is indeed more than, --- 100 If you play the Breiman strategy, [3/20, 3/4], then you can expect to leave the casino (winning!) in 35.08434634, rounds of gambling, with prob., 0.9973784844 even though this is a little riskier than the Kelly prob. 0.9975458702 99 it is still in your safey net below, --- 100 If you are willing to take a chance of completely losing your current capit\ al of, 100, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy with factor, 3/10, that will let you expect to leave the casino in , 20.65959303, rounds of gambling. The probability of winning is, 0.9306418965 which is indeed more than, 9/10 11 If you play the Breiman strategy, [3/10, --], then 20 you can expect to leave the casino (winning!) in 20.24191009, rounds of gambling, with prob., 0.9037298077 even though this is a little riskier than the Kelly prob. 0.9306418965 it is still in your safey net below, 9/10 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 120, dollars, then of course the safest is to play timidly, and then your prob. of exiting a winner is 1.000000000 but the expected time until you exit is 400.0000000, rounds of gambling. If you don't mind taking a small chance of NOT winning, then you may be able\ to leave the casino much sooner as follows. 999 99 For example, with the prob. of winning being at most, ----, ---, 9/10, 1000 100 we can pursue the following strategies. If you are willing to take a chance of completely losing your current capit\ al of, 120, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/10, that will let you expect to leave the casino in , 33.45566162, rounds of gambling. The probability of winning is, 0.9999357266 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [1/10, 4/5], then you can expect to leave the casino (winning!) in 32.27003026, rounds of gambling, with prob., 0.9999289725 even though this is a little riskier than the Kelly prob. 0.9999357266 999 it is still in your safey net below, ---- 1000 If you are willing to take a chance of completely losing your current capit\ al of, 120, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/5, that will let you expect to leave the casino in , 22.34556363, rounds of gambling. The probability of winning is, 0.9900477564 99 which is indeed more than, --- 100 If you are willing to take a chance of completely losing your current capit\ al of, 120, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy with factor, 7/20, that will let you expect to leave the casino in , 12.40417865, rounds of gambling. The probability of winning is, 0.9227364045 which is indeed more than, 9/10 If you play the Breiman strategy, [7/20, 3/5], then you can expect to leave the casino (winning!) in 11.81116216, rounds of gambling, with prob., 0.9123211274 even though this is a little riskier than the Kelly prob. 0.9227364045 it is still in your safey net below, 9/10 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 140, dollars, then of course the safest is to play timidly, and then your prob. of exiting a winner is 1.000000000 but the expected time until you exit is 300.0000000, rounds of gambling. If you don't mind taking a small chance of NOT winning, then you may be able\ to leave the casino much sooner as follows. 999 99 For example, with the prob. of winning being at most, ----, ---, 9/10, 1000 100 we can pursue the following strategies. If you are willing to take a chance of completely losing your current capit\ al of, 140, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 3/20, that will let you expect to leave the casino in , 18.80865831, rounds of gambling. The probability of winning is, 0.9990854211 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [3/20, 4/5], then you can expect to leave the casino (winning!) in 18.18747635, rounds of gambling, with prob., 0.9990001059 even though this is a little riskier than the Kelly prob. 0.9990854211 999 it is still in your safey net below, ---- 1000 If you are willing to take a chance of completely losing your current capit\ al of, 140, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/5, that will let you expect to leave the casino in , 15.79517393, rounds of gambling. The probability of winning is, 0.9936303917 99 which is indeed more than, --- 100 If you play the Breiman strategy, [1/5, 3/4], then you can expect to leave the casino (winning!) in 15.48631685, rounds of gambling, with prob., 0.9933239343 even though this is a little riskier than the Kelly prob. 0.9936303917 99 it is still in your safey net below, --- 100 If you are willing to take a chance of completely losing your current capit\ al of, 140, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy with factor, 9/20, that will let you expect to leave the casino in , 5.672927950, rounds of gambling. The probability of winning is, 0.9084120559 which is indeed more than, 9/10 If you play the Breiman strategy, [9/20, 3/5], then you can expect to leave the casino (winning!) in 5.484733952, rounds of gambling, with prob., 0.9070113255 even though this is a little riskier than the Kelly prob. 0.9084120559 it is still in your safey net below, 9/10 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 160, dollars, then of course the safest is to play timidly, and then your prob. of exiting a winner is 1.000000000 but the expected time until you exit is 200.0000000, rounds of gambling. If you don't mind taking a small chance of NOT winning, then you may be able\ to leave the casino much sooner as follows. 999 99 For example, with the prob. of winning being at most, ----, ---, 9/10, 1000 100 we can pursue the following strategies. If you are willing to take a chance of completely losing your current capit\ al of, 160, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 3/20, that will let you expect to leave the casino in , 12.09026313, rounds of gambling. The probability of winning is, 0.9995010309 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [3/20, 3/4], then you can expect to leave the casino (winning!) in 11.38398565, rounds of gambling, with prob., 0.9993185558 even though this is a little riskier than the Kelly prob. 0.9995010309 999 it is still in your safey net below, ---- 1000 If you are willing to take a chance of completely losing your current capit\ al of, 160, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/5, that will let you expect to leave the casino in , 10.47405457, rounds of gambling. The probability of winning is, 0.9963789549 99 which is indeed more than, --- 100 If you play the Breiman strategy, [1/5, 3/4], then you can expect to leave the casino (winning!) in 9.752323538, rounds of gambling, with prob., 0.9958875094 even though this is a little riskier than the Kelly prob. 0.9963789549 99 it is still in your safey net below, --- 100 If you are willing to take a chance of completely losing your current capit\ al of, 160, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy with factor, 4/5, that will let you expect to leave the casino in , 1.730084121, rounds of gambling. The probability of winning is, 0.9016881726 which is indeed more than, 9/10 If you play the Breiman strategy, [4/5, 2/5], then you can expect to leave the casino (winning!) in 1.698627729, rounds of gambling, with prob., 0.9005833957 even though this is a little riskier than the Kelly prob. 0.9016881726 it is still in your safey net below, 9/10 This took, 154.849, seconds.