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, 3/4, 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 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 least, ----, ---, 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, 2/5, that will let you expect to leave the casino in , 6.438592382, rounds of gambling. The probability of winning is, 0.9991366765 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [2/5, 3/5], then you can expect to leave the casino (winning!) in 5.514582662, rounds of gambling, with prob., 0.9990791489 even though this is a little riskier than the Kelly prob. 0.9991366765 999 it is still in your safey net, being above, ---- 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, 1/2, that will let you expect to leave the casino in , 5.193885019, rounds of gambling. The probability of winning is, 0.9931569459 99 which is indeed more than, --- 100 11 If you play the Breiman strategy, [1/2, --], then 20 you can expect to leave the casino (winning!) in 5.068647243, rounds of gambling, with prob., 0.9916411732 even though this is a little riskier than the Kelly prob. 0.9931569459 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- 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 160.0000000, 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 least, ----, ---, 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, 2/5, that will let you expect to leave the casino in , 4.731866134, rounds of gambling. The probability of winning is, 0.9994949724 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [2/5, 3/5], then you can expect to leave the casino (winning!) in 4.340948394, rounds of gambling, with prob., 0.9994585253 even though this is a little riskier than the Kelly prob. 0.9994949724 999 it is still in your safey net, being above, ---- 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 11 with factor, --, that will let you expect to leave the casino 20 in , 4.230140220, rounds of gambling. The probability of winning is, 0.9926021779 99 which is indeed more than, --- 100 11 If you play the Breiman strategy, [--, 3/5], then 20 you can expect to leave the casino (winning!) in 4.210828449, rounds of gambling, with prob., 0.9925753231 even though this is a little riskier than the Kelly prob. 0.9926021779 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- 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 120.0000000, 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 least, ----, ---, 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, 9/20, that will let you expect to leave the casino in , 2.854016275, rounds of gambling. The probability of winning is, 0.9992285648 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [9/20, 3/5], then you can expect to leave the casino (winning!) in 2.783206052, rounds of gambling, with prob., 0.9992086252 even though this is a little riskier than the Kelly prob. 0.9992285648 999 it is still in your safey net, being above, ---- 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, 3/5, that will let you expect to leave the casino in , 2.418887570, rounds of gambling. The probability of winning is, 0.9924557402 99 which is indeed more than, --- 100 If you play the Breiman strategy, [3/5, 1/2], then you can expect to leave the casino (winning!) in 2.393296091, rounds of gambling, with prob., 0.9918835917 even though this is a little riskier than the Kelly prob. 0.9924557402 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- 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 80.00000000, 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 least, ----, ---, 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, 9/20, that will let you expect to leave the casino in , 2.088823130, rounds of gambling. The probability of winning is, 0.9997050478 999 which is indeed more than, ---- 1000 11 If you play the Breiman strategy, [9/20, --], then 20 you can expect to leave the casino (winning!) in 1.987840763, rounds of gambling, with prob., 0.9993698645 even though this is a little riskier than the Kelly prob. 0.9997050478 999 it is still in your safey net, being above, ---- 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, 7/10, that will let you expect to leave the casino in , 1.697843610, rounds of gambling. The probability of winning is, 0.9916466299 99 which is indeed more than, --- 100 If you play the Breiman strategy, [7/10, 1/2], then you can expect to leave the casino (winning!) in 1.677719897, rounds of gambling, with prob., 0.9910066076 even though this is a little riskier than the Kelly prob. 0.9916466299 99 it is still in your safey net, being above, --- 100 This took, 18.184, seconds.