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 , 360, dollars and for each gamble you have a prob. of, 5/9, of winning. According to your initial capital, we have the following options ----------------------------------------------------------------------------\ ----------- If the initial capital is, 180, 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 1620.000000, 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, 180, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/18, that will let you expect to leave the casino in , 144.8299194, rounds of gambling. The probability of winning is, 0.9998862981 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [1/18, 8/9], then you can expect to leave the casino (winning!) in 143.5429073, rounds of gambling, with prob., 0.9998853035 even though this is a little riskier than the Kelly prob. 0.9998862981 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, 180, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/18, that will let you expect to leave the casino in , 144.8299194, rounds of gambling. The probability of winning is, 0.9998862981 99 which is indeed more than, --- 100 If you play the Breiman strategy, [1/18, 8/9], then you can expect to leave the casino (winning!) in 143.5429073, rounds of gambling, with prob., 0.9998853035 even though this is a little riskier than the Kelly prob. 0.9998862981 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 216, 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 1296.000000, 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, 216, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/18, that will let you expect to leave the casino in , 107.3159775, rounds of gambling. The probability of winning is, 0.9999398048 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [1/18, 8/9], then you can expect to leave the casino (winning!) in 106.0371101, rounds of gambling, with prob., 0.9999388473 even though this is a little riskier than the Kelly prob. 0.9999398048 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, 216, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/9, that will let you expect to leave the casino in , 72.57174411, rounds of gambling. The probability of winning is, 0.9908338731 99 which is indeed more than, --- 100 If you play the Breiman strategy, [1/9, 5/6], then you can expect to leave the casino (winning!) in 72.15572607, rounds of gambling, with prob., 0.9906497562 even though this is a little riskier than the Kelly prob. 0.9908338731 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 252, 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 972.0000000, 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, 252, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/18, that will let you expect to leave the casino in , 75.33929909, rounds of gambling. The probability of winning is, 0.9999678163 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [1/18, 8/9], then you can expect to leave the casino (winning!) in 74.08006428, rounds of gambling, with prob., 0.9999668368 even though this is a little riskier than the Kelly prob. 0.9999678163 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, 252, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/9, that will let you expect to leave the casino in , 51.28138769, rounds of gambling. The probability of winning is, 0.9941130307 99 which is indeed more than, --- 100 If you play the Breiman strategy, [1/9, 5/6], then you can expect to leave the casino (winning!) in 50.83981433, rounds of gambling, with prob., 0.9938806686 even though this is a little riskier than the Kelly prob. 0.9941130307 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 288, 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 648.0000000, 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, 288, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/18, that will let you expect to leave the casino in , 47.78021480, rounds of gambling. The probability of winning is, 0.9999838852 999 which is indeed more than, ---- 1000 If you play the Breiman strategy, [1/18, 8/9], then you can expect to leave the casino (winning!) in 46.28151217, rounds of gambling, with prob., 0.9999827828 even though this is a little riskier than the Kelly prob. 0.9999838852 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, 288, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/9, that will let you expect to leave the casino in , 32.80361509, rounds of gambling. The probability of winning is, 0.9965721842 99 which is indeed more than, --- 100 If you play the Breiman strategy, [1/9, 5/6], then you can expect to leave the casino (winning!) in 32.00762028, rounds of gambling, with prob., 0.9963967965 even though this is a little riskier than the Kelly prob. 0.9965721842 99 it is still in your safey net, being above, --- 100 This took, 33.474, seconds.