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 , 800, 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, 400, 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 800.0000000, 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, 400, 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 , 5.663200588, rounds of gambling. The probability of winning is, 0.9996609020 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 5.436943076, rounds of gambling, with prob., 0.9996430995 even though this is a little riskier than the Kelly prob. 0.9996609020 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, 400, 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 , 5.434946188, rounds of gambling. The probability of winning is, 0.9915410431 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 5.252497692, rounds of gambling, with prob., 0.9911606467 even though this is a little riskier than the Kelly prob. 0.9915410431 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 480, 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 640.0000000, 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, 480, 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 , 4.414985020, rounds of gambling. The probability of winning is, 0.9997879325 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 4.115951551, rounds of gambling, with prob., 0.9995220238 even though this is a little riskier than the Kelly prob. 0.9997879325 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, 480, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 9/20, that will let you expect to leave the casino in , 4.414985020, rounds of gambling. The probability of winning is, 0.9997879325 99 which is indeed more than, --- 100 11 If you play the Breiman strategy, [9/20, --], then 20 you can expect to leave the casino (winning!) in 4.115951551, rounds of gambling, with prob., 0.9995220238 even though this is a little riskier than the Kelly prob. 0.9997879325 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 560, 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 480.0000000, 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, 560, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/2, that will let you expect to leave the casino in , 2.831970530, rounds of gambling. The probability of winning is, 0.9993185982 999 which is indeed more than, ---- 1000 11 If you play the Breiman strategy, [1/2, --], then 20 you can expect to leave the casino (winning!) in 2.732478658, rounds of gambling, with prob., 0.9991352108 even though this is a little riskier than the Kelly prob. 0.9993185982 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, 560, 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 , 2.445889791, rounds of gambling. The probability of winning is, 0.9907731652 99 which is indeed more than, --- 100 11 If you play the Breiman strategy, [7/10, --], then 20 you can expect to leave the casino (winning!) in 2.440076611, rounds of gambling, with prob., 0.9907376960 even though this is a little riskier than the Kelly prob. 0.9907731652 99 it is still in your safey net, being above, --- 100 ----------------------------------------------------------------------------\ ----------- If the initial capital is, 640, 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 320.0000000, 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, 640, 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 , 2.103570694, rounds of gambling. The probability of winning is, 0.9999469831 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 2.028618970, rounds of gambling, with prob., 0.9998805059 even though this is a little riskier than the Kelly prob. 0.9999469831 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, 640, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 3/4, that will let you expect to leave the casino in , 1.726911066, rounds of gambling. The probability of winning is, 0.9922111019 99 which is indeed more than, --- 100 If you play the Breiman strategy, [3/4, 1/2], then you can expect to leave the casino (winning!) in 1.723220348, rounds of gambling, with prob., 0.9921505311 even though this is a little riskier than the Kelly prob. 0.9922111019 99 it is still in your safey net, being above, --- 100 This took, 147.806, seconds.