Explorations of the Discrete Kelly Strategy for Leaving the Casino (With Hig\ h 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 , 1200, 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, 600, 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 3000.000000, 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. For example, let's see what is going on with the 999 99 following list of probabilities cut-offs, [----, ---, 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, 600, 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 , 37.21864983, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 600, 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 , 33.73174101, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 600, 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 , 22.52423438, rounds of gambling. ----------------------------------------------------------------------------\ ----------- If the initial capital is, 720, 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 2400.000000, 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. For example, let's see what is going on with the 999 99 following list of probabilities cut-offs, [----, ---, 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, 720, 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 , 27.51240708, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 720, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/4, that will let you expect to leave the casino in , 23.15768226, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 720, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy with factor, 2/5, that will let you expect to leave the casino in , 13.56321501, rounds of gambling. ----------------------------------------------------------------------------\ ----------- If the initial capital is, 840, 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 1800.000000, 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. For example, let's see what is going on with the 999 99 following list of probabilities cut-offs, [----, ---, 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, 840, 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 , 19.48396796, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 840, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/4, that will let you expect to leave the casino in , 16.27260846, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 840, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy with factor, 1/2, that will let you expect to leave the casino in , 5.590587618, rounds of gambling. ----------------------------------------------------------------------------\ ----------- If the initial capital is, 960, 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 1200.000000, 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. For example, let's see what is going on with the 999 99 following list of probabilities cut-offs, [----, ---, 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, 960, dollars, with a chance of up to, 1/1000 then you can pursue a Kelly-type strategy with factor, 1/5, that will let you expect to leave the casino in , 11.44214178, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 960, dollars, with a chance of up to, 1/100 then you can pursue a Kelly-type strategy with factor, 1/4, that will let you expect to leave the casino in , 10.21119218, rounds of gambling. If you are willing to take a chance of completely losing your current capit\ al of, 960, dollars, with a chance of up to, 1/10 then you can pursue a Kelly-type strategy 17 with factor, --, that will let you expect to leave the casino 20 in , 1.722879728, rounds of gambling. This took, 550.201, seconds.