---------------------------------------------------- Some Kelly Strategy Profiles By Shalosh B. Ekhad You start with a certain initial capital, and toss a loaded coin with probab\ ility, 3/5, of lending Heads. At every round, you can stake any amount (not exceeding your current fortune\ ). If the coin lands Heads, you win the stake, otherwise you lose it. Suppose that your goal in life is to leave the casino with a fortune of , 800, dollars. Since the probability is superfair, if you are a chicken, and want to absolutely maximize your probability of leaving the casino as a winner, then, as Lester Dubins and Jimmie Savage tell us, your best strategy is to play timidly. Alas, this may take a very LONG time! If you are willing to take a small chance of NOT winning, you could expect to leave the casino (as a winner) much sooner, by pursuing a Kelly-style strategy with factor f, for various f (NOT necessa\ rily the Kelly recommendation of 2p-1=, 1/5, .) Below we will list, for each f from 0 (timid strategy) to 1 (bold strategy) \ with resolution, 1/20, the prob. of leaving the casino a winner, followed by the expected time it would take \ to get there, for various starting capitals. Note that it is not always the best to take Kelly's recommendation mentioned\ above f=, 1/5, since this may be too risky for you, or conversely would take too long, and you are willing to take a higher risk of losing, but to reduce the expected time that it would take to win. So YOU\ can decide for yourself, looking at the data what suits you best in this trade-off between ultimately winning and wi\ nning as soon as possible. ----------------------------------------------------------------------------\ ----------- If the initial capital is, 400, dollars, then the Kelly Profile is [[0., 1.000000000, 2000.000000], [0.05000000000, 1.000000000, 78.47029721], [0.1000000000, 0.9999979302, 46.33444742], [0.1500000000, 0.9997484427, 37.07294540], [0.2000000000, 0.9961962690, 33.18235248], [0.2500000000, 0.9827452297, 29.95846595], [0.3000000000, 0.9589999535, 26.08052320], [0.3500000000, 0.9234914187, 21.18644481], [0.4000000000, 0.8837069844, 17.06555829], [0.4500000000, 0.8490578113, 13.29959698], [0.5000000000, 0.8121674841, 9.927153982], [0.5500000000, 0.7863546686, 8.389217809], [0.6000000000, 0.7583726407, 6.921415666], [0.6500000000, 0.7381507257, 5.716949216], [0.7000000000, 0.7183628563, 4.873202787], [0.7500000000, 0.6983624524, 4.268565881], [0.8000000000, 0.6799783136, 3.724412302], [0.8500000000, 0.6643984084, 3.525654377], [0.9000000000, 0.6462589280, 3.165215691], [0.9500000000, 0.6261106415, 2.933705136], [1., 0.6000000000, 1.]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 440, dollars, then the Kelly Profile is [[0., 1.000000000, 1800.000000], [0.05000000000, 1.000000000, 67.83145749], [0.1000000000, 0.9999985193, 40.08190177], [0.1500000000, 0.9998029092, 32.09968573], [0.2000000000, 0.9968988320, 28.82266232], [0.2500000000, 0.9856234106, 25.82560779], [0.3000000000, 0.9653532163, 22.79581287], [0.3500000000, 0.9338322287, 18.24845238], [0.4000000000, 0.9009615797, 14.35599521], [0.4500000000, 0.8710997530, 11.71340121], [0.5000000000, 0.8364835662, 9.017477571], [0.5500000000, 0.8129789002, 7.516305841], [0.6000000000, 0.7894892052, 6.226108690], [0.6500000000, 0.7694373995, 5.308968828], [0.7000000000, 0.7492719407, 4.617395967], [0.7500000000, 0.7286145332, 4.084669654], [0.8000000000, 0.7094082922, 3.741883285], [0.8500000000, 0.6971274092, 2.036116795], [0.9000000000, 0.6907665804, 1.852228671], [0.9500000000, 0.6835236120, 1.684786603], [1., 0.6770118846, 1.515321776]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 480, dollars, then the Kelly Profile is [[0., 1.000000000, 1600.000000], [0.05000000000, 1.000000000, 58.05183758], [0.1000000000, 0.9999989289, 34.35804572], [0.1500000000, 0.9998453003, 27.42833939], [0.2000000000, 0.9974774793, 24.71701704], [0.2500000000, 0.9881100794, 22.38633401], [0.3000000000, 0.9707271284, 19.48891490], [0.3500000000, 0.9449917464, 15.88342402], [0.4000000000, 0.9161017354, 12.79566931], [0.4500000000, 0.8893201886, 10.13297800], [0.5000000000, 0.8598297573, 7.848977319], [0.5500000000, 0.8376086873, 6.642986647], [0.6000000000, 0.8134662812, 5.683793342], [0.6500000000, 0.7932753816, 5.115774909], [0.7000000000, 0.7768064901, 3.265577450], [0.7500000000, 0.7670031710, 2.646170597], [0.8000000000, 0.7589540432, 2.393115498], [0.8500000000, 0.7519227224, 2.130341314], [0.9000000000, 0.7439854534, 1.971746607], [0.9500000000, 0.7357905414, 1.738955838], [1., 0.7283531409, 1.622104551]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 520, dollars, then the Kelly Profile is [[0., 1.000000000, 1400.000000], [0.05000000000, 1.000000000, 49.10309423], [0.1000000000, 0.9999992218, 29.04001852], [0.1500000000, 0.9998793511, 23.50889770], [0.2000000000, 0.9979790346, 20.97911006], [0.2500000000, 0.9901909611, 18.88265125], [0.3000000000, 0.9760577820, 16.67546763], [0.3500000000, 0.9540393167, 13.55015957], [0.4000000000, 0.9298030532, 10.88676763], [0.4500000000, 0.9064307407, 8.981470922], [0.5000000000, 0.8790985707, 7.118243031], [0.5500000000, 0.8589829225, 5.058211384], [0.6000000000, 0.8437665250, 4.144832312], [0.6500000000, 0.8317775202, 3.517276851], [0.7000000000, 0.8205932890, 3.057053081], [0.7500000000, 0.8094577593, 2.676533954], [0.8000000000, 0.7994226237, 2.431256184], [0.8500000000, 0.7918463548, 1.980413927], [0.9000000000, 0.7865286949, 1.850000720], [0.9500000000, 0.7820194223, 1.713737750], [1., 0.7748047538, 1.591576572]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 560, dollars, then the Kelly Profile is [[0., 1.000000000, 1200.000000], [0.05000000000, 1.000000000, 40.84489604], [0.1000000000, 0.9999994373, 24.14611182], [0.1500000000, 0.9999072827, 19.41784179], [0.2000000000, 0.9983837973, 17.35893592], [0.2500000000, 0.9922209471, 15.76145788], [0.3000000000, 0.9806628450, 13.96496804], [0.3500000000, 0.9623525727, 11.65570410], [0.4000000000, 0.9409311470, 9.645700900], [0.4500000000, 0.9209436820, 7.085230074], [0.5000000000, 0.9014749219, 5.331548731], [0.5500000000, 0.8877193339, 4.539214892], [0.6000000000, 0.8743116143, 3.552698566], [0.6500000000, 0.8649645659, 3.131496184], [0.7000000000, 0.8556894632, 2.691555725], [0.7500000000, 0.8480817526, 2.440388023], [0.8000000000, 0.8410232889, 2.236347278], [0.8500000000, 0.8345259004, 2.090456921], [0.9000000000, 0.8258777589, 1.971990738], [0.9500000000, 0.8209096763, 1.763280676], [1., 0.8139219015, 1.665007830]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 600, dollars, then the Kelly Profile is [[0., 1.000000000, 1000.000000], [0.05000000000, 1.000000000, 33.02700796], [0.1000000000, 0.9999995977, 19.66114497], [0.1500000000, 0.9999297987, 16.13018161], [0.2000000000, 0.9987701737, 14.18830645], [0.2500000000, 0.9939219121, 13.01813956], [0.3000000000, 0.9844316596, 11.79965653], [0.3500000000, 0.9693965675, 9.073269755], [0.4000000000, 0.9534827938, 7.326680735], [0.4500000000, 0.9396231245, 5.807087613], [0.5000000000, 0.9248669936, 4.486992932], [0.5500000000, 0.9145418674, 3.885335631], [0.6000000000, 0.9033490563, 3.324245424], [0.6500000000, 0.8952602903, 2.885471860], [0.7000000000, 0.8873451425, 2.578068197], [0.7500000000, 0.8793449809, 2.356012123], [0.8000000000, 0.8719913255, 2.161717794], [0.8500000000, 0.8657593634, 2.082258769], [0.9000000000, 0.8585035712, 1.953076478], [0.9500000000, 0.8504442566, 1.863936227], [1., 0.8400000000, 1.285714286]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 640, dollars, then the Kelly Profile is [[0., 1.000000000, 800.0000000], [0.05000000000, 1.000000000, 25.77803528], [0.1000000000, 0.9999997236, 15.44887116], [0.1500000000, 0.9999496741, 12.32721805], [0.2000000000, 0.9990827738, 11.32084090], [0.2500000000, 0.9952440317, 9.890347117], [0.3000000000, 0.9882908514, 8.657024600], [0.3500000000, 0.9779966985, 7.138959211], [0.4000000000, 0.9664406942, 5.851672714], [0.4500000000, 0.9557280754, 4.771558935], [0.5000000000, 0.9439319029, 3.859860650], [0.5500000000, 0.9350434749, 3.380305718], [0.6000000000, 0.9253865125, 2.998548356], [0.6500000000, 0.9173101526, 2.769616648], [0.7000000000, 0.9107225960, 2.114157821], [0.7500000000, 0.9068012684, 1.895288222], [0.8000000000, 0.9035816173, 1.804029054], [0.8500000000, 0.9007690890, 1.711326381], [0.9000000000, 0.8975941814, 1.653725625], [0.9500000000, 0.8943162166, 1.572284046], [1., 0.8913412564, 1.530196459]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 680, dollars, then the Kelly Profile is [[0., 1.000000000, 600.0000000], [0.05000000000, 1.000000000, 19.09338695], [0.1000000000, 0.9999998202, 11.27341851], [0.1500000000, 0.9999655328, 9.491803361], [0.2000000000, 0.9993535189, 7.936836679], [0.2500000000, 0.9968883788, 7.275065091], [0.3000000000, 0.9922651380, 6.520671750], [0.3500000000, 0.9849410291, 5.555357731], [0.4000000000, 0.9763724588, 4.718228768], [0.4500000000, 0.9683774728, 3.695270411], [0.5000000000, 0.9605899688, 3.001377334], [0.5500000000, 0.9550877336, 2.687614103], [0.6000000000, 0.9497246457, 2.308238396], [0.6500000000, 0.9459858264, 2.145316626], [0.7000000000, 0.9422757853, 1.977690782], [0.7500000000, 0.9392327011, 1.881420994], [0.8000000000, 0.9364093156, 1.803417955], [0.8500000000, 0.9338103602, 1.747278256], [0.9000000000, 0.9303511036, 1.700218782], [0.9500000000, 0.9283638705, 1.623675356], [1., 0.9255687606, 1.585666413]] ----------------------------------------------------------------------------\ ----------- If the initial capital is, 720, dollars, then the Kelly Profile is [[0., 1.000000000, 400.0000000], [0.05000000000, 1.000000000, 12.92457232], [0.1000000000, 0.9999998943, 8.068952670], [0.1500000000, 0.9999798696, 5.930738325], [0.2000000000, 0.9996331095, 5.525843339], [0.2500000000, 0.9980976127, 4.944828167], [0.3000000000, 0.9953163406, 4.438367427], [0.3500000000, 0.9911986794, 3.817549573], [0.4000000000, 0.9865762777, 3.292897070], [0.4500000000, 0.9822912302, 2.857010507], [0.5000000000, 0.9775727612, 2.490813064], [0.5500000000, 0.9740173900, 2.298019045], [0.6000000000, 0.9701546050, 2.144071756], [0.6500000000, 0.9669240610, 2.051001861], [0.7000000000, 0.9642890384, 1.798686378], [0.7500000000, 0.9627205074, 1.714080463], [0.8000000000, 0.9614326469, 1.678191029], [0.8500000000, 0.9603076356, 1.642090034], [0.9000000000, 0.9590376725, 1.619109985], [0.9500000000, 0.9577264866, 1.587273773], [1., 0.9565365025, 1.570360767]] This took, 87.812, seconds of CPU time