How many times should a risk-averse gambler inisist on playing with Probabi\ lity table [[-1,(i-1)/i],[i,1/i]] for i from 2 to 10 with risk-aversenes\ s 10^(-j) for j from 1 to , 4 By Shalosh B. Ekhad The above values are based on the Central Limit theorem approximations, and \ are slightlt higher than necessary for the stated risk-averseness ------------------------------------ For the St. Petersburg finite version with, 3, rounds and entrance fee, 2, so the expected gain of one shot is 1, the probability table is [[0, 1/2], [2, 1/4], [2, 1/4]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 2, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 6, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 10, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 14, times ------------------------------------ For the St. Petersburg finite version with, 4, rounds and entrance fee, 3, so the expected gain of one shot is 1, the probability table is [[-1, 1/2], [1, 1/4], [5, 1/8], [5, 1/8]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 10, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 33, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 58, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 83, times ------------------------------------ For the St. Petersburg finite version with, 5, rounds and entrance fee, 4, so the expected gain of one shot is 1, the probability table is [[-2, 1/2], [0, 1/4], [4, 1/8], [12, 1/16], [12, 1/16]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 35, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 114, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 201, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 291, times ------------------------------------ For the St. Petersburg finite version with, 6, rounds and entrance fee, 5, so the expected gain of one shot is 1, the probability table is [[-3, 1/2], [-1, 1/4], [3, 1/8], [11, 1/16], [27, 1/32], [27, 1/32]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 96, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 314, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 554, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 803, times ------------------------------------ For the St. Petersburg finite version with, 7, rounds and entrance fee, 6, so the expected gain of one shot is 1, the probability table is [[-4, 1/2], [-2, 1/4], [2, 1/8], [10, 1/16], [26, 1/32], [58, 1/64], [58, 1/64] ] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 232, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 764, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 1347, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 1951, times ------------------------------------ For the St. Petersburg finite version with, 8, rounds and entrance fee, 7, so the expected gain of one shot is 1, the probability table is [[-5, 1/2], [-3, 1/4], [1, 1/8], [9, 1/16], [25, 1/32], [57, 1/64], [121, 1/128 ], [121, 1/128]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 523, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 1721, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 3037, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 4399, times ------------------------------------ For the St. Petersburg finite version with, 9, rounds and entrance fee, 8, so the expected gain of one shot is 1, the probability table is [[-6, 1/2], [-4, 1/4], [0, 1/8], [8, 1/16], [24, 1/32], [56, 1/64], [120, 1/128 ], [248, 1/256], [248, 1/256]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 1126, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 3708, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 6542, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 9475, times ------------------------------------ For the St. Petersburg finite version with, 10, rounds and entrance fee, 9, so the expected gain of one shot is 1, the probability table is [[-7, 1/2], [-5, 1/4], [-1, 1/8], [7, 1/16], [23, 1/32], [55, 1/64], [119, 1/ 128], [247, 1/256], [503, 1/512], [503, 1/512]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 2356, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 7761, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 13695, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 19834, times ------------------------------------ For the St. Petersburg finite version with, 11, rounds and entrance fee, 10, so the expected gain of one shot is 1, the probability table is [[-8, 1/2], [-6, 1/4], [-2, 1/8], [6, 1/16], [22, 1/32], [54, 1/64], [118, 1/ 128], [246, 1/256], [502, 1/512], [1014, 1/1024], [1014, 1/1024]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 4844, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 15960, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 28162, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 40788, times ------------------------------------ For the St. Petersburg finite version with, 12, rounds and entrance fee, 11, so the expected gain of one shot is 1, the probability table is [[-9, 1/2], [-7, 1/4], [-3, 1/8], [5, 1/16], [21, 1/32], [53, 1/64], [117, 1/ 128], [245, 1/256], [501, 1/512], [1013, 1/1024], [2037, 1/2048], [2037, 1/2048 ]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 9851, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 32461, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 57279, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 82959, times ------------------------------------ For the St. Petersburg finite version with, 13, rounds and entrance fee, 12, so the expected gain of one shot is 1, the probability table is [[-10, 1/2], [-8, 1/4], [-4, 1/8], [4, 1/16], [20, 1/32], [52, 1/64], [116, 1/ 128], [244, 1/256], [500, 1/512], [1012, 1/1024], [2036, 1/2048], [4084, 1/4096 ], [4084, 1/4096]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 19901, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 65576, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 115712, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 167592, times ------------------------------------ For the St. Petersburg finite version with, 14, rounds and entrance fee, 13, so the expected gain of one shot is 1, the probability table is [[-11, 1/2], [-9, 1/4], [-5, 1/8], [3, 1/16], [19, 1/32], [51, 1/64], [115, 1/ 128], [243, 1/256], [499, 1/512], [1011, 1/1024], [2035, 1/2048], [4083, 1/4096 ], [8179, 1/8192], [8179, 1/8192]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 40038, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 131932, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 232799, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 337175, times ------------------------------------ For the St. Petersburg finite version with, 15, rounds and entrance fee, 14, so the expected gain of one shot is 1, the probability table is [[-12, 1/2], [-10, 1/4], [-6, 1/8], [2, 1/16], [18, 1/32], [50, 1/64], [114, 1/ 128], [242, 1/256], [498, 1/512], [1010, 1/1024], [2034, 1/2048], [4082, 1/4096 ], [8178, 1/8192], [16370, 1/16384], [16370, 1/16384]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 80354, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 264777, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 467212, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 676686, times ------------------------------------ For the St. Petersburg finite version with, 16, rounds and entrance fee, 15, so the expected gain of one shot is 1, the probability table is [[-13, 1/2], [-11, 1/4], [-7, 1/8], [1, 1/16], [17, 1/32], [49, 1/64], [113, 1/ 128], [241, 1/256], [497, 1/512], [1009, 1/1024], [2033, 1/2048], [4081, 1/4096 ], [8177, 1/8192], [16369, 1/16384], [32753, 1/32768], [32753, 1/32768]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 161029, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 530615, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 936294, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 1356083, times ------------------------------------ For the St. Petersburg finite version with, 17, rounds and entrance fee, 16, so the expected gain of one shot is 1, the probability table is [[-14, 1/2], [-12, 1/4], [-8, 1/8], [0, 1/16], [16, 1/32], [48, 1/64], [112, 1/ 128], [240, 1/256], [496, 1/512], [1008, 1/1024], [2032, 1/2048], [4080, 1/4096 ], [8176, 1/8192], [16368, 1/16384], [32752, 1/32768], [65520, 1/65536], [65520 , 1/65536]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 322427, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 1062447, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 1874737, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 2715278, times ------------------------------------ For the St. Petersburg finite version with, 18, rounds and entrance fee, 17, so the expected gain of one shot is 1, the probability table is [[-15, 1/2], [-13, 1/4], [-9, 1/8], [-1, 1/16], [15, 1/32], [47, 1/64], [111, 1 /128], [239, 1/256], [495, 1/512], [1007, 1/1024], [2031, 1/2048], [4079, 1/ 4096], [8175, 1/8192], [16367, 1/16384], [32751, 1/32768], [65519, 1/65536], [ 131055, 1/131072], [131055, 1/131072]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 645273, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 2126280, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 3751918, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 5434096, times ------------------------------------ For the St. Petersburg finite version with, 19, rounds and entrance fee, 18, so the expected gain of one shot is 1, the probability table is [[-16, 1/2], [-14, 1/4], [-10, 1/8], [-2, 1/16], [14, 1/32], [46, 1/64], [110, 1/128], [238, 1/256], [494, 1/512], [1006, 1/1024], [2030, 1/2048], [4078, 1/ 4096], [8174, 1/8192], [16366, 1/16384], [32750, 1/32768], [65518, 1/65536], [ 131054, 1/131072], [262126, 1/262144], [262126, 1/262144]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 1291020, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 4254123, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 7506595, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 10872188, times ------------------------------------ For the St. Petersburg finite version with, 20, rounds and entrance fee, 19, so the expected gain of one shot is 1, the probability table is [[-17, 1/2], [-15, 1/4], [-11, 1/8], [-3, 1/16], [13, 1/32], [45, 1/64], [109, 1/128], [237, 1/256], [493, 1/512], [1005, 1/1024], [2029, 1/2048], [4077, 1/ 4096], [8173, 1/8192], [16365, 1/16384], [32749, 1/32768], [65517, 1/65536], [ 131053, 1/131072], [262125, 1/262144], [524269, 1/524288], [524269, 1/524288]] If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 2582572, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 8509999, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 15016283, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 21748856, times -------------------------------------- This ends this paper that took, 14.573, seconds to produce. ----------------------------- This took, 14.708, seconds.