What is the most you would be willing to pay to play the St Petersburg Paradox if you were focused on EG without regard to utility? Or better phrased, at what price does the game become EG=0?
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Let f = fraction of bankroll initially in pot
Let C = cost of entering game as fraction of initial bankroll
Then for EG to from the game to be zero:1/2 * log(1 + f - C) + 1/4 * log(1 + 2*f - C) + 1/8 * log(1 + 4*f - C) + ... = 0So assuming my Excel calculations correct, the following chart shows, for several given values of f, the value of C that would imply 0 EG (i.e., indifference between playing and not playing for a full-Kelly bettor):
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[nbtable][tr][td][/td][td] [ 2^-i * log(1 + f * 2^(i-1) - C) ] = 0[/td][/tr][/nbtable]
One will note that for values of f > 50%, C > 1. This implies that given a sufficiently large base payout, the player would optimally borrow in order to play the game. This isn't a problem for Kelly as the worst possible outcome for the player is (1 + f - C).Code:f C -------------- 1% 4.36% 5% 16.53% 10% 28.84% 25% 59.06% 33[size=1][frac=1]3[/frac][/size]% 73.61% 50% 100.00% 66[size=1][frac=2]3[/frac][/size]% 123.95% 75% 135.28% 100% 167.38%
Hence, in the face of zero borrowing costs, the usual constraint of C < 1 may be relaxed in favor of the less restrictive C < 1 + f.Comment