The fraction that maximises long-run growth is exactly the edge, 2p-1, which is 0.2 on this coin, and one derivative gets you there. Double it and the growth rate is -0.0024469 a flip, negative on a game that leans your way three hundred times in a row, and the crossing happens at 0.3894 rather than at 0.4. The article carries the exact median over 300 flips, 25 dollars to 10504.19 at the optimum and to 12.00 at double, the reason about 48 percent of overbettors still finish ahead anyway, and the place where the textbook approximation mean minus half the variance returns the opposite sign.
También se pregunta comokelly criterion · kelly formula · kelly bet sizing · optimal bet size · what fraction to bet on a biased coin · biased coin betting game
Valor esperado y market making →
Both seats in the marble game average a dollar a play, and that arithmetic stays true to the last line. Seat A carries variance 3/2 against seat B's 1, and seat A's law turns out to be seat B's law with one prize smeared outward, so every concave utility prefers B without variance ever being mentioned. Once both players stop flipping coins, seat B is ahead on the average too, at 1 against 3/4.
Acceptance restricts the value to below your bid, where a uniform variable averages half of it, and doubling half your bid returns exactly your bid. The expected profit is therefore identically zero at every bid up to 100 and 100 minus b above it, so there is no optimal bid to find. With a general multiplier the profit is b squared times k minus 2, over 200, making doubling the exact break-even multiple, and the article shows a value distribution starting at 50 where the same bidder profits.
Four settlements in five come back below the 1.50 outlay, and the average payoff is still 1.80, an edge of 0.30 a contract or twenty percent of the money at risk. The reflex is not bad arithmetic, it is the mode standing in for the mean. The article carries the tally over one full cycle, the threshold saying you need the large outcome more often than one time in eight, and the reason waiting longer can leave you less likely to be ahead.