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William Poundstone wrote an extensive popular account of the history of Kelly betting. The behavior of the test subjects was far from optimal:. If losing, the size of the next bet gets cut; if winning, the stake increases.

For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff oddsthe Kelly bet is:. If the gambler has zero edge, i. There is no explicit anti-red bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing. For even-money bets i. In this case, as is proved in the next section, the Kelly criterion turns out to be the relatively simple expression.

Thus, using too much margin is not a good investment strategy when the cost of capital is high, even when the opportunity appears promising. Heuristic proofs of the Kelly criterion are straightforward. This gives:. Some corrections have been published.

The resulting wealth will be:. After the same series of wins and losses as the Kelly bettor, they will have:. This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time otherwise one could cheat and, for example, bet zero after the K th win knowing that the rest of the bets will loseone will end up with the most money if one bets:.

The heuristic proof for the general case proceeds as follows. Edward O. Thorp provided a more detailed discussion of this formula for the general case.

In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. In a article, Daniel Bernoulli suggested that, when one has a choice of bets or investments, one should choose that with the highest geometric mean of outcomes.

This is mathematically equivalent to the Kelly criterion, although the motivation is entirely different Bernoulli wanted to resolve the St.

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Petersburg paradox. An English-language translation of the Bernoulli article was not published until[14] but the work was well-known among mathematicians and economists.

Suppose there are several mutually exclusive outcomes. The algorithm for the optimal set of outcomes consists of four steps. One may prove [15] that. The binary growth exponent is. In this case it must be that. In mathematical finance, a portfolio is called growth optimal if security weights maximize the expected geometric growth rate which is equivalent to maximizing log wealth. Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems.

Ex-post performance of a supposed growth optimal portfolio may differ fantastically with the ex-ante prediction if portfolio weights are largely driven by estimation error.

Dealing with parameter uncertainty and estimation error is a large topic in portfolio theory. The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be easily estimated from existing historical data — expected value and variance. Стратегия в баскетбол Овечкин.

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Kelly criterion

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