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In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run (that is, over a span of time in which the observed fraction of bets that are successful equals the probability that any given bet will be successful). It was described by J. L. Kelly, Jr in 1956. The practical use of the formula has been demonstrated. Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times〔). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations. In recent years, Kelly has become a part of mainstream investment theory and the claim has been made that well-known successful investors including Warren Buffett and Bill Gross use Kelly methods. William Poundstone wrote an extensive popular account of the history of Kelly betting.〔 == Statement == 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 odds, the Kelly bet is: : where: * ''f'' * is the fraction of the current bankroll to wager, i.e. how much to bet; * ''b'' is the net odds received on the wager ("''b'' to 1"); that is, you could win $b (on top of getting back your $1 wagered) for a $1 bet * ''p'' is the probability of winning; * ''q'' is the probability of losing, which is 1 − ''p''. As an example, if a gamble has a 60% chance of winning (''p'' = 0.60, ''q'' = 0.40), and the gambler receives 1-to-1 odds on a winning bet (''b'' = 1), then the gambler should bet 20% of his bankroll at each opportunity (''f'' * = 0.20), in order to maximize the long-run growth rate of the bankroll. If the gambler has zero edge, i.e. if ''b'' = ''q'' / ''p'', then the criterion recommends the gambler bets nothing. If the edge is negative (''b'' < ''q'' / ''p'') the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in standard American roulette, the bettor is offered an even money payoff (b = 1) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (p = 18/38). The Kelly bet is -1/19, meaning the gambler should bet one-nineteenth of his bankroll that red will ''not'' come up. Unfortunately, the casino doesn't allow betting ''against'' something coming up, so a Kelly gambler cannot place a bet. The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win $''b'' with probability ''p'', or lose the $1 wagered, i.e. win $-1, with probability ''q''. Hence: : For even-money bets (i.e. when ''b'' = 1), the first formula can be simplified to: : Since q = 1-p, this simplifies further to : A more general problem relevant for investment decisions is the following: 1. The probability of success is . 2. If you succeed, the value of your investment increases from to . 3. If you fail (for which the probability is ) the value of your investment decreases from to . (Note that the previous description above assumes that a is 1). In this case, the Kelly criterion turns out to be the relatively simple expression : Note that this reduces to the original expression for the special case above () for . Clearly, in order to decide in favor of investing at least a small amount , you must have : which obviously is nothing more than the fact that your expected profit must exceed the expected loss for the investment to make any sense. The general result clarifies why leveraging (taking a loan to invest) decreases the optimal fraction to be invested, as in that case . Obviously, no matter how large the probability of success, , is, if is sufficiently large, the optimal fraction to invest is zero. Thus, using too much margin is not a good investment strategy, no matter how good an investor you are. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kelly criterion」の詳細全文を読む スポンサード リンク
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