Kelly criterion (or Kelly strategy or Kelly bet) is a formula for bet sizing that leads almost surely to higher wealth. It was described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive. In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. 18 of the participants bet everything on one toss, while two-thirds gambled on tails. The average payout was just $91.36.

Kelly bet: Bet one-nineteenth of the bankroll that red will not come up. If gamble has 60% chance of winning, gambler should bet 20% of bankroll at each opportunity. In American roulette, the bettor is offered an even money payoff (even money payoff, with a 95% probability of reaching the cap) If edge is negative, then the formula gives a negative result, indicating that the gambler shouldn't 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 with probability or lose the $1 wagered, i.e. win $ + $1, with probability $+.1. For even-money bets, the formula can be simplified to the first simplified formula: b=p-q-q/p, b=1, and b=0.20.

Probability of success is: If you succeed, the value of your investment increases. If you fail (for which the probability is): In that case, the price of the investment decreases. Kelly criterion maximizes the expected value of the logarithm of wealth. Using too much margin is not a good investment strategy when the cost of capital is high, even when the opportunity appears promising. The general result clarifies why leveraging (taking out a loan that requires paying interest in order to raise investment capital) decreases the optimal fraction to be invested, as in that case the odds of winning are less than 0.1. The expected profit must exceed the expected loss for the investment to make any sense. The Kelly criterion can be used to determine the optimal amount of money to invest.

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 Kth win, that the rest of the bets will lose), one will end up with the most money if one bets. In the long run, final wealth is maximized by setting the derivative to zero, which means following the Kelly strategy. The function is maximised when this derivative is equal to zero. For a rigorous and general proof, see Kelly's original paper[1] or some of the other references listed below. Some corrections have been published[12] .

In the long run, Kelly always wins. This is true whether N is small or large. 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. Betting each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and the odds of winning are the same for each trial. The heuristic proof for the general case proceeds as follows. In a single trial, if you invest the fraction f of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor 1-f+f(1+b) Exponential mechanism (differential privacy)