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All Forums > WHATS ON YOUR MIND > SPORTS SHIT > A quantitative introduction to the Kelly criterion
A quantitative introduction to the Kelly criterion
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bagz

Joined: Apr 30, 2006
Posts: 11443

bagz

Joined: Apr 30, 2006
Posts: 11443

bagz

Joined: Apr 30, 2006
Posts: 11443

 Posted: Sat Feb 15, 2014 5:43 pm    Post subject: Re: A quantitative introduction to the Kelly criterion The difference between expected value and growth is the same as that between an arithmetic and a geometric mean. You can think of the expected value of a bet as the arithmetic mean of all outcomes were you to repeat the same dollar-value bet an infinite number of times. Expected growth, on the other hand, corresponds to the geometric mean outcome you'd obtain were you to repeat the same percentage-of-bankroll bet an infinite number of times. This is a subtle but extremely importance difference. The best way to see the difference is by considering a bet of 100% of one's bankroll. The win probability and payout odds are irrelevant to the discussion, just long as they're less than 100% and infinity respectively. For the sake of this discussion we'll assume the win probability is 99% and the bet is made at +100 (decimal: 2.0000). This bet has an expected value of 2*99% - 1 * 100% = 98% of bankroll, and corresponds to expected growth of (1+(2-1)*100%)99% * (1-100%)1% = -100% (this latter figure implies that as your number of sequential bets increases, your probability of going bankrupt approaches certainty). We'll assume the starting bankroll is \$100. After 1 bet, there's a 99% probability of winning and ending up with a bankroll of \$200, and a 1% probability of ending up with a bankroll of zero (in which case betting would stop as the player would have no more money with which to play). After 10 bets, there's a 1-99% ≈ 90.4% probability of ending up with a bankroll of 210* \$100 = \$102,400 and a roughly 9.6% probability of ending up bankrupt. After 1,000 bets, there's a 1-99%1,000 ≈ 0.00431712% probability of ending up with a bankroll of 21,000* \$100 ≈ \$1.07151 × 10303 and a 99%1,000 ≈ 99.995683% probability of ending up bankrupt. What we see is that the expected return per bet is always constant at 98%. However, the expected average growth rate per bet is -100%. Another way to think of expected growth after a large number of bets is that it represents the single most likely outcome. Expected value, on the other hand, represents the average outcome regardless of its relative likelihood. _________________“We make a living by what we get. We make a life by what we give.”
bagz

Joined: Apr 30, 2006
Posts: 11443

 Posted: Sat Feb 15, 2014 5:46 pm    Post subject: Re: A quantitative introduction to the Kelly criterion Great set of posts here, but a quick question: how do you take into account the probability of a push for calculating the optimal bet amount? As it stands now the formula simply assumes no win is a loss, but when dealing with point spreads without a half point this isn't always the case. In the case of a single bet it's very straightforward. Every time win probability is used it's simply replaced with the probability of winning conditioned on not pushing. In general, given a win probability of PW, a loss probability of PL, and a push probability of PT (where PW + PL + PT = 1), then the probability of winning conditioned on not losing would be: P*W = PW / (1 - PT) and the probability of losing conditioned on not pushing would be: P*L = PL / (1 - PT) So assuming decimal odds of O, Edge would be: Edge = O × PW / (1 - PT) - 1 -or- Edge = O × PW - (1 - PT) which in either case is just the same as: Edge = O × P*W - 1 And the Kelly stake would remain unchanged as: Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0 _________________“We make a living by what we get. We make a life by what we give.”
bagz

Joined: Apr 30, 2006
Posts: 11443

 Posted: Sat Feb 15, 2014 5:47 pm    Post subject: Re: A quantitative introduction to the Kelly criterion if your really into it i suggest reading this... www.amazon.com/exec/ob...s&n=507846 _________________“We make a living by what we get. We make a life by what we give.”
bagz

Joined: Apr 30, 2006
Posts: 11443

 Posted: Sat Feb 15, 2014 5:50 pm    Post subject: Re: A quantitative introduction to the Kelly criterion I have thought about this and came up with the following formulation, for 3 simultaneous bets (that are mutually exclusive) E = (1 + W1*F1)^(P1) * (1 + W2*F2)^(P2) * (1 + W3*F3)^(P3) * (1 - (1-F1-F2-F3))^(1-P1-P2-P3) - 1 W1 = odds-1 on bet 1, P1 = probability that bet 1 wins, F1 the stake on this bet in % of total BR. W2 = odds-1 on bet 2, P2 = probability that bet 2 wins, F2 the stake on this bet in % of total BR. W3 = odds-1 on bet 3, P3 = probability that bet 3 wins, F3 the stake on this bet in % of total BR. Maximize E (geometric growth) subject to 0<=F1<=1, 0<=F2<=1, 0<=F3<=1, 0<=F1+F2+F3<=1. (W and P constant of course) (easy to generalise to 1500 simultaneous bets) This is unfortunately a rather complex optimization problem to solve, especially as the number of bets grow large. Are there a simpler way of expressing the problem that I am missing? I suspect this formulation with 1500 variables will be impossible to solve in practice, definitely with excel but also with an optimzation program such as Cplex. Any approximation that errs on the conservative side would be useful too. _________________“We make a living by what we get. We make a life by what we give.”
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