Adjustments to the Kelly Criterion for Correlation
September 27th, 2010 at 3:15:47 PM
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In sports betting and blackjack, the correct fraction of one's bankroll to wager is based on the probability of winning (p) and the odds of the bet (b). We all know mathematically this is:
Kelly fraction = ((b * p) - (1 - p)) / b
Now, my question is how to figure out a) the correlation of the concurrent wagers and b) how to adjust the Kelly fraction based on the correlation.
An obvious example is how much to bet when spreading two hands given a certain true count in blackjack. The example I'm more interested in is how to adjust the Kelly Criterion for Wong teasers.
Wong Example:
Four teams meet the criteria for Wonging (six point teaser that passes through both 3 and 7). The Teams are W, X, Y, and Z. That means we can make four 3-team teasers A = {WXY}, B = {WXZ}, C = {WYZ}, D = {XYZ} (we want to make the 3-teamers at +180 because they have the highest individual EV).
P(A) = P(B) = P(C) = P(D) = 43.22%
Ordinarily, for a 3-team teaser that pays +180, the Kelly fraction for each teaser would be 11.68%. However, since the bets are obviously correlated, I'm sure I shouldn't be wagering 11.68% of my bankroll on each of the four teaser combinations. I don't think this problem is linear, since it is dependent on correlation. These conditional probabilities may be useful, too.
P(A|B) = P(A|C) = P(A|D) = P(B|A), etc. = 75.6%
Finally, there are three possible outcomes to betting on A, B, C, and D. Either you can win all four teasers (p = 32.68%), win one and lose three teasers (p = 42.17%), or lose all four teasers (p = 25.15%). If you bet $100 on each teaser, the EV of the four wagers is $84.11
Thanks for any help that could be provided. Cheers.
Kelly fraction = ((b * p) - (1 - p)) / b
Now, my question is how to figure out a) the correlation of the concurrent wagers and b) how to adjust the Kelly fraction based on the correlation.
An obvious example is how much to bet when spreading two hands given a certain true count in blackjack. The example I'm more interested in is how to adjust the Kelly Criterion for Wong teasers.
Wong Example:
Four teams meet the criteria for Wonging (six point teaser that passes through both 3 and 7). The Teams are W, X, Y, and Z. That means we can make four 3-team teasers A = {WXY}, B = {WXZ}, C = {WYZ}, D = {XYZ} (we want to make the 3-teamers at +180 because they have the highest individual EV).
P(A) = P(B) = P(C) = P(D) = 43.22%
Ordinarily, for a 3-team teaser that pays +180, the Kelly fraction for each teaser would be 11.68%. However, since the bets are obviously correlated, I'm sure I shouldn't be wagering 11.68% of my bankroll on each of the four teaser combinations. I don't think this problem is linear, since it is dependent on correlation. These conditional probabilities may be useful, too.
P(A|B) = P(A|C) = P(A|D) = P(B|A), etc. = 75.6%
Finally, there are three possible outcomes to betting on A, B, C, and D. Either you can win all four teasers (p = 32.68%), win one and lose three teasers (p = 42.17%), or lose all four teasers (p = 25.15%). If you bet $100 on each teaser, the EV of the four wagers is $84.11
Thanks for any help that could be provided. Cheers.
September 27th, 2010 at 3:26:13 PM
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I am admittedly out of my depth here, but I believe that regarding multiple bets with interdependent outcomes, the Kelly calculation should be done with those wagers as a single aggregate bet.
I defer to the Wizard here, as he can give you a much more cogent and mathy answer than I can :)
I defer to the Wizard here, as he can give you a much more cogent and mathy answer than I can :)
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
