Thursday, January 21, 2010
Gamblers Ruin at Casino Incroyable
A nice mixture of mathematics and humor - Gamblers Ruin at Casino Incroyable with passing reference to "Bold Play", "Minimum Gain", and "Gamblers Ruin".
Saturday, January 09, 2010
ZEER: Zero Evens-Equivalent Rate
When comparing handicapper performances across different circuits, it is better to zero the tax rate at evens using ZEER:
ZEER = (StrikeRate * (Odds + Tax - (StrikeRate * Tax))) / 2
Edge = ZEER - (1 - ZEER)
For example, comparing handicappers (see above) at Aqueduct, Gulfstream, and Santa Anita gives ZEERs of 52.94%, 54.04%, and 53.00% respectively.
This yields equivalent edges for the three handicappers of 5.89%, 8.08%, and 6.00%, identifying the Gulfstream handicapper as the most successful.
In summary, ZEER allows a group of handicappers to compare their performances on a common scale (evens at zero tax level)!
ZEER = (StrikeRate * (Odds + Tax - (StrikeRate * Tax))) / 2
Edge = ZEER - (1 - ZEER)
For example, comparing handicappers (see above) at Aqueduct, Gulfstream, and Santa Anita gives ZEERs of 52.94%, 54.04%, and 53.00% respectively.
This yields equivalent edges for the three handicappers of 5.89%, 8.08%, and 6.00%, identifying the Gulfstream handicapper as the most successful.
In summary, ZEER allows a group of handicappers to compare their performances on a common scale (evens at zero tax level)!
Friday, January 01, 2010
Kelly Horse Race or Ziemba Roulette
In the spirit of the New Year, I offer the following thought-provoking argument:
Betting is often described as a competition between you and the "Crowd" (i.e. Pari-Mutuel market) on which you can better estimate the true distribution of odds in a particular sporting event. In that context, a good starting point is to ask Bill Benter's fundamental question of handicapping: what additional variables (if any) explain a significant proportion of the variance in results to date that is not already accounted for by the public odds (Wisdom of Crowds)? By keeping records, it is possible to determine whether or not you have been successful over time in so doing. However, it is not possible to know (in advance) if you have an overlay in an upcoming event. This is the fundamental flaw of handicapping and, eo ipso, the flaw of fundamental handicapping.
The alternative approach is technical trading and an often underestimated strategy is "Bold Play" and its variants. Bold play is recommended for subfair games (i.e. p < 0.5, assuming even money bet) and, given the high-level of taxation, one can argue that horse-racing qualifies. Without going into the mathematical details and assuming a little "poetic licence", bold play reduces to an algorithm comprising two rules:
__a) Bet amount to reach target bankroll in single event (e.g. one race); or
__b) Bet amount in current event to reach bankroll level from which it is possible to attain target bankroll in next event.
You will no doubt immediately recognize that it is possible to iterate rule b) over many events (e.g. complete race card) but, ideally, you want to minimize the number of iterations. This technical trading approach only requires information on current bankroll, target bankroll, number of remaining events, house limits, and access to the public odds.
In summary, the advice to use fundamental handicapping to find overlays which you then exploit using edge-based staking (Kelly, 1956) assumes a superfair game but if, in fact, the game is essentially subfair then technical trading is recommended using bold play (Ziemba, 2002) or one of its variants in as few iterations as is feasible given whatever constraints are in place (e.g. maximum stakes).
Betting is often described as a competition between you and the "Crowd" (i.e. Pari-Mutuel market) on which you can better estimate the true distribution of odds in a particular sporting event. In that context, a good starting point is to ask Bill Benter's fundamental question of handicapping: what additional variables (if any) explain a significant proportion of the variance in results to date that is not already accounted for by the public odds (Wisdom of Crowds)? By keeping records, it is possible to determine whether or not you have been successful over time in so doing. However, it is not possible to know (in advance) if you have an overlay in an upcoming event. This is the fundamental flaw of handicapping and, eo ipso, the flaw of fundamental handicapping.
The alternative approach is technical trading and an often underestimated strategy is "Bold Play" and its variants. Bold play is recommended for subfair games (i.e. p < 0.5, assuming even money bet) and, given the high-level of taxation, one can argue that horse-racing qualifies. Without going into the mathematical details and assuming a little "poetic licence", bold play reduces to an algorithm comprising two rules:
__a) Bet amount to reach target bankroll in single event (e.g. one race); or
__b) Bet amount in current event to reach bankroll level from which it is possible to attain target bankroll in next event.
You will no doubt immediately recognize that it is possible to iterate rule b) over many events (e.g. complete race card) but, ideally, you want to minimize the number of iterations. This technical trading approach only requires information on current bankroll, target bankroll, number of remaining events, house limits, and access to the public odds.
In summary, the advice to use fundamental handicapping to find overlays which you then exploit using edge-based staking (Kelly, 1956) assumes a superfair game but if, in fact, the game is essentially subfair then technical trading is recommended using bold play (Ziemba, 2002) or one of its variants in as few iterations as is feasible given whatever constraints are in place (e.g. maximum stakes).
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