Benfords Law Favorites and Exotic Bets

In various horse-racing jurisdictions (e.g. Australia, UK, Ireland, France), there is a very strong correlation between the winning rates of favorites and Benford's Law. In other words, favorites win approximately 30% of races, second favorites approximately 18% and third favorites approximately 12%. One could conceivably use this information to generate some tickets for Daily Double, Pick 3, 4, 5, or 6 exotic pools by using a random number generator and a "Benford distribution" of win rates. Though unscientific in validation, this method proved invaluable to me over the Breeders Cup weekend (given many upsets to expected outcomes)!

Information Calibration And Confidence

In 1979 [Studies in Intelligence, Vol. 23, No. 1 (Spring 1979)], a study of expert handicappers demonstrated an interesting interaction between information and confidence. There were two key findings. First, as soon as an experienced handicapper has the minimum information (seven plus or minus two variables) necessary to make an informed judgment, obtaining additional information generally does not improve the accuracy of his selections. Second, additional information does, however, lead the handicapper to become more confident in his judgments, to the point of overconfidence. It appears that handicappers have an imperfect understanding of what information they actually use in making judgments. They are unaware of the extent to which their judgments are determined by a few dominant factors, rather than by the systematic integration of all available information.
As ever, if the handicapper cannot find variables that account for sufficient variance in outcomes over and above that provided by market prices then he will not have an edge and will lose his bankroll.

Betfair Pari-Mutuel Equivalence

For those handicappers fortunate enough to have access to both Betfair and Pari-Mutuel markets for the same events and who wish to arbitrage their positions for a "no loss" outcome, they should add the following formulae to their toolset:

• o = 1 - (d * 1/(x - 1)) and
• d = -((o - 1) * (x - 1))

where o = betfair decimal odds, d = pari-mutuel dollar payoff, and x = betfair tax (combination of commission and discount). These prices are equivalent in terms of expectation and volatility..

Betfair InPlay Hedge Stake

Trading an event in-play on Betfair is not for the feint of heart as, ultimately, no position is safe until it is successfully hedged. Psychologically, however, if you have carried out a fundamental analysis of the event then you want to be paid a premium for that analysis should your selection prove to be successful. On the other hand, Cumulative Prospect Theory confirms that we hate losing (loss aversion >= 2.25) more than we enjoy winning. In order to balance those conflicting forces, you could calculate a hedge stake to green-up your position, as follows:

  z = (s*(o+m-1))/(h+m-1)
  where  z = hedge stake
        s = original stake
        o = original price (back)
        m = win multiple (ratio of win payout to loss payout, if greened up)
        h = hedge price (lay)

For example, if I back a selection for $100 @ 6.00 and wish to green-up at 2.00 then the default option is to lay $300 @ 2.00 for a guaranteed $190 whatever the result of the event. By contrast, the above calculation (e.g., m = 2.25), gives a stake of $223.08 with a win payout of $264.74 and a loss payout of $117.66 giving you a win premium!

Trailing Low Threshold (Max Drawdown)

Given Loss Aversion, the painful effect of an unexpected loss is at least twice the joyful effect of an unexpected gain. One of the most frustrating scenarios in sports trading is moving into the black early in the day only to finish it in the red. Sound familiar? Psychologically, we reset the baseline to zero each day even though intellectually we may be focused on generating an annual income. Because we are loss averse it is not possible to simply ignore these daily downswings as it impacts our overall confidence level. I would recommend (even to those handicappers who do not accept session handicapping) setting a trailing low threshold. For example (specific numbers used for illustrative purposes only):

• Day Bankroll: $1000
• Max Drawdown: 20%
• Low Threshold: $800 = (80% * $1000)
• Day High: $1350
• Trailing Low Threshold: $1080 = (80% * $1350)

In other words, even though you set an initial low threshold of $800, should you go into profit on the day ($1350) the low threshold is increased to maintain the 20% drawdown from the day high. This allows sufficient flexibility to continue trading without suffering the negative emotional impact of losing all your profit on the day.

Betfair In-Play Trading (Minimax Regret)

Opportunity Loss (Regret) plays havoc with the emotions of In-Play Traders. One psychologically valid approach is to use Minimax Regret. For example, in a horse-race, assume your selection (AtTheWire) is on offer to back at 3.50 (Win Market) and to lay at 1.80 (TBP Market) and your calculation of edge dictates a stake of 100. In the Win Market, at what price and with what stake should you trade out In-Play to minimize regret? As the above table shows, trading out at less than or equal to 1.80 for 100 is the optimal choice! Note that backing your selection in the Win Market is equivalent to stating that, at a minimum, you expect your selection to contest the finish. Marked-to-Market (TBP Market), your selection is on offer pre-race at 1.80 to contest the finish and this price represents your best exit point In-Play (Win Market).

Discounted Harville v1.17 (VBA Functions - Excel 2007+)

Discounted Harville spreadsheet (Discounted_Harville v1.17) with VBA Functions for Excel 2007+ (DHExactaOdds, DHTrifectaOdds, and experimental DHSuperfectaOdds). Change the values of lambda, and rho to approximate the Henery (lambda = 0.76, rho = 0.62) and/or Stern models (See Donald B. Hausch, Victor S. Y. Lo, and William T. Ziemba, Efficiency Of Racetrack Betting Markets, London, Academic Press, 1994, pp. 478).