Beat the Racetrack by William T. Ziemba and Donald B. Hausch offers a system to arbitrage interrelated betting pools. The Dr. Z system involves betting in the less efficient place and show pools using prices from the more accurate win pool. Place and show probabilities are determined using Harville (1973) formulas. The full-blown version of their system involves adjustments for coupled entries, pool size, extreme favorites and multiple wagers in a race. The Kelly Criterion, maximizing log wealth by betting a fraction of your bankroll equal to edge/odds, determines bet size.

The basic equations for the expected return on place and show wagers are

E(RET_Place) = 0.319 + 0.559(winshare/placeshare)+(2.22-1.29(winshare))(1-take-0.829)

E(RET_Show) = 0.543 + 0.369(winshare/showshare)+(3.60-2.13(winshare))(1-take-0.829)

where winshare (placeshare, showshare) equals the percent of the win (place, show) pool bet on a particular horse and take is the tracks takeout rate.

A simple version of the system is to bet if the expected return is greater than 1.15 and the odds are less than 8-1. With online betting, the Dr. Z system is easy to do and worth trying. Just cut and paste the tote data into an excel spreadsheet embedded with the needed calculations. One thing that you will immediately notice is how much money comes in at the last possible moment. Almost 40% of pool totals are posted after the race has begun and, unfortunately for the arbitrageur, this often dramatically alters the expected returns on place and show wagers. When betting opens there appear to be quite a few exploitable inefficiencies but as post time approaches these disappear.

To test the Dr. Z system I ran three betting simulations. In each simulation, 500 races with arbitrage opportunities are randomly drawn from the full sample (two different data sets discussed below). The random ordering of the races allows for multiple runs that can be used to estimate an average growth equation for the bankroll. There is no reason to believe that time or order would be a factor in outcome. In each race, an imaginary wager is placed based upon the criterion set forth in Beat the Racetrack. The amount wagered is adjusted for the size of the bankroll, betting pool, and track takeout, as well as for coupled entries, multiple bets and heavy favorites. Bets are not made when there is a minus pool. The initial bankroll is set at $2500 and after each race the bankroll changes based upon the size of the wager and payoff. While bets are fictitious, the amount that would have been bet is added to the final pool total so as to alter payouts as if the bet was actually made. Each trial ends after the 500^th race.

The following three graphs represent simulations of the Dr. Z system under three different scenarios.

Simulation 1 involves imaginary bets randomly drawn from 11,361 races at 36 tracks from the Fall of 2002. The bets are made based on full information with 100% of pool totals reported (I only had final pool totals). Of course this is unrealistic, but if this isn’t profitable, there would be no reason to investigate the system in real time. The results are promising. Through the Dr. Z system, an initial bankroll of $2,500 would grow on average to $7,003 after 500 wagers. The best replication increases the bankroll to $21,516. Positive gains are seen in 87.2% of the replications and in only 1.3% does the bankroll fall to zero. The bankroll grows at an average rate of 0.223% per wager.

 

Simulation 1

Simulation 2 and 3 use 1,664 races from 64 racetracks in the winter of 2003 and the winter and spring of 2005. I have both final pool totals and post time pool totals (the amount revealed by the tote at the last possible moment of betting). This allows for a "real-time" test of the Dr. Z system.

Simulation 2 makes bets based on post time calculations and also pays off based on the pool totals at post time. It is the “what I would have gotten had all that late money not come in” scenario. Once again things look quite good.  The average bankroll after the 1000 trials is $9,420, or almost four times the original. The best trial ends with a bankroll of $20,637. Only 0.3% of the trials experience a loss with 0.2% losing the entire bankroll. The bankroll grows at an average rate of 0.312% per wager.

Simulation 2


 

So can I just sit back and watch my bankroll grow while brainlessly cutting and pasting data into excel? Should I automate the system and let it continually build?

Unfortunately, as Simulation 3 shows, nothing is so easy. In this most realistic simulation, bets are made at post time and prices are calculated using final pool totals. None of the trials are profitable. 93.9% of the trials end when the bankroll is depleted. The average final bankroll is $12 with the highest being only $888 of the original $2500.On average, the bankroll declines 0.838% per wager. The bankroll goes to zero as quickly as the 61^st bet and half of the trials have zero dollars after the 303^rd bet. While these bets look attractive at post time, late money eliminates arbitrageable opportunities and renders the Dr. Z system unprofitable.

Simulation 3





Does the late money come from other people engaged in arbitrage? No doubt. On the optimal place bets, the share of the place pool increased from 15.6% at post time to 20.6%. For optimal show bets the increase was from 12.8% to 17.1%.

Aside from the late money there are two other important factors that cause these disappointing results.

The win probability (established by the public and equal to the fraction of the win pool bet on a particular horse) while fairly efficient in portraying a horse's true chance of winning, does not do as well for the full ordering of a race. There are certain speedballs who may be the most likely horse to win, but if they don't they will be as likely to finish in last as in the money. On the other hand, there are horses like Perfect Drift who always hit the board but never seem to find the winners circle. Estimates using win probabilities will over-predict place/show probabilities for the speedball and under-predict place/show probabilities for Perfect Drift. Of the 326 horses that represent optimal place bets, 83 finish first and 67 finish second. Of 612 optimal show bets, 159 finish first, 117 finish second, and 89 finish third. 

The money management strategy also breaks down in real time. Since there is no accounting for the increased risk associated with the variability created by late money, adhering to the Kelly criterion results in overly aggressive play. Simulations using a flat bet amount faired much better.

Can the system be altered to make us winners? Absolutely. Accounting profit yes. Economic profit (subtracting out the opportunity cost of my time) probably not. Perhaps the cost of improving on the system (or the your opportunity cost) is lower for you. Even if it is not the system is a fun to way to play if you can tinker it a bit. I have done better when I avoid speedballs, small track (as defined by their betting volume), and horses whose odds start out above their morning line odds but are falling.