Numb3rs 306: Longshot

In this episode the team investigates the death of a man (Danny) at a horce racing track. Danny had been able to pick the winning horse in the last 30 races he had bet on. Had there not been any foul play involved, this would be extremely unlikely (there are typically 10 horses in each race). Danny had been using a system of betting that his girlfriend (who had a Masters degree in Mathematics from Stanford) had developed to pick out the second most likely horse to win a race. She developed this system in order to take advantage of the higher expectation of betting on the second best horse arising from parimutual betting. When Danny noticed that it was the second best horse that had been winning consistantly, he suspected that the races were being fixed, and took advantage of it.

Parimutual Betting

Parimutual betting is performed at horse tracks all over the globe. Rather than a professional gambler determining the proper odds for a race (based on knowledge of the horses and experience), this system allows bettors to determine the odds by themselves. We'll give a simple example of a payout system where only bets for the winning horse pay off money. Suppose that the track is running a 5 horse race, and say that bets are placed as follows. horse

Horse 1 $4000 Horse 2 $1000 Horse 3 $9000 Horse 4 $2000 Horse 5 $14000

The total amount bet is $30,000. First the house takes its cut (to be concrete we'll assume this is 15%, leaving $27,000). Then anyone who bet $1 on horse 3 will get $27000/9000 = $3. We can see that Parimutual betting guarantees the house a profit, since it always pays out exactly what it took in minus the cut (of course, we're assuming the cut covers the cost of operation and no laws interfere). Now, why is it that Danny and his girlfriend were concerned with finding a way to pick the second most likely to win? Since the odds are set by the bettors they may not actually reflect the true odds of a horse winning a race. Most bettors will concentrate on the best horse and place there bets there, often forgetting about the second best. Suppose that this had happened in our race, and that the actual odds for each horse to win had been

Horse 1 13% Horse 2 4% Horse 3 35% Horse 4 8% Horse 5 40%

Then our expectation of a $1 bet on Horse 5 is 27000/14000*.4 - 1*.6= $.171 and the expected value of a bet on Horse 3 is $27000/9000*.35-.65*1 = $.4, which is far greater than that of Horse 5. (See here or here for a reminder of what an expected value is.) Now, when the track owners were fixing the races, they would pick the second best horse to be the winner because it would cause less suspiscion than making a worse horse win.

The parimutuel system was invented by Parisian perfume maker Joseph Oller in 1865 when asked by a bookmaker friend to devise a fair system for bettors which guarantees a fixed profit for the bookmaker.


Simply put, arbitrage is risk-free profit. That is, when we invest some amount of money, we will lose money with probability 0 and have some positive chance of gaining money. A market that is free of arbitrage opportunities is called an Efficient Market, and markets with arbitrage opportunities are called Inefficient Market. In Longshot Charlie claims (correctly) that the horse race gambling market is inefficient. Example, suppose for simplicity that there are only 2 horses in some race and that two bookmakers disagree on the odds of the outcomes, say

Outcome 1
Outcome 2
Bookmaker 1
Bookmaker 2

For each bookmaker, the sums of the inverse of all outcomes will always be greater than one. This is the bookmakers rate of return. Each of these bookmakers have a rate of return of about 5%. In order for us to find an arbitrage opportunity we must make bets such that no matter what outcome happens we fail to lose money. The idea is to find odds at the two bookmakers such that we are betting on each outcome and the sum of the inverses are less than one. One opportunity is if we bet on Outcome 1 with Bookmaker 2 and Outcome 2 with Bookmaker 1. The sum of the inverses is .9478.

Now if we place a $1000 bet on Outcome 1 with Bookmarker 2 and a bet of $1000*1.5/3.56=$421.35 on Outcome 2 with Bookmaker 1, we will have taken up an arbitrage strategy. Let's make sure. If Outcome 1 happens we win $1000*1.5 from Bookmaker 2, if Outcome 2 happens we win $$421.35*3.56=$1500. Since we invested $1421.35 we are guaranteed a profit of $88.65 either way.

Activity: Find a table with three Bookmakers and three horses with odds such that each bookmarker's rate of return is no more that 15% and such that there exist an Arbitrage opportunity.

Every wonder why the price of a stock on the NYSE and its corresponding futures contrace on the Chicago Mercantile Exchange are the same? The answer is arbitrage. If there were ever a difference in the price of some particular stock, one could buy that stock in the cheaper marker and sell it at the same time in the more expensive market. Many investment companies have complicated computer programs that track opportunities for arbitrage (many are more complicated than this simple example) and try to take advantage of them as quickly as possible. For more info on arbitrage see here.