Understanding the Mathematics of Personal Finance

PARI-MUTUEL BETTING

The pari-mutuel machine system used at race tracks offers a different approach to gambling. A roulette wheel owner is never sure what his or her daily operating cost will be; he or she only knows statistical generalities. On some nights, almost no one could win a lot of money; on some nights, there could be several lucky people. Also, the probabilities are fixed going into the game. The probability of the roulette wheel ball landing in any one slot is just 1/(number of slots) and the payoffs on any type of bet are published. The pari-mutuel machine system, on the other hand, adjusts the payoffs based on the bets, and the track owner can take a fixed amount or a percentage of the bets from each race. The easiest way to explain this is to work through an example.

Look at Table 14.1. The numbers are a bit contrived to make the example easy to follow, but the same calculations will work for any numbers. Six horses are racing, and people have placed bets on these six horses. Horse number 1 has almost one - third the total dollar amount of bets placed on it and is clearly “the favorite to win.” Horse number 6 has very little money bet on it and is a “long shot” to win. The total amount of bets placed is $10,000.

This race track management has decided to distribute 90% of the bets to the bettors, keeping 10% for costs and profits. This leaves $9,000 to distribute. The numbers in the column labeled multiplier are calculated by dividing the amount to distribute by the amount bet on each horse: $9,000/$2,200 = 4.1; $9,000/$1,450 = 6.2; and so on. If horse number 1 wins, each dollar bet returns $4.10. If the favorite wins,

each dollar bet on it returns $2.50. If the long shot wins, each dollar bet on it returns $20.50 and so on.

The next set of calculations is not necessary for understanding the basic idea of pari-mutuel betting. It’s intended for those with some linear algebra background who are interested in how more involved pari-mutuel calculations are made. If you don’t like the algebra, skip over it and just take a look at the examples.

At a race track, where you can bet on a specific horse winning (coming in first), placing (coming in first or second), or showing (coming in first, second, or third), the calculation of payback is, in principle, the same as above but gets a bit more complicated. I’ll set up the case for just winning or placing and limit the race to only three horses.

In a three-horse race, there are six possible sets of results. Table 14.2 shows these six situations, with the horses labeled #1, #2, and #3. The table just shows the horse who won and the horse who placed—clearly the remaining horse finished last.

Let

N = the number of dollars bet on horse #1 to win,

M1 = the number of dollars bet on horse #1to place,

N2 = the number of dollars bet on horse #2 to win, and so on,

P1 = the payoff multiplier for a winning bet on horse #1,

Q1 = the payoff multiplier for a placing bet on horse #1, and so on, where payoff multiplier is the number of dollars paid for each dollar bet. Let T be the total amount bet, which is calculated by adding N1 + N2 + N3 + M1 + M2 + M3.

The amount of money paid to people who bet that horse #1 would win if it does win is P1N1. The amount of money paid to people who bet that horse #1 would place if it wins is Q1M1. The amount of money paid to people who bet that horse #2 would place if it wins is Q2M2 . The total amount of money paid out for the first line of Table 14.2 (horse #1 wins and #2 places) is therefore

P N1 + Q1M1 + Q2 M2 = T.

(I’m ignoring the track owner’s cut of the money.) Six such equations, one for each of the six possible results of the race, are shown in the table.

In algebraic terms, using matrix notation, this is a set of six coupled linear equations in the variables Pi and Qi(

that is easily solved by any number of standard techniques for the six unknowns, Pi, and Qi. The payoff amounts may be scaled by any fraction to account for the track owner’s cut, that is, if the track owner takes 10% then scale the results by a factor of 0.9.

Table 14.3 shows several examples of winning and placing of bets. I’ve used very small dollar amounts for these examples. Remember however that everything scales; the resulting multipliers for everybody betting a total of $10 are the same as the multipliers for everybody betting a total of $1,000,000.

In the first two examples, all the winning bets are the same and all the placing bets are the same. The result is that someone betting to win gets three times his or her money back, while someone betting to place only gets 1.5 times his or her money back. This makes sense. When betting to win in this situation, exactly one-third of the bettors will collect; when betting to place, two-third of the bettors will collect.

The third example is just a check on the calculation. In this example, no one bets to place, three-sixth of the bets are on #1 to win, two-sixth on #2 to win, and one - s ixth on #3 to win. The resulting multipliers duplicate the simple situation already described.

In the fourth (last) example, I kept the winning bets the same as in the third example but changed the placing bets. Looking at the results, you’ll see that even though the winning bets stayed the same, the winning multipliers changed. This is because the placing bets shifted the overall odds of the race, and the winning mul­tipliers had to adjust accordingly. There is no obvious relationship between the winning multipliers and the placing multipliers, but the overall logic of betting on a horse that fewer people think will do well give you higher odds (a higher multi­plier) and betting to win is still a riskier bet than betting to place, so the winning

multiplier for a given horse is always higher than the placing multiplier for that same horse.

It’s easy to extend this system to cover winning, placing, and showing multipli­ers, and also trifectas (predicting exactly which horse will win, which will place, which will show) and other betting options. At no time is the track owner in risk of losing money. If the track owners choose to take a fixed amount (subtract a fixed amount rather than calculate a percentage) of the bets placed, as long as enough people are showing up and betting, then the track owners know to the penny how much they’ll bring in every night.

Horse racing is interesting because bettors can decide for themselves whether this is gambling or a game of skill. From the track owners’ point of view, the debate is moot. Their expected value of return is positive as long as people show up and place bets. This is very different from if someone really could predict the results of a roulette wheel spin (or just kept betting on the correct results by dumb luck). The roulette wheel winner cannot afford a significant long-term deviation from the sta­tistical predictions. The race track owner doesn’t care.

What about expected value of return in a pari-mutuel betting game? In order to calculate an expected value, I need to know the probability of a horse winning. I don’t know this; I only know the perceived probability based on the bets placed. If everyone betting is truly an expert on the horses, then everybody would bet on the horse that’s about to win and everyone would lose money. If there are indeed true experts, then they can only win if there are also a bunch of people betting who aren’t experts at all.

Finally, what about games such as poker? If you are really the most skillful player at the table, then in the long run, you will win. Staying in the game for the long run, however, could involve a lot of hands played, which of course means that you might need very deep pockets to cover your losses and keep you in the game long enough for your skills to prevail.

PROBLEMS

The problems for Chapter 14 are just for fun. There’s nothing to learn about games of chance other than that you can’t beat the house. Games like poker combine chance, actual skill at playing hands, and psychological effects such as bluffing; they’re very hard to quantify.

1. The coin flip game described in this chapter is mathematically equivalent to a “one t dimensional random walk.” What this means is that you flip the coin; if you get heads, you take a step forward; if you get tails, you take an equahsized step backward. The number of steps forward you get is equivalent to dollars won; the number of steps back­ward is equivalent to dollars lost. If you start at position +1, it’s equivalent to starting with $1 in your pocket; if you start at +2, it’s equivalent to starting with $2 in your pocket; and so on.

Try it. You can either actually flip a coin and take steps or you can do it on a spread­sheet. Most spreadsheet programs have a random number generator that gives you a random number between 0 and 1. Generate a column of, say, 100 of these. Then in the next column, use an if statement: If your random number is >0.5, put +1; otherwise, put -1. In the third column, keep a running total of all the numbers in the second column.

Start the column with a number which is the amount of money you started the game with. Show how long you can play without getting wiped out.

2. A state lottery offers a prize of $400 million. A single ticket’s probability of winning is 1/400,000,000. Assume that the state wants to make $100 million for holding the lottery and that tickets cost $1 a piece.

(a) What’s the expected value of a lottery ticket?

(b) Just to help get a feeling for the odds here, about how long (seconds in years) is 1 second in 500 million seconds?