## Implied Sports Probabilities From Vegas Moneylines: Introduction and Methodology

Summer is ending and all the obvious signs are there. Days are shorter, nights are cooler, and football season has arrived!

With the beginning of football season comes the millions and millions of dollars that flow into the gambling industry. Most people like betting on spreads, but others prefer the moneyline.

We always see numbers attached to teams which tell betters who the favored team is in a matchup. For instance, last Wednesday during the New York Giants vs. Dallas Cowboys game, betting websites had the New York Giants as the moneyline favorite at -200 and the Dallas Cowboys as the underdog at +170. This essentially means that if someone put \$100 on the Giants and the won, they would’ve won \$50. Meanwhile, if one bet on the Cowboys, they would’ve won \$170.

This is interesting: We can use these returns and the moneylines to formulate probability values for each team winning the game.

Let’s elaborate:

Let p be the probability that the Cowboys win and assume the only other game outcome is that the Giants win. (We ignore the unlikely event of a tie.) In static equilibrium, the expected payoff (based on the market’s subjective measure over the outcome space) should be the same for both bets, otherwise money would keep flowing to the side that is seen to be more profitable and prices would adjust to make the market indifferent. Of course, this indifference condition is a theoretical point and we need it to pin down p.

If you ever took a probability class, you know that the expected payoff for any event is the sum of the probabilities of each individual event occurring (or not occurring), multiplied by the payoff return one would gain (or lose) from the event occurring or not occurring.

For instance, if one bet \$100 on Cowboys, the expected payoff would have been p(170) + (1-p)(-100), which then equals, 170p – 100 + 100p = 270p – 100. (Note that 170 is the payoff from the Dallas win and -100 is the loss if Dallas loses.)

Similarly, if one bets \$100 on Giants, the expected payoff is p(-100) + (1-p)(50) = 50-150p. (Note that 50 is the payoff from a Giants win, and -100 is the loss from a Giants loss.)

In equilibrium, 270p-100=50-150p. Solving for p, we get approximately 0.35. This is the implied probability that the Cowboys won the game.

If the market is calibrated, then teams that are assessed to have 30-39 percent chance at winning, like the case of the Cowboys, should win roughly 30-39 percent of the time.

If there are deviations, then one can argue that there exist profitable opportunities which one can exploit in the gambling market. If the market is poorly calibrated in a particular percentage range, say whenever the assessment is 40-49 percent, the teams wins 70% of the time, then one can make money by betting on teams priced in that particular range.

Every week we will compile data to see the probability of each team winning and we will group the data by intervals of ten percent:

0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99

From this, we will be able to compare the winning percentage of each interval and see if it is theoretically consistent.