Playing (with) the Odds


Excerpt from the NFL report, Syracuse Post-Standard, 1/8/98.

NEWS SERVICE REPORTS --  Some Steelers were suprised to learn the Las Vegas betting line, which previously listed them as a 1-1/2 point favorites [sic], now favors Denver by one point.

The line is set mostly to draw an equal number of bets on each team and isn't always an indicator of which team truly should be favored. But, some Steelers took umbrage at being a home-field underdog to a team they defeated last month.

"It motivates you," [Steelers' running back Jerome] Bettis said.


Discussion

Before saying anything about the content of this very brief report, let me compliment its writer (who is uncredited) for understanding exactly what odds are and are not! The press is usually as confused as anyone when writing about this sort of thing.

Technically, the point spread is not set to "even the two teams," nor is it set so that the probability of winning a bet is 1/2 no matter which team you choose. Instead, it is set so that (approximately) equal amounts of money are wagered on each of the two teams. The description that follows makes the following point: If public opinion is wrong and you know it, then (perhaps) it is actually in your favor to place a wager! One other thing you should always be aware of: Las Vegas (the bookmaker) always makes money on the deal. These two statements go hand in hand: Since the book always wins, for you to win also you must be smarter (not than the book, who's basically neutral on the outcome of the game and only cares about the aggregate outcome of the wagering) than. . .Yes! You must be smarter than the other bettors. Here's how it all works.

The oddsmakers begin by setting a "line." In the above example, the line was intitially set at Steelers favored by 1-1/2 points. How was this set? It has nothing to do with how good the two teams are...it's set as the book's initial guess as a figure to split the betting. That is, the book thinks about half of the wagers will be for the Steelers (you lose 1-1/2 points if you take this option) and half for the Broncos (you get 1-1/2 points). (In fact, a survey might be taken to estimate the the spread before offering it to the betting public.)

Suppose that the line remains at this figure and that approximately half the money is bet on Pittsburgh. How does Las Vegas make money? In fact, Las Vegas can't help but make money! Each bettor risks more than can be gained by winning. Suppose the bet is $10 to win an additional $9 (it always works this way) and 100 of 200 bettors wager on the Steelers. If the Steelers beat the spread (win by more than 1-1/2 points) then these 100 each win $9 (a total of $900) while the other 100 each lose $10 (a total of $1000) and the house comes up with $100 profit. It's easy to see it doesn't matter who wins--try it yourself assuming the Steelers don't beat the spread. And, in fact, the betting doesn't have to be exactly equal. If 105 of the 200 people bet on the Steelers and the Steelers beat the spread the house pays back $945 in winnings to the 105 winners, while taking in $950 from the 95 losers; still managing a slight $5 profit. If the Steelers don't beat the spread then the house does better, taking in $1050 from the Steelers money and losing $855 on the Bronco money, for a profit of $195. (It's no coincidence that the average profit is $100.) However, if the betting is too far from even then the house loses the sure profit; the profit depends on the outcome of the game. In Las Vegas you can rest assured this never happens--the house never plays favorites, it always strives for the sure profit (it may seem small, but when millions are wagered it adds up).

Why does the point spread change?

It changes when the betting is not split evenly on the two choices. So, when the Steelers were initially favored by 1-1/2 points we can assume that substantially more money was being wagered against them than for them. The spread was changed to favor the Broncos to ensure more money would be bet on the Steelers. Note that it had absolutely nothing to do with the quality of the two teams (which presumably remained the same during this period).

However, the point spread can indirectly reflect the quality of the two teams. Suppose for instance that the initial spread is set to Steelers by 1-1/2. Then, later in the week the Steelers' starting quarterback (Kordell Stewart) is injured. Clearly this gives a large edge to the Bronco's. The oddsmakers don't change the spread because of the injury that weakens the Steelers. . .they change the spread because the betting public will recognize that the injury favors the Broncos and will overwhelmingly bet against the Steelers if the spread is unchanged. This will ruin the "balance" as well as Las Vegas' sure profit. So the spread will change. Injuries often change the spread. A good example is the recent Citrus Bowl, where Penn State played Florida. Initially the spread favored Florida by about 7 points. Then Penn State's two best offensive players were suspended from the team. The spread quickly adjust to favor Florida by about 13 points. Events that affect the game only indirectly influence the point spread through the amounts of money being wagered on each of the teams.

How can you beat the spread? Only by being smarter than the average wager. With the spread set at Broncos by 1 point you can assume that half the wagered money is on each team. Suppose (to be extreme) you know Bronco quarterback John Elway, and know that he's been sick all week (this information might not be publicized, depending on the psychology the team is employing prior to the game). You know that the chance of the Broncos winning by a point or more are really only about 1/4. By all means make the bet! There's a 3/4 chance you'll make $9 and a 1/4 chance you'll lose the $10. That's an expected (mean) gain of

.0.75(9) - 0.25(10) = 6.75 - 2.50 = 4.25

While you might lose this time, bets such as this will win you $4.25 on the average.


Questions

The point spread is set and a bet of $A can win you $B where B < A.

  1. If n bets are made on each team (the betting is even), how much money will the bookie make? (Note: this is not a random quantity.)

Take p to be the probability that you win your bet.

  1. What is your expected (mean) gain (in terms of A, B and p).
  2. For what p is the expected (mean) gain in your favor? For these p you would place the bet, otherwise you would hold on to your money.
  3. What is the variance of your gain?