How to bet on sports and the Super Bowl using game theory

The 2010 Super Bowl matchup is set. The Indianapolis Colts will face the New Orleans Saints. Already there is much speculation about who is the favorite, which quarterback will perform, which defense or offense will show up, and how these dome teams will play in an outdoor stadium.

In fact, none of these things really matter. It is possible to be ignorant and yet make a sure-fire winning gamble, both mathematically and practically.

My goal is not really to gamble. I neither gamble regularly nor do I encourage it. But I wish to illustrate an important point of gambling.

One may recall that gambling is an application of probability. In fact, it was a gambling wager in 1654 that led to the mathematical development of probability theory. The historical details are captured in letters between Blaise Pascal and Pierre de Fermat.

The idea then is the same as it it now: it’s the odds that matter. If you can play the odds, you can win. Here is how to do it in a way similar to the odds line set by Las Vegas bookies.

And yes, this works for almost any event betting on sports, politics, or otherwise.

How to make a guaranteed winning bet

Think about two people who are die-hard fans of their respective teams. Their beliefs can be written in probability terms, but keep in mind this level of precision doesn’t really matter. So let’s get started.

John believes the Colts will win with probability 3/4.

On the other hand, Michael believes the Saints will win with probability 3/5.

Both John and Michael enjoy friendly bets. And they are willing to bet as long as they think it’s a fair deal–that is, as long as the bet has a positive expectation.

Now here’s the fun part. Given these facts, it is possible to design a sure-fire bet that makes you money. It doesn’t matter which team actually wins–you will get money coming to you!

The idea is to make sure your loss for an event is always less than the gain.

Here is a specific way you can construct the bet:

–You bet with John that you will pay him $4 if the Colts win and he will pay you $5 if they lose. John will take this bet because his expected value is positive: (3/4) 4 + (1/4) * -5 = $1.75

–At the same time you bet with Michael. You bet with Michael that you will pay him $4 if the Saints win, and he will pay you $5 if the Saints lose. Michael will want to take this bet because his expected value is positive: (3/5) 4 + (2/5) * -5 = $0.40

The genius part is that both John and Michael believe they are making good bets. They will want to take these bets because their expected winnings are positive. You aren’t doing anything to pressure them or conceal the facts.

So how do you fare? The amazing part is that you net $1 regardless of who wins the game!

See this for yourself:

–If the Colts win, you win $5 from Michael but pay $4 to John. The net is you win $1.

–If the Saints win, you win $5 from John but pay $4 to Michael. Again, the net is you win $1.

You can obviously optimize the bets so you gain maximally, and you can increase the bet in multiples as long as both parties agree.

In financial terms, you have capitalized on differing beliefs to create an arbitrage opportunity. This is similar to what intelligent investors could do when mutual funds and stock options were in their infancy.

This is also something like what Las Vegas bookies do: they set a betting line to encourage both sides of a bet to have even money, and they pay out the winners a share from the pool of the losers’ money. This doesn’t always work in practice, but it is usually profitable over time.

In your case, it might be safer to hide your scheme, or John and Michael might not bet with you out of spite.

Reference:

Su, Francis E., et al. “Sure Betting on Different Beliefs.” Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Discussion questions:

1. Above, John has a winning bet. So does Michael. In the long-run, that should mean both make money. But you too are making money for sure. How is it possible that everyone is making money? This is a betting paradox–how to resolve it?

2. You can design a winning bet as long as the beliefs are different. Construct a winning bet if Alice believes an event will happen with probability p and Bob believes it happens with q > p, and both will accept gambles that have positive expectations.

3. Research project: how do professional bookmakers make money on sports bets? Consider various types of bets like proposition or parlays.