*I am about to lose a subset of my audience by writing about football, so I will note up front that: (a) this post is actually mostly about probability, although this may be a worse admission if I’m trying to retain readers; and (b) an erudite friend offers a similar discussion in the context of nominees for Golden Globes, in case football isn’t enough of a spoonful of sugar to help the algebra go down.*

As the New York Jets continue their improbable NFL play-off run, I find myself grateful that I don’t bet on sports. This is not to say that I don’t enjoy gambling, in tightly-managed moderation; and I should point out for good public-sphere hygiene that my completely-legal conduit for such hypothetical wagers would be my parents, as proud residents of the greater Las Vegas area. Rather, even though I consider myself a loyal fan, I’ve been deeply skeptical that the Jets were a playoff-caliber team, and I was fairly sure that bets against them in each of their last two contests were the closest thing to free money that I could imagine. (There’s also a bit of a “macro hedge” at work in betting against one’s preferred team, as one has a reason to be happy with either outcome; fans loyal to a bad enough team may even find this a profitable strategy.)

I wouldn’t have said it was impossible that the Jets might win against the Patriots this past Sunday, but my intuition was that the probability couldn’t have been more than 20%, and was likely even closer to 10%. I attempt to practice what I preach with respect to evaluating the process of risk-taking rather than the outcome, so in the days before the game I turned to the Las Vegas bookmakers to get a feel for what probability was reflected in the market.

Bets on American football come in two flavors: the **money line bet**, in which odds for the favorite and underdog are explicitly defined; and the **point-spread bet**, where a handicap is applied in favor of the underdog in an effort to make a “fair” contest out of unevenly matched teams. A variety of **proposition bets** may also be offered, such as whether the final score of the game will be greater or less than a given number of points, although analytically these are subsets of the above. The odds and spreads are generally set by professional bookmakers, in contrast to a sport like horse racing in the U.S. where a parimutuel wagering system sets the odds.

Although the size of the point spread is a crude measure of how much stronger the favorite is than the underdog (i.e., do you need to spot the underdog a field goal, or two touchdowns, to make it an even match?) it is much less precise than the money line as a method for deriving a market-implied probability that either team will win.

As of January 10th, bookmakers quoted the Patriots (-425) as favorites against the Jets (+325). These figures are interpreted as: if you *bet* 100 units on the Jets, you will earn a profit of 325 units if they win; but if you want *a profit of* 100 units from betting on the Patriots, you will need to risk 425 units. The size of the wager doesn’t really matter here. As a concrete example, a $10 bet on the Jets would have returned $42.50 for a win ($10 original investment + $32.50 profit). A $10 bet on the Patriots would have returned $12.35 ($10 original investment + $2.35 profit). Clearly, the Patriots were heavy favorites.

Using some basic algebra, one can convert money line odds into probabilities. Odds of “X to 1” imply a probability of 1/(1+X):

In 1/(1+X) of cases, your payoff is X

In the remaining X/(1+X) of cases, your payoff is -1

Therefore, your expected payoff is 0, making it a fair wager

The money line on the underdog is a bit easier to understand intuitively. In my example the Jets were listed at odds of 3.25 to 1, which implied a 24% probability that they would win, and by extension a 76% probability that the Patriots would win. (Ties are not be possible.)

Note, however, that you can only bet on this probability if you want to express the view that the Jets will win. To bet on the Patriots, you must take the -425 line. Odds of “1 to X” can be shown to imply a probability of X/(1+X) using the same analysis of expected payoff as above. So, in this example, the money line implies an 81% probability that the Patriots will win and obviously a 19% probability that the Jets will win – different from the probabilities implied by a bet on the Jets.

If you believe the Jets will win, you will clearly prefer a bet that is priced at a 19% probability that they will win over one that is priced at a 24% probability, but that is not the bet you’re permitted to take; vice versa for the Patriots. This disconnect is exactly where bookies should make their money. An analogy in financial markets is the bid-ask spread: a dealer may offer to sell you an asset for $101, but may only bid you $99 if you try to sell that same asset to him. Wider spreads generally indicate a greater risk to holding an inventory in the asset, for which a market-maker demands compensation. I won’t discuss bid-ask spreads here other than to note that the money line on a sporting event implicitly makes a bid-ask spread out of the probability of either team winning.

I say bookies “should” make their money based on this spread because it’s certainly possible for bookmakers to lose money – for example, if the book is heavily skewed towards one team, the market clearly believes the bookmaker’s implied probabilities are incorrect, and he stands to make or lose significant money based on the outcome of the game. This is not a particularly desirable position for a bookmaker. I assume that bookmakers must have mechanisms to trade risk among themselves, perhaps at ‘wholesale’ spreads; but it is also possible that Las Vegas as a whole may be significantly exposed to one outcome over another. This also happens in financial markets, but that is another story for another time!

*Addenda
*1. I’ve not looked at the academic literature on sports-betting, but I’d imagine there are troves of cognitive biases and market distortions that have been documented. The paper linked here seems like a good starting point for going into a bit more detail, but I’m not knowledgeable enough about the field to say that with confidence.

2. As of Jan 10th, the consensus line on the Jets winning the Super Bowl was approximately 15-1, which implied about a 6% chance that the Jets would go all the way. Starting from a “mid-market” probability of, say, 21% that they would beat the Patriots, this could have implied that the Jets were 50/50 or better to win their division and the Super Bowl if they got past the Patriots. These odds don’t necessarily reflect exactly these probabilities – it could have been, say, 75% to win the division, but only 40% to win the Super Bowl – but given that the teams for those contests weren’t known as of Jan 10th, and my general intuition that the Jets would be legitimate Super Bowl contenders in the state of the world where they beat the Patriots, these odds made sense to me (and I wish I had taken them!).