### Are the NFL gambling lines consistent with each other?

According to this old Boston Globe article, Daryl Morey discovered that the Pythagorean Projection for the NFL should use the exponent 2.37. That means that from the Vegas betting line and the over/under, we should be able to come up with an estimate of the probability of winning the game.

For instance, last night, the Patriots were favored by 13.5 points over the Giants. And the over/under was 46.5 points. That means that the expected score, in a sense, was Patriots 30, Giants 16.5.

Using Pythagoras on that score, we get that New England should have had a 80.5% chance of winning the game.

But the market prediction, from tradesports.com, was 88.0%. (Sorry, no link.)

So why the difference? I can think of a couple of possible reasons:

1. Pythagoras doesn't work well on such heavy favorites;

2. The distributions are not symmetrical, so even though the *median* score is 30-16.5, the *mean* score is something else;

3. The market for outright wins is less efficient than the point-spread market.

I'd bet #2 is the correct answer, that strategic differences (such as the leading team taking time off the clock instead of going for more points) make the comparison inaccurate. In any case, I doubt #3: if it were that easy to make money by betting the heavy underdog to win, someone would have noticed by now.

P.S. This NYT article says that as of last week, the Patriots were 1:8 favorites to win the Super Bowl. That can't be right – those would be the odds of winning one game against a mediocre team, not three straight against quality opponents. The betting is at about even odds at TradeSports.

Labels: football, forecasting, gambling, NFL, pythagoras

## 7 Comments:

As a pro gambler, I can tell you that some sportsbooks are completely uninterested in booking bets on the favorites in futures markets. I haven't been to Vegas since November, but even then many books had the Patriots at 1-5 to win the Super Bowl. Only a complete novice would bet them at that price, but apparently this is one proposition where the bookie prefers to get a large commission from a small number of bets, rather than vice versa.

I'm not sure Pythagorean estimates are valid for football, especially for single games.

Scoring in football is different than most other sports. It comes in irregular increments of 2,3,6,7,and 8--but mostly 3 and 7. Scoring distributions are non-normal, i.e. there are more 21- or 24-point scores than 22-point scores.

Also, although I'm not certain if this affects the Pythagorean, but unlike baseball offense and defense are not independent in football. The better my offense, the less likely you are to score (because of field position considerations).

I think the season win estimates based on Pythagorean estimates are more valid than for single games. The irregularity of the scoring smooths the more games you consider.

But I agree. The betting markets don't always appear very efficient to me. As I recall, the Patriots were at 1-3 odds to win the Super Bowl right before the season started. I calculated what their probability to win the three playoff/SB games they'd have to win would have to be 0.91 for each game. That's way out of whack, especially for before the first snap of the season. An efficient strategy might be to bet counter-hype.

j holz: makes sense. I wonder why competition hasn't reduced those prices? Although offering 1:3 instead of 1:1 is "only" a 33% house take, if my math is right.

That's like offering 3:1 when the correct odds are 5:1. It's a gouge, but not as big a gouge as it looks ...

Brian: Agreed. My thinking was that it would be a decent first approximation, but perhaps not.

Still, though. Imagine that the Patriots outscored their opponents *over the season* with an *average* score of 30-16.5. Then, you'd expect their winning percentage for the season to be .805. Then, the probability that they won *any given game* should also be .805.

By this logic, it seems that it *should* work. Doesn't it?

I recently posted a Patriots' analysis on my Hot Hand website. I tried to estimate the probability of the Pats' going 16-0, based on my estimates of their game-specific win probabilities, which I then multiplied together. For the game-specific probabilities, I used a much simpler system than Pythagorean analysis. My method involves a lot of arbitrary assumptions, but my bottom-line estimate -- that a perfect 16-0 season like the Pats' would occur about once every 85 years -- seems to be in ballpark. Thus far, 30 seasons of the NFL's 16-game format have taken place, and New England was the first to complete one unscathed.

http://thehothand.blogspot.com/

Yes, a decent approximation, definitely. But a 88% vs 80% probably qualifies.

I wonder if it's systematic. Perhaps the bigger the mismatch between teams, the farther the divergence between Pythagoras and the consensus win probability.

Alan-I regularly do NFL win probabilities based on a logistic regression of team efficiencies. Your estimates of A/B teams at home/road etc. are frighteningly accurate.

But I might dispute the estimate of a typical schedule a contending undefeated team would face.

NE was in 1st place last year, so their strength of schedule games included SD and IND, both teams that repeated as division champs. The one division champ that faltered was BAL, but NE happened to draw the AFC N as their in-conference division match-up, so they faced PIT too. NE also happened to draw the toughest division in the league as its inter-conference opponent which included 3 playoff teams (DAL, NYG, WAS) plus PHI, which was probably the best team not to make the playoffs.

Also, there's the possibility that a fairly average team could, by luck of the draw, face a schedule as weak as the Pats was tough.

I just read this and can tell you that 2.37 is a good number to use year to year. If you use the average win margin as your exponent, your numbers are closer. I used it on your example, and came up with about 87%.

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