I heard a great interview last week on NPR with Justin Wolfers, a professor at the Wharton School of Business. The story was also covered by papers such as the San Jose Mercury News. Wolfers has done statistical research that concludes at least approximately 500 games in NCAA basketball over the past 16 years have involved point-shaving: 6% of games with strong favorites which represents 1% of all games. I like his application of simple mathematics to analyze something that everyone can understand, and I also like the name for the field of this application: forensic mathematics.

Wolfers analyzed the game scores of tens of thousands of NCAA division I games and compared their point difference to the Vegas’ betting line for those games. He discovered, in a way that can only be done with a huge set of data, that the games in which one team was favored by over 12 points had a statistically significant anomaly in the distribution of difference in points scored, indicative of point-shaving.

Point-shaving is where a key team member plays poorer so that their team’s score is on the lower-side of the Vegas odds, but typically not so much that his team loses. This is most easily done by a key contributor to a dominant team in a game that is expected to win by a large score: that key player could turn over the ball a couple of times or clunk a few shots off the rim near the end of a game where they will clearly win. For example, a player ensures that they only win the game by 10 points when the Vegas spread had then winning by at least 12 points.

How can statistics tell whether such a thing happened? By looking at conditional probabilities. Wolfers looked at the final point spread in games where the predicted spread was 12 points or more and found that the winning team scored less than predicted. The obvious skepticisms to this are: (i) maybe Vegas was wrong, and the betting line typically over-estimates the spread in lopsided match-ups, or (ii) maybe the overwhelming favorites tend to not play as hard when they are the clearly dominant team so they tend not to score as much as expected. Wolfers raised these reasonable objections in the NPR interview and explained how his analysis avoided these possibilities, but I’m guessing that his quick verbal explanation wasn’t so clear to NPR listeners, so I’m going to provide a visual of what he found.

Wolfers looked at the distributions of game score differences conditioned on the predicted Vegas spread. The figures at the right are my representations of distributions for three different hypotheses. **Figure B** in the center shows what such a distribution probably looks like if the predicted spread of 12 points was accurate on average. The Central Limit Theorem tells us that this distribution will be Gaussian if the individual points scored are independent samples with equal probability distributions--certainly not true but perhaps not an unreasonable assumption. In this case, the Vegas adds were accurate, and the same number of games that had point differences above 12 points equaled the number of games that had point differences below 12 points. **Figure A** at the top shows what this distribution would look like if Vegas tended to over-estimate the point spread for these games, or if the favored teams tended to slack off when the spread is this high and under-achieve. The distribution is approximately the same shape as Figure B, but shifted to the left. **Figure C** at the bottom shows what you would expect if favored teams cut a few points out of their win when the game is close to the spread: the number of games that barely beat the spread would be lower than expected and the number of games that barely missed the spread would be greater than expected. The distribution found by Wolfers for lopsided games resembled Figure C.

The mathematical analysis isn’t too rigorous but is compelling nonetheless. What does this all mean? Point shaving probably occurs in NCAA basketball more than some of us thought. More importantly, however, if you are going to bet on a game with a large point spread, bet against the spread!

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