Did Tim Donaghy really win 70% of his bets against the spread?

According to disgraced NBA referee Tim Donaghy’s new book, Donaghy won 70 to 80 percent of his NBA bets. (The link is to an excellent espn.com article by TrueHoop's Henry Abbott, which I recommend highly.)

70 to 80 percent is huge: these were bets against the spread, so you’d have expected that Donaghy's winning percentage would only be around 50 percent, unless he had some kind of edge.

What was his edge? Did he fix the outcomes of games with biased refereeing? Donaghy says no: the way he won so many games was by knowing which *other* referees were biased. Not corruptly biased, mind you, sometimes just unconciously biased. He says,

"I listened to the directives from the NBA office, I considered the vendettas and grudges referees had against certain players or coaches, and I focused in on the special relationships that routinely influenced the action on the court. Throw in some quirks and predictable tendencies of veteran referees and the recipe was complete. All I had to do was call it in and let the law of averages take over. During the regular season, I was right on the money seven out of 10 times. There was even a streak when I simply couldn’t miss, picking 15 winners out of 16 games. No one on the planet could be that lucky. Of course, luck had little to do with it."

Does that make sense? I don't think it does. I don't think that even perfect knowledge of the tendencies of referees can get you a winning percentage of .700.

According to basketball researcher Wayne Winston, a reasonable pythagorean exponent for basketball is 14. (That's from his book "Mathletics," which I've been planning to review for a while now -- I'll do it this week, I swear.) To increase your winning percentage from .500 to .700, then, requires an extra 6% of points, or about 6 points in a 100-point game.

Six points is huge. It's twice what home field advantage is worth. If you assume that teams score an average one point per possession when they're not fouled, but 1.5 points per two-shot foul, it would take 24 extra foul shots (12 fouls) in a game to account for six points.

Teams shoot about 25 foul shots a game on average. 24 extra foul shots would basically double the number of foul calls per game. If you assume that each of the three referees call 8 foul shots each, one biased referee would have to call *four times as many* foul shots for one team to raise its winning percentage to .700.

Of course, the biased ref could also *refrain* from calling foul shots for the other team ... but, with only 8 shots called per game per ref, there's a natural limit to what you can *not* call.

It just doesn't seem at all plausible that one "vendetta" or "special relationship" could have such a large effect. Especially considering that the examples Donaghy provides are pretty weak. For instance:

"Referee Joe Crawford had a grandson who idolized [Allen] Iverson," writes Donaghy. "I once saw Crawford bring the boy out of the stands and onto the floor during warm-ups to meet the superstar. Iverson and Crawford’s grandson were standing there, shaking hands, smiling, talking about all kinds of things. If Joe Crawford was on the court, I was pretty sure Iverson’s team would win or at least cover the spread."

Doesn't that sound pretty much impossible? First, could any professional referee call an extra 24 foul shots a game for Allen Iverson without drawing some kind of attention? And, second, isn't it completely implausible that anyone could rise to the ranks of NBA referee with judgment so bad that he would be *that biased* in favor of his grandson's idol?

Anyway, we don't have to take Donaghy's word for it: we can check the records. Actually, ESPN's Abbott already checked. It turns out that Donaghy's accusations are completely false. With Crawford refereeing, Iverson's teams went 5-9 against the spread. That's .357, not .700.

Abbott checks a few other of Donaghy's claims, and references others who have done similar checks. The bottom line: absolutely no evidence of any bias at all, much less the kind of bias that would let you pick 70% winners.

That leaves at least three possibilities:

1. Donaghy was not specific enough in describing the circumstances in which he knew he had a .700 chance of winning. Maybe, for instance, that only happened when Joe Crawford's grandson was actually at the game, rather than watching at home.

2. There was other information Donaghy used to make his picks, not just his knowledge of referee bias. As he said: "There were other factors that came into play. Inside information about injuries. Home game or away game. Home crowd. Many more factors to take into consideration."

3. Donaghy himself rigged the games in order to win his bets.

4. Donaghy didn't actually win 70% of his bets.


Number 1 still doesn't seem very plausible: as I argued, it's very hard for a referee to make a team lose 20% of the games it otherwise would have won, and make it look natural. This is especially the case if the grudge the referee holds is against a player, and not a team: can anyone really cause Allen Iverson to lose 6 points a game, without making it obvious?

Number 2 is implausible too: there are thousands of bookies and gamblers analyzing basketball much more thoroughly than Donaghy did. If most of the information he claims to have used was public, the betting line would have adjusted for those factors already.

Number 3 is implausible, for similar reasons. Actually, it's a bit more plausible than number 1, because, for one thing, Donaghy would care about the team, not any individual player, so he could spread out his biased calls. Secondly, he could concentrate his fixes in close games, so that it may not take 6 points a game, but perhaps only 1 or 2 points in a game that's tied in the last minute.

But, to me, Number 4 is the most plausible. It does require you to assume that Donaghy and the FBI are incorrect about the results of the wagers: but if Donaghy can be so wrong about the results of his strategies (which can be verified), why can't he also be wrong about the results of his bets (which cannot)?

Anyway, there would be easier ways to figure this stuff out, if there were a list of games that Donaghy bet on. Just check those games, and the betting lines, and see if there was anything unusual there, either by Donaghy himself, or the other referees at the game. But, according to the article, there is no such list. It seems like the NBA and FBI are taking Donaghy's word for how big his bets were ($2000 each) and how many he made (more than 125 over four seasons).

In that case, isn't it more plausible that the $100,000 Donaghy made came from sources other than winning 70% of his $2,000 bets? Maybe he won the lottery, or he was peddling confidential information about Tiger Woods, or he was selling illegal MRIs to Canadian patients with sore knees. Any of those seem more plausible than being able to go .700 against the spread.

In any case, I'd be willing to put Donaghy to the test. He says he can pick 70% winners. I think he can pick 50% winners. Let's set the bar at 60%. I'm willing to bet him even money that he can't go better than .600 in his choice of 30 NBA games this season.