## The nature of "conspiracy" in the NBA

Kirby Lee-USA TODAY Sports

Retraction. In this post I quote stats on Lebron James playoff games officiated by Joey Crawford. However after doing further digging I now believe they are too good to be true. I'm leaving the part about Crawford and Lebron up because we should all be familiar with the application of the binomial distribution as it relates to life. The Tony Brothers stats I still believe to be genuine and the larger point that the NBA should be more transparent in their referee selection is as valid as ever.

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(I took this from an email i wrote a friend about NBA officiating as it applies to Clipper v Thunder Game 5)

Did you see my text where I showed you Lebron's playoff record when Joey Crawford is one of the refs?
Lebron James' playoff record: 66-53.
With Joe Crawford: 25-3
W/o ref Joe Crawford: 41-50!

Well I decided to plug those numbers into a binomial distribution to calculate if Joey Crawford were a neutral ref (ie his presence doesn't help or hurt lebron) what are the chances that Lebron's team would go 25-3 when he is the ref.
n = 28 ...number of overall games Crawford refereed
p = .45 ....opponents expected winning percentage, we will use 53/(66+53) as an approximation
Rather than solve this by hand, I will use a binomial calculator to get the result.

The probability question is, given a winning probability of .45, what is the chance that an opponent of Lebron James will win 3 games or fewer (ie less than 4) out of 28 attempts? Here are the results https://tinyurl.com/pd8zmgs
P(less than 4 wins) = 1.1113x10-4, or .01%.
The probability that Crawford officiated games would randomly have the outcome that they did is only 0.01% !!!1!!!!1!

Obviously, this is a back-of-the-envelope calculation. It doesn't take into account things like which round the games were in (presumably Lebron's team has a higher winning percentage vs 1st round opponents) or whether these games were home or road games. Did Crawford only officiate Lebron's first round home games? Unlikely, but if you change the numbers so that the Lebron's opponents only has a 20% chance of winning any one particular game, the chance that over 28 game a Lebron opponent wins 3 or fewer games is still just 16%.

The logical conclusion is that when Joe Crawford officiates a Lebron game, Lebron's team's winning percentage goes way up. That doesn't mean that Crawford is necessarily a dirty ref (ie colluding with the league to throw games for Lebron), it just means that there is something about his officiating style (ie the number and distribution of fouls that he calls) that tends to favor a Lebron James led team.

Why is this relevant to game 5 Clippers vs OKC? Well Tony Brothers was one of the officials on last night's game. After last nights games, he has officiated 12 OKC playoff games. OKC is 12-0 in those games! This is Bill Simmons take on the referee selection BEFORE the game was played.

This is the nature of conspiracy in the NBA. Conspiracy is the wrong word, however. Conspiracy mean collusion between multiple parties. Brothers and Crawford almost certainly never colluded with the NBA league office. In fact, Brothers and Crawford probably sleep the sleep of innocent babies because they are probably doing their best to call the games straight up. However their best isn't good enough. Something about their refereeing style inherently biases the games they officiate. Anyone who looks at the numbers can see that Crawford's officiating style favors Lebron and Brothers' style favors OKC.

And it is reasonable to assume that the NBA employee who selects officials for playoff games know this too. They would be wildly incompetent at their jobs if they did not. Thus to not outright fix a game, but to just give the desired team - say, for example, the team with the reigning MVP and not the team with an ongoing PR nightmare of an owner - a healthy statistical boost, all a single employee at the NBA league office has to do is select the right officials. It's not a "conspiracy" because it doesn't involve cloak-and-dagger, smoke-filled-rooms collusion between multiple parties. It's much better than a conspiracy in fact, because the malfeasance has been reduced to the action of one man, a single apparatchik in the league office, and there will never be any evidence of wrong doing. But it is every bit as unethical.

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