I have a new essay, Refereekonomics (cleverly titled by the editor, not me!), at TCS Daily which discusses economic research on referee decisions. The essay makes the point that improved incentives can affect the behavior of players and referees in a positive way, and thus improve the game.
One paper which estimates a model of referee decisions that is quite interesting but didn't quite fit the theme of the essay is Jason Abrevaya's "Reversal of Fortune." Along with Robert McCulloch, Jason shows that in the NHL, referees have a strong tendency to offset a penalty called on team A by making the next call against team B. Holding numerous relevant factors constant, the probability that the next penalty will be called on team B is 2/3. That's huge, and far from random. To me, this is telling evidence that NHL referees have considerable latitude when blowing the whistle. And given the importance of penalties to goal scoring opportunities in the NHL, referee discretion - to blow or not to blow? - can have a big impact on game outcomes.