The hot hand and playing hurt

So, was chatting with someone the other day and it came up that I sometimes do sports statistics, and he told me how he read that someone did some research finding that the hot hand in basketball isn’t real . . .

I replied that the hot hand is real, and I recommended he google “hot hand fallacy fallacy” to find out the full story.

We talked a bit about that, and then I was thinking of something related, which is that I’ve been told that professional athletes play hurt all the time. Games are so intense, and seasons are so long, that they just never have time to fully recover. If so, I could imagine that much of the hot hand has to do with temporarily not being seriously injured, or with successfully working around whatever injuries you have.

I have no idea; it’s just a thought. And it’s related to my reflection from last year:

The null model [of “there is no hot hand”] is that each player j has a probability p_j of making a given shot, and that p_j is constant for the player (considering only shots of some particular difficulty level). But where does p_j come from? Obviously players improve with practice, with game experience, with coaching, etc. So p_j isn’t really a constant. But if “p” varies among players, and “p” varies over the time scale of years or months for individual players, why shouldn’t “p” vary over shorter time scales too? In what sense is “constant probability” a sensible null model at all?

I can see that “constant probability for any given player during a one-year period” is a better model than “p varies wildly from 0.2 to 0.8 for any player during the game.” But that’s a different story. The more I think about the “there is no hot hand” model, the more I don’t like it as any sort of default.