So, if LeBron (or any player) presents the best chance of winning at the end of the game, then he’s the option to take, regardless of any game theory considerations that might govern the midgame.

That being said, I think LeBron has as good a mix as anyone, which is to say, still too predictable. But no more predictable than any of the other “closers.”

Therefore he’s making the right basketball decision most of the time.

Check out what Brian Windhorst wrote years ago. http://sports.espn.go.com/nba/playoffs2008/columns/story?page=WizCavsGame4-080427

To that end, shooting percentage is a crude parameter as an objective to aim for. It may be that 79 percent of teams with the higher shooting percentage win, but a much higher percentage of teams with the higher points per possession win. (Not quite 100 percent, but it’s darned close.) So the game theory approach looks for strategies that optimize for PPP, rather than shooting percentage (or even true shooting percentage). My guess is that the current state of the art isn’t advanced enough to support a full analysis this way, but even an examination of the starts of possessions is feasible and might open some eyes.

It must be emphasized that this is a midgame consideration, not an endgame one. It is the midgame that aims to maximize expected PPP; the endgame aims to maximize probability of winning from specific game positions. So in the endgame you may choose tactics that would be distinctly suboptimal from a PPP perspective, but improve your chances of winning the game. You’d only do that in the midgame if you were much, much worse of a team than your opponents; in fact, you might consider that to be the start of a very long endgame. The classic example is taking the trey at the end of regulation with a two-point deficit, even when the trey is less than two-thirds as likely to go in as the deuce (and therefore sub-optimal from a PPP point of view).

Skinner does acknowledge that he’s not talking about true Nash equilibria, and throughout his paper he uses the term “Nash equilibrium” to refer to this pseudo-Nash equilibrium. However, his paper is considerably weakened by not actually taking a game-theoretical approach to the problem. He assumes that any player’s efficiency is a decreasing function of usage rate. While that’s plausible (though Berri has conducted some studies that appear to contradict it), using that as a fixed condition obscures some more interesting relationships between the defense and offense.

I took some time out yesterday to suss some of these out and for those of you with a mathematical bent, you can find it at

http://www.astronomycorner.net/misc/skinner.pdf

Very quick and dirty, so some errors still floating around in there, and some shorthand notation I don’t take any time to explain. Hopefully they won’t be too difficult to interpret.