Minimax It

January 29, 2011

by Gene Callahan

I’m always shocked by the idea that “the star has to take the final shot” in basketball. I just watched UConn, down one point with eight seconds left against Louisville, force the ball to Kemba Walker for what must have been about a thirty-foot shot. They had the ball in the hands of Shabazz Napier, a dynamic, fast point guard, who can drive, and they only needed two to win. When Napier saw Louisville was focused on Walker, why in the world wasn’t he given the green light to go to the basket? (And I’m sure he wasn’t: UConn Coach Calhoun has publicly declared that Walker will always take the last shot in tight games: and publicly stating that, by the way, seems even more dull than secretly determining to do that… unless Calhoun was bluffing!) In any case, this seems a clear failure to implement a minimax strategy, which says your shots should be distributed amongst all players so that the odds of any of them hitting the final shot equalize (otherwise, you have a superior strategy available that you’re not employing).

In other words, if the defense double-teams your star, someone else should take the shot. But, time and again, we find the star forcing a shot while his supporting cast stands around open.

Is this simply a failure of rationality on the part of basketball coaches, or is there some other explanation? Pete B., you got anything for me here?

13 Responses to “Minimax It”

  1. przemek Says:

    I know that explanations of the sort “they do it because they have a weird utility function” aren’t terribly useful in general, but I do think that this is evidence not of failure of rationality but of the fact that coaches maximize something that has more terms in it than just the probability of winning.

  2. Current Says:

    I think that in a lot of areas it’s useful to take a particular dogmatic approach to something in order to gather data. If you do X for a while dogmatically, then do Y for a while dogmatically then you can gather some (admittedly imperfect) data about how to do it.

    But it may just be that the coach hasn’t thought through the strategy for games like this.

  3. Gene Callahan Says:

    Good one, John G! More evidence that there is a real problem with this strategy.

  4. Gene Callahan Says:

    Phil Jackson, during Jordan’s retirement, once tried to use Scottie Pippen (who was then the star) as a decoy for a last second shot. Pippen refused to return to the court from the timeout! Perhaps this is why coaches do this: otherwise the star will quit!

  5. Gene Callahan Says:

    And przemek, the previous comment may explain what coaches are maximizing: not having the star go into a funk.

  6. Bob Murphy Says:

    Gene, not only do actual coaches apparently understand their sport more than you do, but so do movie-makers.

  7. Bob Murphy Says:

    In any case, this seems a clear failure to implement a minimax strategy, which says your shots should be distributed amongst all players so that the odds of any of them hitting the final shot equalize (otherwise, you have a superior strategy available that you’re not employing).

    Joking aside (now), can you spell this out a little more Gene?

  8. Fuguez Says:

    Did he score?

  9. Gene Callahan Says:

    No, fuguez, he did not. And my anger at that ending motivated this post.

  10. Gene Callahan Says:

    Bob: You should choose your mix of strategies in a mixed-strategy
    two-person, zero-sum game with finite strategies so as to maximize the expected payoff from the mix of strategies. So, if strategy A is “the star takes the shot” and strategy B is “he is a decoy and someone else takes it,” a “rational coach should adjust the probability of playing A or B to maximize the expected payoff. In this specific case, if your star (strategy A) is hitting a third of his shots in the last minute (as apparently Kobe Bryant does!) and the rest of the team (strategy B) is hitting 50% on the rare occasions they get the ball, you’d want to get your star to pass more, as strategy B is giving a much higher payoff, but is being chosen far less often — Kobe had 56 game-winning shots versus one game-winning assist over five years!

  11. Gene Callahan Says:

    Ooh, sorry about the mangled formatting and punctuation on that previous comment!

  12. Eric Hosemann Says:

    From the Kobe article: “Yet we get things wrong all the time anyway, for the simple reason that a lot more happens in the NBA than anybody can catalog in any objective way.”
    TRUE DAT, for a whole host of things!


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