## On the Practical Uselessness of Minimax Prescriptions

25 October 2021[I posted the following as an entry to Facebook two years ago.]

In game theory, there's a proposal that, in selecting amongst options, a player should choose that option with the least-bad worst case. So, for example, if the worst that could happen if you go to the bistro is that they get your order wrong, while the worst that could happen if you go to the bar is that you get knifed, then you should go the bistro.

Such a strategy is usually called minimax

. (I think that it should have been called maximin

, and indeed some people call it that, but they're very much bucking convention.)

The proposal seems plausible, especially when measures of probabilities are unknown (so that expected values cannot be calculated). But I think that notions of probabilities as measures inform and thus disinform the apparent reasonableness of minimax strategy.

When one conceptualizes probability as a measure, it is all too easy to think that all events with probability measure 0 might as well be treated as impossible. But they're not quite the same thing if one has to deal with infinitely many possible cases; some precisely specified possibilities would each have to have a probability measure of 0.

Possible events with truly horrible consequences can have a probability measure of zero, and will be disregarded if one treats that probability as equivalent to impossibility.

And, in the absence of actual measures of probability, then from the habit of treating events with very low measures of probability as if impossible can slide into the practice of more generally disregarding events that have low probability rank.

In a textbook game of poker or of dice or whatever, the worst possible outcome in the model is most often a loss of funds. In a real-world game of poker or of dice or whatever, the worst possible outcome is something far more dire, and hard to identify; we can think of awful possibilities, and then think of something still worse. Puffy may get angry, and shoot us; or he may shoot not only us but in his rage go shoot our loved ones as well.

Tags: game theory, maximin, minimax

Originally the criterion was MinMax loss. I love Dan Ellsberg’s AER Theory of the Reluctant Duelist. Dan asked a game theorist that if optimal play guaranteed Zero (zero sum games) why not just guarantee zero by not playing. The game theorist replied that sometimes we must cannot choose not to play. That gave Dan his title and topic!

Thank you for explaining the etymology!

Of course, even in the lab, the other player or perhaps the researcher may be the aforementioned Puffy.

A duel might seem to have the worst outcome in one's own death, but duels could could become melees involving more than two persons. The Bowie knife became famous as a result of one such case.