Friday, October 21, 2005

Satisficing and Open Intervals

What's the solution to this very simple game?
P1 picks some x from the open interval (0,1). The payoff to P1 is x.
Okay, so it's sort of a stupid game. In fact, there is no Nash equilibrium. That is, there is no strategy that P1 can follow such that there is no strictly better strategy available. This is easy enough to see: if P1 picks x, then (x+1)/2 is a strictly better strategy.

What would happen if this game were posed to experimental subjects? Suppose that they get to pick any value between $0 and $10, but not $0 or $10. Most subjects would just pick $9.99 without worrying about the extra fractions of a cent that could be extracted. So what's wrong? Why didn't game theory predict this? Isn't this an example of satisficing and not maximizing behavior?

The problem is that the game is misspecified. That is, the real payoff for the experimental subject is not an increasing function of how much money they receive. It's just not worth it to even say "$9.99999". So it's not that game theory doesn't make a prediction in this case. Game theory doesn't make a prediction in the game with the payoffs described as above. But if you include the, albeit very small, cost of choosing a number with a lot of digits then there can easily be a realistic solution.

Now consider a similar game in which player P1 picks some number x from (0,∞), with payoff x. That is, the player can pick a number as high as they want. There's no Nash equilibrium here, either, for the same reason. Now consider implementing this game in an experiment. (Not actually possible, but....) What would happen? How many zeros would you add if there were no restriction whatsoever?

Unfortunately this is one of those economic experiments that will never happen, artificially or naturally. But we can still think about it. Would there be a point at which you'd just stop thinking of bigger and bigger numbers? Maybe this is a game where no equilibrium is actually a prediction of confusion and indecision. But I suspect that there would be an outcome. My guess is that decreasing returns set in rather quickly and you'd probably settle for something like "10 to a billion dollars," but why not say "10 to a googol?" How would you make up your mind? Does the difference between those numbers matter?

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