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could be 100, but only if everyone in the game guessed 100. (Don't laugh; as we will
see later, there are people who do this.) If the average cannot be above 100, then the
two-thirds point cannot be above 66.67. It would therefore be irrational to guess
anything above 66. So, cross off a big chunk of the possibilities.
Now, because we are working from the premise that everyone is rational, we
must assume that they know that guessing above 66 is irrational. Therefore, we also
know that no one is going to guess above 66. Well, if no one is going to guess above
66, then we can also assert that the two-thirds point will not be above 44 (two-thirds
of 66). Because none of our rational co-players will guess above 44, we know the
two-thirds point will not be above 29.48 (two-thirds of 44). Lather, rinse, repeat…
(Figure 6.1).
FIGURE 6.1 By iteratively eliminating the strictly dominated strategies
(i.e., those that have no possibility of winning), we determine
that the only pure strategy is that of guessing zero.
Eventually, by eliminating the possible guesses of rational players, we get to the
point where any guess above 0 is irrational. That means, in the Guess Two-Thirds
of the Average Game, the pure strategy guess is… zero. Of course, this answer only
exists in the world of superrationality. Everyone has to be playing by the same
purely rational strategy, and the odds of that happening are fairly slim.
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