2, 5, 9, 25, 69, 73, 82, 96, 100, 126, 150
Two people are separately confronted with the list of numbers shown above and offered a reward if they independently choose the same number. If the two are mathematicians, it is likely that they will both choose 2 – the only even prime. Non-mathematicians are likely to choose 100 – a number which seems, to the mathematicians, no more unique than the other two exact squares. Illiterates might agree on 69, because of its peculiar symmetry – as would, for a different reason, those whose interest in numbers is more prurient than mathematical.
There are three things worth noting about this simple problem in coordination without communication. The first is that each pair of players is looking for a number that is in some way unique. To a mathematician, all three squares are special numbers, as are the three primes. But if they try to coordinate on a square or a prime, they have only one chance in three of success—and besides, one may be trying primes and the other squares. 2 is unique. If the set of numbers did not contain 2 but did contain only one prime (or only one square, or one perfect number) they would choose that.
The second thing to note is that there is no single right answer; the number chosen by one player, and hence the number that ought to be chosen by the other, depends on the categories that the person choosing uses to classify the alternatives. The right strategy is to find some classification in terms of which there is a unique number, then choose that number – a strategy whose implementation depends on the particular classifications that pair of players uses. Thus the right answer depends on subjective characteristics of the players.
The third point, which follows from this, is that it is possible to succeed in the game because of, not in spite of, the bounded rationality of the players. To a mind of sufficient scope every number is unique. It is only because the players are limited to a small number of the possible classification schemes for numbers, and because the two players may be limited to the same schemes, that a correct choice may exist. In this respect the theory of this game is radically different from conventional game theory, which assumes players with unlimited ability to examine alternatives and so abstracts away from all subjective characteristics of the players except those embodied in their utility functions.
Gets bonus points for including a 69 joke in an academic journal paper.
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