Synthese 136 (2):135-157 (
2003)
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Abstract
According to the so-called “Folk Theorem” for repeated games, stable cooperative relations can be sustained in a Prisoner’s Dilemma if the game
is repeated an indefinite number of times. This result depends on the possibility of applying strategies that are based on reciprocity, i.e., strategies that reward cooperation with subsequent cooperation and punish defectionwith subsequent defection. If future interactions are sufficiently important, i.e., if the discount rate is relatively small, each agent may be motivated to cooperate by fear of retaliation in the future.
For finite games, however, where the number of plays is known beforehand,
there is a backward induction argument showing that rational agents
will not be able to achieve cooperation.
On behalf of the Hobbesian “Foole”, who cannot see any advantage
in cooperation, Gregory Kavka (1983, 1986) has presented an argument
that significantly extends the range of the backward induction argument.
He shows that, for the backward induction argument to be effective, it is
not necessary that the precise number of future interactions be known. It
is sufficient that there is a known definite upper bound on the number of
interactions. A similar argument is developed by John W. Carroll (1987).
We will here question the assumption of a known upper bound. When
the assumption is made precise in the way needed for the argument to go
through, its apparent plausibility evaporates. We then offer a reformulation of the argument, based on weaker, and more plausible, assumptions.