What about Monte Carlo methods? There are many problems for which Monte Carlo integration is the most efficient method available.
(you are of course free to suggest and to suspect anything you like; I will, however, point out that suspicion is no substitute for building something that actually works...)
This “perfectly rational” game-theoretic solution seems to be fragile, in that the threshold of “irrationality” necessary to avoid N out of N rounds of defection seems to be shaved successively thinner as N increases from 1.
Also, though I don’t remember the details, I believe that slight perturbations in the exact rules may also cause the exact game-theoretic solution to change to something more interesting. Note that adding uncertainty in the exact number of rounds has the effect of removing your induction premise: e.g., a 1% chance of ending the iteration each round has the effect of making the hanging genuinely unexpected.
Anyway, the iterated prisoner’s dilemma is a better approximation of our social intuition, as in a social context, we expect at least the possibility of having to deal repeatedly with others. The alternate framing in the previous article seems to have been designed to remove such a social context, but in the interests of Overcoming Bias, we should probably avoid such spin-doctoring in favor of an explicit, above-board articulation of the problem.