I THINK rational agents will defect 100 times in a row, or 100 million times in a row for this specified problem. But I think this problem is impossible. In all cases there will be uncertainty about your opponent/partner - you won't know its utility function perfectly, and you won't know how perfectly it's implemented. Heck, you don't know your OWN utility function perfectly, and you know darn well you're implemented somewhat accidentally. Also, there are few real cases where you know precisely when there will be no further games that can be affected by the current choice.
In cases of uncertainty on these topics, cooperation can be rational. Something on the order of tit-for-tat with an additional chance of defecting or forgiving that's based on expectation of game ending with this iteration might be right.
Followup to: The True Prisoner's Dilemma
For everyone who thought that the rational choice in yesterday's True Prisoner's Dilemma was to defect, a follow-up dilemma:
Suppose that the dilemma was not one-shot, but was rather to be repeated exactly 100 times, where for each round, the payoff matrix looks like this:
As most of you probably know, the king of the classical iterated Prisoner's Dilemma is Tit for Tat, which cooperates on the first round, and on succeeding rounds does whatever its opponent did last time. But what most of you may not realize, is that, if you know when the iteration will stop, Tit for Tat is - according to classical game theory - irrational.
Why? Consider the 100th round. On the 100th round, there will be no future iterations, no chance to retaliate against the other player for defection. Both of you know this, so the game reduces to the one-shot Prisoner's Dilemma. Since you are both classical game theorists, you both defect.
Now consider the 99th round. Both of you know that you will both defect in the 100th round, regardless of what either of you do in the 99th round. So you both know that your future payoff doesn't depend on your current action, only your current payoff. You are both classical game theorists. So you both defect.
Now consider the 98th round...
With humanity and the Paperclipper facing 100 rounds of the iterated Prisoner's Dilemma, do you really truly think that the rational thing for both parties to do, is steadily defect against each other for the next 100 rounds?