(**UPDATE**: I think there’s a mistake in the argument here – see Bob Stalnaker’s comment 11 below.)

Row and Column are going to play 100 rounds of Prisoners Dilemma. At each round they can either play Co-op or Defect, with standard rules. (So the payoffs are symmetric, and on each round Defect dominats Co-op for each player, but each playing Co-op is Pareto superior to each playing Defect.) The following is true at the start of the game.

- Each player is rational.
- No player loses a belief that they have if they receive evidence that is consistent with that belief.
- For any
*r*, if it is true that if a player were to play Co-op on round*r*, the other player would play Defect on every subsequent round, then it is also true that if the first player were to play Defect on round*r*, then the other player would still play Defect on every subsequent round. - The first three premises are matters of common belief.

Call a strategy that a player can play consistent with those four assumptions an approved strategy. (Note that one of the assumptions is that the player is rational, so these will all be rational strategies.) Assume for reductio that there are approved strategies S1 and S2 such that if Column plays S2, then Row can play S1, and this will involve sometimes playing Co-op. I will try to derive a contradiction from that assumption.

Let *r* be the largest number such that there are approved strategies S1, S2 and if Column is playing S2, and Row plays S1, then Row plays Co-op on round *r*. I will now argue that it is irrational for Row to play Co-op on round *r*, contradicting the assumption that S1 is an approved strategy.

Since both players are playing approved strategies, they are both acting consistently with the initial assumptions. So by premise 2, the initial assumptions still hold, and this is a matter of common belief. So it is still a matter of common belief that each player is playing an approved strategy.

If Row plays Co-op on round *r*, that is still, by hypothesis, an approved strategy, so Column would react by sticking to her approved strategy, by another application of premise 2. Since *r* is the last round under which any playing an approved strategy against an approved strategy co-operates, and Column is playing an approved strategy, Row believes that if she were to play Co-op, Column would play Defect on every subsequent round. By premise 3 (or, more precisely, by her belief that premise 3 still holds), Row can infer that Column will also play Defect on every subsequent round if she plays Defect on this round.

Putting these two facts together, Row believes prior to playing this round that whatever she were to to, Column would react by playing Defect on every subsequent round. If that’s the case, then she would get a higher return by playing Defect this round, since the only reason to ever play Co-op is that it has an effect on play in later rounds. But it will have no such effect. So it is uniquely rational for Row to play Defect at this round. But this contradicts our assumption that S1 is a rational strategy, and according to it Row plays Co-op on round *r*.

If our assumption is true, then there can be no approved strategy that ever co-operates before observing the other player co-operate. If there were such a strategy, call it S3, then we can imagine a game where both players play S3. By hypothesis there is a round *r* where the player playing S3 co-operates before the other player co-operates. So if both players play S3, which is approved, then they will both play Defect up to round *r*, then play Co-op on that round. But that’s to say that they will play Co-op while (a) playing an approved strategy and (b) believing that the other playing will play an approved strategy. And this contradicts our earlier result.

This does not mean that a rational player can never co-operate, but it does mean that they can never co-operate while the initial assumptions are in place. A rational player might, for instance, co-operate on seeing that her co-player is playing tit-for-tat, and hence that the initial assumptions are not operational.

Nor does it mean, as I think some theorists have been too quick to conclude, that playing Defect all the time is an approved, or even a rational, strategy. Assume that there are approved strategies, and that (as we’ve shown so far) they all involve playing Defect on the first round. Now the familiar objections to backward induction reasoning, tracing back at least to Philip Pettit and Robert Sugden’s “The Backward Induction Paradox”, become salient objections.

If Row holds all the initial assumptions, she may also believe that if she were to play Co-op on the first round, then Column would infer that she is an irrational agent, and that as such she’ll play Tit-for-Tat. (This isn’t built into the original assumptions, but it is consistent with them.) And if Row believes that is how Column would react, then Row is rational to play Co-op, or at least more rational on this occasion than playing Defect. Indeed, even if Row thinks there is a small chance that if she plays Co-op, Column will conclude that she is irrationally playing Tit-for-Tat, then the expected return of playing Co-op will be higher, and hence it will be rational. I conclude that, given any kind of plausible assumptions Row might have about Column’s beliefs, playing Co-op on the first round is rational.

In their paper, Pettit and Sugden try to make two arguments. The first I’ve very quickly excerpted here – namely that the assumption that always Defect is uniquely rational leads to contradiction given minimal assumption about Row’s beliefs about how Column would react. The second, if I’m reading them correctly, is that rational players may play some strategy other than always Defect. The argument for the second conclusion involves rejecting premise 2 of my model. They rely on cases where players react to rational strategies by inferring the other player is irrational, or believes they are irrational, or believes they believe that they are irrational etc. Such cases are not altogether implausible, but it is interesting to think about what happens without making such a possibility.

And I conclude that given my initial assumption, there is no approved strategy. And I’m tempted to think that’s because there is no rational strategy to follow. Just like in Death in Damascus, any strategy a player might follow, they have reason to believe is an irrational strategy when they are playing it. This is a somewhat depressing conclusion, I think, but causal decision theory sometimes doesn’t give us straightforward advice, and I suspect finite iterated Prisoners Dilemma, at least given assumptions like my premise 2, is a case where causal decision theory doesn’t give us any advice at all.