I don't know enough math and I don't know if this is important, but in the hopes that it helps someone figure something out that they otherwise might not, I'm posting it.

In Soares & Fallenstein (2015), the authors describe the following problem:

Consider a simple two-player game, described by Slepnev (2011), played by a human and an agent which is capable of fully simulating the human and which acts according to the prescriptions of UDT. The game works as follows: each player must write down an integer between 0 and 10. If both numbers sum to 10 or less, then each player is paid according to the number that they wrote down. Otherwise, they are paid nothing. For example, if one player writes down 4 and the other 3, then the former gets paid $4 while the latter gets paid $3. But if both players write down 6, then neither player gets paid. Say the human player reasons as follows:

"I don’t quite know how UDT works, but I remember hearing that it’s a very powerful predictor. So if I decide to write down 9, then it will predict this, and it will decide to write 1. Therefore, I can write down 9 without fear."

The human writes down 9, and UDT, predicting this, prescribes writing down 1. This result is uncomfortable, in that the agent with superior predictive power “loses” to the “dumber” agent. In this scenario, it is almost as if the human’s lack of ability to predict UDT (while using correct abstract reasoning about the UDT algorithm) gives the human an “epistemic high ground” or “first mover advantage.” It seems unsatisfactory that increased predictive power can harm an agent.

More precisely: two agents A and B must choose integers m and n with 0 ≤ m, n ≤ 10, and if m + n ≤ 10, then A receives a payoff of m dollars and B receives a payoff of n dollars, and if m + n > 10, then each agent receives a payoff of zero dollars. B has perfect predictive accuracy and A knows that B has perfect predictive accuracy.

Consider a variant of the aforementioned decision problem in which the same two agents A and B must choose integers m and n with 0 ≤ m, n ≤ 3; if m + n ≤ 3, then {A, B} receives a payoff of {m, n} dollars; if m + n > 3, then {A, B} receives a payoff of zero dollars. This variant is similar to a variant of the Prisoner's Dilemma with a slightly modified payoff matrix:

Likewise, A reasons as follows:

If I cooperate, then B will predict that I will cooperate, and B will defect. If I defect, then B will predict that I will defect, and B will cooperate. Therefore, I defect.

And B:

I predict that A will defect. Therefore, I cooperate.

I figure it's good to have multiple takes on a problem if possible, and that this particular take might be especially valuable, what with all of the attention that seems to get put on the Prisoner's Dilemma and its variants.