I’m not sure I see how you’re modeling the trusted mediator and the option of early stopping. Shouldn’t you extend the payoff matrix to 3 dimensions (including the trusted moderator) with 3 outcomes for the players (C/D/Stopped) and 2 for the moderator (stop or continue, with payouts that switch based on whether a player defected in the last round)?
I think this reduces, with proper moderator payouts, to an overall Nash equilibrium that’s the same as a standard 2-player series with communication: cooperate on all, but defect on the last iteration. And if you can add significant uncertainty to which iteration is last (so EV of possible future iterations is greater than the difference between min and max payouts), that goes away too.
Although the original paper didn’t mention this, I think your description makes sense! Indeed “the game must go on” even in case of stopping. One simpler idea of how to model the situation with moderator: there are just two players who have actions (1) adhere to (crypto) mediator’s advice (2) deviate from advice.
Thanks for the insight about adding uncertainty to avoid ending with defection!
I’m not sure I see how you’re modeling the trusted mediator and the option of early stopping. Shouldn’t you extend the payoff matrix to 3 dimensions (including the trusted moderator) with 3 outcomes for the players (C/D/Stopped) and 2 for the moderator (stop or continue, with payouts that switch based on whether a player defected in the last round)?
I think this reduces, with proper moderator payouts, to an overall Nash equilibrium that’s the same as a standard 2-player series with communication: cooperate on all, but defect on the last iteration. And if you can add significant uncertainty to which iteration is last (so EV of possible future iterations is greater than the difference between min and max payouts), that goes away too.
Although the original paper didn’t mention this, I think your description makes sense! Indeed “the game must go on” even in case of stopping. One simpler idea of how to model the situation with moderator: there are just two players who have actions (1) adhere to (crypto) mediator’s advice (2) deviate from advice.
Thanks for the insight about adding uncertainty to avoid ending with defection!