Very interesting! I’m eager to see where this will lead.
If the PD has different values (thus is non-symmetric), but still has the same ordinal relations on outcomes (player 1 prefers (D,C) to (C,C) to (D,D) to (C,D), etc...), is (C,C) still the unique level 1 metathreat equilibrium?
The metathreat equilibrium definitely depends on the cardinal utilities, for example if uCC<uCD+uDC2 then the “nicer bots” are not an equilibrium since both agents want to lower the probability of cooperation.
The thermodynamic approach is needed since we need a way to rule out some of the metathreat Nash equilibria. A pair of defect bots is a Nash equilibrium in the level 1 metathreat game but this equilibrium is “unstable” since there are bots that are equally good against the defect bot but do better against other bots. So we need some notion of equilibrium stability, but the usual trembling hand is not good enough since for trembling hand equilibrium stability wrt an arbitrary totally mixed perturbation is sufficient (and it is possible to construct perturbations that stabilize “bad” equilibria). The thermodynamic equilibrium works by considering perturbations in which lower utility alternatives get higher order infinitesimal probabilities. Together with the ease to achieve this equilibrium in a natural decision theory, it becomes an attractive choice.
Very interesting! I’m eager to see where this will lead.
If the PD has different values (thus is non-symmetric), but still has the same ordinal relations on outcomes (player 1 prefers (D,C) to (C,C) to (D,D) to (C,D), etc...), is (C,C) still the unique level 1 metathreat equilibrium?
And what inspired this “thermodynamic” approach?
The metathreat equilibrium definitely depends on the cardinal utilities, for example if uCC<uCD+uDC2 then the “nicer bots” are not an equilibrium since both agents want to lower the probability of cooperation.
The thermodynamic approach is needed since we need a way to rule out some of the metathreat Nash equilibria. A pair of defect bots is a Nash equilibrium in the level 1 metathreat game but this equilibrium is “unstable” since there are bots that are equally good against the defect bot but do better against other bots. So we need some notion of equilibrium stability, but the usual trembling hand is not good enough since for trembling hand equilibrium stability wrt an arbitrary totally mixed perturbation is sufficient (and it is possible to construct perturbations that stabilize “bad” equilibria). The thermodynamic equilibrium works by considering perturbations in which lower utility alternatives get higher order infinitesimal probabilities. Together with the ease to achieve this equilibrium in a natural decision theory, it becomes an attractive choice.
Does this equilibrium notion differ meaningfully from proper equilibria?