My preferred way of resolving it is treating the process of “arguing over which equilibrium to move to” as a bargaining game, and just find a ROSE point from that bargaining game. If there’s multiple ROSE points, well, fire up another round of bargaining. This repeated process should very rapidly have the disagreement points close in on the Pareto frontier, until everyone is just arguing over very tiny slices of utility.
This is imperfectly specified, though, because I’m not entirely sure what the disagreement points would be, because I’m not sure how the “don’t let foes get more than what you think is fair” strategy generalizes to >2 players. Maaaybe disagreement-point-invariance comes in clutch here? If everyone agrees that an outcome as bad or worse than their least-preferred ROSE point would happen if they disagreed, then disagreement-point-invariance should come in to have everyone agree that it doesn’t really matter exactly where that disagreement point is.
Or maybe there’s some nice principled property that some equilibria have, which others don’t, that lets us winnow down the field of equilibria somewhat. Maybe that could happen.
I’m still pretty unsure, but “iterate the bargaining process to argue over which equilibria to go to, you don’t get an infinite regress because you rapidly home in on the Pareto frontier with each extra round you add” is my best bad idea for it.
EDIT: John Harsanyi had the same idea. He apparently had some example where there were multiple CoCo equilibria and his suggestion was that a second round of bargaining could be initiated over which equilibria to pick, but that in general, it’d be so hard to compute the n-person Pareto frontier for large n, that an equilibria might be stable because nobody can find a different equilibria nearby to aim for.
So this problem isn’t unique to ROSE points in full generality (CoCo equilibria have the exact same issue), it’s just that ROSE is the only one that produces multiple solutions for bargaining games, while CoCo only returns a single solution for bargaining games. (bargaining games are a subset of games in general)
They approach the Pareto frontier, but never in fact reach it, from what I understand. So wouldn’t this just move the problem into the meta level? What’s stopping the Players from going through the threat escalation spiral to force their ideal ROSE point?
Sure they may be very tiny slices of utility in contention, but then it only takes one obstinate Player to threaten all utility to ensure they lose zero.
What I mean is, the players hit each other up, are like “yo, let’s decide on which ROSE point in the object-level game we’re heading towards” Of course, they don’t manage to settle on what equilibrium to go for in the resulting bargaining game, because, again, multiple ROSE points might show up in the bargaining game. But, the ROSE points in the bargaining game are in a restricted enough zone (what with that whole “must be better than the random-dictator point” thing) to seriously constrain the possibilities in the object-level game. “Worst ROSE point for Alice in the object-level-game” is a whole lot worse for her than “Worst ROSE point for Alice in the bargaining game about what to do in the object-level-game”.
So, the players should be able to ratchet up their disagreement point and go “even if the next round of bargaining fails, at least we can promise that everyone does this well, right? Sure, everyone’s going for their own idea of fairness, but even if Alice ends up with her worst ROSE point in the bargaining game, her utility is going to be at least this high, and similar for everyone else.”
And so, each step of bargaining that successfully happens ratchets up the disagreement point closer to the Pareto frontier, in a way that should quickly converge. If someone disagrees on step 3, then the step-3 disagreement point gets played, which isn’t that short of the Pareto frontier. And if someone doesn’t have time for all this bargaining, they can just break things off at step 10 or something, that’s just about as good as going all the way to infinity.
Or at least, it should work like this. I haven’t proved that it does, and it depends on things like “what does ROSE bargaining look like for n players” and “does the random-dictator-point-dominance thing still hold in the n-player case” and “what’s the analogue of that strategy where you block your foe from getting more than X utility, when there are multiple foes?”. But this disagreement-point ratcheting is a strategy that address your worries with “ever-smaller pieces of the problem live on higher meta-levels, so the process of adding layers of meta actually converges to solving the problem, and breaking it off early solves most of the problem”
Regarding your last comment, yes, you could always just have a foe that’s a jerk, but you can at least act so they don’t gain from being an jerk, in a way robust against you and a foe having slightly different definitions of “jerk”.
Regarding your last comment, yes, you could always just have a foe that’s a jerk, but you can at least act so they don’t gain from being an jerk, in a way robust against you and a foe having slightly different definitions of “jerk”.
How would someone ensure that the jerk does not gain from their threats regarding the choice of ROSE points, assuming Players cannot exit the game?
Reading Clarifications 1, 2, and 3 seems to imply that it would not be useful or applicable in this scenario.
Clarification 1: Yes, utilities are invariant up to a positive affine transformation so there’s no canonical way to split utilities evenly. Hence the part about “Assume a magical solution N which gives us the fair division.” If we knew the exact properties of how to implement this magical solution, taking it at first for magical, that might give us some idea of what N should be, too.
Clarification 2: The way this might work is that you pick a series of increasingly unfair-to-you, increasingly worse-for-the-other-player outcomes whose first element is what you deem the fair Pareto outcome: (100, 100), (98, 99), (96, 98). Perhaps stop well short of Nash if the skew becomes too extreme. Drop to Nash as the last resort. The other agent does the same, starting with their own ideal of fairness on the Pareto boundary. Unless one of you has a completely skewed idea of fairness, you should be able to meet somewhere in the middle. Both of you will do worse against a fixed opponent’s strategy by unilaterally adopting more self-favoring ideas of fairness. Both of you will do worse in expectation against potentially exploitive opponents by unilaterally adopting looser ideas of fairness. This gives everyone an incentive to obey the Galactic Schelling Point and be fair about it. You should not be picking the descending sequence in an agent-dependent way that incentivizes, at cost to you, skewed claims about fairness.
Clarification 3: You must take into account the other agent’s costs and other opportunities when ensuring that the net outcome, in terms of final utilities, is worse for them than the reward offered for ‘fair’ cooperation. Offering them the chance to buy half as many paperclips at a lower, less fair price, does no good if they can go next door, get the same offer again, and buy the same number of paperclips at a lower total price.
Can you show how this would lead to a ‘jerk’ never gaining in the aforementioned scenario?
My preferred way of resolving it is treating the process of “arguing over which equilibrium to move to” as a bargaining game, and just find a ROSE point from that bargaining game. If there’s multiple ROSE points, well, fire up another round of bargaining. This repeated process should very rapidly have the disagreement points close in on the Pareto frontier, until everyone is just arguing over very tiny slices of utility.
This is imperfectly specified, though, because I’m not entirely sure what the disagreement points would be, because I’m not sure how the “don’t let foes get more than what you think is fair” strategy generalizes to >2 players. Maaaybe disagreement-point-invariance comes in clutch here? If everyone agrees that an outcome as bad or worse than their least-preferred ROSE point would happen if they disagreed, then disagreement-point-invariance should come in to have everyone agree that it doesn’t really matter exactly where that disagreement point is.
Or maybe there’s some nice principled property that some equilibria have, which others don’t, that lets us winnow down the field of equilibria somewhat. Maybe that could happen.
I’m still pretty unsure, but “iterate the bargaining process to argue over which equilibria to go to, you don’t get an infinite regress because you rapidly home in on the Pareto frontier with each extra round you add” is my best bad idea for it.
EDIT: John Harsanyi had the same idea. He apparently had some example where there were multiple CoCo equilibria and his suggestion was that a second round of bargaining could be initiated over which equilibria to pick, but that in general, it’d be so hard to compute the n-person Pareto frontier for large n, that an equilibria might be stable because nobody can find a different equilibria nearby to aim for.
So this problem isn’t unique to ROSE points in full generality (CoCo equilibria have the exact same issue), it’s just that ROSE is the only one that produces multiple solutions for bargaining games, while CoCo only returns a single solution for bargaining games. (bargaining games are a subset of games in general)
They approach the Pareto frontier, but never in fact reach it, from what I understand. So wouldn’t this just move the problem into the meta level? What’s stopping the Players from going through the threat escalation spiral to force their ideal ROSE point?
Sure they may be very tiny slices of utility in contention, but then it only takes one obstinate Player to threaten all utility to ensure they lose zero.
What I mean is, the players hit each other up, are like “yo, let’s decide on which ROSE point in the object-level game we’re heading towards”
Of course, they don’t manage to settle on what equilibrium to go for in the resulting bargaining game, because, again, multiple ROSE points might show up in the bargaining game.
But, the ROSE points in the bargaining game are in a restricted enough zone (what with that whole “must be better than the random-dictator point” thing) to seriously constrain the possibilities in the object-level game. “Worst ROSE point for Alice in the object-level-game” is a whole lot worse for her than “Worst ROSE point for Alice in the bargaining game about what to do in the object-level-game”.
So, the players should be able to ratchet up their disagreement point and go “even if the next round of bargaining fails, at least we can promise that everyone does this well, right? Sure, everyone’s going for their own idea of fairness, but even if Alice ends up with her worst ROSE point in the bargaining game, her utility is going to be at least this high, and similar for everyone else.”
And so, each step of bargaining that successfully happens ratchets up the disagreement point closer to the Pareto frontier, in a way that should quickly converge. If someone disagrees on step 3, then the step-3 disagreement point gets played, which isn’t that short of the Pareto frontier. And if someone doesn’t have time for all this bargaining, they can just break things off at step 10 or something, that’s just about as good as going all the way to infinity.
Or at least, it should work like this. I haven’t proved that it does, and it depends on things like “what does ROSE bargaining look like for n players” and “does the random-dictator-point-dominance thing still hold in the n-player case” and “what’s the analogue of that strategy where you block your foe from getting more than X utility, when there are multiple foes?”. But this disagreement-point ratcheting is a strategy that address your worries with “ever-smaller pieces of the problem live on higher meta-levels, so the process of adding layers of meta actually converges to solving the problem, and breaking it off early solves most of the problem”
Regarding your last comment, yes, you could always just have a foe that’s a jerk, but you can at least act so they don’t gain from being an jerk, in a way robust against you and a foe having slightly different definitions of “jerk”.
How would someone ensure that the jerk does not gain from their threats regarding the choice of ROSE points, assuming Players cannot exit the game?
With this.
Reading Clarifications 1, 2, and 3 seems to imply that it would not be useful or applicable in this scenario.
Can you show how this would lead to a ‘jerk’ never gaining in the aforementioned scenario?