I tried to generalize Eliezer’s outcomes to functions, and realized if both agents are unexploitable, the optimal functions to pick would lead to Stuart’s solution precisely. Stuart’s solution allows agents to arbitrarily penalize the other though, which is why I like extending Eliezer’s concept better. Details below, P.S. I tried to post this in a comment above, but in editing it I appear to have somehow made it invisible, at least to me. Sorry for repost if you can indeed see all the comments I’ve made.
It seems the logical extension of your finitely many step-downs in “fairness” would be to define a function f(your_utility) which returns the greatest utility you will accept the other agent receiving for that utility you receive. The domain of this function should run from wherever your magical fairness point is down to the Nash equilibrium. As long as it is monotonically increasing, that should ensure unexploitability for the same reasons your finite version does. The offer both agents should make is at the greatest intersection point of these functions, with one of them inverted to put them on the same axes. (This intersection is guaranteed to exist in the only interesting case, where the agents do not accept as fair enough each other’s magical fairness point)
Curiously, if both agents use this strategy, then both agents seem to be incentivized to have their function have as much “skew” (as EY defined it in clarification 2) as possible, as both functions are monotonically increasing so decreasing your opponents share can only decrease your own. Asymptotically and choosing these functions optimally, this means that both agents will end up getting what the other agent thinks is fair, minus a vanishingly small factor!
Let me know if my reasoning above is transparent. If not, I can clarify, but I’ll avoid expending the extra effort revising further if what I already have is clear enough. Also, just simple confirmation that I didn’t make a silly logical mistake/post something well known in the community already is always appreciated.
I tried to generalize Eliezer’s outcomes to functions, and realized if both agents are unexploitable, the optimal functions to pick would lead to Stuart’s solution precisely. Stuart’s solution allows agents to arbitrarily penalize the other though, which is why I like extending Eliezer’s concept better.
I concur, my reasoning likely overlaps in parts. I particularly like your observation about the asymptotic behaviour when choosing the functions optimally.
I tried to generalize Eliezer’s outcomes to functions, and realized if both agents are unexploitable, the optimal functions to pick would lead to Stuart’s solution precisely. Stuart’s solution allows agents to arbitrarily penalize the other though, which is why I like extending Eliezer’s concept better. Details below, P.S. I tried to post this in a comment above, but in editing it I appear to have somehow made it invisible, at least to me. Sorry for repost if you can indeed see all the comments I’ve made.
It seems the logical extension of your finitely many step-downs in “fairness” would be to define a function f(your_utility) which returns the greatest utility you will accept the other agent receiving for that utility you receive. The domain of this function should run from wherever your magical fairness point is down to the Nash equilibrium. As long as it is monotonically increasing, that should ensure unexploitability for the same reasons your finite version does. The offer both agents should make is at the greatest intersection point of these functions, with one of them inverted to put them on the same axes. (This intersection is guaranteed to exist in the only interesting case, where the agents do not accept as fair enough each other’s magical fairness point)
Curiously, if both agents use this strategy, then both agents seem to be incentivized to have their function have as much “skew” (as EY defined it in clarification 2) as possible, as both functions are monotonically increasing so decreasing your opponents share can only decrease your own. Asymptotically and choosing these functions optimally, this means that both agents will end up getting what the other agent thinks is fair, minus a vanishingly small factor!
Let me know if my reasoning above is transparent. If not, I can clarify, but I’ll avoid expending the extra effort revising further if what I already have is clear enough. Also, just simple confirmation that I didn’t make a silly logical mistake/post something well known in the community already is always appreciated.
I concur, my reasoning likely overlaps in parts. I particularly like your observation about the asymptotic behaviour when choosing the functions optimally.