2. In the Destruction Game, does everyone get the ability to destroy arbitrary amounts of utility, or is how much utility they are able to destroy part of the setup of the game, such that you can have games where e.g. one player gets a powerful button and another player gets a weak one?
For 1, it’s just intrinsically mathematically appealing (continuity is always really nice when you can get it), and also because of an intution that if your foe experiences a tiny preference perturbation, you should be able to use small conditional payments to replicate their original preferences/incentive structure and start negotiating with that, instead.
I should also note that nowhere in the visual proof of the ROSE value for the toy case, is continuity used. Continuity just happens to appear.
For 2, yes, it’s part of game setup. The buttons are of whatever intensity you want (but they have to be intensity-capped somewhere for technical reasons regarding compactness). Looking at the setup, for each player pair i,j, Di,j is the cap for how much of j’s utility that i can destroy. These can vary, as long as they’re nonnegative and not infinite. From this, it’s clear “Alice has a powerful button, Bob has a weak one” is one of the possibilities, that would just mean Da,b>>Db,a. There isn’t an assumption that everyone has an equally powerful button, because then you could argue that everyone just has an equal strength threat and then it wouldn’t be much of a threat-resistance desideratum, now would it? Heck, you can even give one player a powerful button and the other a zero-strength button that has no effect, that fits in the formalism.
So the theorem is actually saying “for all members of the family of destruction games with the button caps set wherever the heck you want, the payoffs are the same as the original game”.
OK, thanks! Continuity does seem appealing to me but it seems negotiable; if you can find an even more threat-resistant bargaining solution (or an equally threat-resistant one that has some other nice property) I’d prefer it to this one even if it lacked continuity.
Awesome work!
Misc. thoughts and questions as I go along:
1. Why is Continuity appealing/important again?
2. In the Destruction Game, does everyone get the ability to destroy arbitrary amounts of utility, or is how much utility they are able to destroy part of the setup of the game, such that you can have games where e.g. one player gets a powerful button and another player gets a weak one?
For 1, it’s just intrinsically mathematically appealing (continuity is always really nice when you can get it), and also because of an intution that if your foe experiences a tiny preference perturbation, you should be able to use small conditional payments to replicate their original preferences/incentive structure and start negotiating with that, instead.
I should also note that nowhere in the visual proof of the ROSE value for the toy case, is continuity used. Continuity just happens to appear.
For 2, yes, it’s part of game setup. The buttons are of whatever intensity you want (but they have to be intensity-capped somewhere for technical reasons regarding compactness). Looking at the setup, for each player pair i,j, Di,j is the cap for how much of j’s utility that i can destroy. These can vary, as long as they’re nonnegative and not infinite. From this, it’s clear “Alice has a powerful button, Bob has a weak one” is one of the possibilities, that would just mean Da,b>>Db,a. There isn’t an assumption that everyone has an equally powerful button, because then you could argue that everyone just has an equal strength threat and then it wouldn’t be much of a threat-resistance desideratum, now would it? Heck, you can even give one player a powerful button and the other a zero-strength button that has no effect, that fits in the formalism.
So the theorem is actually saying “for all members of the family of destruction games with the button caps set wherever the heck you want, the payoffs are the same as the original game”.
OK, thanks! Continuity does seem appealing to me but it seems negotiable; if you can find an even more threat-resistant bargaining solution (or an equally threat-resistant one that has some other nice property) I’d prefer it to this one even if it lacked continuity.