I’m hard-pressed to this of any more I could want from [the coco-value] (aside from easy extensions to bigger classes of games).
Invariance to affine transformations of players’ utility functions. This solution requires that both players value outcomes in a common currency, plus the physical ability to transfer utility in this currency outside the game (unless there are two outcomes o_1 and o_2 of the game such that A(o_1) + B(o_1) = A(o_2) + B(o_2) = max_o A(o) + B(o), and such that A(o_1) >= A’s coco-value >= A(o_2), in which case the players can decide to play the convex combination of these two outcomes that gives each player their coco-value, but this only solves the utility transfer problem, it doesn’t make the solution invariant under affine transformations).
Invariance to affine transformations of players’ utility functions.
This is done by the transfer function between the players, since if I redefine my utility to be 10 times its previous value, then it takes only one of your utility to give me 10, and 10 of my utility to give you one.
Now, of course, you want to lie about the transfer function instead of your utility; “no, I don’t like dollars you’ve given me as much as dollars I’ve earned myself.”
Oh, I definitely agree. I meant it’s hard to hope for anything more inside environments with transferable/quasilinear utility. It’s a big assumption, but I’ve resigned myself to it somewhat since we need it for most of the positive results in mechanism design.
Invariance to affine transformations of players’ utility functions. This solution requires that both players value outcomes in a common currency, plus the physical ability to transfer utility in this currency outside the game (unless there are two outcomes o_1 and o_2 of the game such that A(o_1) + B(o_1) = A(o_2) + B(o_2) = max_o A(o) + B(o), and such that A(o_1) >= A’s coco-value >= A(o_2), in which case the players can decide to play the convex combination of these two outcomes that gives each player their coco-value, but this only solves the utility transfer problem, it doesn’t make the solution invariant under affine transformations).
Invariance of the players’ utility functions by the same affine transformation, or by independent transformations?
Independent.
This is done by the transfer function between the players, since if I redefine my utility to be 10 times its previous value, then it takes only one of your utility to give me 10, and 10 of my utility to give you one.
Now, of course, you want to lie about the transfer function instead of your utility; “no, I don’t like dollars you’ve given me as much as dollars I’ve earned myself.”
Oh, I definitely agree. I meant it’s hard to hope for anything more inside environments with transferable/quasilinear utility. It’s a big assumption, but I’ve resigned myself to it somewhat since we need it for most of the positive results in mechanism design.