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.”
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.”