In the first part, the two respective properties of the two definitions of chaaness you mentioned apply after rescaling and shifting of utility functions is done, right? I.e., the properties actually say “after rescaling and shifting the points, if you move the Pareto-frontier points for a player up, they should get more utility” and “untaken options are irrelevant if you don’t change the scale after removing them”. Now, I don’t see why these properties are interesting and what they correspond to in real life. In contrast, if they applied before rescaling and shifting, then they would be quite interesting. So, can you please elaborate why they are interesting as they are and what they actually mean as they are?
Actually, they apply anyways in all circumstances, not just after the rescaling and shifting is done! Scale-and-shift invariance means that no matter how you stretch and shift the two axes, the bargaining solution always hits the same probability-distribution over outcomes, so monotonicity means “if you increase the payoff numbers you assign for some or all of the outcomes, the Pareto frontier point you hit will give you an increased number for your utility score over what it’d be otherwise” (no matter how you scale-and-shift). And independence of irrelevant alternatives says “you can remove any option that you have 0 probability of taking and you’ll still get the same probability-distribution over outcomes as you would in the original game” (no matter how you scale-and-shift)
In the first part, the two respective properties of the two definitions of chaaness you mentioned apply after rescaling and shifting of utility functions is done, right? I.e., the properties actually say “after rescaling and shifting the points, if you move the Pareto-frontier points for a player up, they should get more utility” and “untaken options are irrelevant if you don’t change the scale after removing them”. Now, I don’t see why these properties are interesting and what they correspond to in real life. In contrast, if they applied before rescaling and shifting, then they would be quite interesting. So, can you please elaborate why they are interesting as they are and what they actually mean as they are?
Actually, they apply anyways in all circumstances, not just after the rescaling and shifting is done! Scale-and-shift invariance means that no matter how you stretch and shift the two axes, the bargaining solution always hits the same probability-distribution over outcomes, so monotonicity means “if you increase the payoff numbers you assign for some or all of the outcomes, the Pareto frontier point you hit will give you an increased number for your utility score over what it’d be otherwise” (no matter how you scale-and-shift). And independence of irrelevant alternatives says “you can remove any option that you have 0 probability of taking and you’ll still get the same probability-distribution over outcomes as you would in the original game” (no matter how you scale-and-shift)