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)
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)