Suppose the axes are not showing the utilities of independent agents, but a single agent’s utility from two arguments, say ice cream and sleep. We see that the (2,2) point with its total utility of 4 is preferred by this single agent. Halve the utility from ice cream; the agent now prefers the former (1.2, 2.6) point, which has been transformed to a (0.6, 2.6) point with a total utility of 3.2. Clearly then this is not the same utility function. This demonstrates that scaling one axis in a combined utility function is not an affine transformation: It does not preserve ordering.
I don’t see why you expect two independant agents with their own coherent utility functions to have a coherent combined utility function, and it seems that the point of Stuart’s argument is to show that there are cases where they can’t.
Ok; I thought of a better way to phrase Stuart’s point. Suppose there are five alternatives, and I rank them 1-2-3-4-5, but you rank them 5-4-3-2-1. If we are equal in power we will compromise on 3. (Well… given some simplifying assumptions, anyway. It’s quite possible that you are almost indifferent between 2 and 3, but I care a lot about that gap. If so, even if we are equal in power I will likely commit a lot more resources to the fight, and drag the compromise up to 2.) But if the available options had been 1, 2, and 3, we would instead have compromised on 2. This demonstrates that removing options changes the outcome.
However, I think there is a problem with carrying the “irrelevant alternatives” axiom into a two-agent problem. If I have A>B>C, then I should choose A whether or not C is an option; fine. But this needn’t be true of problems with multiple agents, because that phrase “we will compromise on” is hiding rather a lot of complexity that doesn’t have anything to do with utility functions, per se. Options 4 and 5 are not, in fact, irrelevant; they are bargaining chips. Removing one side’s bargaining chips breaks the symmetry; it is equivalent to giving the other side more power. Suppose I had left the options as they were, but specified that the agent whose utility is on the y axis suddenly gets a lot more bargaining power; would we then expect the decision to be option 3? Surely not. And this is exactly what is accomplished by asymmetrically removing options.
The problem rises from breaking the game-theoretic symmetry and asserting that only the utility symmetry is important.
The most common formulation of IIA precisely assumes that there is no such thing as “bargaining chips”. So yes, you could rewrite the point of my post as: any symmetric bargaining solution will have bargaining chips.
The point is that if the transformation that Stuart uses were applied to a single agent, it would convert a coherent utility function into an incoherent one; therefore it cannot demonstrate anything about the incoherence of combined utility functions. It is too general—in fact, it is a Fully General Counterargument to the existence of utility functions with more than one input. It could well be the case that independent agents cannot have a coherent combined utility function, but this argument does not demonstrate it unless you also wish to assert that single-agent utility functions cannot consist of linear additions of sub-utilities.
in fact, it is a Fully General Counterargument to the existence of utility functions with more than one input
No, it’s not, because it is not even talking about a utility function with more than one input. It is talking about two completely seperate utility functions. A single utility function with multiple inputs has to include a scaling between the inputs and therefore is not described by Stuart’s argument, which exploits the lack of such scaling between two seperate utility functions.
Suppose the axes are not showing the utilities of independent agents, but a single agent’s utility from two arguments, say ice cream and sleep. We see that the (2,2) point with its total utility of 4 is preferred by this single agent. Halve the utility from ice cream; the agent now prefers the former (1.2, 2.6) point, which has been transformed to a (0.6, 2.6) point with a total utility of 3.2. Clearly then this is not the same utility function. This demonstrates that scaling one axis in a combined utility function is not an affine transformation: It does not preserve ordering.
I don’t see why you expect two independant agents with their own coherent utility functions to have a coherent combined utility function, and it seems that the point of Stuart’s argument is to show that there are cases where they can’t.
Ok; I thought of a better way to phrase Stuart’s point. Suppose there are five alternatives, and I rank them 1-2-3-4-5, but you rank them 5-4-3-2-1. If we are equal in power we will compromise on 3. (Well… given some simplifying assumptions, anyway. It’s quite possible that you are almost indifferent between 2 and 3, but I care a lot about that gap. If so, even if we are equal in power I will likely commit a lot more resources to the fight, and drag the compromise up to 2.) But if the available options had been 1, 2, and 3, we would instead have compromised on 2. This demonstrates that removing options changes the outcome.
However, I think there is a problem with carrying the “irrelevant alternatives” axiom into a two-agent problem. If I have A>B>C, then I should choose A whether or not C is an option; fine. But this needn’t be true of problems with multiple agents, because that phrase “we will compromise on” is hiding rather a lot of complexity that doesn’t have anything to do with utility functions, per se. Options 4 and 5 are not, in fact, irrelevant; they are bargaining chips. Removing one side’s bargaining chips breaks the symmetry; it is equivalent to giving the other side more power. Suppose I had left the options as they were, but specified that the agent whose utility is on the y axis suddenly gets a lot more bargaining power; would we then expect the decision to be option 3? Surely not. And this is exactly what is accomplished by asymmetrically removing options.
The problem rises from breaking the game-theoretic symmetry and asserting that only the utility symmetry is important.
The most common formulation of IIA precisely assumes that there is no such thing as “bargaining chips”. So yes, you could rewrite the point of my post as: any symmetric bargaining solution will have bargaining chips.
The point is that if the transformation that Stuart uses were applied to a single agent, it would convert a coherent utility function into an incoherent one; therefore it cannot demonstrate anything about the incoherence of combined utility functions. It is too general—in fact, it is a Fully General Counterargument to the existence of utility functions with more than one input. It could well be the case that independent agents cannot have a coherent combined utility function, but this argument does not demonstrate it unless you also wish to assert that single-agent utility functions cannot consist of linear additions of sub-utilities.
No, it’s not, because it is not even talking about a utility function with more than one input. It is talking about two completely seperate utility functions. A single utility function with multiple inputs has to include a scaling between the inputs and therefore is not described by Stuart’s argument, which exploits the lack of such scaling between two seperate utility functions.