The way Wallace expresses the theorem in the paper is misleading. The theorem does rule out utility functions that recover preferences if expected utility is calculated using non-Born probabilities. I think many people, on first glance, interpret the theorem the way you did, which makes it much less impressive, and not really a justification of the Born probabilities at all.
The way to read the theorem is not ”… there is a unique utility function with the property that...”, It is ”...there is a unique utility function and it has the property that...”
Ah, I see. Yes, that kind of result is remarkable.
I don’t know what you mean, though, by “there is a unique (up to affine transformations) utility function over the rewards”. If you mean there is a unique utility function on rewards that recovers the agent’s preferences on rewards, that’s false. But I don’t know what else you could mean.
Ah, I thought of a charitable interpretation of “there is a unique (up to affine transformations) utility function over the rewards”. Given a preference ordering on sequences of rewards, there is a unique utility function on individual rewards that recovers that preference ordering. I believe this because if rewards are repeatable, the diachronicity hypothesis implies that any utility function on sequences of rewards must be additive. (We also need a hypothesis ruling out lexicographically-ordered preferences.)
You’re right. Diachronic consistency is required to establish the uniqueness of the utility function. Also, Wallace does include continuity axioms that rule out lexically ordered preferences, but I left them out of my summary for the sake of simplicity.
The way Wallace expresses the theorem in the paper is misleading. The theorem does rule out utility functions that recover preferences if expected utility is calculated using non-Born probabilities. I think many people, on first glance, interpret the theorem the way you did, which makes it much less impressive, and not really a justification of the Born probabilities at all.
The way to read the theorem is not ”… there is a unique utility function with the property that...”, It is ”...there is a unique utility function and it has the property that...”
Ah, I see. Yes, that kind of result is remarkable.
I don’t know what you mean, though, by “there is a unique (up to affine transformations) utility function over the rewards”. If you mean there is a unique utility function on rewards that recovers the agent’s preferences on rewards, that’s false. But I don’t know what else you could mean.
EDIT: See my comment below.
Ah, I thought of a charitable interpretation of “there is a unique (up to affine transformations) utility function over the rewards”. Given a preference ordering on sequences of rewards, there is a unique utility function on individual rewards that recovers that preference ordering. I believe this because if rewards are repeatable, the diachronicity hypothesis implies that any utility function on sequences of rewards must be additive. (We also need a hypothesis ruling out lexicographically-ordered preferences.)
You’re right. Diachronic consistency is required to establish the uniqueness of the utility function. Also, Wallace does include continuity axioms that rule out lexically ordered preferences, but I left them out of my summary for the sake of simplicity.