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.
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.