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pragmatist comments on Problems of the Deutsch-Wallace version of Many Worlds - Less Wrong Discussion
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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.