Ihor Kendiukhov

If your behavior violates the axioms, you can be Dutch-booked, turned into a money pump, exploited by anyone who notices the inconsistency.

You don't want to be a money pump, do you?

Then you must maximize expected utility.

QED There are four axioms in the von Neumann-Morgenstern framework.

Completeness, transitivity, continuity, and independence.

Three of them are relatively uncontroversial.

The fourth, independence, does enormous structural work, it is the axiom that forces preferences to be linear in probabilities, which is mathematically equivalent to requiring that preferences be representable as the expected value of a utility function.

Without independence, you still have a well-defined preference functional, by Debreu's theorem, given the other axioms, you can still order outcomes, you can still make consistent choices, but you are no longer constrained to maximize expected utility specifically.

Independence is the fifth postulate of decision theory.

And just as with Euclid's fifth, I believe, the resolution is not to keep trying harder to justify it but to ask.

What happens when we drop it?

Does it perhaps describe actual rational behavior better?

The answer, I will argue, is yes on all three counts.

Dropping independence does not lead to irrationality or exploitability.

Several well-known alternatives to expected utility theory exist precisely because they relax independence, and they do so for a reason.

Ergodicity economics, in particular, offers a principled and parsimonious replacement that derives the appropriate evaluation function from the dynamics of the stochastic process the agent is embedded in, rather than postulating an ad hoc utility function and taking its expectation.

And the less wrong community's own research into updateless decision theory has been converging on the same conclusion from a completely different direction.

That the most reflectively stable agents may be precisely those who violate the independence axiom.

Heading.

A tale of two utilities.