We’re not actually talking about the VNM formalism here. That’s why the “in the absence of uncertainty” part is important.
We have a finite set of world-states and preferences over those world-states. We do not care about preferences over random mixtures of world-states, we don’t even have a notion of random mixtures, just the deterministic states themselves. We want a utility function which encodes our preferences over those deterministic world-states.
In the absence of uncertainty, we don’t actually need the continuity assumption or the independence assumption for anything. They don’t even make sense; we need a notion of random mixtures just to state those assumptions. VNM utility needs those because it’s trying to get expected utility maximization right out the door. But we’re not starting from VNM utility, we’re starting from deterministic utility.
Whether we need completeness or not is more debatable. It depends on how we’re interpreting missing preferences. If we interpret missing preferences as “I don’t know”, then it seems natural to allow the utility function to give any possible preference for that pair. In that case, lack of completeness may mean our utility function isn’t unique, but it won’t prevent a utility function from existing.
It’s exactly the same in Eliezer’s post. His circular preferences argument comes before random outcomes are even introduced. There’s no notion of randomness at that point, no notion of lotteries, so he’s not talking about VNM utility. The circular preferences argument is not the VNM utility theorem, it is a separate thing which makes a different claim under weaker assumptions. That does not make it incorrect.
We’re not actually talking about the VNM formalism here. That’s why the “in the absence of uncertainty” part is important.
We have a finite set of world-states and preferences over those world-states. We do not care about preferences over random mixtures of world-states, we don’t even have a notion of random mixtures, just the deterministic states themselves. We want a utility function which encodes our preferences over those deterministic world-states.
In the absence of uncertainty, we don’t actually need the continuity assumption or the independence assumption for anything. They don’t even make sense; we need a notion of random mixtures just to state those assumptions. VNM utility needs those because it’s trying to get expected utility maximization right out the door. But we’re not starting from VNM utility, we’re starting from deterministic utility.
Whether we need completeness or not is more debatable. It depends on how we’re interpreting missing preferences. If we interpret missing preferences as “I don’t know”, then it seems natural to allow the utility function to give any possible preference for that pair. In that case, lack of completeness may mean our utility function isn’t unique, but it won’t prevent a utility function from existing.
It’s exactly the same in Eliezer’s post. His circular preferences argument comes before random outcomes are even introduced. There’s no notion of randomness at that point, no notion of lotteries, so he’s not talking about VNM utility. The circular preferences argument is not the VNM utility theorem, it is a separate thing which makes a different claim under weaker assumptions. That does not make it incorrect.