I might have missed something but isn’t the “solution” to the concerns about the completeness money pump equivalent to the agent becoming complete.
E.g. after the agent has chose B over A it now effectively has a preference of B over A-.
I haven’t worked this through e.g. the proof of VNM etc. but are we sure this weaker notion of completeness might end up being enough to still get the relevant conclusions?
(quite busy might have a bit more of a think about this later)
Nice point. The rough answer is ‘Yes, but only once the agent has turned down a sufficiently wide array of options.’ Depending on the details, that might never happen or only happen after a very long time.
I’ve had a quick think about the more precise answer, and I think it is:
The agent’s preferences will be functionally complete once and only once it is the case that, for all pairs of options between which the agent has a preferential gap, the agent has turned down an option that is strictly preferred to one of the options in the pair.
I had a similar thought to Shiny. Am I correct that an agent following your suggested policy (“‘if I previously turned down some option X, I will not choose any option that I strictly disprefer to X.’”) would never *appear* to violate completeness from the perspective of an observer that could only see their decisions and not their internal state? And assuming completeness is all we need to get to full utility maximization, does that mean an agent following your policy would act like a utility maximizer to an observer?
But suppose we set that complication aside. Suppose we know this about an agent’s preferences:
There is some option A such that the agent strictly prefers A+$1
Then we can observe violations of Completeness. Suppose that we first offer our agent a choice between A and some other option B, and that the agent chooses A. Then we give the agent the chance to trade in A for B, and the agent takes the trade. That indicates that the agent does not strictly prefer A to B and does not strictly prefer B to A. Two possibilities remain: either the agent is indifferent between A and B, or the agent has a preferential gap between A and B.
Now we offer our agent another choice: stick with B, or trade it in for A+$1. If the agent is indifferent between A and B, they will strictly prefer A+$1 to B (because indifference is sensitive to all sweetenings and sourings), and so we will observe the agent taking the trade. If we observe that the agent doesn’t take the trade, then they must have a preferential gap between A and B, and so their preferences must be incomplete.
I might have missed something but isn’t the “solution” to the concerns about the completeness money pump equivalent to the agent becoming complete.
E.g. after the agent has chose B over A it now effectively has a preference of B over A-.
I haven’t worked this through e.g. the proof of VNM etc. but are we sure this weaker notion of completeness might end up being enough to still get the relevant conclusions?
(quite busy might have a bit more of a think about this later)
Nice point. The rough answer is ‘Yes, but only once the agent has turned down a sufficiently wide array of options.’ Depending on the details, that might never happen or only happen after a very long time.
I’ve had a quick think about the more precise answer, and I think it is:
The agent’s preferences will be functionally complete once and only once it is the case that, for all pairs of options between which the agent has a preferential gap, the agent has turned down an option that is strictly preferred to one of the options in the pair.
I had a similar thought to Shiny. Am I correct that an agent following your suggested policy (“‘if I previously turned down some option X, I will not choose any option that I strictly disprefer to X.’ ”) would never *appear* to violate completeness from the perspective of an observer that could only see their decisions and not their internal state? And assuming completeness is all we need to get to full utility maximization, does that mean an agent following your policy would act like a utility maximizer to an observer?
There’s a complication here related to a point that Rohin makes : if we can only see an agent’s decisions and we know nothing about its preferences, all behavior can be rationalized as EU maximization.
But suppose we set that complication aside. Suppose we know this about an agent’s preferences:
There is some option A such that the agent strictly prefers A+$1
Then we can observe violations of Completeness. Suppose that we first offer our agent a choice between A and some other option B, and that the agent chooses A. Then we give the agent the chance to trade in A for B, and the agent takes the trade. That indicates that the agent does not strictly prefer A to B and does not strictly prefer B to A. Two possibilities remain: either the agent is indifferent between A and B, or the agent has a preferential gap between A and B.
Now we offer our agent another choice: stick with B, or trade it in for A+$1. If the agent is indifferent between A and B, they will strictly prefer A+$1 to B (because indifference is sensitive to all sweetenings and sourings), and so we will observe the agent taking the trade. If we observe that the agent doesn’t take the trade, then they must have a preferential gap between A and B, and so their preferences must be incomplete.