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.
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.