Epistemic Status: Really unsure about a lot of this.
It’s not clear to me that the randomization method here is sufficient for the condition of not missing out on sure gains with probability 1.
Scenario: B is preferred to A, but preference gap between A & C and B & C, as in the post.
Suppose both your subagents agree that the only trades that will ever be offered are A->C and C->B. These trades occur with a Poisson distribution, with λ = 1 for the first trade and λ = 3 for the second. Any trade that is offered must be immediately declined or accepted. If I understand your logic correctly, this would mean randomizing the preferences such that
pC→B = 1⁄3,
pA→C = 1
In the world where one of each trade is offered, the agent always accepts A->C but will only accept C->B 1⁄3 of the time, thus the whole move from A->B only happens with probability 1⁄3. So the agent misses out on sure gains with probability 2⁄3.
In other words, I think you’ve sufficiently shown that this kind of contract can taken a strongly-incomplete agent and make them not-strongly-incomplete with probability >0 but this is not the same as making them not-strongly-incomplete with probability 1, which seems to me to be necessary for expected utility maximization.
Epistemic Status: Really unsure about a lot of this.
It’s not clear to me that the randomization method here is sufficient for the condition of not missing out on sure gains with probability 1.
Scenario: B is preferred to A, but preference gap between A & C and B & C, as in the post.
Suppose both your subagents agree that the only trades that will ever be offered are A->C and C->B. These trades occur with a Poisson distribution, with λ = 1 for the first trade and λ = 3 for the second. Any trade that is offered must be immediately declined or accepted. If I understand your logic correctly, this would mean randomizing the preferences such that
pC→B = 1⁄3,
pA→C = 1
In the world where one of each trade is offered, the agent always accepts A->C but will only accept C->B 1⁄3 of the time, thus the whole move from A->B only happens with probability 1⁄3. So the agent misses out on sure gains with probability 2⁄3.
In other words, I think you’ve sufficiently shown that this kind of contract can taken a strongly-incomplete agent and make them not-strongly-incomplete with probability >0 but this is not the same as making them not-strongly-incomplete with probability 1, which seems to me to be necessary for expected utility maximization.
Yeah, the argument is not intended to be “here’s the optimal contract/modification to complete the preferences”. The argument is roughly:
If preferences are strongly incomplete, then there exists at least one contract/modification which is a pareto-improvement.
Therefore, non-dominated strategies must not be strongly incomplete.
So, the argument in the post allows the possibility that the preferences will be completed in some other way which does even better.