MichaelStJules comments on There are no coherence theorems

[MichaelStJules]

Coming back to this, the policy

if I previously turned down some option X, I will not choose any option that I strictly disprefer to X

seems irrational to me if applied in general. Suppose I offer you $X$ and $Y$, where both $X$ and $Y$ are random, and $Y$ is ex ante preferable to $X$, e.g. stochastically dominates $X$, but has some chance of being worse than $X$. You pick $Y$. Then you evaluate $Y$ to get $y$. However, suppose you get unlucky, and $y$ is worse than $X$. Suppose further that there’s a souring of $X$, $X^{-}$, that’s still preferable to $y$. Then, I offer you to trade $y$ for $X^{-}$. It seems irrational to not take $X^{-}$.

Maybe what you need to do is first evaluate according to your multi-utility function (or stochastic dominance, which I think is a special case) to rule out some options, i.e. to rule out not trading $y$ for $X^{-}$ when the latter is better than the former, and then apply your policy to rule out more options.

[EJT]

Ah yes, nice point. The policy should really be something like ‘if I previously turned down some option $X$, then given that no uncertainty has been resolved in the meantime, I will not choose any option that I strictly disprefer to $X$.’ An agent acting in accordance with that policy can trade $y$ for $X^{-}$.

And I think that even agents acting in accordance with this restricted policy can avoid pursuing dominated strategies. As your case makes clear, these agents might end up with $X^{-}$ when they could have had $X$ (because they got unlucky with $Y$ yielding $y$). But although that’s unfortunate for the agent, it doesn’t put any pressure on the agent to revise its preferences.

[MichaelStJules]

I think a multi-step decision procedure would be better. Do what your preferences themselves tell you to do and rule out any options you can with them. If there are multiple remaining incomparable options, then apply your original policy to avoid money pumps.