I think the content of this argument is not that maxmin is fundamental, but rather that simplicity priors “look like” or justify Hurwicz-like decision rules. Simple versions of this are easy to prove but (as far as I know) do not appear in the literature.
What I was saying up there is not a justification of Hurwicz’ decision rule. Rather, it is that if you already accept the Hurwicz rule, it can be reduced to maximin, and for a simplicity prior the reduction is “cheap” (produces another simplicity prior).
Why accept the Hurwicz’ decision rule? Well, at least you can’t be accused of a pessimism bias there. But if you truly want to dig deeper, we can start instead from an agent making decisions according to an ambidistribution, which is a fairly general (assumption-light) way of making decisions. I believe that a similar argument (easiest to see in the LF-dual cramble set representation) would allow reducing that to maximin on infradistributions (credal sets).
To make such an approach even more satisfactory, it would be good to add a justification for a simplicity ambi/infra-prior. I think this should be possible by arguing from “opinionated agents”: the ordinary Solomonoff prior is the unique semicomputable one that dominates all semicomputable measures, which decision-theoretically corresponds to something like “having preferences about as many possible worlds as we can”. Possibly, the latter principle formalized can be formalized into something which ends up picking out an infra-Solomonoff prior (and, replacing “computability” by a stronger condition, some other kind of simplicity infra-prior).
This called a Hurwicz decision rule / criterion (your t is usually alpha).
I think the content of this argument is not that maxmin is fundamental, but rather that simplicity priors “look like” or justify Hurwicz-like decision rules. Simple versions of this are easy to prove but (as far as I know) do not appear in the literature.
Thanks for this!
What I was saying up there is not a justification of Hurwicz’ decision rule. Rather, it is that if you already accept the Hurwicz rule, it can be reduced to maximin, and for a simplicity prior the reduction is “cheap” (produces another simplicity prior).
Why accept the Hurwicz’ decision rule? Well, at least you can’t be accused of a pessimism bias there. But if you truly want to dig deeper, we can start instead from an agent making decisions according to an ambidistribution, which is a fairly general (assumption-light) way of making decisions. I believe that a similar argument (easiest to see in the LF-dual cramble set representation) would allow reducing that to maximin on infradistributions (credal sets).
To make such an approach even more satisfactory, it would be good to add a justification for a simplicity ambi/infra-prior. I think this should be possible by arguing from “opinionated agents”: the ordinary Solomonoff prior is the unique semicomputable one that dominates all semicomputable measures, which decision-theoretically corresponds to something like “having preferences about as many possible worlds as we can”. Possibly, the latter principle formalized can be formalized into something which ends up picking out an infra-Solomonoff prior (and, replacing “computability” by a stronger condition, some other kind of simplicity infra-prior).