An easy way to deal with this difficulty is to replace ‘at least as happy with policy A as with policy B (in any situation that we think might arise in practice)’ with ‘at least as happy with policy A as with policy B (when averaged over the distribution of situations that we expect to arise)’, though this is clearly much weaker.
To me it seems that the reason this stronger sense of ordering is used is because we expect this amplification procedure to be of a sort that produces results such that A+ is strictly better than A but that even if this wasn’t the case, the concept of an obstruction would still be a useful one. Perhaps it would be reasonable to take the more relaxed definition but expect that amplification would produce results that are strictly better.
I also agree with Chris below that defining an obstruction in terms of this ‘better than’ relation brings in serious difficulty. There are exponentially many policies Bthat are no better than A+ and there may well be a subset of these can be amplified beyond A+ but as far as I can tell there’s no clear way to identify these. We thus have an exponential obstacle to progress even within a partition, necessitating a stronger definition.
An easy way to deal with this difficulty is to replace ‘at least as happy with policy A as with policy B (in any situation that we think might arise in practice)’ with ‘at least as happy with policy A as with policy B (when averaged over the distribution of situations that we expect to arise)’, though this is clearly much weaker.
To me it seems that the reason this stronger sense of ordering is used is because we expect this amplification procedure to be of a sort that produces results such that A+ is strictly better than A but that even if this wasn’t the case, the concept of an obstruction would still be a useful one. Perhaps it would be reasonable to take the more relaxed definition but expect that amplification would produce results that are strictly better.
I also agree with Chris below that defining an obstruction in terms of this ‘better than’ relation brings in serious difficulty. There are exponentially many policies B that are no better than A+ and there may well be a subset of these can be amplified beyond A+ but as far as I can tell there’s no clear way to identify these. We thus have an exponential obstacle to progress even within a partition, necessitating a stronger definition.