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by Alex Appel 70 days ago | Abram Demski likes this | link | parent

If you drop the Pareto-improvement condition from the cell rank, and just have “everyone sorts things by their own utility”, then you won’t necessarily get a Pareto-optimal outcome (within the set of cell center-points), but you will at least get a point where there are no strict Pareto improvements (no points that leave everyone better off).

The difference between the two is… let’s say we’ve got a 2-player 2-move game that in utility-space, makes some sort of quadrilateral. If the top and right edges join at 90 degrees, the Pareto-frontier would be the point on the corner, and the set of “no strict Pareto improvements” would be the top and the right edges.

If that corner is obtuse, then both “Pareto frontier” and “no strict Pareto improvements” agree that both line edges are within the set, and if the corner is acute, then both “Pareto frontier” and “no strict Pareto improvements” agree that only the corner is within the set. It actually isn’t much of a difference, it only manifests when the utilities for a player are exactly equal, and is easily changed by a little bit of noise.

The utility-approximation issue you pointed out seems to be pointing towards the impossibility of guaranteeing limiting to a point on the Pareto frontier (when you make the cell size smaller and smaller), precisely because of that “this set is unstable under arbitrarily small noise” issue.

But, the “set of all points that have no strict Pareto improvements by more than \(\delta\) for all players”, ie, the \(\delta\)-fuzzed version of “set of points with no strict pareto improvement”, does seem to be robust against a little bit of noise, and doesn’t require the Pareto-improvement condition on everyone’s ranking of cells.

So I’m thinking that if that’s all we can attain (because of the complication you pointed out), then it lets us drop that inelegant Pareto-improvement condition.

I’ll work on the proof that for sufficiently small cell size \(\epsilon\), you can get an outcome within \(\delta\) of the set of “no strict Pareto improvements available”

Nice job spotting that flaw.



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