Remarks on the tail order on moment sequences

Abstract

We consider positively supported Borel measures for which all moments exist. On the set of compactly supported measures in this class a partial order is defined via eventual dominance of the moment sequences. Special classes are identified on which the order is total, but it is shown that already for the set of distributions with compactly supported smooth densities the order is not total. In particular we construct a pair of measures with smooth density for which infinitely many moments agree and another one for which the moments alternate infinitely often. This disproves some recently published claims to the contrary. Some consequences for games with distributional payoffs are discussed.

On Game Theory Using Stochastic Tail Orders

“…Indeed, the computation of equilibria along known direct or iterative (online-learning) algorithms delivers something that is not a Nash equilibrium. This is shown by an instructive counterexample due to [2], which we will repeat it later as Example 1.…”

“…the supremum norm. Overall, this work presents a correction to [10,12], in response to recent findings of [2].…”

“…The proposal in [12] attempted to classify a set of distributions that are totally ordered under , and for which the ordering was independent of the choice of U. This would have delivered a "natural" lift of games with distributions as payoffs into an analogue of classical matrix games, only played inside * R. However, it was found later in [2] that the conditions were too weak to equate the ≤-order in * R to a -more interpretable -stochastic tail order. Specifically, the examples of unimodal (even monotone) distributions constructed in [2] exhibit some oscillatory behavior that precluded it from comparing in the same ≤-sense under all ultrafilters, thus showing that the dependency on U is still there.…”

“…This would have delivered a "natural" lift of games with distributions as payoffs into an analogue of classical matrix games, only played inside * R. However, it was found later in [2] that the conditions were too weak to equate the ≤-order in * R to a -more interpretable -stochastic tail order. Specifically, the examples of unimodal (even monotone) distributions constructed in [2] exhibit some oscillatory behavior that precluded it from comparing in the same ≤-sense under all ultrafilters, thus showing that the dependency on U is still there. As a second observation of [2], the practical computation of Nash equilibria in * R is more involved, as the usual convergence results known for classical games, upon application to games in * R, may fail (as already recognized earlier in [10]).…”

“…Specifically, the examples of unimodal (even monotone) distributions constructed in [2] exhibit some oscillatory behavior that precluded it from comparing in the same ≤-sense under all ultrafilters, thus showing that the dependency on U is still there. As a second observation of [2], the practical computation of Nash equilibria in * R is more involved, as the usual convergence results known for classical games, upon application to games in * R, may fail (as already recognized earlier in [10]). Indeed, the computation of equilibria along known direct or iterative (online-learning) algorithms delivers something that is not a Nash equilibrium.…”