Uncertainty in Games: Using Probability-Distributions as Payoffs

• Lemma 1 asserts the total ordering only on a dense subset of $$\mathcal{F}$$. Theorem 1 needs the additional hypothesis of $$f_1(a)\neq f_2(a)$$ at $$a=\max(\Omega)$$ and with $$\Omega$$ being the union of both supports. (Thanks to Vincent Bürgin for pointing this out by providing a respective example). Since this condition is satisfiable by truncating the distributions at any smaller value $$a-\varepsilon$$ for some $$\varepsilon>0$$, Theorem 1 only asserts the invariance w.r.t. the ultrafilter on a dense subset of the whole space of probability distributions. See also the respectively updated preprint on arxiv for more details.