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Under dominance reasoning, the fact that we strictly prefer ''A'' to ''B'' for all possible observed values ''a'' should imply that we strictly prefer ''A'' to ''B'' without observing ''a''; however, as already shown, that is not true because <math>E(B)=E(A)=\infty</math>. To salvage dominance reasoning while allowing <math>E(B)=E(A)=\infty</math>, one would have to replace expected value as the decision criterion, thereby employing a more sophisticated argument from mathematical economics.
For example, we could assume the decision maker is an [[expected utility]] maximizer with initial wealth ''W'' whose utility function, <math>u(w)</math>, is chosen to satisfy <math>E(u(W+B)|A=a)<u(W+a)</math> for at least some values of ''a'' (that is, holding onto <math>A=a</math> is strictly preferred to switching to ''B'' for some ''a''). Although this is not true for all utility functions, it would be true if <math>u(w)</math> had an upper bound, <math>\beta<\infty</math>, as ''w'' increased toward infinity (a common assumption in mathematical economics and decision theory).<ref>{{cite book|last1=DeGroot|first1=Morris H.|title=Optimal Statistical Decisions|date=1970|publisher=McGraw-Hill|pages=109}}</ref> [[Michael R. Powers]] provides necessary and sufficient conditions for the utility function to resolve the paradox, and notes that neither <math>u(w)<\beta</math> nor <math>E(u(W+A))=E(u(W+B))<\infty</math> is required.<ref>{{cite journal|last1=Powers|first1=Michael R.|title=Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems|journal=Risks|date=2015|volume=3|issue=1|pages=26–34|doi=10.3390/risks3010026
Some writers would prefer to argue that in a real-life situation, <math>u(W+A)</math> and <math>u(W+B)</math> are bounded simply because the amount of money in an envelope is bounded by the total amount of money in the world (''M''), implying <math>u(W+A) \leq u(W+M)</math> and <math>u(W+B) \leq u(W+M)</math>. From this perspective, the second paradox is resolved because the postulated probability distribution for ''X'' (with <math>E(X)=\infty</math>) cannot arise in a real-life situation. Similar arguments are often used to resolve the [[St. Petersburg paradox]].
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