Logic Seminar

Luke SerafinCornell University
Social welfare without nonprincipal ultrafilters on $\omega$

Friday, August 25, 2023 - 2:55pm
Malott 205

A large population is likely to contain members whose utilities for different policies vary widely; how can the preferences of the society as a whole be derived from the preferences of its component individuals? The theoretical economics literature, and in particular the work of Laguzzi, has modelled this type of scenario by considering a countably infinite collection of individuals each of whom derives some utility in $[0,1]$ from each policy, and looking for linear orders of the space $[0,1]^\omega$ which satisfy various desirable properties. One such property is finite anonymity, which requires that if the natural-number labels of two individuals are interchanged, the resulting policy should be indistinguishable from the original policy. Another is strong equity, which requires that, all else being equal, if given a choice of two policies one individual is always worse off than another, then that worse-off individual gets to choose the policy (and naturally chooses the policy with highest utility). Social welfare orders which are finitely anonymous and strongly equitable can be constructed using a nonprincipal ultrafilter on $\omega$, and Laguzzi asks whether this ultrafilter is necessary. We use techniques of geometric set theory developed by Larson and Zapletal to prove that the ultrafilter is not necessary, by constructing a model in which there is a finitely-anonymous, strongly equitable order on $[0,1]^\omega$ but no nonprincipal ultrafilter on $\omega$.