This paper initiates the reverse mathematics of social choice theory, studying Arrow’s impossibility theorem and related results including Fishburn’s possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in ${\mathsf {RCA}}_0$. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in ${\mathsf {RCA}}_0$. This approach yields a proof of Arrow’s theorem in ${\mathsf {RCA}}_0$, and thus in $\mathrm {PRA}$, since Arrow’s theorem can be formalised as a $\Pi ^0_1$ sentence. Finally we show that Fishburn’s possibility theorem for countable societies is equivalent to ${\mathsf {ACA}}_0$ over ${\mathsf {RCA}}_0$.

Most political liberals argue that only rules, policies and institutions that are part of society’s basic structure need to be justified with so-called public reasons. Laws enacted outside this set are legitimate if and when public reasons can justify the procedure that selects them. I argue that this view is susceptible to known problems from social choice theory. However, there are resources within political liberalism that could address them. If the scope of public reason is extended beyond the basic structure it could order people’s preferences in a way that circumvents the identified issues.

The design process often requires work by teams, rather than individuals. During team based design it is likely that situations will arise in which individual members of the team have different opinions, yet a group decision must still be made. Unfortunately, Arrow’s impossibility theorem indicates that there is no method for aggregating group preferences that will always satisfy a small number of ‘fair’ conditions. This work seeks to identify methods of combining individual preferences that can come close to satisfying Arrow’s conditions, enabling decisions that are fairer in practice. First, experiential conjoint analysis was used to obtain individual empirical utility functions for drinking mug designs. Each empirical utility function represented individual members who were part of a design team. Then, a number of functions for constructing group preference were analysed using both randomly generated preferences and empirical preferences derived from the experiential conjoint survey. The analysis involved checking each of Arrow’s conditions, as well as assessing the potential impact of strategic voting. Based on the results, methods that should be used to aggregate group preference within a design team in practice were identified and recommended.