May's Theorem with an infinite population

In this paper, we investigate majority rule with an infinite number of voters. We use an axiomatic approach and attempt to extend May’s Theorem characterizing majority rule to an infinite population. The analysis hinges on correctly generalizing the anonymity condition and we consider three different versions. We settle on bounded anonymity as the appropriate form for this condition and are able to use the notion of asymptotic density to measure the size of almost all sets of voters. With this technique, we define density q-rules and show that these rules are characterized by neutrality, monotonicity, and bounded anonymity on almost all sets. Although we are unable to provide a complete characterization applying to all possible sets of voters, we construct an example showing that our result is the best possible. Finally, we show that strengthening monotonicity to density positive responsiveness characterizes density majority rule on almost all sets.