Primarily about primaries

Much of the social choice literature examines direct voting systems, in which voters submit their ranked preferences over candidates and a voting rule picks a winner. Real-world elections and decision-making processes are often more complex and involve multiple stages. For instance, one popular voting system filters candidates through primaries: first, voters affiliated with each political party vote over candidates of their own party and the voting rule picks a set of candidates, one from each party, who then compete in a general election.

We present a model to analyze such multi-stage elections, and conduct what is, to the best of our knowledge, the first quantitative comparison of the direct and primary voting systems in terms of the quality of the elected candidate, using the metric of distortion, which attempts to quantify how far from the optimal winner is the actual winner of an election. Our main theoretical result is that voting rules (which are independent of party affiliations, of course) are guaranteed to perform in the primary system within a constant factor of the direct, single stage setting. Surprisingly, the converse does not hold: we show settings in which there exist voting rules that perform significantly better under the primary system than under the direct system. Using simulations, we see that plurality benefits significantly from using a primary system over a direct one, while Condorcet-consistent rules do not.