Replicating Electoral Success

A core tension in the study of plurality elections is the clash between the classic Hotelling-Downs model, which predicts that two office-seeking candidates should position themselves at the median voter's policy, and the empirical observation that real-world democracies often have two major parties with divergent policies. Motivated by this tension and drawing from bounded rationality, we introduce a dynamic model of candidate positioning based on a simple behavioral heuristic: candidates imitate the policy of previous winners. The resulting model is closely connected to evolutionary replicator dynamics and exhibits complex behavior, despite its simplicity. For uniformly-distributed voters, we prove that when there are $k = 2$, $3$, or $4$ candidates per election, any symmetric candidate distribution converges over time to a concentration of candidates at the center. With $k \ge 5$, however, we prove that the candidate distribution does not converge to the center. For initial distributions without any extreme candidates, we prove a stronger statement than non-convergence, showing that the density in an interval around the center goes to zero when $k \ge 5$. As a matter of robustness, our conclusions are qualitatively unchanged if a small fraction of candidates are not winner-copiers and are instead positioned uniformly at random. Beyond our theoretical analysis, we illustrate our results in simulation; for five or more candidates, we find a tendency towards the emergence of two clusters, a mechanism suggestive of Duverger's Law, the empirical finding that plurality leads to two-party systems. Our simulations also explore several variations of the model, including non-uniform voter distributions and other forms of noise, which exhibit similar convergence patterns. Finally, we discuss the relationship between our model and prior work on strategic equilibria of candidate positioning games.