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 in part 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, despite its simplicity, exhibits complex behavior and contrasts considerably with existing modeling approaches. For uniformly-distributed voters, we prove in our model 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 ≥ 5 or more candidates per election, however, we prove that the candidate distribution does not converge to the center. For initial distributions of k ≥ 5 candidates 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. As a matter of robustness, our conclusions are qualitatively unchanged (though require different analyses) if a small fraction of candidates are not winner-copiers and are instead positioned uniformly at random in each election. Beyond our theoretical analysis, we illustrate our results in extensive simulations; 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, other forms of noise, and replication with memory of earlier rounds of elections. In these simulated variants, we find the same general pattern: convergence to the center with four or fewer candidates, but not with five or more. Finally, we discuss the relationship between our replicator dynamics model and prior work on strategic equilibria of candidate positioning games.