Replicating Electoral Success

Kiran Tomlinson, Tanvi Namjoshi, Johan Ugander, Jon M. Kleinberg

Abstract: 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 cater to the median voter, and the empirical observation that democracies often have two major parties with divergent policies. Motivated in part by this tension, we introduce a dynamic model of candidate positioning based on a simple bounded rationality heuristic: candidates imitate the policy of previous winners. The resulting model is closely connected to evolutionary replicator dynamics. For uniformly-distributed voters, we prove in our model that with k = 2, 3, or 4 candidates per election, any symmetric candidate distribution converges over time to the center. With k ≥ 5 candidates per election, however, we prove that the candidate distribution does not converge to the center and provide an even stronger non-convergence result in a special case with no extreme candidates. Our conclusions are qualitatively unchanged 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, where 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.