  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.
