Replicating Electoral Success

TL;DR

This work develops a simple bounded-rationality model of candidate positioning in a one-dimensional policy space, where candidates imitate the position of previous election winners across generations, connecting to replicator dynamics. The authors prove a sharp phase transition: symmetric, atomless initial distributions converge to the center $1/2$ for $k=2,3,4$ candidates per election, but fail to converge to the center for $k\ge 5$, often producing two clusters near $1/4$ and $3/4$; these results remain qualitatively robust under noise and several model variants. Simulations corroborate the theory, showing fast center convergence for small $k$ and persistent non-central clustering as $k$ grows, a phenomenon reminiscent of Duverger's Law. The paper also situates these dynamic results relative to static Nash equilibria, illustrating that focusing on equilibria alone can be misleading and that dynamic imitation can generate qualitatively different outcomes.

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 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.

Figures (18)

• Figure 1: Replicator dynamics for candidate positioning with $k=3$ candidates per election. The top row shows the winner distributions $F_{k, t}$ for each generation $t$, starting from a uniform distribution at $t = 0$, while the bottom row shows four example elections per generation. In each generation, candidates sample their positions from the winner distribution from the previous generation. Plurality winners (with voters uniform over $[0, 1]$) are indicated in green.
• Figure 2: Replicator dynamics runs for $k = 2, \dots, 7$ and 200 generations. Each plot shows 50 runs layered on top of each other, where each run simulates 100,000 elections per generation. We also use enhanced symmetry, a trick to keep the symmetry of the analytical model by reflecting copied points across $1/2$ (discussed further in \ref{['sec:simulations']}). Darker regions indicate higher candidate density; we use a log-scaled colormap to make low-density regions visible. As our theory establishes, the candidate distribution converges to the center for $k = 2, 3, 4$, but does not for $k \ge 5$. The convergence is very fast for $k = 2$ and $3$, but much slower for $k = 4$.
• Figure 3: Replicator dynamics runs with $0.01$-uniform noise for $k = 2, \dots, 7$ and 200 generations, using enhanced symmetry, 50 trials per plot, and 100,000 elections per generation. The behavior is qualitatively identical to the model without noise (\ref{['fig:main-sim']}).
• Figure 4: Replicator dynamics runs with no noise (top row) and $0.01$-uniform noise (bottom row) for larger candidate counts $k$ and using enhanced symmetry. Other settings are identical to \ref{['fig:small-k-symmetry']}, with 50 runs shown in each plot. As the theory predicts, the candidate distribution does not converge to the center; but the exact behavior varies.
• Figure 5: Simulations demonstrating our convergence results \ref{['thm:k-2-convergence', 'thm:k-3-convergence', 'thm:k-4-convergence']}, showing the simulated candidate distribution CDF at various points $x$ alongside the theoretical predictions. The simulations use 50 trials with 100,000 elections per generation, no noise, and enhanced symmetry. The theorems get progressively weaker: \ref{['thm:k-2-convergence']} provides an exact characterization of the two-candidate dynamics, while \ref{['thm:k-3-convergence', 'thm:k-4-convergence']} give upper bounds that converge to 0.

Theorems & Definitions (83)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Corollary 1
• Theorem 3
• Corollary 2
• Lemma 1
• Theorem 4
• Corollary 3