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Replicating Electoral Success
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Introduces a replicator-dynamics model where candidates imitate previous plurality winners, proving that candidates converge to the center for k≤4 per election but not for k≥5, with simulations showing two-cluster emergence (Duverger-like).
Problem
Hotelling-Downs predicts candidates converge to the median voter, yet real plurality democracies have two divergent major parties; existing analyses focus on rational one-shot equilibria or two-candidate dynamics. The paper asks how candidate positions evolve when candidates imitate past winners acr
Approach
Model candidate positioning as replicator dynamics in a continuous strategy space with k-way plurality interactions: each generation's candidates copy positions of previous-generation winners. Under uniform voters, the winner distribution converges to the center for k=2,3,4 but not for k≥5, with robustness to uniform noise.
Key results
- For k=2, the paper derives the exact closed form F_{2,t}(x)=[2F_0(x)]^{2^t}/2, so the distribution converges to a point mass at 1/2.
- For k=3 and k=4 the CDF is bounded above by expressions decaying to 0, so candidates converge to the center; convergence is much slower for k=4.
- For k≥5, the candidate distribution does not converge to 1/2; with support on (1/4,3/4), the density at 1/2 goes to 0 and simulations show two clusters near 1/4 and 3/4 for k=5,6,7.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use winner-copying replicator dynamics with 2, 3, or 4 candidates per election | Replicator dynamics for candidate positioning · Uniform voters; symmetric atomless initial distribution positive near 1/2 | candidate distribution convergence | Helped |
| Increase the number of candidates per election to 5 or more | Replicator dynamics for candidate positioning · Uniform voters; symmetric atomless initial distribution positive near 1/2 | candidate distribution convergence to center | Hurt |
| Add ε-uniform random positioning noise (ablation) | Replicator dynamics for candidate positioning · 2 candidates; any x ∈ [0, 1/2) | limiting CDF F₂,t^ε(x): ≤ ε | Mixed |
| Add ε-uniform random positioning noise (ablation) | Replicator dynamics for candidate positioning · 3 candidates; ε ∈ (0, 1/3) | limsup CDF F₃,t^ε(x): ≤ 1.5ε | Mixed |
| Add ε-uniform random positioning noise (ablation) | Replicator dynamics for candidate positioning · 4 candidates | candidate distribution convergence to center | Mixed |
| Add ε-uniform random positioning noise (ablation) | Replicator dynamics for candidate positioning · 5 or more candidates | candidate distribution convergence to center | Hurt |
| Restrict the initial candidate distribution to support (1/4, 3/4) (ablation) | Replicator dynamics for candidate positioning · 5 or more candidates; x ∈ (F₀⁻¹(ℓ), 1/2) | CDF Fₖ,t(x): lim Fₖ,t(x) = 1/2 | Hurt |
| Use Uniform(1/4, 3/4) as the initial distribution (ablation) | Replicator dynamics for candidate positioning · 5 or more candidates; central interval [0.34, 0.66] | candidate density: goes to 0 | Hurt |
| Use symmetric unimodal or bimodal non-uniform voter distributions (ablation) | Replicator dynamics for candidate positioning · Non-uniform voters; varying candidate counts | qualitative convergence pattern | No clear effect |
| Add memory of winners from the last 2 or 3 generations (ablation) | Replicator dynamics for candidate positioning · Memory variant; varying candidate counts | qualitative convergence pattern | No clear effect |
| Add Gaussian perturbation noise with variance σ² = 0.005 (ablation) | Replicator dynamics for candidate positioning · Varying candidate counts | qualitative convergence pattern: σ² = 0.005 | No clear effect |
| Increase Gaussian perturbation noise to variance σ² = 0.01 (ablation) | Replicator dynamics for candidate positioning · 6 candidates | formation of a central cluster: σ² = 0.01 | Helped |
Surprising findings
- The tipping point between centripetal convergence and centrifugal flanking-driven divergence occurs between four and five candidates per election.
- Top-h copying (h=2 or 3) breaks the convergence dichotomy, with k=3,4 behaving like k=5 under h=2.
Limitations
- Asymptotic characterization of the limiting distribution for k≥5 is not derived in general.
- Results are proved for symmetric atomless initial distributions and uniform voters; other settings are only explored in simulation.
- Model abstracts a population of elections as infinite and uses continuous distributions rather than individual candidates.
Experimental setup
Metrics: candidate distribution CDF over generations convergence to center vs non-convergence
Baselines: Hotelling-Downs model Nash equilibria with equal split tie-breaking (complete plurality maximization)
Builds on: Hotelling-Downs model replicator dynamics iterative voting Duverger's Law
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