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Strategic Partitioning and Manipulability in Two-Round Elections

Emilio De Santis, Antonio Di Crescenzo, Verdiana Mustaro

Abstract

We consider a two-round election model involving $m$ voters and $n$ candidates. Each voter is endowed with a strict preference list ranking the candidates. In the first round, the candidates are partitioned into two subsets, $A$ and $B$, and voters select their preferred candidate from each. Provided there are no ties, the two respective winners advance to a second round, where voters choose between them according to their initial preference lists. We analyze this scenario using a probabilistic framework based on a spatial voting model with cyclically constructed preference lists and uniformly distributed ideal points. Our objective is to determine the optimal initial partition of $A$ and $B$ that maximizes a target candidate's probability of winning. We analytically evaluate this success probability and derive its asymptotic behavior as the number of candidates $n \to \infty$. A key finding is that the asymptotically optimal relative width of the main discrete cluster converges precisely to one-fifth of the total number of candidates. Finally, we provide computational results and confidence intervals derived from simulation algorithms that validate the analytical framework. Specifically, we demonstrate that the probability of the universal victory event rapidly approaches $1$ as the electorate size increases.

Strategic Partitioning and Manipulability in Two-Round Elections

Abstract

We consider a two-round election model involving voters and candidates. Each voter is endowed with a strict preference list ranking the candidates. In the first round, the candidates are partitioned into two subsets, and , and voters select their preferred candidate from each. Provided there are no ties, the two respective winners advance to a second round, where voters choose between them according to their initial preference lists. We analyze this scenario using a probabilistic framework based on a spatial voting model with cyclically constructed preference lists and uniformly distributed ideal points. Our objective is to determine the optimal initial partition of and that maximizes a target candidate's probability of winning. We analytically evaluate this success probability and derive its asymptotic behavior as the number of candidates . A key finding is that the asymptotically optimal relative width of the main discrete cluster converges precisely to one-fifth of the total number of candidates. Finally, we provide computational results and confidence intervals derived from simulation algorithms that validate the analytical framework. Specifically, we demonstrate that the probability of the universal victory event rapidly approaches as the electorate size increases.
Paper Structure (17 sections, 17 theorems, 122 equations, 4 figures, 7 tables)

This paper contains 17 sections, 17 theorems, 122 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Optimal Agenda Setting and Asymptotic Winning Probabilities
  4. A Continuous Limit of the Discrete Model
  5. Formalization of the Continuous Model
  6. Asymptotic Behavior of the Universal Victory Event
  7. Numerical Analysis of the Target Candidate's Victory
  8. Continuous Limit and Empirical Validation
  9. The Law of Exponential Decay: The 4/5 Constant
  10. High-Precision Validation in the Extreme Asymptotic Regime
  11. Observational Trends and Heuristic Considerations
  12. Numerical Analysis of the Universal Victory Event
  13. Discrete Structural Constraints and High-Resolution Optimization
  14. Convergence to the Continuous Limit and Computational Evaluation
  15. Conclusions
  16. ...and 2 more sections

Key Result

Lemma 1

Let $m,n \in \mathbb{N}$. For any $R \in O_{m,n}$ and for any non-empty $A \subseteq [n]$, if $i\in A$, then

Figures (4)

  • Figure 1: Graph $G_{14}$, with $A=\{1,2,3,4,6,8,10\}$ and A-open (non A-open) edges drawn with continuous (dashed) lines.
  • Figure 2: Comparison of optimization regimes for $n=5000$. Left: Peaked regime at $m=20$, where the probability follows a parabolic trend and the Wilson boundaries form a tight interval around the theoretical vertex. Right: Saturated regime at $m=100$, where the emergence of the "Wilson Plateau" expands the stability region across hundreds of values. In both cases, the optimal strategy $l_{opt}$ is universally identified as the centroid of this region.
  • Figure 3: Heatmap of the optimized winning probability $\hat{p}_1(n, m, l_{opt}(n, m))$.
  • Figure 4: Semi-log plot of the distance from certainty ($1 - p_1$). The numerical values (blue dots) align with the theoretical $(4/5)^m$ law (red dashed line) across 43 orders of magnitude.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 33 more