Strategic Partitioning and Manipulability in Two-Round Elections

TL;DR

This paper analyzes a two-round election model with strategic partitioning of candidates, using probabilistic spatial voting to determine optimal initial splits and asymptotic success probabilities.

Contribution

It introduces an analytical framework for optimal partitioning in two-round elections and characterizes the asymptotic behavior of success probabilities as candidates grow large.

Findings

Optimal relative width of candidate clusters converges to one-fifth of total candidates.

Success probability approaches 1 as electorate size increases.

Analytical results are validated through simulations.

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.