Probability of a Condorcet Winner for Large Electorates: An Analytic Combinatorics Approach

TL;DR

The paper develops an analytic-combinatorics framework to compute the probability that a given candidate is an alpha-winner under General Independent Culture, with independent voter preferences. By encoding voter distributions into a multivariate generating function and applying Cauchy- and saddle-point analyses, it derives exact asymptotics and higher-order terms, capturing subcritical, critical, mixed, and supercritical regimes. The framework recovers classical Condorcet results in Impartial Culture and Mallows models, providing explicit constants, parity-dependent corrections, and efficient computational methods, validated by numerical simulations. The approach advances precise convergence-rate understanding in large electorates and offers tools extendable to other voting-events and cultures. The work thus bridges combinatorial generating-function techniques with social choice theory to yield refined probabilistic insights for voting outcomes.

Abstract

We study the probability that a given candidate is an alpha-winner, i.e. a candidate preferred to each other candidate j by a fraction alpha_j of the voters. This extends the classical notion of Condorcet winner, which corresponds to the case alpha = (1/2, ..., 1/2). Our analysis is conducted under the general assumption that voters have independent preferences, illustrated through applications to well-known models such as Impartial Culture and the Mallows model. While previous works use probabilistic arguments to derive the limiting probability as the number of voters tends to infinity, we employ techniques from the field of analytic combinatorics to compute convergence rates and provide a method for obtaining higher-order terms in the asymptotic expansion. In particular, we establish that the probability of a given candidate being the Condorcet winner in Impartial Culture is a_0 + a_{1, n} n^{-1/2} + O(n^{-1}), where we explicitly provide the values of the constant a_0 and the coefficient a_{1, n}, which depends solely on the parity of the number of voters n. Along the way, we derive technical results in multivariate analytic combinatorics that may be of independent interest.

Figures (7)

• Figure 1: Definition of the characteristic polynomial $P(x_1, x_2)$ encoding the probability distribution for a single voter when $m = 3$. The notation $p_{123}$, for example, is a shorthand for $p_{(1, 2, 3)}$. The formal variable $x_j$ marks rankings where candidate $j$ is preferred to candidate $3$.
• Figure 2: Tree representing the algebraic expansion of $P(x_1, x_2)^2$. The edges correspond to multiplications, and each path from the root to a leaf represents a term in the expansion. For example, the highlighted paths correspond to the terms involving ${x_1}^2 {x_2}^{1}$, with a total coefficient given by $[{x_1}^2 {x_2}^{1}]P(x_1, x_2)^2 = 2 p_{\{1\}} p_{\{1, 2\}}$. This coefficient extraction corresponds to the probability that, among the two voters, exactly two prefer candidate $1$ over candidate $3$, and exactly one prefers candidate $2$ over candidate $3$.
• Figure 3: Probability that candidate $3$ is the Condorcet winner as a function of $n$ in a culture $\mathcal{M}_{3 \textnormal{ last}}$ with $\rho = \log(2)$, shown on a semilog scale. The theoretical equivalent is based on \ref{['th_mallows-3-last']}, while exact results rely on \ref{['eq_alpha-winner-coeff-extraction']}. Monte Carlo simulations use 10,000 profiles per point, yielding an error of order $10^{-2}$. For $n \geq 30$, they return zero, which is not visible due to the logarithmic scale.
• Figure 4: Probability that a given candidate is the Condorcet winner as a function of $n$ in the Impartial Culture with three candidates. The theoretical equivalent is based on \ref{['th_impartial_culture']}, while exact results rely on \ref{['eq_alpha-winner-coeff-extraction']}. Monte Carlo simulations use 10,000 profiles per point, yielding an error of order $10^{-2}$. When the exact results curve is not visible, it is overlapped by the Monte Carlo results curve.
• Figure 5: Probability that candidate $4$ is the Condorcet winner as a function of $n$ in a culture $\mathcal{M}_{4 \textnormal{ last}}$ with $\rho = \log(2)$, shown on a semilog scale. The theoretical equivalent is based on \ref{['th_mallows-4-last']}, while exact results rely on \ref{['eq_alpha-winner-coeff-extraction']}. Monte Carlo simulations use 10,000 profiles per point, yielding an error of order $10^{-2}$. For $n \geq 14$, they return zero, which is not visible due to the logarithmic scale. Exact results are not computed for $n \geq 26$ due to prohibitive runtime (e.g., 7 minutes for $n = 25$).

Theorems & Definitions (46)

• Definition 1
• Definition 2
• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: may1971some
• Theorem 6
• Theorem 7