Computing Equilibrium Nominations in Presidential Elections

TL;DR

This paper studies strategic candidate nomination in Plurality elections when voters exhibit party-aligned single-peaked preferences. It introduces a polynomial-time recognition algorithm for party-aligned single-peakedness and develops polynomial-time methods to decide equilibrium existence and to determine whether a given party can win under some or all nominations, as well as whether a Nash equilibrium can exist. The authors prove that pure strategy Nash equilibria always exist for up to three parties in this domain, but may fail for four; they also show NP-hardness outside this domain, such as in 1-D Euclidean elections. A central contribution is a dynamic-programming framework that partitions voters and computes viable score pairs to efficiently find Nash equilibria and winner possibilities under party-aligned single-peaked preferences. Overall, the work provides a tractable algorithmic toolkit for analyzing strategic nominations in structured ideological domains and clarifies the boundaries of tractability versus hardness when moving beyond party-aligned single-peakedness.

Abstract

We study strategic candidate nomination by parties in elections decided by Plurality voting. Each party selects a nominee before the election, and the winner is chosen from the nominated candidates based on the voters' preferences. We introduce a new restriction on these preferences, which we call party-aligned single-peakedness: all voters agree on a common ordering of the parties along an ideological axis, but may differ in their perceptions of the positions of individual candidates within each party. The preferences of each voter are single-peaked with respect to their own axis over the candidates, which is consistent with the global ordering of the parties. We present a polynomial-time algorithm for recognizing whether a preference profile satisfies party-aligned single-peakedness. In this domain, we give polynomial-time algorithms for deciding whether a given party can become the winner under some (or all) nominations, and whether this can occur in some pure Nash equilibrium. We also prove a tight result about the guaranteed existence of pure strategy Nash equilibria for elections with up to three parties for single-peaked and party-aligned single-peaked preference profiles.

Figures (6)

• Figure 1: Illustration for Theorem \ref{['thm:No-NE-partySP-twoparties']}. The left figure shows the voters supporting candidates of $A$ and $B$ in all nomination schemes. On the right, the corresponding winners and non-winners are indicated by "1" and "0", respectively.
• Figure 2: Location of voters and candidates on the real line in the election constructed in Theorem \ref{['thm:NoNE']}. In all figures, voters are indicated by arrows, and candidates by black circles.
• Figure 3: The left figure shows the numbers of voters supporting candidates of $P_1$ and $P_2$ in all possible nomination schemes. On the right, the corresponding winners and non-winners are indicated by "$1$" and "$0$", respectively.
• Figure 4: The location of voters and candidates within the triple segment corresponding to $S_j=\{a_u,a_v,a_z\} \in \mathcal{S}$.
• Figure 5: The location of voters and candidates within the $i^\textrm{th}$ auxiliary segment for some $i \in [m-n]$. The numbers indicating locations are relative to the start of the segment.

Theorems & Definitions (46)

• Remark 1
• Remark 2
• Definition 1
• Example 1
• Theorem 1
• Lemma 1: sec:proof-of-lemvotePSP
• Corollary 1
• Lemma 2: sec:proof-oflembottomparties
• Lemma 3: sec:proof-of-partyfixedplacement
• Lemma 4: sec:proof-of-reducingproblem