Necessary President in Elections with Parties

TL;DR

The paper analyzes the Necessary President problem under a party-nomination model across a spectrum of voting rules, providing a comprehensive complexity landscape. It establishes polynomial-time solvability for Borda, Copeland$^\alpha$, and Maximin, while proving $\mathsf{coNP}$-completeness for short and veto-like scoring rules even with tiny party sizes, and for Ranked Pairs under small-party settings. The authors advance a rich parameterized complexity analysis, showing $\mathsf{W}[2]$-hardness in the number of parties, but fixed-parameter tractability when gauged by the number of voter types, and they obtain $\mathsf{FPT}$ results for combined parameters. These results illuminate the computational barriers parties face in strategic nomination scenarios and offer efficient algorithms under favorable parameter regimes, contributing to the broader understanding of election control and strategic candidacy. The work also connects to prior findings on Possible President and strategic nomination, highlighting the nuanced differences between existence and robustness of winners under nominee uncertainty.

Abstract

Consider an election where the set of candidates is partitioned into parties, and each party must choose exactly one candidate to nominate for the election held over all nominees. The Necessary President problem asks whether a candidate, if nominated, becomes the winner of the election for all possible nominations from other parties. We study the computational complexity of Necessary President for several voting rules. We show that while this problem is solvable in polynomial time for Borda, Maximin, and Copeland$^α$ for every $α\in [0,1]$, it is $\mathsf{coNP}$-complete for general classes of positional scoring rules that include $\ell$-Approval and $\ell$-Veto, even when the maximum size of a party is two. For such positional scoring rules, we show that Necessary President is $\mathsf{W}[2]$-hard when parameterized by the number of parties, but fixed-parameter tractable with respect to the number of voter types. Additionally, we prove that Necessary President for Ranked Pairs is $\mathsf{coNP}$-complete even for maximum party size two, and $\mathsf{W}[1]$-hard with respect to the number of parties; remarkably, both of these results hold even for constant number of voters.

Theorems & Definitions (11)

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