An Axiomatic Characterization of Split Cycle

TL;DR

This paper provides a complete axiomatic characterization of Split Cycle, the margin-based rule for resolving majority cycles, in both variegated decision frameworks: VCCR and VSCC. It shows that Split Cycle is uniquely determined by combining the standard axioms with Coherent IIA and two new defeat-focused axioms (Coherent Defeat and Positive Involvement in Defeat), and, in the presence of ties, by a tolerant variant of Positive Involvement. The work also extends the characterization to social-choice correspondences, introducing a strengthened version called Tolerant Positive Involvement for VSCCs and demonstrating the uniqueness of the Split Cycle VSCC under analogous axioms. Additionally, the paper discusses the necessity of the three special axioms, provides tie-handling extensions, and outlines future directions for relating Split Cycle to other margin-based methods and exploring recovery of VCCRs from VSCCs.

Abstract

A number of rules for resolving majority cycles in elections have been proposed in the literature. Recently, Holliday and Pacuit (Journal of Theoretical Politics 33 (2021) 475-524) axiomatically characterized the class of rules refined by one such cycle-resolving rule, dubbed Split Cycle: in each majority cycle, discard the majority preferences with the smallest majority margin. They showed that any rule satisfying five standard axioms plus a weakening of Arrow's Independence of Irrelevant Alternatives (IIA), called Coherent IIA, is refined by Split Cycle. In this paper, we go further and show that Split Cycle is the only rule satisfying the axioms of Holliday and Pacuit together with two additional axioms, which characterize the class of rules that refine Split Cycle: Coherent Defeat and Positive Involvement in Defeat. Coherent Defeat states that any majority preference not occurring in a cycle is retained, while Positive Involvement in Defeat is closely related to the well-known axiom of Positive Involvement (as in J. Perez, Social Choice and Welfare 18 (2001) 601-616). We characterize Split Cycle not only as a collective choice rule but also as a social choice correspondence, over both profiles of linear ballots and profiles of ballots allowing ties.

Figures (7)

• Figure 1: A margin graph (left) with three cycles: $(a,d,b,a)$, $(a,d,c,a)$, and $(a,d,b,c,a)$. Deleting the weakest edge in each cycle (namely $(b,a)$ in the first, $(c,a)$ in the second, and $(b,c)$ in the third) results in the Split Cycle defeat graph (right).
• Figure 2: Profiles (left) and their margin graphs (right) illustrating the "Fallacy of IIA."
• Figure 3: An example of generating a linear ballot $L$ as in Lemma \ref{['lem:add-ballot']}. Given $x, y$, we first identify (a) a minimal cut in the margin graph using only edges with margin smaller than $\mathrm{Margin}(x, y)$ and (b) the nodes that are not reachable from $y$ in one step (all highlighted on the left). Then we form the graph with the cut inverted and with arrows from $y$ to those highlighted nodes. This graph must be acyclic. Then we linearize this acyclic graph to obtain $L$.
• Figure 4: An example of eliminating all paths from $y$ to $x$ by adding ballots that put $y$ in a sufficiently high position. For each of the first two margin graphs, a minimal cut from $y$ to $x$ is highlighted by thickened arrows. There are many ways to eliminate all paths from $y$ to $x$, and the key is to do this quickly before the positive margin from $x$ to $y$ runs out.
• Figure 5: the profile $\mathbf{Q}$.

Theorems & Definitions (102)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Definition 2.6
• Definition 2.7
• Lemma 2.8
• Definition 2.9
• Example 2.10