Simple Stable Voting (SSV)

• Simple Stable Voting is a ranking-based, recursive single-winner method that ensures stability-for-winners by neutralizing spoiler effects.
• The algorithm recursively eliminates candidates via the largest head-to-head margin, guaranteeing a unique winner when margins are distinct.
• Empirical studies indicate SSV achieves tie rates below 1%, outperforming methods like IRV, Minimax, and cycle-resolving procedures in stability and tie avoidance.

Simple Stable Voting (SSV) is a ranking-based, single-winner voting method developed as a simplification of the Stable Voting rule. SSV is designed to satisfy the "Stability for Winners" principle—mandating that the presence of a "spoiler" candidate, one defeated head-to-head by a winner, cannot alter that winner's outcome, unless another candidate has exactly the same claim—while also exhibiting strong avoidance of ties among winners in empirical settings. SSV operates by recursively identifying the first (largest-margin) head-to-head pair (A, B) such that, if B is eliminated, A is the (recursive) SSV-winner in the reduced profile. When margins are all distinct, SSV winner uniqueness is guaranteed. SSV further refines the undefeated (Condorcet) set and compares with cycle-resolving Condorcet methods such as Split Cycle and Ranked Pairs (Holliday et al., 2021, Holliday et al., 29 Nov 2025).

1. Formal Definition and Theoretical Guarantees

Let $C$ be a finite candidate set and $N = \{1, ..., n\}$ the voter set. Each voter $i$ submits a ranking $\succ_i$ (possibly partial) over $C$. For $A, B \in C$, the margin is defined as: $$m(A, B) := |\{ i : A \succ_i B \}| - |\{ i : B \succ_i A \}|$$ Define $A \triangleright B$ if $m(A, B) > 0$. Let $P_{-B}$ denote the profile with $B$ removed from all ballots.

SSV(P):

1. If $|C| = 1$, return the unique candidate.
2. List all ordered pairs $(A, B)$, $A \ne B$, sorted by non-increasing $m(A,B)$.
3. For each $(A, B)$, if $SSV(P_{-B}) = A$, declare $A$ as the winner and stop.
4. If multiple $(A, B)$ have the same top margin, admit all corresponding $A$ as tied winners (subject to tiebreaking).

Stability-for-Winners (with Tiebreaking):

For any $A, B \in C$, if $m(A, B) > 0$ and $A$ would win $P_{-B}$, then $A$ wins $P$ (or is tied with others with the same claim). SSV ensures the SSV winner(s) always witness this property via the first qualifying $(A, B)$ pair (Holliday et al., 2021).

2. Algorithmic Structure and Complexity

A summary of the SSV algorithm is as follows:

• Compute all $m(m-1)$ margins for candidates ($m = |C|$).
• Sort by decreasing margin: $O(m^2 \log m)$.
• For each pair, recursively apply SSV to $P_{-B}$.
• Terminate when the recursion certifies a winner $A$ for pair $(A, B)$.

Complexity:

• Each call: $O(m^2 \cdot n)$ to compute margins; $O(m^2 \log m)$ sort; up to $m(m-1)$ recursive calls to $SSV(m-1, n)$.
• Worst-case:

$$T(m, n) = O\left( \sum_{k=1}^m k^2 (n + k \log k) \prod_{j=k+1}^m j^2 \right)$$

• $T(m) = O((m!)^2)$ for fixed $n$; polynomial in $n$, but super-polynomial in $m$.
• In practice, restricting to the Smith set or using memoization is common for $m > 10$.

3. Illustrative Examples and Empirical Tie Avoidance

Example A: 4 Candidates, 5 Voters

Given $C = \{A, B, C, D\}$, margin computations and candidate eliminations yield a unique SSV winner with a well-defined recursive path.

Example B: Majority Cycle (4-cycle)

With ballots tailored to produce a cycle ($$W \triangleright X \triangleright Y \triangleright Z \triangleright W$$), SSV resolves the cycle by identifying the marginally weakest link and recursively eliminating candidates along margin order.

Empirical simulations (for up to 7 candidates, thousands of voters, uniformly sampled linear ballots) show tie rates below 1%, outperforming standard methods such as IRV, Plurality, Minimax, Beat Path, and Ranked Pairs (Holliday et al., 2021).

4. Relationship to Split Cycle and Condorcet Methods

SSV is closely related to Split Cycle (SC), which resolves cycles by discarding the weakest margin in any violating cycle.

• SC-winner: An alternative not SC-defeated by any other, with defeat defined via cycles and edge strengths.
• Conjecture: For strictly ordered (distinct) margins and up to 6 alternatives, every SSV winner is an SC winner ($SSV(\mathcal{M}) \subseteq SC(\mathcal{M})$) (Holliday et al., 29 Nov 2025). Proven for $n \leq 6$ by induction and SAT (for $n=6$).
• Counterexample for $n = 7$: Explicitly constructed margin graphs show SSV can select a winner outside the SC undefeated set. For $n \geq 7$, SSV and SC can diverge.

Like Minimax and Beat-Path, SSV always elects a Condorcet winner if one exists, but SSV is distinguished by enforcing the Stability-for-Winners property. Other Condorcet methods can suffer from instability to spoilers and ties at various cycle-breaking steps.

5. Proof Structure for Stability-For-Winners

The stability property underpinning SSV is established by induction on the number of candidates:

• Lemma: In any non-trivial profile, at least one $(A, B), m(A, B) > 0$ with $SSV(P_{-B})=A$.
• Proof insight: The absence of such a pair would contradict via cycle construction, violating the inductive assumption. This guarantees that, at each step, recursive elimination proceeds via a pair satisfying the stability-defining property, and that winners are not sensitive to addition of defeated "spoiler" candidates, barring tied claims.

6. Comparison with IRV, Ranked Pairs, and Related Methods

Method	Condorcet Consistency	Stability-for-Winners	Tie Avoidance
SSV	Yes	Yes	Strong, statistically rare
IRV	No	No	Poor, frequent with few voters
Minimax	Yes	No	Moderate
Beat-Path	Yes	No	Moderate
Split Cycle	Yes	No	Variable

• IRV: Eliminates last-place candidate by first-place votes; susceptible to spoilers and frequent ties.
• Minimax/Beat-Path/SC: All Condorcet-consistent, resolve cycles via alternative margin path rules; can violate Stability-for-Winners and tie more often than SSV.

7. Complexity, Scalability, and SAT-Based Findings

While practical for small $m$ (up to 20--30 with preprocessing), SSV's factorial (or worse) growth renders it infeasible for large-scale elections without Smith-set restriction or heavy memoization (Holliday et al., 2021, Holliday et al., 29 Nov 2025). The worst-case decision complexity for general inputs remains an open theoretical question.

The SAT-encoding technique developed for comparing SSV and SC generalizes to other majority-cycle-resolving rules, providing a tool for automated theorem-proving and counterexample generation in social choice theory. For $n\leq6$, SSV is guaranteed to refine SC for strictly ordered margins; this fails for $n\geq7$ by explicit counterexample. This delineates the operational range where SSV can reliably be presented as a refinement of SC, and underscores the power and limitations of SSV in the broader landscape of Condorcet methods (Holliday et al., 29 Nov 2025).