Probabilistic voting model upends classic social choice with Copeland beating Random Dictator
New research shows Copeland's rule has distortion ≤2 under Plackett-Luce model, outperforming Random Dictator in large elections.
Metric distortion evaluates how well voting rules minimize social cost when voters and candidates exist in a shared metric space — a voter's cost is their distance to the winner. Traditionally, distortion is studied under deterministic voter preferences. In this paper, Goyal and Sarmasarkar extend the framework to probabilistic voting, where voters exhibit randomness in their rankings. They focus on the Plackett-Luce (PL) model, widely used in economics and machine learning, where each voter's probability of ranking a candidate depends on that candidate's “strength” (inversely proportional to the voter's metric distance to the candidate).
Key results flip conventional wisdom. Under PL with strength proportional to 1/distance², Copeland's rule achieves a distortion of at most 2, while Random Dictator's distortion grows as Ω(√m) with candidate count m (as n → ∞). This contrasts sharply with classical deterministic results, where Random Dictator beats Copeland (distortion 3 vs 5). For Borda, the distortion generalizes to Θ(m^{1-2/θ}) when θ>2 and Θ(1) otherwise, generalizing the classical bound of 2m−1. The proofs use asymptotic duality with Lagrange multipliers. The work paves the way for analyzing voting rules under more realistic voter behavior.
- Under Plackett-Luce model with inverse square distance, Copeland's distortion ≤2 — far better than the classical bound of 5.
- Random Dictator's distortion becomes Ω(√m) in large elections under the same probabilistic model, losing to Copeland.
- Borda's distortion generalizes to Θ(m^{1-2/θ}) for θ>2, smoothly interpolating between constant and linear growth.
Why It Matters
Probabilistic voting models better reflect real-world voter randomness, offering more accurate guidance for election design.