Condorcet-loser dominance among scoring rules

This paper demonstrates that in voting models with three or more alternatives, the Borda rule is the unique scoring rule that minimizes the selection of Condorcet losers, as it strictly dominates all other scoring rules in this regard and is the only one capable of dominating any other rule.

The Gist

Imagine you are organizing a tournament to pick the best player from a group of athletes. You have a few different ways to decide the winner:

1. The "Popularity Contest" (Plurality Rule): You just ask everyone, "Who is your favorite?" The person with the most "favorite" votes wins.
2. The "All-Rounder" (Borda Rule): You ask everyone to rank everyone. The person ranked #1 gets 10 points, #2 gets 9 points, #3 gets 8, and so on. The person with the highest total points wins.

There is a famous problem with the "Popularity Contest." Sometimes, it picks a winner who is actually the worst player in the room. This worst player is called the Condorcet Loser.

What is a Condorcet Loser?
Imagine a player named "Bob." If you put Bob in a one-on-one race against any other player, Bob loses every single time. He is the worst of the worst. Yet, if you only count who people put in the #1 spot, Bob might still win because he has a small group of die-hard fans who love him, while everyone else just tolerates him.

The paper you asked about is a mathematical investigation into which voting rules are best at avoiding picking this "Bob" (the Condorcet Loser).

The Main Characters

• The Plurality Rule: The popular, simple method. It often picks "Bob" by mistake.
• The Borda Rule: The careful, detailed method. It almost never picks "Bob."
• The "Middle Ground" Rules: There are many other ways to count votes that sit somewhere between Plurality and Borda.

The Big Question

The authors asked: "Is there a hierarchy among these 'Middle Ground' rules?"

Imagine you have a rule that is 99% like the Borda rule (very careful) and another rule that is only 10% like the Borda rule (very sloppy). Intuitively, you'd think the "99% Borda" rule is always better at avoiding "Bob" than the "10% Borda" rule.

The paper's shocking answer: No.

The Analogy: The "Bad Luck" Lottery

Think of avoiding the Condorcet Loser like avoiding a specific bad outcome in a lottery.

• The Borda Rule is like a magician who has a magic wand. No matter what lottery ticket you buy, the magician never picks the losing ticket. It is perfect.
• Every other rule (Plurality, and all the "Middle Ground" ones) is like a regular person flipping a coin. Sometimes they pick the winner, sometimes they pick the loser.

The paper proves a fascinating fact: There is no "better" regular person.

If you take two regular people (two non-Borda rules), say "Alice" and "Charlie":

• There will be a specific scenario (a specific set of voter preferences) where Alice accidentally picks the loser, but Charlie gets it right.
• But there will be a different scenario where Charlie accidentally picks the loser, and Alice gets it right.

You cannot say Alice is "better" than Charlie, or that Charlie is "better" than Alice. They are equally flawed in different ways. One might fail in a storm, while the other fails in a drought.

The "Special Status" of the Borda Rule

The paper concludes that the Borda Rule is the only rule that is truly "dominant."

• The Borda rule is the only one that is "better" than any other rule. It beats everyone else because it never makes the specific mistake of picking the worst player.
• No other rule can claim to be better than another. They are all in a tie for "imperfect."

Why Does This Matter?

In the real world, we often try to tweak voting systems to make them "more fair." We might think, "If we just move our scoring system 1% closer to the Borda rule, we'll be safer."

This paper says: Don't get too comfortable.
Unless you are using the full Borda rule, you are still vulnerable. Moving slightly closer to the "perfect" rule doesn't guarantee you won't pick the worst candidate in a specific, tricky situation. The only way to be 100% safe from picking the "Condorcet Loser" is to use the Borda rule itself.

Summary in One Sentence

The Borda voting rule is the only "perfect" guard against picking the worst candidate, and among all the imperfect rules, none is consistently better than another; they just fail in different, unpredictable ways.

Technical Summary

1. Problem Statement

The paper investigates the comparative performance of scoring rules in social choice theory, specifically focusing on their ability to avoid selecting a Condorcet loser.

• Context: A Condorcet loser is an alternative that loses in a pairwise majority vote against every other alternative. While the Borda rule is known to never select a Condorcet loser, other scoring rules (such as the Plurality rule) can and do select them under certain preference profiles.
• The Gap: Previous probabilistic studies (e.g., Lepelley et al., 2000) suggested that scoring rules "closer" to the Borda rule (in terms of their score vectors) are less likely to select a Condorcet loser. However, these studies relied on average probabilities over specific distributions.
• The Core Question: Is there a strict dominance hierarchy among non-Borda scoring rules regarding the avoidance of Condorcet losers? Specifically, if a rule $f$ is "closer" to the Borda rule than rule $g$, does $f$ always avoid selecting a Condorcet loser whenever $g$ does?
• Formal Definition: The authors define Condorcet-loser dominance (CL-dominance). A scoring rule $f$ CL-dominates $g$ if the set of preference profiles where $f$ selects a Condorcet loser is a proper subset of the set where $g$ selects one.

2. Methodology

The authors employ a constructive, mathematical approach using induction on the number of alternatives ($m$).

• Model Setup:
  • Alternatives $X = \{x_1, \dots, x_m\}$ with $m \geq 3$.
  • Scoring rules are defined by score vectors $s = (s(1), \dots, s(m))$ where $1 = s(1) \geq \dots \geq s(m) = 0$.
  • The Borda rule corresponds to a linearly decreasing score vector.
• Proof Strategy (Proposition 1):
  The central technical contribution is proving Proposition 1: For any non-Borda scoring rule $f$ and any other scoring rule $f'$, there exists a preference profile where $f$ selects a Condorcet loser, but $f'$ does not.
  • Base Case ($m=3$): The authors explicitly construct preference profiles for all possible relative positions of the score vectors $s$ and $s'$. They divide the analysis into six sub-cases based on the value of the second-ranked score $s(2)$ relative to $0.5$ (the Borda value) and relative to $s'(2)$. By manipulating the number of voters in specific preference orderings, they demonstrate that the "threshold" for selecting the Condorcet loser can be crossed by one rule but not the other.
  • Inductive Step ($m > 3$): To extend the result to $m$ alternatives, the authors use an embedding technique.
    • They show that for most pairs of score vectors, one can construct a valid $(m-1)$-alternative or 3-alternative profile that satisfies the conditions of the inductive hypothesis.
    • They then "embed" this smaller profile into the $m$-alternative space by adding new alternatives and a "dummy" population of voters.
    • Crucially, they construct a "uniform" sub-profile where new alternatives are inserted at various ranks to ensure that the pairwise majority relationships (specifically making the target alternative a Condorcet loser) are preserved or adjusted as needed, while the score differences between the original alternatives remain dominant.
  • Exceptional Cases: The proof handles specific edge cases where score vectors have equalities (e.g., $s(2) = s(m-1) = 0.5$) by explicitly constructing tailored profiles rather than relying on the general embedding lemma.

3. Key Results

The paper establishes two main theorems:

1. Theorem 1 (Reiteration of Known Result): The Borda rule is the unique scoring rule that never selects a Condorcet loser ($L(f_B) = \emptyset$). Consequently, the Borda rule CL-dominates all other scoring rules.
2. Theorem 2 (Main Result): No dominance relationship exists between any pair of non-Borda scoring rules.
   • Formally: A scoring rule $f$ CL-dominates a non-Borda rule $f'$ if and only if $f$ is the Borda rule.
   • Implication: There is no "better" non-Borda rule. For any two distinct non-Borda rules $f$ and $g$, there is always a scenario where $f$ fails to avoid a Condorcet loser while $g$ succeeds, and vice versa.
   • This contradicts the intuition derived from probabilistic analyses that "closer to Borda" implies "strictly better." While rules closer to Borda may perform better on average under specific distributions, they are not universally superior in terms of the set of profiles they handle correctly.

4. Significance and Contributions

• Theoretical Clarification: The paper resolves the question of whether a hierarchy exists among scoring rules based on the Condorcet loser criterion. It definitively shows that the Borda rule stands alone as the unique maximal element; the rest of the class of scoring rules is an antichain (no element dominates another) with respect to CL-dominance.
• Methodological Innovation: The paper introduces a powerful embedding method (Lemma 1 and the construction of $\succ^*$ and $\succ'$) to generalize properties from small numbers of alternatives ($m=3$) to arbitrary $m$. This technique allows researchers to extend results derived in simple cases to general voting models without needing to solve complex high-dimensional optimization problems for every $m$.
• Characterization of Borda Rule: While previous literature characterized the Borda rule via axioms (e.g., Young's properties, Smith's condition) or by its unique ability to avoid Condorcet losers entirely, this paper characterizes it via a dominance relation. It highlights that the Borda rule is the only rule that dominates any other rule in this specific context.
• Comparison with Literature: The work contrasts with probabilistic studies (Lepelley et al., 2000; Kamwa and Valognes, 2017) by shifting the focus from "average performance" to "worst-case/profile-specific performance." It demonstrates that average superiority does not imply dominance in the set-theoretic sense.

Conclusion

Doi and Nakamura demonstrate that while the Borda rule is uniquely robust against selecting Condorcet losers, no other scoring rule can be ranked as "better" or "worse" than another in a strict dominance sense. The set of non-Borda scoring rules is flat regarding this criterion; any perceived advantage of a rule being "closer" to Borda is distribution-dependent and does not hold universally across all preference profiles.