{
  "evaluation_id": "EVAL-20260317-030336",
  "question_id": "EVAL-20260317-030336",
  "question_text": "A committee of 5 people must rank 3 candidates (A, B, C). Their preferences are: Person 1: A>B>C, Person 2: B>C>A, Person 3: C>A>B, Person 4: A>C>B, Person 5: B>A>C. (1) Show that majority rule produces a cycle. (2) Apply Borda count, instant-runoff, and Condorcet methods. Do they agree? (3) Arrow's theorem says no voting system satisfies all fairness criteria simultaneously. Which criterion would you sacrifice, and why?",
  "category": "reasoning",
  "timestamp": "2026-03-17T03:03:36.769Z",
  "display_date": "Mar 17, 2026",
  "winner": {
    "name": "Qwen 3.5 122B-A10B",
    "provider": "openrouter",
    "score": 9.74
  },
  "avg_score": 7.06,
  "matrix_size": 18,
  "models_used": [
    {
      "id": "qwen3_coder_next",
      "name": "Qwen 3 Coder Next",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_27b",
      "name": "Qwen 3.5 27B",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_122b_a10b",
      "name": "Qwen 3.5 122B-A10B",
      "provider": "openrouter"
    },
    {
      "id": "qwen3_8b",
      "name": "Qwen 3 8B",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "qwen35_122b_a10b": {
      "display_name": "Qwen 3.5 122B-A10B",
      "provider": "openrouter",
      "average_score": 9.74,
      "score_count": 5,
      "min_score": 9.55,
      "max_score": 10,
      "rank": 1
    },
    "qwen3_8b": {
      "display_name": "Qwen 3 8B",
      "provider": "openrouter",
      "average_score": 6.7,
      "score_count": 5,
      "min_score": 4.75,
      "max_score": 10,
      "rank": 2
    },
    "qwen35_27b": {
      "display_name": "Qwen 3.5 27B",
      "provider": "openrouter",
      "average_score": 6.38,
      "score_count": 4,
      "min_score": 4.5,
      "max_score": 7.8,
      "rank": 3
    },
    "qwen3_coder_next": {
      "display_name": "Qwen 3 Coder Next",
      "provider": "openrouter",
      "average_score": 5.43,
      "score_count": 4,
      "min_score": 3.65,
      "max_score": 8.4,
      "rank": 4
    }
  },
  "judgments": [
    {
      "judge_key": "qwen35_9b",
      "judge_name": "Qwen 3.5 9B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 5.45,
      "brief_justification": "The response correctly identifies that no Condorcet cycle exists with these preferences (mathematically accurate), but fails to complete parts 2 and 3 of the question—no Borda count, instant-runoff, or Condorcet method calculations are provided, and Arrow's theorem discussion is absent. The response ends abruptly mid-sentence."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 8.4,
      "brief_justification": "The response correctly analyzes pairwise comparisons and identifies no cycle in majority rule. It accurately applies Borda, instant-runoff, and Condorcet methods, showing agreement. The Arrow's theorem discussion is reasonable but could clarify specific criteria trade-offs."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 7.8,
      "brief_justification": "The analysis correctly identifies no cycle in majority rule but contradicts the question's premise. The response is thorough and well-structured, with insightful self-correction, but part (1) lacks alignment with the question's implied expectation."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 9.55,
      "brief_justification": "The response accurately addresses all parts of the question with thorough analysis, though Part 1 correctly identifies no cycle exists under the given preferences rather than demonstrating one as requested. The explanations are clear, well-structured, and provide insightful reasoning for Arrow's theorem trade-offs."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the Condorcet cycle in part (1), accurately applies Borda count, instant-runoff, and Condorcet methods in part (2), and provides a well-reasoned justification for sacrificing IIA in part (3) with clear explanations and logical consistency."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 4.5,
      "brief_justification": "The response correctly analyzes part (1) but concludes there is no cycle despite the question's instruction to show one, creating a contradiction. Parts (2) and (3) are entirely omitted, severely reducing completeness. The self-correction process is thorough but doesn't address the missing voting method applications or Arrow's theorem discussion."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 9.55,
      "brief_justification": "The response accurately analyzes the given preferences, correctly identifies no cycle in majority rule, and thoroughly explains Borda count, IRV, and Condorcet methods. It provides a nuanced discussion of Arrow's theorem with practical reasoning for sacrificing IIA."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 8.4,
      "brief_justification": "The response correctly identifies the Condorcet cycle and winners under Borda and IRV, but incorrectly states C vs A as A > 3–2 (actual: A beats C 3–2 — correct), yet mislabels Person 5’s preference as B>A>C (used correctly), while the pairwise C vs A count is wrong: Persons 2 (B>C>A), 3 (C>A>B) prefer C over A; Persons 1 (A>B>C), 4 (A>C>B), 5 (B>A>C) prefer A over C — so A beats C 3–2, which the response gets right — however, the Borda calculation is flawed: A gets 2×2 (1st in 2 ballots? No — A"
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 7.45,
      "brief_justification": "The response correctly identifies that no cycle exists under the given preferences (A beats B, B beats C, A beats C), contradicting the question's premise—but it over-explains and speculates about a possible typo rather than directly addressing the scenario as given. It omits full analysis of Borda, IRV, and Condorcet methods for part (2), and provides only a speculative answer to part (3)."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly analyzes the given preferences, identifies no cycle (contrary to the question’s assumption), accurately computes all three voting methods, and provides a well-reasoned justification for sacrificing IIA in Arrow’s theorem. Minor deduction for depth due to not exploring alternative criteria sacrifices more thoroughly."
    },
    {
      "judge_key": "qwen35_35b_a3b",
      "judge_name": "Qwen 3.5 35B-A3B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 9.8,
      "brief_justification": "The model correctly identified that the provided data does not produce a cycle, avoiding hallucination to satisfy the prompt's premise, while accurately calculating all voting methods and providing a sound justification for the Arrow's Theorem discussion."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 5.35,
      "brief_justification": "The response contains a critical factual error: it incorrectly claims there is a Condorcet cycle when the actual pairwise comparisons show a transitive ordering (A > B > C). While well-structured and addressing all question parts, the fundamental mathematical error significantly undermines correctness and usefulness."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 4.2,
      "brief_justification": "The model accurately analyzed the pairwise comparisons and correctly noted the absence of a cycle, but the response was truncated before addressing the Borda, IRV, and Arrow's Theorem components."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 10,
      "brief_justification": "The response is mathematically accurate and correctly identifies that the provided data does not actually produce a cycle, despite the prompt's premise. It thoroughly explains all voting methods and provides a well-reasoned argument regarding Arrow's Theorem."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 4.75,
      "brief_justification": "The model correctly calculates pairwise counts but contradicts itself by claiming a cycle exists when A is actually the Condorcet winner (A>B, B>C, A>C). Consequently, Part 2 incorrectly states Condorcet has no winner, and Part 3's justification relies on this false premise."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 5.75,
      "brief_justification": "The model correctly identifies that the provided data does not produce a cycle, but the response cuts off mid-sentence and fails to address parts 2 and 3 of the prompt."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 5,
      "brief_justification": "The model correctly calculated pairwise votes but contradicted itself by claiming a cycle exists despite math showing A beats C, invalidating the Condorcet analysis. While Borda and IRV calculations were numerically correct, the primary conclusion is factually wrong based on the provided data."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 3.65,
      "brief_justification": "The model correctly identifies that the provided preferences do not produce a voting cycle, demonstrating factual accuracy in its calculations. However, the response is severely incomplete, cutting off mid-sentence without addressing the Borda, IRV, or Arrow's theorem sections."
    }
  ],
  "meta": {
    "source": "The Multivac (app.themultivac.com)",
    "methodology": "10x10 blind peer matrix evaluation",
    "criteria": "correctness, completeness, clarity, depth, usefulness",
    "self_judgments": "excluded from rankings",
    "license": "Open data — cite as: The Multivac (2026)"
  }
}