{
  "evaluation_id": "EVAL-20260402-163912",
  "question_id": "REASON-014",
  "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": "2026-04-02T16:39:12.442Z",
  "display_date": "Apr 02, 2026",
  "winner": {
    "name": "DeepSeek V4",
    "provider": "openrouter",
    "score": 8.71
  },
  "avg_score": 7.5025,
  "matrix_size": 70,
  "models_used": [
    {
      "id": "mimo_v2_flash",
      "name": "MiMo-V2-Flash",
      "provider": "Xiaomi"
    },
    {
      "id": "gemini_25_flash",
      "name": "Gemini 2.5 Flash",
      "provider": "openrouter"
    },
    {
      "id": "deepseek_v4",
      "name": "DeepSeek V4",
      "provider": "openrouter"
    },
    {
      "id": "claude_opus_46",
      "name": "Claude Opus 4.6",
      "provider": "openrouter"
    },
    {
      "id": "gpt_5_4",
      "name": "GPT-5.4",
      "provider": "openrouter"
    },
    {
      "id": "grok_420",
      "name": "Grok 4.20",
      "provider": "openrouter"
    },
    {
      "id": "claude_sonnet_46",
      "name": "Claude Sonnet 4.6",
      "provider": "openrouter"
    },
    {
      "id": "gemini_31_pro",
      "name": "Gemini 3.1 Pro",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "deepseek_v4": {
      "display_name": "DeepSeek V4",
      "provider": "openrouter",
      "average_score": 8.71,
      "score_count": 8,
      "min_score": 6.9,
      "max_score": 9.45,
      "rank": 1
    },
    "gpt_5_4": {
      "display_name": "GPT-5.4",
      "provider": "openrouter",
      "average_score": 8.59,
      "score_count": 9,
      "min_score": 6.45,
      "max_score": 10,
      "rank": 2
    },
    "claude_sonnet_46": {
      "display_name": "Claude Sonnet 4.6",
      "provider": "openrouter",
      "average_score": 8.54,
      "score_count": 9,
      "min_score": 5.3,
      "max_score": 9.85,
      "rank": 3
    },
    "grok_420": {
      "display_name": "Grok 4.20",
      "provider": "openrouter",
      "average_score": 8.42,
      "score_count": 9,
      "min_score": 4.9,
      "max_score": 9.8,
      "rank": 4
    },
    "gemini_25_flash": {
      "display_name": "Gemini 2.5 Flash",
      "provider": "openrouter",
      "average_score": 8.32,
      "score_count": 9,
      "min_score": 5.65,
      "max_score": 9.45,
      "rank": 5
    },
    "claude_opus_46": {
      "display_name": "Claude Opus 4.6",
      "provider": "openrouter",
      "average_score": 7.28,
      "score_count": 9,
      "min_score": 5.55,
      "max_score": 8.65,
      "rank": 6
    },
    "mimo_v2_flash": {
      "display_name": "MiMo-V2-Flash",
      "provider": "Xiaomi",
      "average_score": 7.12,
      "score_count": 9,
      "min_score": 5.35,
      "max_score": 9.45,
      "rank": 7
    },
    "gemini_31_pro": {
      "display_name": "Gemini 3.1 Pro",
      "provider": "openrouter",
      "average_score": 3.04,
      "score_count": 8,
      "min_score": 2,
      "max_score": 5.4,
      "rank": 8
    }
  },
  "judgments": [
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.45,
      "brief_justification": "Accurate, thorough, and insightful with clear explanations."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.45,
      "brief_justification": "Highly accurate, thorough, and insightful with clear explanations."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 6.9,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 7.35,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.8,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 6.9,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 7.25,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 5.35,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 6.9,
      "brief_justification": ""
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 5.4,
      "brief_justification": "Correctly identifies no cycle but misses answering all parts of the question."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.65,
      "brief_justification": "Detailed and accurate analysis with clear structure, though slightly verbose."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.45,
      "brief_justification": "Accurate, thorough, and clear with insightful analysis."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.45,
      "brief_justification": "Accurate, thorough, and insightful analysis with clear explanations and practical implications."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.45,
      "brief_justification": "Accurate, thorough, and insightful with clear structure."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.75,
      "brief_justification": "The response begins by claiming the preferences don't produce a cycle, but let's check: A vs B (persons 1,4,5 prefer A? No - person 5 prefers B>A>C, so A>B supporters are persons 1,4 = 2, B>A supporters are persons 2,5 = 2, person 3 prefers C>A>B so A>B, giving A>B 3-2). A vs C: persons 1,5 prefer A over C = 2, persons 2,3 prefer C over A = 2, person 4 prefers A>C giving A>C 3-2. B vs C: persons 2,5 prefer B>C = 2, persons 3,4 prefer C>B = 2, person 1 prefers B>C giving B>C 3-2. So A>B, A>C, B>C"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9,
      "brief_justification": "The response is highly accurate across all three parts. The pairwise comparisons in part (1) are correctly computed, the Borda count and IRV calculations in part (2) are correct, and the Condorcet cycle is properly identified. The discussion of Arrow's theorem in part (3) is well-reasoned and correctly identifies IIA as the most commonly sacrificed criterion with good justification. The only minor issue is that the Copeland method analysis could have been slightly more nuanced, and the depth of "
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9,
      "brief_justification": "The response is exceptionally well-structured and clear. Critically, the model correctly identifies that the given preferences do NOT produce a Condorcet cycle—A is actually a Condorcet winner—rather than blindly following the question's incorrect premise. This demonstrates strong mathematical reasoning and intellectual honesty. The pairwise comparisons, Borda count, IRV, and Condorcet analyses are all computed correctly. The discussion of Arrow's theorem and the choice to sacrifice IIA is well-"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.3,
      "brief_justification": "The response is well-structured, clear, and thorough. However, there are correctness issues: (1) In B vs C, Person 4 (A>C>B) prefers C>B, not B>C, so B beats C 3-2, not 4-1. The cycle conclusion still holds but the margin is wrong. (2) The claim that the Condorcet method satisfies IIA is incorrect - pairwise majority rule does not satisfy IIA in general. (3) The C vs A check: Person 5 (B>A>C) prefers A>C, not C>A, so A beats C 3-2, breaking the cycle entirely. This means there IS a Condorcet win"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 8.95,
      "brief_justification": "The analysis is thorough, well-structured, and largely correct. The pairwise comparisons are accurately computed, the Condorcet cycle is correctly identified, and the Borda/IRV calculations are correct. One minor issue: in the A vs C comparison, Person 5 (B>A>C) prefers A over C, giving A voters {1,4,5}=3 and C voters {2,3}=2, so A beats C 3-2, not C beats A. This means there is actually NO cycle — A is the Condorcet winner. This is a significant factual error that undermines Part 1 and the Cond"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 6.1,
      "brief_justification": "The response is very well-structured and clearly written, but contains a critical error in Part 1. The head-to-head for C vs A: P2 (B>C>A) prefers C over A, P3 (C>A>B) prefers C over A, P4 (A>C>B) prefers A over C, P1 (A>B>C) prefers A over C, P5 (B>A>C) prefers A over C. So A beats C 3-2, meaning A is actually a Condorcet winner and there is NO cycle. The response claims C beats A to form a cycle (A>B>C>A), which is factually wrong. This undermines the entire premise of Part 1. Consequently, Pa"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.75,
      "brief_justification": "The response is well-structured, clearly presented, and mostly correct. The pairwise comparison for C vs A contains an error: Person 4 (A>C>B) prefers A over C, not C over A. Despite this, the final result (C>A 3:2 becoming actually A>C 3:2 if corrected... wait, let me recheck. Person 2: B>C>A means C>A, Person 3: C>A>B means C>A, Person 4: A>C>B means A>C. So C>A supporters: P2, P3 = 2. A>C supporters: P1, P4, P5 = 3. So actually A beats C 3:2, meaning there IS no cycle - A would be the Condorc"
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.7,
      "brief_justification": "The response only asserts that the profile does not produce a cycle, which is correct, but it fails to actually work through the pairwise comparisons, does not answer the requested voting-method analysis, and does not address the Arrow criterion tradeoff. Clear opening, but highly incomplete and minimally useful."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9,
      "brief_justification": "Accurately identifies the majority cycle, computes Borda and IRV correctly, and correctly notes no Condorcet winner. Clear and well-structured, with a sensible discussion of sacrificing IIA. Slightly limited because 'Condorcet method' is treated generically rather than distinguishing among specific Condorcet completion rules."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 5.65,
      "brief_justification": "The response correctly finds that the given profile does not produce a majority cycle and that Borda, IRV, and Condorcet all select A. However, it directly contradicts part (1) of the prompt rather than addressing the likely intended result, and part (3) is cut off and incomplete. Presentation is organized and clear, but the unfinished Arrow discussion significantly reduces completeness and practical usefulness."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9,
      "brief_justification": "Accurately identifies the majority cycle and correctly computes Borda and IRV. Clear, well-structured, and gives a thoughtful Arrow-theorem discussion. Minor issue: saying the 'Condorcet method' satisfies IIA is too broad/inexact, and no specific Condorcet completion method is discussed."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 8.4,
      "brief_justification": "Accurately identifies the majority cycle, computes Borda and IRV correctly, and notes that Condorcet has no winner. Clear and thorough. Minor issue: saying the Condorcet method fails unrestricted domain is debatable, since many Condorcet methods still return a winner via tie-breaking/resolution despite no Condorcet winner existing."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 5.75,
      "brief_justification": "Well-structured and mostly complete, but it makes a major factual error: there is no majority cycle here because A beats both B and C, making A the Condorcet winner. It also misstates Arrow’s theorem criteria and incorrectly suggests the Condorcet method satisfies IIA."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.4,
      "brief_justification": "Accurately shows the majority cycle and correctly computes Borda and IRV. It appropriately notes there is no Condorcet winner, though 'Condorcet method' is underspecified because specific Condorcet completion methods could select a winner. The Arrow discussion is thoughtful and well justified, with only minor conceptual looseness around linking the example to IIA."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.4,
      "brief_justification": "The response correctly identifies that the given preferences do not produce a cycle under majority rule (A beats B 3-2, B beats C 3-2, A beats C 4-1), but it fails to actually demonstrate or compute the cycle as requested in part (1), instead dismissing the question. It promises to proceed with other methods and Arrow's theorem but provides no actual calculations or answers for (2) or (3), making it incomplete and low in usefulness."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.2,
      "brief_justification": "Accurate calculations and logical reasoning; comprehensive coverage of all three methods and Arrow's theorem; clear structure; insightful justification for sacrificing IIA."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.45,
      "brief_justification": "The response is factually accurate on all voting calculations and Arrow's criteria. It provides thorough step-by-step explanations with correct pairwise, Borda, and IRV results. Minor incompleteness in noting that standard Condorcet requires a winner (cycle means failure), but overall very clear, insightful on trade-offs, and practically useful."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.25,
      "brief_justification": "The response accurately identifies no cycle in the given profile (correcting the question's assumption), correctly applies all three voting methods showing A wins, and provides a solid explanation of Arrow's theorem (though truncated). Tables and calculations are clear and well-structured. Depth is good but slightly reduced for not fully addressing the cycle as posed and for the incomplete final section."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 8.65,
      "brief_justification": "The response accurately identifies that the given preferences do not produce a cycle (A is Condorcet winner), correctly applies Borda, IRV, and Condorcet (all select A), and provides a reasoned choice to sacrifice IIA in Arrow's theorem. Minor deduction for depth as it doesn't explore alternative interpretations or the exact cycle profile beyond an offer."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 8.65,
      "brief_justification": "The response is factually accurate, logically sound, and covers all parts thoroughly with clear structure, tables, and explanations. It provides insightful analysis on Arrow's theorem and practical tradeoffs, making it highly useful, though the depth in philosophical aspects could be slightly more nuanced."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 6.75,
      "brief_justification": "Response is mostly clear and structured but contains factual errors in the Condorcet analysis (incorrectly claims A beats C and is Condorcet winner when data shows C beats A 3-2, creating a true cycle). Borda and IRV calculations are accurate. Arrow's theorem criteria are slightly misstated (misses transitivity, includes unanimity instead). Depth on Arrow's trade-off is reasonable but not profound."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.65,
      "brief_justification": "Response is factually accurate, logically sound, and covers all parts thoroughly with clear structure and good explanations. Minor issue: Arrow's criteria list slightly misstates 'universal domain' vs. the standard 'unrestricted domain' but overall excellent."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.85,
      "brief_justification": "The response makes an incorrect claim that the given preferences do not produce a cycle, when in fact they do. A vs B: A wins (persons 1,4,5 prefer A; persons 2,3 prefer B) — wait, let me verify: A>B: persons 1,4,5 (3 votes); B>A: persons 2,3 (2 votes). B>C: persons 1,2,5 (3 votes); C>B: persons 3,4 (2 votes). A>C: persons 1,3(?no),4,5 — person 3 prefers C>A, so A>C: persons 1,4,5 (3 votes); C>A: persons 2,3 (2 votes). Actually A beats B, A beats C, B beats C — no cycle. The response starts to m"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.8,
      "brief_justification": "The response is highly accurate in all three parts. The pairwise comparisons are correctly computed, Borda count arithmetic is verified, IRV elimination is properly executed, and the Condorcet cycle is correctly identified. The explanation of Arrow's theorem criteria is accurate and the reasoning for sacrificing IIA is well-argued. Minor gap: the depth on Arrow's theorem could explore the tension between IIA and other criteria more philosophically, and the Copeland tie-breaking note is a nice to"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 7.2,
      "brief_justification": "The response is well-structured and clearly presented, but makes a critical error: it correctly identifies that the given 5-voter profile produces no cycle (A is a Condorcet winner), yet the question explicitly asks to 'show that majority rule produces a cycle.' Rather than questioning the premise or resolving the contradiction cleanly, the response pivots to a different 3-voter example to demonstrate a cycle, which sidesteps the actual question. The Borda, IRV, and Condorcet analyses are accura"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 7.75,
      "brief_justification": "The response is exceptionally clear and well-structured, with accurate calculations for Borda, IRV, and Condorcet methods. However, it fails on part (1): the question explicitly asks to 'show that majority rule produces a cycle,' but the given preferences actually produce A as a Condorcet winner with no cycle. The AI correctly identifies this contradiction rather than fabricating a cycle, which shows intellectual honesty, but it doesn't attempt to reconcile the question's premise (e.g., noting t"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.8,
      "brief_justification": "The response is highly accurate, correctly identifying the Condorcet cycle, computing Borda scores, applying IRV properly, and noting the absence of a Condorcet winner. The Arrow's theorem discussion is well-reasoned with a defensible choice to sacrifice IIA. Minor issue: the B vs C pairwise comparison appears incorrect (Person 4 prefers C>B, not B>C, making it 3-2 not 4-1), but this doesn't affect the cycle conclusion. Overall excellent structure, depth, and practical reasoning."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.3,
      "brief_justification": "The response is well-structured and covers all three parts thoroughly. However, there is a significant error in Part 1: the head-to-head result for C vs. A is incorrectly calculated. C beats A (P2, P3, P4 prefer C over A = 3 votes vs P1, P5 = 2 votes), meaning the cycle should be A>B, B>C, C>A — which actually IS a cycle. The response states A beats C (3-2) which is wrong (P4 prefers C over A based on A>C>B... wait, P4: A>C>B means A preferred over C). Let me recheck: P1(A>B>C): A>C ✓, P2(B>C>A)"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9,
      "brief_justification": "Highly accurate pairwise calculations, correct Borda scores, proper IRV execution, and correct identification of no Condorcet winner. All three methods clearly explained and compared. The Arrow's theorem discussion is solid with a well-reasoned argument for sacrificing IIA. Minor limitation: the depth on Arrow's theorem could explore the trade-offs more rigorously (e.g., discussing what IIA violation means normatively), but overall this is an excellent, well-structured response."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.2,
      "brief_justification": "The response fails to answer the prompt by claiming the provided preferences do not produce a cycle, which is factually incorrect (A beats B 3-2, B beats C 3-2, C beats A 3-2). It stops before applying any voting methods or discussing Arrow's theorem."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.4,
      "brief_justification": "Accurate and thorough analysis of all parts, clear structure, insightful discussion of Arrow's theorem trade-offs, and practical conclusions."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 7.75,
      "brief_justification": "Strong clarity and structure, but marked down on correctness for initially misdiagnosing the cycle (it does exist) and on completeness for cutting off mid-sentence."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 7.25,
      "brief_justification": "Response is clear and well-structured, but contains a critical factual error in part (1) by incorrectly stating no cycle exists when the given preferences do produce a Condorcet cycle (A>B, B>C, C>A)."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.4,
      "brief_justification": "Excellent response. Correctly identifies the Condorcet cycle, accurately applies Borda and IRV, and provides a sophisticated, well-reasoned answer to the Arrow's theorem question. The analysis is thorough and clearly structured."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.4,
      "brief_justification": "The response is highly accurate, covering all parts of the question with exceptional clarity and structure. The analysis of Arrow's theorem and the justification for sacrificing IIA are particularly insightful and well-reasoned."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2,
      "brief_justification": "The response fails to answer the asked questions, providing no analysis of the cycle, voting methods, or Arrow's criterion trade-off; it is incomplete and lacks depth, though the writing is somewhat clear."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 5.55,
      "brief_justification": "The answer is clearly presented and well-structured, but it misstates the majority-cycle result for the given preferences and leaves the Arrow's theorem discussion unfinished, reducing overall accuracy and usefulness."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 6.45,
      "brief_justification": "The response correctly computes pairwise results and explains voting methods, but it does not fulfill the prompt's request to show a majority cycle, making the core answer incorrect. It is thorough, clear, and reasonably insightful, though its practical usefulness is reduced by the key mismatch."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 4.9,
      "brief_justification": "The answer is clearly written and covers all parts, but it contains critical errors in the pairwise majority calculations, leading to an incorrect claim of a Condorcet cycle. The rest of the analysis is superficial, limiting its practical value."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 5.3,
      "brief_justification": "The answer is clearly presented and covers all requested parts, but it incorrectly computes the A vs C pairwise result, leading to a false claim of a Condorcet cycle. The rest of the methods are applied correctly, and the discussion of Arrow's theorem is reasonable, yet the core error reduces overall correctness and practical usefulness."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 5.85,
      "brief_justification": "The answer is well-structured and covers all parts, but it incorrectly claims a majority-cycle when A actually beats both B and C, reducing factual accuracy. Other sections are clear and reasonably detailed, though the discussion of Arrow's theorem lacks deeper insight."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 5.65,
      "brief_justification": "The response is well organized and covers all requested parts, but it gets the core majority-rule analysis wrong (no cycle) and misstates the existence of a Condorcet winner, undermining factual accuracy. Clarity and coverage are good, but depth and practical usefulness suffer due to the critical errors."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.45,
      "brief_justification": "The response is highly accurate, clear, and provides a comprehensive breakdown of each voting method. The explanation of Arrow's theorem and the justification for sacrificing IIA are well-reasoned and insightful."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.45,
      "brief_justification": "The response correctly identifies the initial miscalculation and self-corrects, demonstrating a strong understanding of the concepts. It then accurately applies the voting methods and discusses Arrow's Theorem, though the explanation of the criteria is cut off."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "The response is exceptionally thorough, accurate, and well-structured. It correctly identifies the absence of a cycle in the given scenario, accurately applies all three voting methods, and provides a well-reasoned justification for sacrificing IIA in Arrow's Theorem. The step-by-step breakdown enhances clarity and understanding."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.45,
      "brief_justification": "The response is highly accurate, clear, and provides a thorough explanation for each part of the question. The depth of analysis, especially in justifying the sacrifice of IIA, is excellent."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.85,
      "brief_justification": "The response is exceptionally thorough, accurate, and well-structured. It correctly identifies and explains all aspects of the problem, including the nuances of Arrow's theorem and the practical implications of sacrificing IIA. The use of tables and clear explanations makes it very easy to follow."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.65,
      "brief_justification": "The response accurately calculates and explains the voting methods, clearly identifies the cycle, and provides a thoughtful justification for sacrificing IIA. The Condorcet winner identification in part 2.C is slightly off, as the cycle means there is no Condorcet winner, but the rest is solid."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.05,
      "brief_justification": "Response is incomplete and contains a critical error: it claims no cycle exists, but the pairwise comparisons (A beats B 3-2, B beats C 3-2, C beats A 3-2) clearly form a Condorcet cycle. The response cuts off before answering any of the three sub-questions."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.65,
      "brief_justification": "Accurately demonstrates the Condorcet cycle, correctly applies all three voting methods with clear calculations, and provides a well-reasoned explanation for sacrificing IIA. Well-structured with good depth, though could explore practical implications further."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 6.7,
      "brief_justification": "Accurate voting method calculations and honest identification that the given profile doesn't actually cycle (correct but doesn't answer the question as posed). Part 3 is incomplete - Arrow's criteria table starts but cuts off mid-sentence without answering which criterion to sacrifice and why."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9,
      "brief_justification": "The response correctly identifies that majority rule does NOT produce a cycle with these preferences (A is Condorcet winner with 4-1 over B and 3-2 over C). It then accurately computes Borda count (A=6, B=5, C=4), IRV (A wins after C elimination), and Condorcet (A wins all pairwise contests). All methods agree on A. The explanation of which Arrow criterion to sacrifice (IIA) is well-reasoned. The response is thorough, well-structured, and insightful."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.8,
      "brief_justification": "Comprehensive and accurate response. Part 1 correctly shows the Condorcet cycle with proper pairwise calculations. Part 2 correctly applies all three methods with full transparency (Borda: A=6, B=5, C=4; IRV: A wins after C elimination; Condorcet: no winner due to cycle), and clearly states agreement between Borda and IRV. Part 3 demonstrates strong understanding of Arrow's theorem with well-reasoned argument for sacrificing IIA. Well-structured with clear headings and step-by-step calculations."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.65,
      "brief_justification": "Factually accurate throughout with correct calculations for all three methods. Thoroughly covers all parts of the question. Exceptionally clear with excellent tables and structure. Deep analysis explains the philosophical reasoning behind sacrificing IIA. Highly useful with practical insights and clear conclusions."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 7.9,
      "brief_justification": "Mostly accurate but contains a critical error in part 1: claims a cycle A>B>C>A when actual head-to-head results show A beats C (3-2), not C beats A. Parts 2 and 3 are thorough, well-organized, and insightful. The Borda, IRV, and Condorcet calculations are correct, and the discussion of Arrow's theorem and IIA sacrifice is thoughtful and practical."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.65,
      "brief_justification": "Accurate and thorough analysis of all three parts. Clear structure with correct calculations for Borda count and IRV. Good depth in explaining the cycle and Arrow's theorem. The choice to sacrifice IIA is well-reasoned with practical justifications."
    }
  ],
  "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)"
  }
}