{ "evaluation_id": "EVAL-20260318-163448", "question_id": "EVAL-20260318-163448", "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": "2026-03-18T16:34:48.960Z", "display_date": "Mar 18, 2026", "winner": { "name": "GPT-5.4", "provider": "openrouter", "score": 9.07 }, "avg_score": 8.37, "matrix_size": 29, "models_used": [ { "id": "minimax_m25", "name": "MiniMax M2.5", "provider": "openrouter" }, { "id": "minimax_01", "name": "MiniMax-01", "provider": "openrouter" }, { "id": "judge_gpt54", "name": "GPT-5.4", "provider": "openrouter" }, { "id": "judge_claude_sonnet", "name": "Claude Sonnet 4.6", "provider": "openrouter" }, { "id": "minimax_m27", "name": "MiniMax M2.7", "provider": "openrouter" } ], "rankings": { "judge_gpt54": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 9.07, "score_count": 7, "min_score": 7.9, "max_score": 10, "rank": 1 }, "minimax_m25": { "display_name": "MiniMax M2.5", "provider": "openrouter", "average_score": 9.03, "score_count": 6, "min_score": 7.4, "max_score": 9.85, "rank": 2 }, "minimax_m27": { "display_name": "MiniMax M2.7", "provider": "openrouter", "average_score": 8.39, "score_count": 5, "min_score": 5.15, "max_score": 10, "rank": 3 }, "minimax_01": { "display_name": "MiniMax-01", "provider": "openrouter", "average_score": 8.35, "score_count": 7, "min_score": 5.35, "max_score": 9.45, "rank": 4 }, "judge_claude_sonnet": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 7.01, "score_count": 4, "min_score": 6.55, "max_score": 7.85, "rank": 5 } }, "judgments": [ { "judge_key": "minimax_m27", "judge_name": "MiniMax M2.7", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9.8, "brief_justification": "The response accurately demonstrates the Condorcet cycle with correct pairwise comparisons, properly calculates Borda count (A=6, B=5, C=4) and IRV (A wins after C elimination), correctly identifies Condorcet failure, and provides a thoughtful explanation for sacrificing the Condorcet criterion based on practical considerations about cycle rarity and method reliability." }, { "judge_key": "minimax_m27", "judge_name": "MiniMax M2.7", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 9.45, "brief_justification": "The response correctly demonstrates the majority rule cycle with accurate pairwise calculations, thoroughly applies all three voting methods (Borda, IRV, and Condorcet) with correct arithmetic, and provides a well-reasoned argument for sacrificing IIA. The only minor gap is that it doesn't explicitly state whether the methods 'agree'—Borda and IRV both select A while Condorcet finds no winner, which would strengthen the conclusion." }, { "judge_key": "minimax_m27", "judge_name": "MiniMax M2.7", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "The response correctly identifies that the given preferences do not produce a majority cycle (A beats B, B beats C, and A beats C all 3-2, making A the transitive winner), demonstrating strong critical thinking in questioning the premise. All three voting methods are correctly applied and yield consistent results (A wins), and the discussion of Arrow's theorem is thoughtful and well-reasoned. The clarity and structure are excellent." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 9, "brief_justification": "The response correctly identifies the majority cycle, accurately calculates all three voting methods (A wins Borda and IRV, Condorcet has no winner), and provides thoughtful analysis of Arrow's theorem. Minor deduction for slightly imprecise naming of Arrow's criteria (listing Unanimity instead of Pareto efficiency). Overall thorough and well-structured." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 6.65, "brief_justification": "Major error in Part 1: The response initially claims a cycle exists but then correctly shows A beats everyone (3-2 in all pairwise matchups) - no cycle exists with this preference data. Parts 2 and 3 are accurate and well-explained, but the fundamental error in demonstrating the cycle undermines the response's correctness." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.4, "brief_justification": "The response correctly identifies that the given preferences do NOT produce a cycle (A beats B, B beats C, A beats C - transitive), then correctly applies all three voting methods which all yield A as winner. Part (3) provides thoughtful analysis of Arrow's theorem criteria. The only minor issue is that the question premise was flawed (no cycle exists), which the response properly addresses." }, { "judge_key": "minimax_m21", "judge_name": "MiniMax M2.1", "respondent_key": "minimax_m27", "respondent_name": "MiniMax M2.7", "weighted_score": 10, "brief_justification": "The response is comprehensive, accurate, and well-structured. It correctly calculates all voting method results, explains the Condorcet paradox, and provides nuanced reasoning for sacrificing IIA. The step-by-step explanations with tables enhance clarity and practical applicability." }, { "judge_key": "minimax_m21", "judge_name": "MiniMax M2.1", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9.25, "brief_justification": "The response correctly identifies the Condorcet cycle, accurately computes all three voting methods (A wins under Borda and IRV, no Condorcet winner), and provides a reasonable argument for sacrificing the Condorcet criterion. Minor depth limitation: could elaborate more on which specific fairness criteria from Arrow's theorem are being traded off." }, { "judge_key": "minimax_m21", "judge_name": "MiniMax M2.1", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 9.25, "brief_justification": "The response correctly identifies the Condorcet cycle, accurately calculates all three voting methods with proper methodology, and thoughtfully addresses Arrow's theorem trade-offs. The IRV redistribution was clearly explained. Minor扣分 for slight ambiguity in Borda scoring convention and one miscount in the IRV redistribution (Person 4's redistribution was technically unnecessary since A was already their first choice), but overall the analysis is thorough and pedagogically sound." }, { "judge_key": "minimax_m21", "judge_name": "MiniMax M2.1", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.8, "brief_justification": "Response is mathematically rigorous and completely accurate, correctly identifying that no cycle exists with these preferences. All three voting methods are properly calculated, and Arrow's theorem discussion provides substantive justification for sacrificing IIA. Well-organized with clear notation and thorough explanations." }, { "judge_key": "minimax_m2", "judge_name": "MiniMax M2", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9.85, "brief_justification": "The response correctly identifies the Condorcet cycle, accurately calculates Borda (A=6, B=5, C=4), IRV (A wins after C eliminated), and Condorcet (no winner). It clearly explains why Borda and IRV agree on A while Condorcet fails, and provides a thoughtful analysis of which fairness criterion to sacrifice, explaining the trade-offs in voting systems." }, { "judge_key": "minimax_m2", "judge_name": "MiniMax M2", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 8.95, "brief_justification": "The response correctly identifies the majority cycle and provides accurate Borda count and Condorcet analysis, but contains a minor calculation error in IRV (shows final round as 5-1 instead of 4-1). Excellent coverage of all methods with clear explanations and insightful discussion of Arrow's theorem trade-offs." }, { "judge_key": "minimax_m2", "judge_name": "MiniMax M2", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 7.9, "brief_justification": "The response correctly identifies that there is NO majority cycle in this preference profile (which contradicts the question's premise), and handles parts (2) and (3) excellently. However, it fails to show a cycle as explicitly requested in part (1), though its analysis of why no cycle exists is accurate and insightful." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "minimax_m27", "respondent_name": "MiniMax M2.7", "weighted_score": 9.2, "brief_justification": "Response is thorough, well-structured, and technically accurate. The majority cycle is correctly demonstrated, all three voting methods are properly applied, and the recommendation to sacrifice IIA is well-justified with practical reasoning." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 8.9, "brief_justification": "The response correctly identifies the voting cycle in part 1 and applies Borda and Condorcet methods accurately. However, there is an error in the IRV calculation - the final round shows A with 5 votes and C with 1 vote, but based on proper redistribution from B's votes (Person 2 goes to C, Person 5 goes to A), it should be A=4 and C=1. Part 3 provides good depth on Arrow's theorem with reasonable justification for sacrificing IIA." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 6.55, "brief_justification": "Response correctly calculates all three voting methods (Borda=6, IRV=A, Condorcet=A) and explains Arrow's theorem well, but fails the core question: it shows NO cycle exists (A beats all 3-2) rather than demonstrating the requested cycle, and the cycle explanation becomes confusing." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 10, "brief_justification": "The response correctly identifies that with these preferences, majority rule does NOT produce a cycle (the question was flawed or testing detection of this). All three voting methods (Borda, IRV, Condorcet) are correctly calculated and all agree on winner A. The discussion of Arrow's theorem is insightful, correctly explaining IIA and giving reasonable justification for sacrificing it." }, { "judge_key": "minimax_01", "judge_name": "MiniMax-01", "respondent_key": "minimax_m27", "respondent_name": "MiniMax M2.7", "weighted_score": 9.45, "brief_justification": "The response provides a thorough, accurate, and well-structured analysis of the voting methods and their outcomes, with insightful reasoning for the choice of sacrificing IIA under Arrow's theorem. Minor improvements in clarity could be made in the explanation of the tie-breaking scenario in IRV." }, { "judge_key": "minimax_01", "judge_name": "MiniMax-01", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9.45, "brief_justification": "The response is factually accurate, thoroughly explains the voting methods, and provides a clear analysis of the results. It demonstrates strong reasoning but could benefit from slightly more elaboration on the implications of Arrow's theorem and the trade-offs involved." }, { "judge_key": "minimax_01", "judge_name": "MiniMax-01", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7.85, "brief_justification": "The response demonstrates a strong grasp of voting theory and provides detailed calculations for majority rule, Borda count, IRV, and Condorcet methods. However, it contains minor errors (e.g., misinterpreting the cycle initially) and could improve in clarity by better structuring the explanation of Arrow's theorem and its implications." }, { "judge_key": "minimax_01", "judge_name": "MiniMax-01", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 8.6, "brief_justification": "The response is factually accurate and provides a comprehensive breakdown of the voting methods, but the explanation of the majority rule section is slightly unclear in its wording, and the depth of analysis for Arrow's theorem could be expanded further." }, { "judge_key": "judge_claude_sonnet", "judge_name": "Claude Sonnet 4.6", "respondent_key": "minimax_m27", "respondent_name": "MiniMax M2.7", "weighted_score": 8.15, "brief_justification": "The response is well-structured and covers all three parts thoroughly. However, there are notable errors: in part (1), the cycle is stated incorrectly at the end (says 'C defeats A' but the pairwise comparison showed A defeats C with 3 votes), and in IRV, Person 3's first choice is C (not A), so after C is eliminated, Person 3's vote goes to A (correct result but the reasoning about 'Person 3's vote' going to A needs clarification since C was Person 3's first choice). The IIA sacrifice argument " }, { "judge_key": "judge_claude_sonnet", "judge_name": "Claude Sonnet 4.6", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 8.45, "brief_justification": "The response is factually accurate and well-structured, correctly demonstrating the Condorcet cycle, applying all three voting methods with correct results, and noting their disagreement. The discussion of Arrow's theorem is reasonable but somewhat surface-level—it conflates 'sacrificing the Condorcet criterion' (a property of voting systems) with the Condorcet method failing due to cycles, and the philosophical justification for which criterion to sacrifice could be more nuanced and rigorous." }, { "judge_key": "judge_claude_sonnet", "judge_name": "Claude Sonnet 4.6", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 7.55, "brief_justification": "The response correctly demonstrates the voting cycle and applies Borda count and Condorcet methods accurately. However, the IRV section contains errors in vote redistribution logic (Person 4's vote shouldn't be redistributed since C wasn't their first choice, and the round 2 tallies are confused), leading to an incorrect final count though the winner (A) happens to be correct. The Arrow's theorem discussion is reasonable but could explore the trade-offs more deeply." }, { "judge_key": "judge_claude_sonnet", "judge_name": "Claude Sonnet 4.6", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 8.15, "brief_justification": "The response is mathematically accurate and well-structured, correctly identifying that the given preferences do NOT produce a Condorcet cycle (A is the Condorcet winner), which is an important correction to the question's premise. However, the question explicitly asks to 'show that majority rule produces a cycle,' suggesting the question designer may have intended different preferences or expected the student to demonstrate the concept differently—the response could have been more helpful by no" }, { "judge_key": "judge_gpt54", "judge_name": "GPT-5.4", "respondent_key": "minimax_m27", "respondent_name": "MiniMax M2.7", "weighted_score": 5.15, "brief_justification": "The response is well organized and discusses all requested methods, but it contains major factual errors: majority rule does not produce a cycle here (A beats both B and C, so A is the Condorcet winner), and the IRV first-choice tally is miscounted. The Arrow-theorem discussion is reasonable, but the flawed core computations substantially reduce correctness and practical usefulness." }, { "judge_key": "judge_gpt54", "judge_name": "GPT-5.4", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 7.4, "brief_justification": "The pairwise comparisons, Borda totals, and IRV outcome are computed correctly, and the writeup is clear and well organized. However, the Arrow-theorem discussion is conceptually shaky because the Condorcet criterion is not one of Arrow’s fairness axioms, so the final part does not fully answer the question in the proper theoretical frame." }, { "judge_key": "judge_gpt54", "judge_name": "GPT-5.4", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 5.35, "brief_justification": "The pairwise majority cycle and Borda result are correct, and the Arrow discussion is reasonable, but the IRV calculation is seriously wrong: only Person 3’s ballot transfers from C to A, so A wins 3-2 in round 2 and B is not subsequently eliminated. The response covers all requested parts, but the flawed IRV tabulation materially reduces correctness and usefulness." }, { "judge_key": "judge_gpt54", "judge_name": "GPT-5.4", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7, "brief_justification": "The response correctly computes the actual pairwise outcomes and voting-method winners, but it fails the prompt’s request to 'show that majority rule produces a cycle' by not explicitly resolving that the given profile does not produce one. It is well organized and offers a thoughtful Arrow-theorem discussion, though the final section is cut off and thus somewhat incomplete." } ], "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)" } }