{ "evaluation_id": "EVAL-20260402-113127", "question_id": "REASON-001", "question_text": "You're given two sealed envelopes. You're told one contains twice as much money as the other, but you don't know which is which.\n\nYou pick envelope A and find $100.\n\nYou reason: \"Envelope B either has $50 or $200. If I switch, I have a 50% chance of getting $50 and 50% chance of getting $200. Expected value of switching = 0.5($50) + 0.5($200) = $125. That's more than $100, so I should switch.\"\n\nBut wait - this logic would apply no matter what amount you found. That can't be right.\n\nWhat's the flaw in this reasoning? Provide a rigorous explanation.", "category": "reasoning", "timestamp": "2026-01-14T00:00:00.000Z", "display_date": "Jan 14, 2026", "winner": { "name": "GPT-5.4", "provider": "openrouter", "score": 9.54 }, "avg_score": 8.161111, "matrix_size": 80, "models_used": [ { "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": "mimo_v2_flash", "name": "MiMo-V2-Flash", "provider": "Xiaomi" }, { "id": "gpt_oss_120b", "name": "GPT-OSS-120B", "provider": "OpenAI" }, { "id": "gemini_25_flash", "name": "Gemini 2.5 Flash", "provider": "openrouter" }, { "id": "gemini_31_pro", "name": "Gemini 3.1 Pro", "provider": "openrouter" } ], "rankings": { "gpt_5_4": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 9.54, "score_count": 9, "min_score": 8.85, "max_score": 10, "rank": 1 }, "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 9.19, "score_count": 8, "min_score": 8.2, "max_score": 10, "rank": 2 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 9.13, "score_count": 9, "min_score": 8.45, "max_score": 10, "rank": 3 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 8.82, "score_count": 9, "min_score": 8, "max_score": 9.65, "rank": 4 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 8.78, "score_count": 9, "min_score": 8, "max_score": 9.65, "rank": 5 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 8.41, "score_count": 9, "min_score": 6.85, "max_score": 9.65, "rank": 6 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 7.74, "score_count": 9, "min_score": 5.25, "max_score": 9, "rank": 7 }, "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 6.81, "score_count": 9, "min_score": 3.2, "max_score": 9.65, "rank": 8 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 5.03, "score_count": 9, "min_score": 2.05, "max_score": 8.25, "rank": 9 } }, "judgments": [ { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 5.25, "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": 10, "brief_justification": "The response provides a perfectly rigorous, mathematically sound explanation of the Two Envelope Paradox. It correctly identifies the implicit assumption of an improper prior, uses Bayes' theorem to prove why the conditional probability isn't always 0.5, and offers a clear concrete example to resolve the paradox." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 10, "brief_justification": "The response perfectly identifies the flaw, provides a rigorous Bayesian mathematical explanation, includes a concrete example, and correctly notes the impossibility of a uniform prior over all positive numbers, demonstrating exceptional depth and clarity." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.5, "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": 10, "brief_justification": "The response provides a mathematically rigorous and exceptionally clear explanation of the Two Envelopes Paradox, correctly identifying the implicit use of an improper prior as the core logical flaw." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.6, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 3.2, "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": 9.45, "brief_justification": "The response correctly identifies the flaw as an unjustified assumption of a uniform prior distribution. It provides a clear, rigorous explanation with helpful examples." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.25, "brief_justification": "Accurate and insightful, but clarity could be improved with better structure." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.9, "brief_justification": "Rigorous and accurate explanation of the paradox with clear mathematical reasoning." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "Rigorous and correct explanation with deep analysis and clear structure." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.65, "brief_justification": "Rigorous and correct explanation of the flaw, with thorough coverage and insightful analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.65, "brief_justification": "Rigorous and correct explanation of the flaw, with thorough coverage and insightful analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.65, "brief_justification": "Rigorous explanation covering Bayesian and symmetry approaches, flaw clearly identified." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.65, "brief_justification": "The response is factually accurate, thoroughly covers the topic with a rigorous derivation, and provides insightful analysis. The clarity is high, though slightly technical, and the practical value is excellent for understanding the paradox." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 9.65, "brief_justification": "Rigorous and accurate explanation of the flaw, with clear breakdown of the probabilistic reasoning and its assumptions." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.1, "brief_justification": "The response correctly identifies the Two Envelopes Paradox and begins to outline two relevant flaws (the improper prior assumption and the algebraic equivocation), but it is dramatically incomplete - it cuts off mid-sentence before actually delivering the rigorous explanation promised. The initial framing is reasonable, mentioning Bayesian priors and variable equivocation, but without the actual mathematical arguments, the response fails to provide the rigorous explanation requested. It cannot " }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 6.2, "brief_justification": "The response correctly identifies the core issue: the 50/50 assumption is unjustified and requires an implicit prior distribution over x. However, the claim that p must equal 1/3 to make E(B)=100 is problematic. This value of p is not derived from any principled argument—it's just the value that makes switching neutral, presented as if it's 'the correct' probability. In reality, p depends on the prior distribution, and for some priors switching could favor you while for others it wouldn't. The c" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.4, "brief_justification": "The response provides a rigorous, well-structured Bayesian analysis that correctly identifies the core flaw: the assumption that after observing $100, the other envelope is equally likely to contain $50 or $200. The Bayesian formulation is precise, the concrete example is illuminating, and the impossibility argument about no proper prior satisfying the 50-50 condition for all amounts is an excellent addition. The explanation is exceptionally clear with good mathematical notation. One minor point" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.2, "brief_justification": "The response correctly identifies the two envelopes paradox and provides a solid Bayesian analysis showing that the 50/50 assumption is unjustified under any proper prior. The Bayes' rule calculation has a minor technical issue with the density transformation (the factor of 1/2 in the denominator needs more careful justification), but the core argument is sound: no proper probability distribution satisfies f(a) = (1/2)f(a/2) for all a, so the equal probability assumption fails. The response coul" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.2, "brief_justification": "The response correctly identifies the core flaw: the implicit assumption of p=1/2 requires an improper prior that cannot be normalized. The Bayesian analysis is rigorous and well-structured. The reductio ad absurdum argument (both parties would always want to switch) is a powerful complementary point. The explanation that with any proper prior, the switching recommendation becomes amount-dependent is exactly right. The summary table is excellent. Minor quibble: the response could have mentioned " }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.2, "brief_justification": "The response correctly identifies the core flaw: the assumption that the two scenarios are equally likely given the observed value. The Bayesian analysis is well-done and shows how the prior distribution f(x) determines the posterior probabilities. The symmetry argument is also valid. However, there's a subtle issue: the response could have more clearly addressed why a uniform prior over all positive reals is improper and what that implies, and could have discussed the self-referential nature of" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 4.55, "brief_justification": "The response correctly identifies the core flaw: the assumption that the two cases have equal probability 1/2 is unjustified without specifying a prior. The Bayesian framework setup is correct. However, the critical claim that E[Z|Y=100]=100 for every proper prior is simply false. The expression (2p_A + 0.5p_B)/(p_A+p_B) equals 1 only when p_A = p_B/3, not for all priors. The response attempts to 'prove' this equality but the proof is circular nonsense - it just repeats the expression multiple t" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6.85, "brief_justification": "The response correctly identifies that the flaw involves unjustified probability assignments and that the prior distribution matters. However, the analysis is somewhat muddled and circular. The concrete example actually shows a case where 50/50 IS correct, which undermines the argument. The deeper mathematical point—that no proper prior distribution over all positive reals can make the 50/50 assumption hold for every possible observed value simultaneously (which is the real reason the 'always sw" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 2.05, "brief_justification": "The response starts in the right direction by identifying the two-envelope paradox and hinting at the issue with assigning 50/50 probabilities after observing $100, but it is incomplete and cuts off before giving the rigorous explanation requested." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.95, "brief_justification": "Clear and mostly correct explanation identifying the key flaw: the unjustified 50/50 conditional probabilities and need for a prior. Good rigor and structure. Main issue: it incorrectly states the expected value of switching is zero or that E(B)=100 only if p=1/3 as a general resolution; in fact the switching advantage depends on the chosen prior after conditioning on the observed amount, and not all consistent priors make it zero." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.2, "brief_justification": "Accurately identifies the key flaw as the unjustified conditional 1/2 assumption and gives a solid Bayesian/prior-based resolution with good depth. Minor issue: the displayed expected-value formula is malformed/redundant, and the claim that effects exactly cancel for any proper prior is a bit too sweeping without qualification." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.65, "brief_justification": "Accurately identifies the core flaw: the posterior probabilities after observing $100 are not generally 50-50, and supports this with a Bayesian treatment and symmetry argument. It is thorough and rigorous. Minor issue: the density-transform/Bayes derivation is a bit compressed and could be presented more cleanly to avoid confusion." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.45, "brief_justification": "Accurately identifies the core flaw: the 50/50 conditional probabilities are unjustified and depend on a prior over amounts. Gives a rigorous Bayes-based treatment and explains the improper-prior issue well. Minor weakness: one displayed formula in the 'Correct Framework' section is a bit muddled/not the cleanest conditional expectation statement, but the overall analysis is strong, clear, and practically resolves the paradox." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8, "brief_justification": "Accurately identifies the key flaw: the 50/50 conditional assumption is unjustified without a prior. Good Bayesian framing and useful symmetry discussion. Minor issue: it claims an improper uniform prior yields equal expected value and presents E[B|A]=A too loosely; rigorously, conditioning on an observed value requires a proper prior or measure-theoretic care." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 5.15, "brief_justification": "Correctly identifies the key flaw as unjustified 50/50 conditional probabilities and sets up a Bayesian model, but then makes a serious false claim that for every proper prior the conditional expected value equals 100. The derivation does not support that conclusion, and the algebra is invalid. Clear structure, but the final rigorous explanation is mathematically wrong." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.25, "brief_justification": "Accurately identifies the key flaw: the unjustified 50/50 conditional probabilities after observing $100, and correctly ties this to the need for a prior distribution. It gives a reasonably rigorous Bayesian-style explanation and examples. Minor weakness: one example is slightly awkward and it could more explicitly formalize the conditional expectation or note that without a proper prior the expectation may be undefined, but overall it is strong and useful." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 6.2, "brief_justification": "The response correctly identifies the paradox and names two flaws but is incomplete as it cuts off mid-explanation. It starts a rigorous Bayesian argument but doesn't finish, reducing completeness. Clarity and depth are good for what is shown." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.4, "brief_justification": "The response accurately identifies the core flaw (improper prior and unequal probabilities), provides a rigorous Bayesian-style explanation with math, and resolves the paradox correctly. It is clear and well-structured, though the p=1/3 derivation assumes a specific improper prior without fully justifying it from first principles." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.85, "brief_justification": "The response provides a rigorous Bayesian explanation of the paradox, correctly identifying the improper prior issue, with clear math, a concrete example, and a summary table. It fully resolves the flaw while being insightful and practical." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.85, "brief_justification": "The response provides a rigorous Bayesian explanation of the paradox, correctly identifying the flawed 50/50 assumption after observation, with clear math, priors, and a concrete example. It is thorough and insightful, though slightly verbose in places." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.85, "brief_justification": "Rigorous Bayesian explanation correctly identifies the improper prior issue and symmetry paradox; thorough with math, tables, and clear structure, though minor redundancy slightly reduces conciseness." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.65, "brief_justification": "The response accurately identifies the core flaw (unjustified 50/50 assumption), provides two rigorous mathematical approaches with formulas, explains the Bayesian update via prior density f(x), and uses symmetry for the uninformative case. It is thorough, logically sound, and clearly structured, though the uniform prior discussion could note its impropriety more explicitly." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7.8, "brief_justification": "The response correctly identifies the flaw as unjustified equal probabilities and provides a Bayesian derivation showing E[Z|Y=100] depends on the prior. However, it contains a critical error claiming the ratio always equals 1 for any proper prior (the algebra is faked and the claim is false; the ratio equals 1 only under specific priors). The model is otherwise rigorous and clearly explained." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 7.8, "brief_justification": "The response correctly identifies the flaw as assuming equal probabilities without a prior distribution and provides a rigorous Bayesian explanation with examples. It is clear and well-structured but slightly repetitive in places and could more precisely note that a proper prior over the smaller amount resolves the paradox definitively." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 4.35, "brief_justification": "The response starts well by identifying two key flaws and begins a rigorous explanation, but it is severely incomplete - it cuts off mid-sentence before finishing even the first flaw explanation. The two-envelope paradox requires careful treatment of prior distributions and the algebraic equivocation error, neither of which is fully developed here. What exists is directionally correct but the response is essentially unfinished." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.15, "brief_justification": "The response correctly identifies the core flaw (improper prior distribution assumption) and explains it well. However, there's a significant error in point 4: the claim that p=1/3 makes E(B)=100 is mathematically wrong (1/3×200 + 2/3×50 = 66.67+33.33=100, actually that's correct), but the conclusion that 'the correct probability that B is larger given A=100 is 1/3' is presented without justification and is misleading—the actual probability depends entirely on the prior. The final claim that 'un" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 10, "brief_justification": "Exceptional response that rigorously identifies the core flaw (improper prior), uses Bayesian analysis to formalize the argument, demonstrates why a uniform 50/50 assumption for all observed values requires an unnormalizable distribution, provides a concrete example showing the paradox dissolves with a proper prior, and presents everything with excellent clarity and structure. This is a textbook-quality explanation of the two-envelope paradox." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "Exceptionally rigorous and thorough response. Correctly identifies the core flaw (unjustified 50-50 posterior assumption), provides full Bayesian derivation, concrete numerical counterexample, and explains why no proper prior can satisfy the paradoxical reasoning. The 'another way to see the error' section adds valuable insight about the impossibility of a uniform prior over all positive reals. Clarity is excellent with well-structured sections. Minor deduction for clarity only because the LaTeX" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.85, "brief_justification": "The response correctly identifies the core flaw (unjustified equal probability assumption), provides a rigorous Bayesian analysis, and explains why no normalizable distribution can satisfy the required condition. The mathematical treatment is sound and the explanation is well-structured. Minor quibble: the Bayes' rule formula has a slight notational issue in the denominator but the conceptual argument is correct. Overall an excellent, rigorous response." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.4, "brief_justification": "The response correctly identifies the core flaw—unjustified equal probability assumption—and provides two rigorous approaches (Bayesian and symmetry). The Bayesian analysis is sound and well-explained. The symmetry argument correctly shows E[gain]=0. Minor issue: the claim that a uniform prior gives 50/50 is problematic since a uniform prior over all positive reals is improper and doesn't actually resolve the paradox cleanly, but this is a subtle point. Overall, this is a thorough, well-structur" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 5.6, "brief_justification": "Correctly identifies the flaw (unjustified 50/50 prior) and sets up the Bayesian framework well. However, the claimed proof that E[Z|Y=100]=100 for every proper prior is mathematically incorrect and the 'proof' is circular nonsense that just repeats the same expression. The fraction 2p_A+0.5p_B/(p_A+p_B) does not universally equal 1. This significant error undermines the rigor claimed." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.4, "brief_justification": "The response correctly identifies the core flaw—unjustified equal probability assignment to the two cases—and explains it well with concrete examples. It covers the role of prior distributions thoroughly. Minor weakness: it doesn't fully address the deeper mathematical issue that no proper probability distribution can assign equal probability to both cases for all possible observed values simultaneously (the key reason the paradox is universal), but the explanation is rigorous and practically co" }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 4.85, "brief_justification": "Response is incomplete and cuts off mid-explanation, failing to provide the rigorous analysis requested. It identifies the flaw types but does not fully explain them." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9, "brief_justification": "Accurate and thorough explanation of the flaw, with rigorous probabilistic reasoning and clear resolution." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.2, "brief_justification": "Rigorous, accurate, and thorough explanation of the flaw with proper prior distribution analysis, clear structure, and practical insight." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "Rigorous Bayesian explanation correctly identifies the flaw, covers all aspects, and provides concrete examples." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.65, "brief_justification": "Accurate and thorough explanation of the flaw using Bayesian reasoning; clear structure and insightful analysis; slightly less actionable for a general audience." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.2, "brief_justification": "Rigorous, accurate, and thorough explanation of the flaw with proper Bayesian analysis and logical reasoning." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.65, "brief_justification": "Rigorous Bayesian explanation correctly identifies the flaw and derives the expected value, though some steps are slightly repetitive." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.85, "brief_justification": "Accurate and thorough explanation of the conditional probability flaw, with clear structure and concrete examples. Slightly less actionable due to abstract nature of the problem." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5.45, "brief_justification": "The response correctly identifies the key issues (prior distribution and variable misuse) and is clearly written, but it is incomplete and stops before giving a full rigorous explanation, limiting its depth and practical usefulness." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.45, "brief_justification": "Accurately identifies the flawed equal-probability assumption and explains the need for a prior distribution, covering key aspects clearly and with sufficient depth for practical understanding." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.25, "brief_justification": "Accurately identifies the false 50/50 assumption, provides rigorous Bayesian analysis, examples, and explains dependence on prior distribution, all clearly and usefully." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8, "brief_justification": "Accurately identifies the false 50/50 assumption and explains it with Bayesian reasoning and prior distributions, covering the key points clearly and with sufficient depth for practical understanding." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.45, "brief_justification": "Accurately identifies the unjustified 50/50 assumption and explains it via Bayes and improper priors, covering the core issue with good structure and depth, though could include a bit more on alternative priors." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.25, "brief_justification": "Accurate, thorough explanation of the flaw with Bayesian and symmetry arguments, clearly presented and insightful." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.25, "brief_justification": "Accurately identifies the mistaken 50/50 assumption and explains the need for a prior distribution, with clear examples and logical structure, though could delve deeper into Bayesian formalism." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7.25, "brief_justification": "The response correctly identifies the core of the paradox and the two main types of errors. However, it is incomplete as it cuts off before fully explaining the probabilistic error and doesn't even get to the algebraic error. The explanation provided so far is clear and accurate." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.85, "brief_justification": "The response accurately identifies the flaw as a misuse of probability distributions and provides a rigorous explanation. It correctly points out the need for a prior distribution and demonstrates how the 0.5 probability assumption is incorrect. The resolution is clear and the key takeaway summarizes the core issue effectively." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.1, "brief_justification": "The response provides a rigorous and clear explanation of the two-envelope paradox, correctly identifying the flaw in reasoning related to conditional probabilities and improper priors. It uses mathematical notation effectively and offers a concrete example to illustrate the resolution." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.85, "brief_justification": "The response provides a rigorous and comprehensive explanation of the flaw in the reasoning of the Two Envelopes Problem. It correctly identifies the incorrect assumption about the conditional probabilities and uses a Bayesian framework to demonstrate why it's flawed. The concrete example and alternative explanations further enhance the clarity and depth of the answer." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.25, "brief_justification": "The explanation correctly identifies the flaw and provides a rigorous mathematical justification using Bayes' rule and discussing the properties of probability density functions. It also contextualizes the problem as the 'two envelopes paradox.'" }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.1, "brief_justification": "The response accurately identifies the flaw in the reasoning, providing a rigorous explanation using probability theory and Bayes' theorem. It clearly explains why the assumption of p=1/2 is problematic due to an incoherent prior. The summary table is also very helpful." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.65, "brief_justification": "The explanation is rigorously correct, complete, and provides excellent depth using two different analytical approaches. The clarity is very high, making a complex topic understandable." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.45, "brief_justification": "The response correctly identifies the flaw in the reasoning, which is the unjustified assumption of equal posterior probabilities. It provides a rigorous, step-by-step derivation using Bayes' rule to demonstrate how the expected value of switching is actually equal to the amount in the current envelope, given any proper prior distribution. The explanation is clear, well-structured, and deeply insightful, making it highly useful for understanding this classic paradox." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.8, "brief_justification": "Response correctly identifies the Two Envelopes Paradox and mentions key error types, but is severely incomplete - cuts off mid-explanation without providing the actual rigorous mathematical proof promised." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.45, "brief_justification": "Correctly identifies the flaw in assuming equal probabilities without a prior distribution. Provides rigorous mathematical explanation showing the expected value equals $100 when p=1/3. Well-structured with clear notation. Slightly incomplete in exploring different prior scenarios." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.25, "brief_justification": "Mathematically rigorous explanation that correctly identifies the core flaw: assuming a 1/2 conditional probability for every observed amount requires an impossible (improper) prior distribution. Uses Bayes' theorem properly, explains why no valid probability model supports the paradoxical conclusion, and provides a concrete resolution example." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.6, "brief_justification": "Excellent rigorous Bayesian explanation that correctly identifies the flaw (unjustified 50-50 assumption after observation). Thorough coverage includes mathematical derivation, concrete counterexample, and multiple perspectives. Well-structured with clear notation and logical flow." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.25, "brief_justification": "Mathematically rigorous explanation that correctly identifies the core flaw - the illicit 50-50 assumption. Uses proper Bayesian analysis to demonstrate why this assumption is impossible in any valid probability model. Clear structure and good depth." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.25, "brief_justification": "Mathematically rigorous explanation correctly identifies the core issue: assuming p=1/2 requires an impossible uniform prior over all scales. The reductio ad absurdum argument (perpetual switching) is particularly effective. Well-structured with clear Bayes' theorem application and summary table." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9, "brief_justification": "Rigorous and correct explanation using both Bayesian analysis and symmetry arguments. Identifies the key flaw in assuming equal prior probabilities without justification. Well-structured with clear mathematical derivations." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7.2, "brief_justification": "Clear explanation identifying the key flaw about prior probabilities. However, the final claim that the fraction equals 1 for all priors is not properly proven - the algebraic manipulation shown doesn't actually demonstrate this. The depth suffers from this gap in the mathematical reasoning." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.2, "brief_justification": "Correctly identifies the core flaw: the implicit assumption of equal conditional probabilities (50/50) without knowledge of the underlying prior distribution. Well-structured with concrete examples showing how different distributions yield different probabilities. The explanation addresses why the logic incorrectly seems to apply universally. Slightly less complete than ideal as it focuses primarily on the prior distribution resolution rather than also mentioning the self-undermining nature of t" } ], "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)" } }