{ "evaluation_id": "EVAL-20260402-152445", "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": 9.6 }, "avg_score": 8.438889, "matrix_size": 79, "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": 9.6, "score_count": 8, "min_score": 8.85, "max_score": 10, "rank": 1 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 9.25, "score_count": 9, "min_score": 8.45, "max_score": 10, "rank": 2 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 9.1, "score_count": 9, "min_score": 8.45, "max_score": 9.65, "rank": 3 }, "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 9.06, "score_count": 9, "min_score": 8.45, "max_score": 9.85, "rank": 4 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 8.56, "score_count": 9, "min_score": 7.2, "max_score": 9.45, "rank": 5 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 8.47, "score_count": 9, "min_score": 7.65, "max_score": 9, "rank": 6 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 8.31, "score_count": 8, "min_score": 7.65, "max_score": 8.8, "rank": 7 }, "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 8.27, "score_count": 9, "min_score": 4.55, "max_score": 9.8, "rank": 8 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 5.33, "score_count": 9, "min_score": 3.3, "max_score": 8.45, "rank": 9 } }, "judgments": [ { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "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": 8.95, "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": 10, "brief_justification": "The response perfectly identifies and rigorously explains the flaw in the Two Envelopes Paradox using Bayesian probability. It provides a clear mathematical breakdown, a concrete counterexample, and addresses the psychological trap of the paradox." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.6, "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 perfectly accurate explanation of the Two Envelopes Paradox, correctly identifying the implicit assumption of an improper prior. The use of Bayes' theorem and the clear breakdown of the logical errors make it an exceptional answer." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.45, "brief_justification": "The response accurately identifies the Two-Envelope Paradox and rigorously explains the mathematical flaw regarding improper prior distributions. It is clear, well-structured, and insightful." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 4.55, "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": 7.7, "brief_justification": "The response correctly identifies the mathematical flaw regarding improper prior distributions, but it cuts off mid-sentence at the end." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.45, "brief_justification": "Accurate and insightful, but slightly incomplete in the explanation." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.65, "brief_justification": "The response is factually accurate, thoroughly covers the topic with insightful analysis, and clearly explains the flaw in the reasoning. It provides practical value by resolving the paradox rigorously." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "The response thoroughly and accurately explains the flaw in the reasoning, providing a rigorous and detailed analysis with clear examples and Bayesian calculations. It is highly useful and insightful." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.65, "brief_justification": "The response accurately identifies the flaw in the reasoning, provides a rigorous mathematical explanation, and includes a concrete example to illustrate the point. The clarity is high, though slightly technical, and the depth of analysis is excellent." }, { "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 deep mathematical analysis and clear structure." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.1, "brief_justification": "" }, { "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 explanation, and provides insightful analysis. It is clear and well-structured, though slightly technical, and highly useful for understanding the flaw in the reasoning." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.65, "brief_justification": "" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.3, "brief_justification": "The response correctly identifies the Two Envelopes Paradox and begins to touch on the key issues (the implicit prior distribution assumption and the Bayesian perspective), but it is severely incomplete - it cuts off mid-sentence and never delivers the promised rigorous explanation. The mention of 'uniform probability distribution over an infinite range' is on the right track but is stated without justification. The response promises two parts but delivers neither fully. A rigorous answer would " }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.65, "brief_justification": "The response correctly identifies the core flaw: the implicit assumption of an improper uniform prior over all possible amounts, and provides a proper Bayesian framework. The mathematical formulation is correct and the concrete example is helpful. However, there's a visible text corruption/truncation mid-sentence ('а/ авгThe flaw is...') which hurts clarity. The response could also have mentioned alternative perspectives on the paradox (e.g., the symmetry argument that switching cannot help on a" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.85, "brief_justification": "The response is exceptionally rigorous and correct. It precisely identifies the flaw (the unjustified assumption of equal conditional probabilities), provides a full Bayesian derivation, gives a concrete numerical counterexample, explains the symmetry contradiction, and clearly articulates why the paradox is persuasive. The writing is well-structured with progressive levels of detail. The only minor note is that it could briefly mention the impossibility of a uniform prior over all positive real" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.2, "brief_justification": "The response correctly identifies the core flaw: the assumption that the two possibilities are equally likely (50-50) cannot hold for all observed amounts under any proper prior distribution. The Bayesian analysis is rigorous and correct, the functional equation argument (f(a) = 2f(a/2) for all a is impossible for a proper distribution) is the standard resolution, and the concrete example effectively illustrates the point. The clarity is excellent with well-structured sections. Minor quibble: th" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.4, "brief_justification": "This is an excellent, rigorous explanation of the two-envelope paradox. The Bayesian formalization is correct, the identification of the improper prior as the core flaw is spot-on, and the proof that no proper prior can make switching always favorable is well-articulated. The summary table is a nice touch. Minor quibble: the claim that P(X=a)=P(X=a/2) for all a requires a uniform prior is slightly imprecise (it requires a specific log-uniform structure, not exactly a 'uniform prior'), and the re" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 7.2, "brief_justification": "The response correctly identifies the Two-Envelope Paradox and gets at the core issue: the 50/50 probability assumption across all possible amounts is unjustified without a proper prior distribution. The explanation about mixing conditional and unconditional expectations is on the right track. However, the mathematical analysis has some imprecisions. The claim that 'the total expected value of the game is infinite' is presented as a necessary consequence but isn't quite right—the issue is more p" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7.35, "brief_justification": "The response is exceptionally well-structured and clearly written, with rigorous Bayesian formalism. However, there is a significant correctness issue: the claim that E[B|A=a] = a for ANY proper prior is false. The algebraic step showing the numerator simplifies to a times the denominator is incorrect — 2p_X(a) + (1/4)p_X(a/2) does not generally equal p_X(a) + (1/2)p_X(a/2). For many proper priors, switching CAN have positive or negative expected value depending on the observed amount. The core " }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 7.65, "brief_justification": "The response correctly identifies the core flaw: the unjustified assumption that P(B=200|X=100) = P(B=50|X=100) = 0.5, and explains that this requires an unspecified (and often improper) prior distribution over k. The pre-opening expected value symmetry argument is well presented. However, the explanation is somewhat repetitive and could be more concisely structured. The response also gets cut off at the end. A deeper treatment could have more explicitly discussed the improper prior (the prior t" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.3, "brief_justification": "Identifies that the issue involves probability assumptions/Bayesian conditioning, but the response is incomplete and overstates the flaw as requiring an impossible uniform prior. It 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": 7.8, "brief_justification": "Mostly correct and rigorous: it identifies the key flaw as the unjustified 50/50 conditional probabilities after observing the amount and uses a Bayesian prior-based explanation. Good depth and examples. Clarity is reduced by a corrupted sentence and some imprecise formulas missing the 1/2 sampling factors, though the main conclusion remains sound." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9, "brief_justification": "Accurately identifies the key flaw: the unjustified 50/50 conditional probabilities after observing $100, and explains it rigorously via priors and Bayes. Clear, well-structured, and includes both conditional and unconditional perspectives. Minor caveat: the claim that the bad calculation implicitly assumes an improper 1/x prior is a common framing but slightly overstated without qualification." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.25, "brief_justification": "Accurate and rigorous resolution of the paradox using conditional probabilities and priors, with a strong concrete example. Clear overall; only minor quibble is the informal wording about the functional equation implying exponential growth rather than emphasizing impropriety/non-normalizability more directly." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.65, "brief_justification": "Rigorous and largely correct: it identifies the key flaw as the unjustified 50/50 conditional assumption and uses Bayes/prior dependence well. Strong depth and coverage, though a few statements are slightly overstated or imprecise (e.g. phrasing around uniform priors and symmetry/null-operation argument), and the discrete/continuous notation could be cleaner." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.45, "brief_justification": "Accurately identifies the core flaw: the unjustified 50/50 conditional probabilities after observing $100 and the need for a prior over envelope amounts. Clear structure and good explanation, though parts about 'uniform distribution' and 'infinite expected value' are somewhat imprecise or overstated." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5.85, "brief_justification": "Response is incomplete and cuts off mid-explanation, reducing completeness and usefulness. The initial identification of the paradox and the flaw is correct and clear, but the lack of a full rigorous explanation limits depth and overall score." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 5.7, "brief_justification": "Well-structured and rigorous in form, but it makes a key false claim: for any proper prior, E[B|A=a] is exactly a. In general this conditional expectation depends on the prior and need not equal a; only the unconditional expectations are equal, or equality can hold for special priors/examples. The core flaw about confusing prior 50/50 with posterior probabilities is correct, but the conclusion is overstated and mathematically incorrect." }, { "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 key flaws like improper uniform priors over infinite support and equivocation on variables, but is incomplete as it cuts off mid-explanation (stops at \"Let $S$ be the\"), limiting full rigor. Clarity is strong with good structure, but overall coverage of the topic suffers." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.65, "brief_justification": "The response accurately identifies the core flaw in assuming equal probabilities without a proper prior, provides a rigorous Bayesian explanation with formulas and an illustrative example, and clearly structures the paradox resolution. Minor deduction for a small typo in the text ('a/ авг') and slight redundancy in sections." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.65, "brief_justification": "The response provides a thorough, rigorous, and well-structured explanation of the paradox, correctly identifying the improper prior and symmetry issues with clear math. Minor deduction for slightly overemphasizing one resolution approach while covering the core flaw excellently." }, { "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 rigorously identifies the core flaw (unjustified 50/50 assumption), provides a full Bayesian analysis with priors, concrete counterexample, symmetry argument, and clear explanations. Minor deduction on usefulness as it is more theoretical than a quick practical takeaway." }, { "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": "Rigorous Bayesian explanation correctly identifies the improper prior flaw, with clear math and summary table. Highly insightful on why the paradox arises from mixing distributions." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.45, "brief_justification": "The response accurately identifies the core flaw in assuming equal 50/50 probabilities without a proper prior, explains the mathematical contradiction and conditional probabilities rigorously, and covers the paradox comprehensively with clear structure and useful insights." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.85, "brief_justification": "The response provides a rigorous Bayesian explanation with formal derivations, posterior calculations, and an example. It correctly identifies the flaw in assuming equal posterior probabilities independent of a prior. Minor deduction for usefulness as it is more theoretical than immediately actionable for non-technical readers." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.2, "brief_justification": "The response accurately identifies the core flaw in assuming equal 50/50 probabilities after observing the amount, explains the role of the unspecified prior on k, and shows via pre-opening EV that switching is indifferent. It is thorough and structured, though slightly repetitive in places and cuts off at the end." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 4.55, "brief_justification": "The response starts well by identifying the core issue (impossible uniform prior over infinite range) and begins a structured Bayesian analysis, but it is severely incomplete - it cuts off mid-sentence. The partial content shown is on the right track but the response fails to deliver the promised rigorous explanation, making it largely unusable as a complete answer." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.2, "brief_justification": "The response correctly identifies the core flaw—the improper prior assumption—and provides a rigorous Bayesian treatment. The explanation of why a uniform prior over all positive amounts is impossible is sound. The concrete example with discrete X values is helpful. Minor issues: there's a garbled sentence mid-response ('a/ авг'), and the response could more explicitly address the 'self-referential' nature of the paradox (that the logic applies regardless of the observed amount, which is the key" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.85, "brief_justification": "Exceptional response that rigorously identifies the core flaw (unjustified 50/50 prior), demonstrates the reductio ad absurdum argument, provides Bayesian analysis, shows the symmetric expected value calculation, and correctly identifies the improper prior issue. Mathematically precise, well-structured, and covers all major angles of this classic 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, "brief_justification": "The response correctly identifies the core flaw (unjustified equal prior probabilities), provides a rigorous Bayesian treatment, includes a concrete numerical example, and uses the symmetry argument to show the paradox is self-contradictory. Well-structured and thorough." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9, "brief_justification": "Excellent rigorous treatment of the two-envelope paradox. Correctly identifies the flaw as the unjustified 50-50 assumption, derives the correct Bayesian probabilities with proper factors, proves no valid prior can satisfy the required functional equation for all values, and provides a concrete numerical example that clearly illustrates when the naive reasoning fails. Minor quibble: the Bayes formula derivation could be slightly more explicit, but overall this is a thorough, accurate, and well-s" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.8, "brief_justification": "The response correctly identifies the core flaw—the unjustified 50/50 prior—and explains it rigorously through multiple angles: the mathematical contradiction of infinite expected value, the conditional probability issue, and the prior distribution problem. The explanation is well-structured and clear. Minor shortcoming: could more explicitly state that no proper probability distribution over all positive reals can make both P(X=x/2|X=x) and P(X=2x|X=x) equal to 0.5 simultaneously, but this is l" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.4, "brief_justification": "Exceptionally rigorous and thorough response. Correctly identifies the core flaw (misapplication of prior probabilities as posterior probabilities), provides formal Bayesian derivation showing E[B|A=a]=a, includes a concrete illustrative example, and explains why the improper 1/x prior is the root cause. The mathematical treatment is sound and the explanation is well-structured. Minor issue: the Jacobian factor explanation in section 2 is slightly informal but the conclusion is correct." }, { "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—the unjustified assumption that P(B=200|X=100) = P(B=50|X=100) = 0.5—and explains it rigorously using conditional probability and prior distributions. The pre-opening expected value calculation effectively demonstrates indifference. The response is thorough and covers multiple angles including the improper prior problem. Minor issues: the response gets cut off at the end, and could more explicitly state that the 50/50 assumption would require an im" }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9, "brief_justification": "The response correctly identifies the flaw as the assumption of equal probabilities without a proper prior distribution. It provides a rigorous Bayesian explanation and a discrete example to illustrate the concept, demonstrating significant depth and clarity." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.45, "brief_justification": "The response is mathematically rigorous and complete. It correctly identifies the flaw as the assumption of equal probabilities without a prior, explains the reductio ad absurdum, and provides the correct expected value calculation. It goes further by identifying the specific improper prior (1/x) that makes the fallacious argument work, which is a high-level insight." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.8, "brief_justification": "The response is mathematically rigorous and complete. It correctly identifies the flaw as the unjustified assumption of equal conditional probabilities (50/50) and explains this via Bayes' theorem and prior distributions. It provides a concrete counter-example, addresses the symmetry argument, and resolves the paradox fully." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.45, "brief_justification": "Rigorous, correct, and thorough explanation of the paradox, including Bayesian analysis and a concrete example." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.8, "brief_justification": "Excellent, rigorous explanation. Correctly identifies the core flaw (improper prior), uses Bayes' theorem, explains why uniform prior fails, discusses proper priors, and notes the symmetry argument. Very thorough and insightful." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.8, "brief_justification": "The response is mathematically rigorous, correctly identifying the flaw as the confusion between prior and posterior probabilities. It provides a complete formal derivation using Bayes' theorem, a concrete example, and a clear summary." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.8, "brief_justification": "The response correctly identifies the core flaw (unjustified 50/50 probability assumption) and rigorously explains it using conditional probability and prior distributions. It covers the problem thoroughly, discusses the ill-defined nature of a uniform prior, and correctly concludes that the decision is statistically indifferent. The only minor deduction in clarity is for slightly dense phrasing in the middle section." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 4.7, "brief_justification": "The answer identifies the key issue of an improper prior but is cut off and incomplete, limiting its thoroughness and practical guidance." }, { "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 faulty 50/50 assumption, explains need for a prior, gives Bayesian formula and example, clear structure, good depth and practical insight." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.45, "brief_justification": "Accurately identifies the false 50/50 assumption, explains need for a prior, uses Bayes and symmetry, well-structured and insightful, offering practical understanding of the paradox." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.45, "brief_justification": "Accurate Bayesian explanation of the flaw, well-structured with equations and examples, covering key points without unnecessary detail." }, { "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 improper uniform prior assumption, provides formal Bayes analysis, explains why 50/50 is unjustified, and presents the resolution clearly and insightfully." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.45, "brief_justification": "Accurately identifies the faulty 50/50 assumption and explains the need for a prior distribution, with clear, well-structured reasoning and sufficient depth for understanding the paradox." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.45, "brief_justification": "Accurately identifies the mistaken 50/50 assumption and explains the need for a prior distribution, with clear, thorough reasoning." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7.45, "brief_justification": "The response correctly identifies the core of the paradox and begins to explain the flaw. However, it is incomplete as it cuts off mid-explanation, preventing a full assessment of its rigor and completeness." }, { "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 in reasoning, providing a rigorous explanation using Bayesian probability. It clearly explains why the 50/50 probability assignment is incorrect and demonstrates the conditions under which it might appear to hold. The example helps illustrate the concept, and the general solution is well-articulated." }, { "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 accurately identifies the core flaw in the reasoning, providing a rigorous and multi-faceted explanation. It covers the Bayesian perspective, the reductio ad absurdum, and the technical flaw of an improper prior. The explanation is clear and well-structured." }, { "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 comprehensive and rigorous explanation of the flaw in the two-envelope paradox. It clearly identifies the incorrect assumption of equal probabilities and uses Bayesian reasoning and examples to illustrate the dependence on prior distributions. The structure is logical, moving from a short version to a detailed mathematical explanation, and finally summarizing the precise flaw and why the paradox is persuasive. The symmetry argument further strengthens the explanation." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.65, "brief_justification": "The explanation is rigorous, clear, and provides a deep understanding of the paradox using Bayes' rule and a concrete example. It correctly identifies the flaw in the equiprobability assumption." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.65, "brief_justification": "The response provides a rigorous and comprehensive explanation of the flaw in the reasoning, delving into the mathematical and probabilistic underpinnings. It clearly identifies the illegitimate mixing of probability distributions and the issue with assuming a uniform prior. The formal setup, Bayes' theorem application, and the discussion of proper priors are excellent. The summary table is a great addition for clarity." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.85, "brief_justification": "The response provides a comprehensive and rigorous explanation of the Two-Envelope Paradox, clearly identifying the flawed assumption and detailing the mathematical contradiction. It breaks down the problem into understandable scenarios and explains the concept of conditional probability effectively. The explanation is well-structured and easy to follow." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.65, "brief_justification": "The explanation is rigorous, complete, and correct. It clearly identifies the flaw using Bayesian probability and provides a formal model and an illustrative example." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 4.2, "brief_justification": "Response correctly identifies the paradox and hints at the key flaw (uniform distribution over infinite range) but is severely incomplete - it cuts off mid-sentence and never provides the actual rigorous explanation requested." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.85, "brief_justification": "Excellent rigorous explanation identifying the core flaw: incorrectly assuming 50-50 probabilities after observation requires an impossible uniform prior. Provides proper Bayesian analysis with formulas and concrete examples. Well-structured and thorough." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.45, "brief_justification": "Accurately identifies the core flaw using Bayes' theorem and prior distributions. Thoroughly covers the paradox from multiple angles including the reductio ad absurdum argument. Well-structured with clear mathematical notation. Explains the improper prior issue which is the technical root of the paradox." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.8, "brief_justification": "Excellent rigorous explanation identifying the core flaw: assuming 50-50 probability after observing $100 without specifying prior distribution. Provides multiple valid perspectives (Bayesian, example, symmetry), making the mathematical error crystal clear." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.65, "brief_justification": "Accurately identifies the flawed 50-50 assumption, provides rigorous Bayesian analysis, and shows with concrete examples why the naive expected value calculation fails. Clear and insightful." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9, "brief_justification": "Rigorous mathematical explanation using Bayes' theorem to show the flaw is assuming an impossible uniform prior over positive reals. Well-structured with clear equations, logical flow, and a helpful summary table. Demonstrates deep understanding by proving the expected gain from switching is zero under any proper prior." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.3, "brief_justification": "Accurately identifies the paradox and explains the flawed 50/50 probability assumption with proper mathematical derivation showing the 1.25X contradiction. Clear structure with good depth on conditional probability issues. Slightly less complete as it doesn't cover different paradox formulations or resolution approaches." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.45, "brief_justification": "Excellent rigorous explanation using Bayes' theorem. Correctly identifies the flaw as assuming equal posterior probabilities when only prior probabilities are equal. Provides thorough mathematical treatment with clear structure, concrete example, and insightful discussion of prior vs posterior probabilities." }, { "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": "Rigorous explanation correctly identifying the flaw as the unjustified assumption of 50/50 probability after observation, with good use of conditional probability and prior distribution concepts. Well-structured and thorough, though could explore more edge cases." } ], "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)" } }