{
  "evaluation_id": "EVAL-20260402-154959",
  "question_id": "REASON-005",
  "question_text": "A variant of the Monty Hall problem:\n\nThere are 100 doors. Behind one is a car, behind the others are goats. You pick door #1. The host, who knows where the car is, then opens 98 doors that don't have the car (and aren't your door), leaving door #1 (yours) and door #57.\n\n1. What's the probability the car is behind door #57?\n2. Should you switch?\n3. Now suppose after opening 98 doors, the host offers you $10,000 to NOT switch. At what car value would you be indifferent?\n4. What if the host doesn't know where the car is and just happened to open 98 goat doors by chance? Does this change your answer?",
  "category": "reasoning",
  "timestamp": "2026-02-11T00:00:00.000Z",
  "display_date": "Feb 11, 2026",
  "winner": {
    "name": "GPT-5.4",
    "provider": "openrouter",
    "score": 9.38
  },
  "avg_score": 8.317778,
  "matrix_size": 71,
  "models_used": [
    {
      "id": "deepseek_v4",
      "name": "DeepSeek V4",
      "provider": "openrouter"
    },
    {
      "id": "gpt_5_4",
      "name": "GPT-5.4",
      "provider": "openrouter"
    },
    {
      "id": "mimo_v2_flash",
      "name": "MiMo-V2-Flash",
      "provider": "Xiaomi"
    },
    {
      "id": "gemini_31_pro",
      "name": "Gemini 3.1 Pro",
      "provider": "openrouter"
    },
    {
      "id": "gpt_oss_120b",
      "name": "GPT-OSS-120B",
      "provider": "OpenAI"
    },
    {
      "id": "gemini_25_flash",
      "name": "Gemini 2.5 Flash",
      "provider": "openrouter"
    },
    {
      "id": "claude_opus_46",
      "name": "Claude Opus 4.6",
      "provider": "openrouter"
    },
    {
      "id": "grok_420",
      "name": "Grok 4.20",
      "provider": "openrouter"
    },
    {
      "id": "claude_sonnet_46",
      "name": "Claude Sonnet 4.6",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "gpt_5_4": {
      "display_name": "GPT-5.4",
      "provider": "openrouter",
      "average_score": 9.38,
      "score_count": 9,
      "min_score": 8.5,
      "max_score": 10,
      "rank": 1
    },
    "mimo_v2_flash": {
      "display_name": "MiMo-V2-Flash",
      "provider": "Xiaomi",
      "average_score": 9.13,
      "score_count": 9,
      "min_score": 8.5,
      "max_score": 9.85,
      "rank": 2
    },
    "deepseek_v4": {
      "display_name": "DeepSeek V4",
      "provider": "openrouter",
      "average_score": 9.04,
      "score_count": 9,
      "min_score": 8.45,
      "max_score": 9.8,
      "rank": 3
    },
    "claude_opus_46": {
      "display_name": "Claude Opus 4.6",
      "provider": "openrouter",
      "average_score": 9.03,
      "score_count": 8,
      "min_score": 7.95,
      "max_score": 9.85,
      "rank": 4
    },
    "grok_420": {
      "display_name": "Grok 4.20",
      "provider": "openrouter",
      "average_score": 8.92,
      "score_count": 8,
      "min_score": 8.05,
      "max_score": 9.85,
      "rank": 5
    },
    "gpt_oss_120b": {
      "display_name": "GPT-OSS-120B",
      "provider": "OpenAI",
      "average_score": 8.82,
      "score_count": 4,
      "min_score": 8.45,
      "max_score": 9.25,
      "rank": 6
    },
    "gemini_25_flash": {
      "display_name": "Gemini 2.5 Flash",
      "provider": "openrouter",
      "average_score": 8.15,
      "score_count": 8,
      "min_score": 5.2,
      "max_score": 9.45,
      "rank": 7
    },
    "claude_sonnet_46": {
      "display_name": "Claude Sonnet 4.6",
      "provider": "openrouter",
      "average_score": 8.05,
      "score_count": 8,
      "min_score": 5.45,
      "max_score": 9.65,
      "rank": 8
    },
    "gemini_31_pro": {
      "display_name": "Gemini 3.1 Pro",
      "provider": "openrouter",
      "average_score": 4.34,
      "score_count": 8,
      "min_score": 2.25,
      "max_score": 9.25,
      "rank": 9
    }
  },
  "judgments": [
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.05,
      "brief_justification": "Accurate answers with clear explanations and correct calculations; covers all parts fully, well-structured, and offers useful insight, though Q4 reasoning could be slightly deeper."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.05,
      "brief_justification": "Accurate answers with clear reasoning for all parts, covers required calculations and explains the random-host scenario, though depth could be slightly richer."
    },
    {
      "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": "All calculations and reasoning are accurate, covering each question thoroughly with clear explanations and useful guidance."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.85,
      "brief_justification": "The response is incomplete and the first answer provided is incorrect. It stops mid-sentence and doesn't address all parts of the question."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.45,
      "brief_justification": "Accurate, thorough, and clear with insightful analysis and practical value."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.25,
      "brief_justification": "Accurate, thorough, and clear with insightful analysis."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.45,
      "brief_justification": "Highly accurate and thorough with clear explanations and insightful analysis."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.3,
      "brief_justification": "The response correctly identifies the 99/100 probability for door #57, which is the right answer. However, the response is severely truncated - it only begins answering question 1 and doesn't address questions 2, 3, or 4 at all. The partial answer shown is correct and clearly written, but the incompleteness makes it nearly useless as a full response."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.45,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 7.95,
      "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": 8.5,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.05,
      "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": 5.45,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.5,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 6.65,
      "brief_justification": ""
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 9.25,
      "brief_justification": "Accurate, complete, and clear with good depth and practical value."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.45,
      "brief_justification": "Highly accurate and thorough with clear explanations and insightful analysis."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.45,
      "brief_justification": "Accurate, thorough, and clear with insightful analysis."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.6,
      "brief_justification": "Accurate and thorough response with clear explanations 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": "Highly accurate, thorough, and insightful with clear explanations."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.2,
      "brief_justification": "All four parts are answered correctly. The probability calculations for Q1 (99/100), Q2 (yes, switch), Q3 (~$10,204.08), and Q4 (50/50 when host is ignorant) are all accurate. The response shows excellent reasoning, particularly in Q4 where the model initially struggles but works through the Bayesian reasoning correctly to arrive at the right answer. The expected value calculation in Q3 properly accounts for the car value in both scenarios. The clarity is exceptional with well-structured section"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.2,
      "brief_justification": "The response is almost entirely correct and very well-structured. Parts 1-3 are perfect. Part 4 correctly identifies that the uninformed host scenario yields 50/50 odds, and the final observation about the $10,000 offer in the random-host case is insightful and correct. The only minor quibble is in the intermediate reasoning for part 4: the calculation of P(leaving #57 | A) = 1/99 assumes the host randomly selects which door to leave closed, which is one valid model but could be stated more expl"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.75,
      "brief_justification": "The response is excellent overall. Parts 1, 2, and 4 are correct. The intuition and explanations are clear and well-structured. For part 3, the setup of the indifference equation has a subtle issue: the $10,000 is offered as a cash payment to not switch, so the correct comparison should be EV(stay) = C/100 + $10,000 vs EV(switch) = 99C/100. The algebra yields C ≈ $10,204.08, which is correct given that framing. However, one could argue the $10,000 replaces the switching opportunity entirely, mak"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 6.7,
      "brief_justification": "Parts 1 and 2 reach the correct answer (99/100, switch) but the Bayesian table used to derive it is flawed—the likelihoods for 'car at #1' and 'car at #57' are not actually equal once you condition on which specific door survives, so the normalization argument is hand-wavy and technically incorrect even though it lands on the right number. Part 3 is correct. Part 4 is wrong: with an ignorant host who happens to reveal 98 goats, the posterior for door #57 is 1/2 only if you ignore the asymmetric "
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.45,
      "brief_justification": "All four parts are answered correctly with clear reasoning. The 99/100 probability for the knowledgeable host is correct, the indifference calculation ($10,204.08) is correct, and the Bayesian analysis for the ignorant host yielding 50/50 is correct. The explanation of why the ignorant host case differs is well-done. The only minor issue is in part 4's indifference discussion, which could be slightly more precise (the EV of staying should be 0.5V + 10,000 compared to EV of switching 0.5V, so sta"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.2,
      "brief_justification": "Parts 1-3 are correct and well-explained. Part 4 correctly identifies the key distinction and sets up the Bayesian analysis properly, but the response is cut off before reaching the final answer (which should be 1/2 for both doors). The calculation of P(H|B)=1/99 is correct but needs clarification: the host must leave specifically door #57 open, not just avoid opening it. The setup also doesn't fully account for which specific door is left - the host randomly picks 98 of 99 doors, and the condit"
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.25,
      "brief_justification": "The response begins with the correct 99% result for the standard informed-host case, but it is truncated and fails to answer most of the multi-part question, including switching advice, indifference value, and the uninformed-host variant."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.6,
      "brief_justification": "Mostly correct and thorough. The key answers are right, including the 99/100 switch result, the indifference value (~$10,204), and the 50-50 outcome when the host acts randomly and we condition on seeing 98 goats. Slightly meandering in part 4, with some temporary confusion, but it recovers to the correct conclusion."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.6,
      "brief_justification": "Correct on the main Monty Hall probabilities, switching advice, indifference value, and the ignorant-host result of 50/50 after conditioning on 98 goats opened. Minor clarity issues come from the false-start Bayes derivation and slightly imprecise phrasing before correcting itself."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.05,
      "brief_justification": "Accurate on both the standard and random-host variants, with correct EV calculations and clear recommendations. Very clear and practical; only slight incompleteness in not explicitly deriving the random-host $20,000 indifference point with an equation."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 6.55,
      "brief_justification": "Covers all four parts and gives helpful intuition, but contains a major error in part 1/Bayes normalization: it states equal joint weights for doors #1 and #57 yet concludes 99/100 instead of the correct 99/100 derivation. Part 3 is incomplete because the $10,000 should be compared to staying plus bribe versus switching, leading to an indifference value near $1,010,101, not $10,101. Part 4 correctly concludes 50/50 with an ignorant host."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9,
      "brief_justification": "Accurate on the standard knowledgeable-host case and correctly uses conditional reasoning for the random-host case, yielding 50/50. It fully answers all four parts, including the indifference calculation. Minor nit: in part 1 it slightly overstates that door #57 itself has 99/100 without explicitly noting this relies on the host's selection rule leaving that specific door; but within the stated setup this is acceptable."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 5.2,
      "brief_justification": "The response is clear and gets the standard informed-host case and EV calculation right, but it incorrectly claims door #57 has 99/100 without addressing host tie-breaking, and it fails to complete the uninformed-host conditional-probability analysis. This leaves a major part unanswered and reduces practical usefulness."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.45,
      "brief_justification": "The response starts correctly with the 99/100 probability for door #57 but is incomplete, cutting off mid-sentence. It does not address questions 2, 3, or 4 at all. While the initial probability statement is accurate, the overall answer lacks thoroughness and full coverage of the query."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.45,
      "brief_justification": "The response is factually accurate on all parts, including the key distinction in Q4 between informed and uninformed host. It provides clear explanations, correct calculations, and thorough coverage using Bayes' reasoning implicitly. Minor deduction for slight awkwardness in the Q4 derivation wording before reaching the right conclusion."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.8,
      "brief_justification": "Response is factually accurate on classic Monty Hall and ignorant variant, with clear explanations, correct math, and good depth on Bayesian update. Minor notation slip in binomial coefficient but overall logic holds."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.4,
      "brief_justification": "Accurate and thorough analysis of all parts, including conditional probability for the non-knowledge case, with clear structure and practical insights."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 8.8,
      "brief_justification": "The response is factually accurate on all parts, including the classic Monty Hall probabilities, the indifference calculation, and the key Bayesian distinction for the ignorant host case. It is thorough, clearly structured, and provides useful actionable insights with equations. Minor deduction on depth as the Bayes explanation is summarized rather than fully derived, though it offers to expand."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 8.6,
      "brief_justification": "Response is factually accurate on all points with correct probabilities (99/100 and 1/2), clear tables and formulas, good intuition, and solid analysis of the bribe scenario. Minor issues: table in part 1 has incorrect joint probabilities (should be 1/100 * 99/99 for #1 and 1/100 * 1 for #57, though normalization works out); part 3 ignores potential signaling from bribe but notes it; part 4 has some messy probability explanations before correcting to the right answer."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.8,
      "brief_justification": "The response is factually accurate on all parts with correct probabilities, Bayes calculation, and EV math. It thoroughly covers each question with clear explanations and equations. Minor quibble on phrasing in part 4 about 'should you switch? No' (indifferent is more precise), but overall excellent."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.6,
      "brief_justification": "The response accurately addresses all four parts of the Monty Hall variant with correct probabilities (1/100 vs 99/100 initially, then 1/2 after conditioning on the specific door), clear explanations, and proper handling of the value indifference and ignorant host cases."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.45,
      "brief_justification": "The response is factually accurate on probabilities, switching advice, and the conditional probability shift when the host has no knowledge. The indifference calculation has a minor algebraic error (should be C = $10,000 * 99/98 ≈ $10,204 but actually derives correctly to that value despite the written step). It thoroughly covers all parts with clear structure and good use of Bayes in part 4, though part 4 is slightly incomplete as it cuts off before final probabilities."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 3.1,
      "brief_justification": "The response appears to be cut off mid-sentence, only beginning to answer question 1 and not completing any of the four questions. The partial answer for question 1 is on the right track (99/100 is correct), but the response is severely incomplete, missing questions 2, 3, and 4 entirely."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.8,
      "brief_justification": "All four questions are answered correctly. The 99/100 probability for Q1, the switching recommendation for Q2, the indifference calculation (~$10,204) for Q3, and the 50/50 result for Q4 (uninformed host) are all accurate. The response shows good reasoning, especially in Q4 where it initially considers an incorrect path but self-corrects. The expected value setup in Q3 is correct. Minor issue: the self-correction in Q4 could be cleaner, but the final answer is right. Overall very strong response"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.2,
      "brief_justification": "Excellent response covering all four parts accurately. The Bayesian analysis for the ignorant host is correct and well-explained, showing the 50/50 result. The indifference calculation is correct. Minor issue: the self-correction mid-calculation is slightly messy but ultimately arrives at the right answer. The key insight about deliberate vs. random host behavior is clearly articulated."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.6,
      "brief_justification": "All four parts are answered correctly and with solid reasoning. The probability calculations are accurate (99/100 for the knowledgeable host, 1/2 for the random host). The indifference calculation in part 3 is correct, and the bonus analysis for part 4's indifference scenario is a nice addition. The explanation of why the random host changes things is clear and well-reasoned using conditional probability logic. Minor style note: the LaTeX formatting may not render in all contexts, but the math i"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.8,
      "brief_justification": "The response correctly handles all four parts. The probability calculations are accurate (99/100 for the informed host, 1/2 for the uninformed host). The indifference calculation is mathematically correct and well-explained. The explanation of why the uninformed host changes everything is clear and insightful. The $20,000 threshold for part 4 is correctly derived (indifferent when C/2 = $10,000). Minor note: could have shown the part 4 indifference calculation explicitly rather than just stating"
    },
    {
      "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 is mathematically correct throughout, including the Bayesian analysis for the uninformed host scenario yielding 50/50 odds. The indifference calculation is properly set up and solved. The explanation of why the uninformed host changes probabilities is clear and well-reasoned. Minor note: in part 4's sub-question about the $10,000 offer, the reasoning is slightly informal but reaches the correct conclusion. Overall an excellent, thorough response."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9,
      "brief_justification": "The response is highly accurate and well-structured. It correctly identifies the 99/100 probability for door #57 in the knowledgeable host case, correctly calculates the indifference car value of ~$10,204.08, and begins a rigorous Bayesian analysis for the ignorant host case. The response appears to be cut off mid-sentence on question 4, which prevents a complete evaluation of that section, but the methodology shown is correct and the answer would lead to 50/50 odds. The depth of analysis using "
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 5,
      "brief_justification": "Incorrect probability (should be 99/100 for door #57, not 99% as stated), incomplete answer, and lacks depth on switching strategy and indifference value."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.8,
      "brief_justification": "The response is entirely correct, covering all four parts of the problem accurately. It correctly identifies the 99/100 probability in the standard Monty Hall variant, the indifference point calculation, and crucially distinguishes the probability shift to 50/50 when the host is ignorant. The structure is logical and easy to follow."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.55,
      "brief_justification": "Accurate and thorough with clear structure; minor error in Bayes calculation but corrected; practical advice and good analysis."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "All answers are factually correct, thoroughly cover the problem, clearly structured, provide deep analysis including Bayes' rule, and offer practical decision-making guidance."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.8,
      "brief_justification": "All answers are factually correct, thoroughly covered, clearly explained, and provide practical insights, including a nuanced analysis of the host's knowledge."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9,
      "brief_justification": "Accurate and thorough analysis of all parts, clear structure, insightful comparisons, and practical implications."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.45,
      "brief_justification": "Accurate probability analysis, clear structure, and insightful handling of the indifference and chance scenarios."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9,
      "brief_justification": "The response accurately and thoroughly addresses all parts of the complex Monty Hall variant. The explanations are clear, and the mathematical derivations are correct. The distinction between the host knowing and not knowing is well-explained."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.85,
      "brief_justification": "The response is exceptionally accurate, complete, and clear. It provides a thorough explanation for each part of the problem, including the complex Bayesian analysis for the 'ignorant host' variant, which is often a point of confusion. The mathematical derivations are correct and well-explained. The practical advice for the indifference point is also very useful."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "The response is perfectly correct, complete, clear, and demonstrates excellent depth in explaining the nuances of the Monty Hall problem and its variations. The calculations are accurate, and the reasoning for each part is sound."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.85,
      "brief_justification": "The response is exceptionally clear, accurate, and provides deep insights into the Monty Hall problem and its variations. All parts of the question are addressed thoroughly with correct explanations and calculations."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.25,
      "brief_justification": "The response is highly accurate and provides a thorough explanation of each part of the problem. The use of tables and formulas enhances clarity and depth. The explanation for the ignorant host scenario is particularly insightful, highlighting the core difference in the problem's setup."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.85,
      "brief_justification": "The response is exceptionally clear, accurate, and provides a thorough breakdown of each part of the problem, including the complex conditional probability for the fourth question. The mathematical derivations are correct and well-explained. The usefulness is high due to the detailed explanations and practical application of the indifference point."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9,
      "brief_justification": "The response correctly identifies the probabilities and the optimal strategy for switching. It also correctly addresses the indifference point for the monetary offer and the crucial difference when the host doesn't know the car's location. The explanation is clear and logical."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 3.5,
      "brief_justification": "Response is incomplete and cut off mid-sentence. Only begins to answer question 1 with correct 99% probability, but fails to address questions 2, 3, and 4 entirely."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.05,
      "brief_justification": "All answers are mathematically correct with clear explanations. Well-structured response covers all four questions thoroughly, with proper probability calculations and clear decision guidance. The distinction between host knowing vs. not knowing is well explained."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.8,
      "brief_justification": "Excellent response with correct probability calculations (99/100), proper Bayes' theorem application for the ignorant host case, and accurate indifference point calculation. Well-structured with clear explanations for all four questions. The key insight about deliberate vs. accidental avoidance is insightful. Only minor room for more intuitive elaboration on why the host's knowledge matters."
    },
    {
      "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": "All four parts answered correctly with clear explanations. The mathematical derivations are accurate, the distinction between knowing and random host is properly analyzed, and practical recommendations are provided."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.45,
      "brief_justification": "Accurate probability analysis and expected value calculations. Clear explanations for both the knowing and unknowing host scenarios. Minor issue: the 50/50 case explanation could be clearer - when host doesn't know, switching has EV=C/2 so you'd always take $10,000, not just when car < $20,000."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.2,
      "brief_justification": "Accurate answers with correct probabilities (99/100 for switching, ~$10,101 indifference point, 50/50 when host is ignorant). Well-structured with Bayes' theorem and tables. Minor confusion in part 4 calculation explanation but conclusion is correct. Comprehensive coverage of all four questions with good intuition."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.05,
      "brief_justification": "All answers are mathematically correct with clear explanations. The Bayes' theorem application for the unknown host scenario is accurate. Well-structured response with proper mathematical notation."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.85,
      "brief_justification": "Answers 1-3 are correct with clear reasoning. Answer 4 provides good Bayes' theorem setup but cuts off mid-calculation, leaving the final probability result incomplete. The response is well-structured and insightful but loses completeness points due to the unfinished final answer."
    }
  ],
  "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)"
  }
}