{
  "evaluation_id": "EVAL-20260207-133505",
  "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": "Claude Opus 4.5",
    "provider": "Anthropic",
    "score": 9.81
  },
  "avg_score": 8.315,
  "matrix_size": 90,
  "models_used": [
    {
      "id": "gemini_3_pro",
      "name": "Gemini 3 Pro Preview",
      "provider": "Google"
    },
    {
      "id": "gemini_2_5_flash",
      "name": "Gemini 2.5 Flash",
      "provider": "Google"
    },
    {
      "id": "gpt_oss_120b",
      "name": "GPT-OSS-120B",
      "provider": "OpenAI"
    },
    {
      "id": "grok_direct",
      "name": "Grok 3 (Direct)",
      "provider": "xAI"
    },
    {
      "id": "olmo_think",
      "name": "OLMo Think",
      "provider": "Allen AI"
    },
    {
      "id": "gemini_3_flash",
      "name": "Gemini 3 Flash Preview",
      "provider": "Google"
    },
    {
      "id": "claude_sonnet",
      "name": "Claude Sonnet 4.5",
      "provider": "Anthropic"
    },
    {
      "id": "mimo_v2_flash",
      "name": "MiMo-V2-Flash",
      "provider": "Xiaomi"
    },
    {
      "id": "claude_opus",
      "name": "Claude Opus 4.5",
      "provider": "Anthropic"
    },
    {
      "id": "deepseek_v3",
      "name": "DeepSeek V3.2",
      "provider": "DeepSeek"
    }
  ],
  "rankings": {
    "claude_opus": {
      "display_name": "Claude Opus 4.5",
      "provider": "Anthropic",
      "average_score": 9.81,
      "score_count": 7,
      "min_score": 9.05,
      "max_score": 10,
      "rank": 1
    },
    "mimo_v2_flash": {
      "display_name": "MiMo-V2-Flash",
      "provider": "Xiaomi",
      "average_score": 9.74,
      "score_count": 6,
      "min_score": 9.4,
      "max_score": 10,
      "rank": 2
    },
    "gemini_3_flash": {
      "display_name": "Gemini 3 Flash Preview",
      "provider": "Google",
      "average_score": 9.72,
      "score_count": 7,
      "min_score": 9.45,
      "max_score": 10,
      "rank": 3
    },
    "claude_sonnet": {
      "display_name": "Claude Sonnet 4.5",
      "provider": "Anthropic",
      "average_score": 9.57,
      "score_count": 6,
      "min_score": 9,
      "max_score": 10,
      "rank": 4
    },
    "grok_direct": {
      "display_name": "Grok 3 (Direct)",
      "provider": "xAI",
      "average_score": 9.35,
      "score_count": 7,
      "min_score": 7.3,
      "max_score": 10,
      "rank": 5
    },
    "deepseek_v3": {
      "display_name": "DeepSeek V3.2",
      "provider": "DeepSeek",
      "average_score": 8.72,
      "score_count": 6,
      "min_score": 8.2,
      "max_score": 9.6,
      "rank": 6
    },
    "gemini_2_5_flash": {
      "display_name": "Gemini 2.5 Flash",
      "provider": "Google",
      "average_score": 8.65,
      "score_count": 6,
      "min_score": 8.25,
      "max_score": 8.9,
      "rank": 7
    },
    "gpt_oss_120b": {
      "display_name": "GPT-OSS-120B",
      "provider": "OpenAI",
      "average_score": 7.15,
      "score_count": 5,
      "min_score": 0.7,
      "max_score": 10,
      "rank": 8
    },
    "olmo_think": {
      "display_name": "OLMo Think",
      "provider": "Allen AI",
      "average_score": 6.66,
      "score_count": 4,
      "min_score": 0.7,
      "max_score": 9.65,
      "rank": 9
    },
    "gemini_3_pro": {
      "display_name": "Gemini 3 Pro Preview",
      "provider": "Google",
      "average_score": 3.77,
      "score_count": 7,
      "min_score": 2.1,
      "max_score": 6.25,
      "rank": 10
    }
  },
  "judgments": [
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 0,
      "brief_justification": "No response was provided to evaluate. The response field is completely empty, making it impossible to assess any of the criteria."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.55,
      "brief_justification": "Excellent response with correct probability analysis for all scenarios. Minor issue: Question 4's probability calculation could be more rigorously justified using Bayes' theorem to show why it becomes 1/2, though the intuitive explanation is sound."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 0,
      "brief_justification": "No response provided to evaluate."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.55,
      "brief_justification": "The response is factually accurate and logically sound, thoroughly addressing all four parts of the question with clear explanations and practical calculations, though the depth could be slightly enhanced by discussing the conditional probability more formally in part 4."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9,
      "brief_justification": "The response is factually accurate and well-structured, covering all parts of the question with clear explanations; the only minor deduction is for not explicitly mentioning the host's knowledge in the Bayesian update for part 4, though the conclusion is correct."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 9.05,
      "brief_justification": "All four parts are answered accurately with correct calculations and clear explanations; the response is well‑structured and useful, though the probabilistic reasoning for the unknowing host could be a bit deeper."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 3.5,
      "brief_justification": "The response correctly states the probability for part 1 but is incomplete, missing answers to parts 2‑4, and provides only a brief, superficial explanation."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 8.8,
      "brief_justification": "The response is factually accurate and logically sound for the standard Monty Hall variant, correctly applying Bayes' theorem and indifference calculations. It thoroughly addresses all sub-questions, though the analysis for the 'host doesn't know' scenario is cut off and incomplete, slightly reducing completeness and depth."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "The response is factually accurate, thoroughly covers all parts of the question, is exceptionally clear and well-structured, provides insightful analysis (especially on the unknowing host scenario), and offers practical value with actionable conclusions."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 2.65,
      "brief_justification": "The response is factually incorrect for part 1 (probability should be 99/100 for the car being behind door #57, but the logic is incomplete and cuts off), and it fails to address parts 2, 3, and 4 at all, making it severely incomplete and lacking depth or usefulness."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.8,
      "brief_justification": "The response correctly applies Bayesian reasoning to the classic Monty Hall variant, yielding accurate probabilities and a sound indifference calculation. It thoroughly addresses all sub-questions and explains the crucial difference when the host is ignorant, though the final answer for part 4 is cut off."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.6,
      "brief_justification": "The response correctly applies probability theory to the Monty Hall variant, provides clear reasoning for switching, and addresses the indifference point and the host's knowledge scenario, though it could elaborate more on the conditional probability in part 4."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 8.25,
      "brief_justification": "The response correctly applies the Monty Hall logic for 100 doors, identifies the 99% probability for door #57, advises switching, and calculates the indifference car value. It correctly notes the change in probability if the host doesn't know, but could more explicitly state the probability becomes 50% in that case."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.8,
      "brief_justification": "The response is factually accurate, thoroughly addresses all four questions with clear step-by-step reasoning, and provides actionable insights for decision-making."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 10,
      "brief_justification": "The response is perfectly accurate, addressing all four parts of the question with precise mathematical reasoning. It correctly identifies the shift from 99/100 to 50/50 probability when the host's knowledge is removed, and the indifference point calculation is flawless."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 10,
      "brief_justification": "The response is perfectly accurate, correctly identifying the 99/100 probability for the standard variant, solving the indifference point equation correctly, and explaining the Monty Fall (random host) variant accurately."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.25,
      "brief_justification": "The answer provides accurate probabilities, correct expected‑value indifference calculation, and correctly distinguishes the knowledgeable vs. random host scenarios, with clear presentation; Bayesian reasoning for the random host case is correct but could be elaborated further."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 9.6,
      "brief_justification": "The model provides a mathematically rigorous and correct derivation for all parts of the problem. It correctly identifies the probability shift in the standard Monty Hall case (99/100) and the indifference point for the monetary offer. For the final part (host doesn't know), it correctly identifies that the probability becomes 1/2 (though the text cuts off slightly at the very end, the logic established for the 1/99 probabilities for both C1 and C57 leads directly to that conclusion)."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "The response is perfectly accurate, covering the mathematical nuances of both the standard Monty Hall logic and the 'Monty Fall' (unknowing host) variant. The indifference calculation is correctly solved using expected value equations."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 6.25,
      "brief_justification": "The response is incomplete; it cuts off mid-sentence during the explanation of the first part and fails to address questions 2, 3, and 4 entirely."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.65,
      "brief_justification": "The response provides an excellent, mathematically rigorous derivation for the first three parts of the question, correctly identifying the 99/100 probability and the indifference point. However, the response is incomplete as it cuts off mid-sentence during the explanation of the fourth part (the 'naive host' variant)."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 0,
      "brief_justification": "The model provided an empty response, failing to address any part of the multi-part probability problem."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 0,
      "brief_justification": "The model provided no response content to evaluate."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 10,
      "brief_justification": "The response is perfectly accurate across all four parts of the problem. It correctly identifies the probability shift in the standard Monty Hall variant (99/100), accurately calculates the expected value for the indifference point, and correctly explains the 'Monty Fall' variation where the host's lack of knowledge results in a 50/50 probability."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.8,
      "brief_justification": "Exceptionally accurate response with correct probability calculations for all scenarios, clear explanations of the Monty Hall variant, proper mathematical derivations, and insightful distinction between knowledgeable and ignorant host cases. The only minor limitation is the slightly informal treatment of the Bayesian calculation in part 4, though the conclusion is correct."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.8,
      "brief_justification": "Excellent response with correct probabilities, clear mathematical derivation for the indifference point, and proper distinction between informed vs. random host behavior. All four questions answered accurately with strong explanations."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 8.8,
      "brief_justification": "Excellent mathematical analysis with proper Bayesian reasoning for all parts. The response correctly handles the probability calculations and provides accurate answers (99/100, yes switch, $10,204.08). Minor deduction for the abrupt ending on question 4, though the setup was correct."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "Flawless analysis of all four parts with correct probabilities, clear mathematical derivations, and excellent explanation of the crucial distinction between knowing vs unknowing host using proper Bayesian conditioning."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 2.1,
      "brief_justification": "The response is incomplete, cutting off mid-sentence after only addressing the first question. While the initial answer of 99/100 for door #57 appears correct, the explanation is truncated and the remaining three questions are not addressed at all."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.65,
      "brief_justification": "Excellent probabilistic analysis using Bayes' theorem correctly for questions 1-3. The response is cut off at question 4, preventing full evaluation of completeness, but the approach shown is rigorous and the mathematical reasoning is sound throughout."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 0,
      "brief_justification": "No response was provided to evaluate. The response field is completely empty, making it impossible to assess any of the criteria."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.8,
      "brief_justification": "The response is factually accurate and logically valid, thoroughly addresses all sub-questions, is clearly structured with step-by-step reasoning, provides insightful analysis including expected value and conditional probability, and offers practical, actionable advice for decision-making."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.8,
      "brief_justification": "The response is factually perfect, thoroughly addresses all parts of the question with clear logic, and provides actionable calculations; depth is slightly reduced as the random-host scenario could be elaborated with conditional probability formulas."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.55,
      "brief_justification": "The response is factually accurate, thoroughly addresses all four parts of the question, presents the logic with exceptional clarity, demonstrates insightful analysis particularly in the Bayesian update for the random host scenario, and provides directly actionable calculations and conclusions."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "The response is factually perfect, thoroughly addresses all sub-questions, presents clear reasoning with structured tables and formulas, provides insightful analysis of the host's knowledge condition, and offers practical, actionable calculations for decision-making."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 3.7,
      "brief_justification": "The response correctly states the probability for part 1 but is cut off and does not address the other three parts of the question, making it incomplete and of limited value."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.9,
      "brief_justification": "The response provides mathematically rigorous and correct answers to all four parts, including detailed Bayesian analysis for the probability calculation and expected value reasoning for the indifference point, though it cuts off before completing the final scenario."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.4,
      "brief_justification": "The response correctly calculates the 99/100 probability for door #57, recommends switching, and provides a clear expected value analysis for the indifference point. It also accurately explains how the host's knowledge changes the probability, though the indifference calculation could be slightly more detailed."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 7.3,
      "brief_justification": "The response correctly answers parts 1-3 but has a significant error in part 4 regarding probability when the host doesn't know, reducing correctness and depth."
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.4,
      "brief_justification": "The response provides accurate answers to all four parts with clear reasoning. The probability calculations for parts 1-3 are correct, and the crucial distinction between the informed and uninformed host scenarios is well explained. Minor deduction for not fully working through the Bayesian calculation in part 4, though the conclusion is correct. Excellent structure with clear formatting and practical interpretations."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.8,
      "brief_justification": "The response is completely correct on all four parts, including the subtle but crucial distinction in part 4 about host knowledge. The mathematical work is accurate, the explanations are clear and well-structured, and the reasoning demonstrates genuine understanding of conditional probability. The only minor improvement would be a slightly deeper explanation of why the uninformed host scenario creates equal probabilities (Bayesian updating)."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.25,
      "brief_justification": "All four parts are answered correctly with accurate probability calculations. The explanation of why the host's knowledge matters in part 4 is correct but could benefit from a more detailed Bayesian derivation. The response is exceptionally clear and well-structured with practical takeaways."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 8.25,
      "brief_justification": "The response demonstrates excellent mathematical reasoning with correct Bayesian analysis for parts 1-3. The answer of 99/100 for part 1 is correct, the switching recommendation is properly justified, and the indifference calculation is accurate. However, the response is incomplete - part 4 is cut off mid-calculation, leaving the important distinction about the uninformed host scenario unfinished. The work shown for part 4 was heading in the right direction but never reached a conclusion."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 2.1,
      "brief_justification": "The response is severely incomplete, cutting off mid-sentence after only partially addressing question 1. While the initial answer of 99/100 for the probability appears correct, the response fails to complete even the first question's explanation and provides no answers to questions 2, 3, or 4."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.25,
      "brief_justification": "The response correctly solves parts 1-3 with rigorous Bayesian analysis and clear explanations. However, part 4 is incomplete - it cuts off mid-sentence without providing the actual answer for the uninformed host scenario (which should yield 50/50 odds). The mathematical work shown is accurate and well-structured, but the incompleteness significantly impacts the overall evaluation."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 0.7,
      "brief_justification": "The response is completely empty, providing no answer to any of the four parts of the Monty Hall variant question. No analysis, calculations, or explanations were given for the probability questions, the switching decision, the indifference calculation, or the epistemological variant."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 0.7,
      "brief_justification": "The response is completely empty, providing no answer to any of the four parts of the Monty Hall variant question. No analysis, calculations, or explanations were given for the probability questions, decision analysis, or the epistemological variant."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.55,
      "brief_justification": "The response correctly solves all four parts of the problem with accurate probability calculations and clear reasoning. The explanation of why the host's knowledge matters in part 4 is particularly well done. Minor deduction for not explicitly discussing the conditional probability framework that underlies the random host scenario, though the conclusion is correct."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 10,
      "brief_justification": "The model correctly calculates the probabilities for the 100-door Monty Hall variant, accurately solves the expected value equation for the indifference point, and correctly identifies the change in probability for the 'Monty Fall' (random host) scenario."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 0,
      "brief_justification": "No response text was provided to evaluate."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 10,
      "brief_justification": "The response correctly and thoroughly answers all parts of the question, including detailed explanations and calculations for each scenario."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.65,
      "brief_justification": "The response correctly answers all parts of the question with clear explanations and accurate calculations, demonstrating a strong understanding of the Monty Hall problem and its variants."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 10,
      "brief_justification": "The response accurately and clearly answers all parts of the question, demonstrating a deep understanding of the Monty Hall problem and its variations."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 8.2,
      "brief_justification": "The model correctly solves the first three parts of the problem with clear explanations. The fourth part is incomplete and contains a slight error in reasoning about the host's random selection process."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "The response provides a perfectly correct, complete, clear, and in-depth analysis of the Monty Hall variant, including the indifference calculation and the crucial distinction of an unknowing host."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 0,
      "brief_justification": "The response is incomplete, cutting off mid-sentence. Therefore, it cannot be evaluated for correctness, completeness, clarity, depth, or usefulness."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The model provides a perfectly correct and well-explained answer to all parts of the modified Monty Hall problem, including the nuances of the host's knowledge."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 9.65,
      "brief_justification": "The model provides a correct and clear explanation for each part of the problem, including the nuances of the host's knowledge and the financial decision."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 10,
      "brief_justification": "The response provides a perfectly accurate, complete, and well-explained solution to all parts of the complex Monty Hall variant. The mathematical derivations are correct, and the reasoning for each scenario is clear and insightful."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.45,
      "brief_justification": "The response is factually accurate, thoroughly addresses all parts of the question, and provides clear reasoning with detailed calculations. Minor improvements could be made in simplifying explanations for broader accessibility."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.45,
      "brief_justification": "The response is factually accurate across all parts, covers each question thoroughly, and provides clear, logical explanations with insightful expected value calculations. Minor improvements could be made in simplifying the language for broader accessibility."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.65,
      "brief_justification": "The response is factually accurate across all parts, thoroughly addresses each question with clear explanations, and provides insightful analysis, especially in distinguishing the random host scenario. Minor room for deeper exploration of edge cases or assumptions."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 8.65,
      "brief_justification": "The response is highly accurate and detailed in addressing the Monty Hall variant, with correct probability calculations and logical reasoning. It slightly lacks in clarity due to occasional complexity in explanation, but overall provides deep and useful insights."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 9.65,
      "brief_justification": "The response is factually accurate, thoroughly addresses all parts of the question, and is clearly structured. It provides deep insight into the probability dynamics, though the practical usefulness is slightly limited by the theoretical nature of the problem."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 6.1,
      "brief_justification": "The response correctly identifies the probability for door #57 as 99/100 but is incomplete as it does not address all parts of the question. The explanation lacks depth and clarity in fully elaborating the logic behind the Monty Hall variant."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.65,
      "brief_justification": "The response is highly accurate and detailed for questions 1-3, with clear probability calculations and logical reasoning. It is slightly incomplete as question 4's explanation is cut off, though the provided content is correct."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.05,
      "brief_justification": "The response accurately addresses the probability calculations and decision to switch in the Monty Hall variant, though it lacks some depth in explaining the indifference point for the car value. It covers most aspects clearly but could elaborate more on the host's knowledge impacting the probability."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 8.05,
      "brief_justification": "The response accurately addresses the Monty Hall variant with correct probability calculations and logical reasoning for switching, though it lacks some depth in explaining the indifference point for the car value."
    }
  ],
  "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)"
  }
}