{
  "evaluation_id": "EVAL-20260402-163314",
  "question_id": "REASON-013",
  "question_text": "Sleeping Beauty is told: 'We'll flip a coin. If heads, we wake you once (Monday). If tails, we wake you twice (Monday and Tuesday) with memory erased between wakings. Each time you wake, we'll ask: what is your credence that the coin landed heads?' (1) Is the answer 1/2 (halfer position) or 1/3 (thirder position)? (2) Argue for both positions rigorously. (3) What if we change the experiment to 1 waking for heads and 1,000,000 wakings for tails?",
  "category": "reasoning",
  "timestamp": "2026-04-02T16:33:14.612Z",
  "display_date": "Apr 02, 2026",
  "winner": {
    "name": "Grok 4.20",
    "provider": "openrouter",
    "score": 8.9
  },
  "avg_score": 8.072,
  "matrix_size": 88,
  "models_used": [
    {
      "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": "minimax_m25",
      "name": "MiniMax M2.5",
      "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": "deepseek_v4",
      "name": "DeepSeek V4",
      "provider": "openrouter"
    },
    {
      "id": "gemini_31_pro",
      "name": "Gemini 3.1 Pro",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "grok_420": {
      "display_name": "Grok 4.20",
      "provider": "openrouter",
      "average_score": 8.9,
      "score_count": 9,
      "min_score": 7.8,
      "max_score": 9.75,
      "rank": 1
    },
    "claude_sonnet_46": {
      "display_name": "Claude Sonnet 4.6",
      "provider": "openrouter",
      "average_score": 8.9,
      "score_count": 8,
      "min_score": 8.2,
      "max_score": 9.4,
      "rank": 2
    },
    "gpt_5_4": {
      "display_name": "GPT-5.4",
      "provider": "openrouter",
      "average_score": 8.86,
      "score_count": 9,
      "min_score": 8.05,
      "max_score": 9.45,
      "rank": 3
    },
    "minimax_m25": {
      "display_name": "MiniMax M2.5",
      "provider": "openrouter",
      "average_score": 8.81,
      "score_count": 9,
      "min_score": 8,
      "max_score": 9.45,
      "rank": 4
    },
    "claude_opus_46": {
      "display_name": "Claude Opus 4.6",
      "provider": "openrouter",
      "average_score": 8.79,
      "score_count": 9,
      "min_score": 7.65,
      "max_score": 9.65,
      "rank": 5
    },
    "mimo_v2_flash": {
      "display_name": "MiMo-V2-Flash",
      "provider": "Xiaomi",
      "average_score": 8.57,
      "score_count": 9,
      "min_score": 7.4,
      "max_score": 9.45,
      "rank": 6
    },
    "gpt_oss_120b": {
      "display_name": "GPT-OSS-120B",
      "provider": "OpenAI",
      "average_score": 8.53,
      "score_count": 8,
      "min_score": 6.6,
      "max_score": 9.65,
      "rank": 7
    },
    "gemini_25_flash": {
      "display_name": "Gemini 2.5 Flash",
      "provider": "openrouter",
      "average_score": 7.32,
      "score_count": 9,
      "min_score": 4.45,
      "max_score": 8.85,
      "rank": 8
    },
    "deepseek_v4": {
      "display_name": "DeepSeek V4",
      "provider": "openrouter",
      "average_score": 6.54,
      "score_count": 9,
      "min_score": 4.2,
      "max_score": 8.85,
      "rank": 9
    },
    "gemini_31_pro": {
      "display_name": "Gemini 3.1 Pro",
      "provider": "openrouter",
      "average_score": 5.5,
      "score_count": 9,
      "min_score": 2.65,
      "max_score": 8.45,
      "rank": 10
    }
  },
  "judgments": [
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.4,
      "brief_justification": "Comprehensive, rigorous, and clear arguments for both positions with insightful analysis of the variant scenario."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.3,
      "brief_justification": "Comprehensive and accurate explanation of both positions, with clear arguments and insightful analysis of the modified experiment."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.65,
      "brief_justification": "Comprehensive and rigorous treatment of both positions, clear structure, and insightful analysis."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.85,
      "brief_justification": "Comprehensive and rigorous analysis of both positions with clear arguments and depth."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9.3,
      "brief_justification": "Comprehensive and rigorous analysis of both positions, with clear formalizations and thoughtful extension to the extreme case."
    },
    {
      "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": "Comprehensive and rigorous analysis of both positions, with clear arguments and deep philosophical insight."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.45,
      "brief_justification": "Comprehensive and rigorous analysis of both positions, with clear explanations and strong arguments."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.45,
      "brief_justification": "Comprehensive and accurate coverage of both positions with rigorous arguments and clear structure."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 5.8,
      "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": 8.7,
      "brief_justification": "The response provides an excellent, rigorous, and deep analysis of both positions, including the underlying philosophical assumptions (SIA vs SSA) and the extreme variant. However, the text cuts off abruptly at the very end, impacting completeness."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 8.05,
      "brief_justification": "The response accurately and deeply explains both positions and the extreme variant, but the text cuts off abruptly at the end, impacting completeness."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.75,
      "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": 9.25,
      "brief_justification": "The response is exceptionally accurate, deep, and well-structured, covering all aspects of the prompt with rigorous philosophical and mathematical arguments. However, it cuts off mid-sentence at the very end, slightly impacting completeness."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.4,
      "brief_justification": "The response is highly clear, comprehensive, and well-structured. It accurately captures the core philosophical arguments, though the formal Bayesian formulation for the Thirder position is slightly informal."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 6.5,
      "brief_justification": "The response provides solid, well-structured arguments for both the halfer and thirder positions, but cuts off abruptly and completely fails to answer the third part of the prompt."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9.45,
      "brief_justification": "The response accurately and clearly explains both the Halfer and Thirder positions, uses formal probability to illustrate the arguments, and correctly applies the logic to the extreme 1,000,000 wakings variant."
    },
    {
      "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": "Detailed and well-structured, covers both positions rigorously, but slightly incomplete in halfer argument."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.5,
      "brief_justification": "The response is cut off mid-argument, leaving the halfer argument incomplete, the thirder argument entirely missing, and the million-wakings variant (Part 3) completely unaddressed. What is present is accurate and well-structured - the framing of the debate is correct, the attribution to David Lewis is appropriate, and the 'no new information' argument begins correctly. However, the severe incompleteness means it fails to deliver on 2 of the 3 requested parts. The claim that 1/3 is the 'majority"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 5.7,
      "brief_justification": "The response covers both halfer and thirder positions and addresses the million-waking variant, but it is disorganized and incomplete. The text abruptly switches from English to Chinese mid-sentence and then restarts the entire analysis in Chinese, with the Chinese portion cutting off mid-sentence. The arguments presented are mostly correct but lack rigor - the halfer frequency argument is actually wrong (claiming heads awakenings are 1/2 of total awakenings, which contradicts the thirder's corr"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 8.45,
      "brief_justification": "The response is factually accurate, clearly structured, and presents both halfer and thirder positions rigorously with multiple justification approaches (frequency, betting, Bayesian, philosophical). It correctly addresses the 1,000,000 variant. The response appears to be cut off at the end (mid-sentence in the thirder rigor section), which slightly impacts completeness. The depth is strong, covering the key philosophical distinction between centered and uncentered evidence, diachronic consisten"
    },
    {
      "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 well-structured, clearly written, and covers both positions rigorously with multiple arguments for each side. The extreme case analysis is handled well. Minor correctness issues: the attribution of the 'no new information' argument to Elga before he 'switched sides' is somewhat misleading (Elga was always a thirder), and the halfer betting argument presented is not quite right—the standard halfer betting argument is more nuanced. The Lewis centered-worlds argument assigns 1/4 to "
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 8.2,
      "brief_justification": "The response presents both halfer and thirder positions rigorously with formal arguments, betting arguments, and frequency arguments. The thirder position's Elga indifference argument is well-presented. The Dutch book argument for thirders is correct. The halfer position is fairly represented, though the claim that Lewis accepts P(Monday∧Heads) = 1/4 is slightly misleading (Lewis actually assigns 1/2 to Heads∧Monday). The million-wakings variant is handled well with good philosophical discussion"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 7.75,
      "brief_justification": "The response covers both positions and the modified experiment clearly. However, there are some issues: (1) The Bayesian argument for the thirder position is muddled - the standard approach uses centered possible worlds and a halving principle, not the somewhat confused likelihood ratio presented. (2) The halfer position is somewhat strawmanned, especially in the extreme case - sophisticated halfer arguments (like Lewis's or Elga's counterpart) deserve more rigorous treatment. (3) The response s"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 7.2,
      "brief_justification": "The response provides a reasonable overview of both halfer and thirder positions and addresses the million-wakings variant. However, there are some technical issues: (1) The thirder derivation is somewhat confused - the standard thirder argument uses a uniform distribution over the three centered possibilities (each getting 1/3), not the conditional probability calculation shown which actually gives 1/2 not 1/3 with the stated weights (1/2, 1/4, 1/4). The author writes P(H,Mon)=1/2 and P(T,Mon)="
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 6.65,
      "brief_justification": "The response provides a solid and well-structured treatment of both the halfer and thirder positions with multiple arguments for each side. The Bayesian analysis for the halfer position is correctly presented, and the thirder arguments (long-run frequency, betting, indifference) are well articulated. However, the response is clearly incomplete - it cuts off mid-sentence in part (3), failing to address the 1,000,000 wakings variant at all, which was a significant part of the question. The halfer "
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.6,
      "brief_justification": "The response is well-structured, clearly written, and covers both positions rigorously with formalizations. The halfer Bayesian argument is correctly presented, and the thirder argument using the principle of indifference over centered worlds is well-articulated. The extension to N=1,000,000 is handled properly. Minor issues: the halfer formalization somewhat oversimplifies by treating W as a single event rather than grappling with the deeper issue of how to define the sample space for self-loca"
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.65,
      "brief_justification": "The response starts appropriately and correctly notes the halfer/thirder split, but it is truncated mid-sentence and fails to actually present the rigorous arguments, resolve the main question, or address the 1 vs 1,000,000 waking variant. Limited practical value due to incompleteness."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 4.2,
      "brief_justification": "Covers both halfer and thirder intuitions and gives the 1 vs 1,000,000 extension, but the response is flawed and incomplete: it incorrectly treats the three awakening states as simply 'equally likely' without rigorous justification, contains dubious claims about long-run frequency under the halfer view, breaks off mid-sentence, and is partially replaced by unrelated duplicated Chinese text, making the answer confusing and not fully responsive."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 7.65,
      "brief_justification": "Accurately presents the main halfer and thirder arguments with good structure and insight, including the 1 vs. 1,000,000 variant. However, it overstates some points (e.g. SIA framing for thirders, 'internally consistent' without caveats), the thirder derivation is not fully rigorous, and the response is cut off before fully finishing the extreme-variant halfer discussion."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 7.8,
      "brief_justification": "Clear, well-structured, and covers both halfer and thirder arguments plus the extreme case. However, it contains some inaccuracies and imprecisions: the halfer centered-world weights given as 1/2, 1/4, 1/4 are not standard prior probabilities in Lewis-style treatment, some betting claims are oversimplified or misleading, and the suggestion that halfers may update slightly in the 1 vs 1,000,000 case is not generally representative. Still, it is a strong overview with good philosophical depth."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.2,
      "brief_justification": "Clear, well-structured summary of both halfer and thirder views with the million-wakings variant handled correctly in broad terms. Minor weakness: the Bayesian treatment is somewhat loose and the thirder argument relies on counting waking moments without fully formalizing the sampling assumption, but overall it is accurate and useful."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 6.6,
      "brief_justification": "Clear and reasonably rigorous, but it mishandles the thirder derivation by assigning 1/2, 1/4, 1/4 yet concluding 1/3, and the response is truncated before fully arguing both sides or fully addressing the 1,000,000-waking variant."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 4.45,
      "brief_justification": "Clear structure and a decent presentation of standard halfer/thirder arguments, but it is incomplete because it cuts off before answering part (3), and some rigor is lacking or misleading—especially the thirder section's treatment of awakenings as 'equally likely' without proper normalization and the oversimplified halfer conditionalization on waking."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8,
      "brief_justification": "Accurately presents both halfer and thirder views and extends them to the 1 vs 1,000,000 variant. Clear and well-structured. Slightly oversimplifies by treating the thirder's three awakenings as simply 'equally likely' without fully justifying the measure, and the halfer Bayes formalization is somewhat loose about the event conditioned on."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 7.9,
      "brief_justification": "The response accurately describes the debate, provides solid arguments for the halfer position (though incomplete in the given text), and correctly notes the majority thirder view. It is clear and insightful but the provided excerpt cuts off before fully arguing both sides or addressing the 1,000,000 variant, limiting completeness."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 6.8,
      "brief_justification": "The response accurately presents both halfer and thirder positions with structured arguments and addresses the 1M variant, but mixes languages (English/Chinese), has some repetition and incomplete sections, and lacks rigorous mathematical formalization like conditional probabilities or betting arguments for deeper insight."
    },
    {
      "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 accurately presents both halfer and thirder positions with rigorous arguments, formal math, tables, and rebuttals. It covers the core debate, long-run frequencies, Bayesian updates, and the extreme variant thoroughly and clearly, though the halfer section on the extreme case is slightly truncated."
    },
    {
      "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 accurately presents both halfer and thirder positions with rigorous arguments, formalizations, betting justifications, and frequency analogies. It fully addresses all three parts of the query, including the generalized 1M wakings case, while maintaining exceptional clarity and balanced depth on the philosophical disagreement."
    },
    {
      "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": "The response accurately and rigorously presents both halfer and thirder positions with formal arguments, betting considerations, and frequency examples. It clearly addresses all three parts of the query, including the large-number variant, though the halfer section for the variant is slightly truncated. Depth is strong with multiple intuition pumps and philosophical references."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.65,
      "brief_justification": "The response accurately presents both halfer and thirder positions with rigorous arguments, correctly handles the extreme case, and provides clear structure and insightful analysis."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.65,
      "brief_justification": "The response accurately and rigorously presents both halfer and thirder positions with formal probability models, sample spaces, and justifications. It covers the generalized case with 1M awakenings. Minor incompleteness as the response is cut off mid-sentence in the thirder Dutch book argument, but overall excellent."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.3,
      "brief_justification": "The response accurately presents both halfer and thirder positions with rigorous arguments, including Bayesian updates, frequentist long-run frequencies, and betting arguments. It is clear and well-structured. Completeness is slightly reduced as the response is cut off mid-sentence in part (3). Depth is good but could explore more criticisms of each view. Usefulness is solid for understanding the debate."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.65,
      "brief_justification": "The response accurately presents both halfer and thirder positions with formal Bayes arguments and generalizations. It is comprehensive, clearly structured, and insightful on the philosophical tension, though it slightly favors thirders in conclusion without noting that both remain actively debated."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.5,
      "brief_justification": "The response starts well with a clear introduction and begins a rigorous argument for the halfer position, but it is severely truncated and incomplete. It cuts off mid-sentence while presenting the halfer argument, never completes the halfer case, never presents the thirder argument, and never addresses the 1,000,000 wakings variant. While what is present is accurate and clearly written, the response fails to deliver on the core requirements of the question."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 5.65,
      "brief_justification": "The response starts well with a clear explanation of both halfer and thirder positions, including the 1,000,001 variant. However, the response catastrophically breaks down midway, switching to Chinese text and repeating content in a disorganized manner, ending abruptly mid-sentence. This severely damages clarity and usefulness. The mathematical content that is present is correct, and both positions are argued reasonably before the breakdown. The frequency argument for the halfer position is slig"
    },
    {
      "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,
      "brief_justification": "Excellent treatment of the Sleeping Beauty problem. Both positions are argued rigorously with multiple supporting frameworks (centered worlds, frequency arguments, Bayesian formalism, SSA vs SIA). The table summarizing the deep disagreement is particularly insightful. The extreme variant is handled well for both positions. Minor issue: the halfer analysis for the 1,000,000 variant appears cut off at the end, but the core argument is clear. The response correctly identifies this as a genuinely un"
    },
    {
      "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": "Excellent coverage of both halfer and thirder positions with rigorous arguments, frequency-style justifications, betting arguments, and formal Bayesian framing. Clearly explains the core philosophical disagreement about self-locating belief. Addresses the 1,000,000 waking variant. Minor issue: the response appears to be cut off at the end, but the content provided is thorough and well-structured throughout."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.2,
      "brief_justification": "Excellent response covering both positions rigorously with multiple argument types (philosophical, frequentist, betting). Correctly presents Lewis's halfer position with proper probability weighting, Elga's thirder argument, and handles the extreme case well. The summary table is a nice touch. Minor quibble: the 'no new information' argument is slightly simplified, but overall this is a thorough, accurate, and well-structured treatment of a genuinely difficult problem."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.55,
      "brief_justification": "The response accurately presents both halfer and thirder positions with rigorous arguments, correctly handles the modified experiment, and provides intuitive explanations. The Bayesian argument for the thirder position is slightly muddled (the likelihood ratio framing is informal), and the halfer position could be more rigorously defended using Lewis's framework or the notion of centered worlds. The computer simulation analogy is excellent for intuition. Overall a strong, well-structured respons"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 8.65,
      "brief_justification": "The response is mathematically rigorous and well-structured, presenting both halfer and thirder positions with formal probability spaces and Bayesian calculations. It correctly identifies the core disagreement as a reference class/sample space modeling choice. The 1,000,000 waking variant is addressed correctly. Minor deductions: the response appears cut off at the end (Dutch-book argument incomplete), and the completeness of the third question's treatment could be deeper (e.g., exploring what t"
    },
    {
      "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,
      "brief_justification": "The response provides rigorous arguments for both halfer and thirder positions with good mathematical detail, frequentist and Bayesian framings, and a betting argument. The halfer's Bayesian calculation is presented but correctly noted as contested. The clarity and structure are excellent. However, the response is cut off before completing part (3), which is a significant omission that hurts completeness. The betting argument section has a minor inconsistency in framing but is largely correct. O"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.55,
      "brief_justification": "The response accurately presents both halfer and thirder positions with rigorous formalizations, covers the generalized N-waking variant well, and is clearly structured. The halfer formalization correctly shows P(W|C)=P(W|T)=1, though it could note this is the crux of the halfer's error (conflating event probability with indexical probability). The thirder argument is well-explained using the principle of indifference over awakenings. The response acknowledges the ongoing philosophical debate wi"
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.2,
      "brief_justification": "The response is incomplete (cut off mid-sentence) and contains a misleading characterization of the Halfer argument. It claims the 'No New Information' argument is the most famous defense of the Halfer position, but this specific argument is widely considered flawed (it confuses the probability of the coin toss with the probability of the current waking state). The response fails to address the question about the 1,000,000 wakings variant."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 8.45,
      "brief_justification": "Accurate and thorough explanation of both positions and the variant; clear structure. Slight deduction in completeness for not fully addressing the 'what if' scenario's implications, and in depth for lacking advanced nuance (e.g., SIA/SA)."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.85,
      "brief_justification": "Response is factually accurate, covers both positions thoroughly, well-structured, provides deep philosophical analysis, and is practically useful for understanding the debate."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9,
      "brief_justification": "Accurate, thorough, and deep analysis of both positions with strong clarity and practical examples."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9,
      "brief_justification": "Accurate, thorough, and well-structured explanation of both positions and the extreme case, with rigorous arguments and clear analysis."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.4,
      "brief_justification": "Comprehensive, accurate, and well-structured analysis of both positions with rigorous arguments and clear extension to the variant."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.4,
      "brief_justification": "Comprehensive, accurate, and well-structured explanation of both positions with rigorous arguments and extension to the modified experiment."
    },
    {
      "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": "Accurate and thorough explanation of both positions with strong arguments (Bayesian, frequentist, betting). The response is well-structured and highly useful, though it cuts off slightly early."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9,
      "brief_justification": "Accurate and thorough explanation of both positions, rigorous formalization, correct handling of the generalization, and clear discussion of the philosophical conflict."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 3.75,
      "brief_justification": "The response provides some accurate background and partial arguments but is incomplete, cutting off before finishing the halfer argument and omitting the thirder argument and the extreme variant, limiting its overall usefulness."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 5.85,
      "brief_justification": "The response correctly outlines the halfer and thirder positions and gives the right probabilities for the modified experiment, but it stops short of a definitive answer to (1), lacks rigorous formal arguments, mixes languages and formatting, and offers only a superficial depth of analysis."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.65,
      "brief_justification": "Accurately presents both halfer and thirder arguments, covers all parts of the question, and is well organized with clear explanations and deeper philosophical context."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 8.65,
      "brief_justification": "Accurately presents both halfer and thirder arguments, covers the extended case, and is well organized with solid reasoning, though some minor exposition could be clearer."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.25,
      "brief_justification": "Accurately presents both halfer and thirder positions, covers all three sub‑questions, and is well‑structured; depth is solid but not exhaustive, and the answer is practically useful for understanding the debate."
    },
    {
      "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 presents both halfer and thirder arguments, covers the variant, and is well-structured with solid reasoning, though could include more formal detail."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 7.4,
      "brief_justification": "Accurately presents halfer and thirder positions and their core arguments, covers the modified scenario, and is clearly organized, though it lacks deeper philosophical nuance and some formal precision."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 6.65,
      "brief_justification": "The answer accurately presents the main arguments for both halfer and thirder positions and is clearly organized, but it stops short of completing the third part of the question and omits some deeper technical nuances, reducing overall completeness and practical usefulness."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.25,
      "brief_justification": "Accurately presents both halfer and thirder arguments, covers all parts of the prompt, is clearly organized, offers solid but not exhaustive depth, and provides practical insight for understanding the problem."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 8.45,
      "brief_justification": "The response accurately explains the Sleeping Beauty problem and the core arguments for both positions. It clearly states the prevailing view and addresses the extreme variant effectively. The explanation of the 'no new information' argument for the Halfer position is particularly well-articulated."
    },
    {
      "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 provides a very thorough and clear explanation of the Sleeping Beauty problem, covering both the halfer and thirder positions with rigorous arguments. It also effectively addresses the modified experiment."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.1,
      "brief_justification": "The response provides a very thorough and well-structured explanation of the Sleeping Beauty problem, rigorously arguing for both the thirder and halfer positions. It also effectively addresses the extreme variant. The use of tables and formal arguments enhances clarity and depth."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 8.85,
      "brief_justification": "The response provides a comprehensive and clear explanation of both the halfer and thirder positions, including their arguments and the underlying philosophical differences. It also addresses the modified experiment effectively."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.45,
      "brief_justification": "The response accurately and thoroughly explains both the halfer and thirder positions, providing strong arguments for each. The explanation of the extreme case is particularly insightful, highlighting the intuitive shift towards the thirder position in such scenarios. The structure is clear and easy to follow."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 8.85,
      "brief_justification": "The response provides a comprehensive and well-structured explanation of the Sleeping Beauty problem, covering both halfer and thirder positions with rigorous arguments. It effectively addresses the variant with 1,000,000 wakings and highlights the philosophical implications."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.45,
      "brief_justification": "The response provides a comprehensive and accurate explanation of the Sleeping Beauty Problem, rigorously arguing for both the Halfer and Thirder positions. It effectively addresses the modified experiment and highlights the strengths of the Thirder position in that context. The structure is clear 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": 8.85,
      "brief_justification": "The response provides a very clear and detailed explanation of both the Halfer and Thirder positions, including their underlying assumptions and mathematical derivations. It also addresses the altered experiment effectively."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9.45,
      "brief_justification": "The response provides a comprehensive and accurate explanation of the Sleeping Beauty problem, rigorously arguing for both the halfer and thirder positions. It effectively addresses the modified scenario and highlights the core philosophical conflict, making it very useful for understanding this complex topic."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 5.1,
      "brief_justification": "Response provides accurate framing but is severely incomplete - it cuts off mid-sentence and never completes the halfer argument, never presents the thirdder position, and never addresses the 1,000,000 wakings variant. The partial content is accurate but fails to answer the full question."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 7.6,
      "brief_justification": "Accurate explanation of both halfer and thirdr positions with rigorous mathematical reasoning. Well-structured with clear headings and logical flow. Covers the modified experiment correctly. Slight incompleteness in exploring deeper philosophical nuances and counterarguments. Minor formatting issue with Chinese characters at the end."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 8.85,
      "brief_justification": "Comprehensive and rigorous treatment covering both positions with multiple arguments (symmetry, frequency, Bayesian, SIA/SSA). Clear structure with tables and mathematical notation. Correctly identifies the philosophical underpinnings and addresses the extreme variant. Slight deduction in usefulness as the response could better help the reader reach a decision, though it excellently presents both sides."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.45,
      "brief_justification": "Comprehensive and accurate coverage of both positions with rigorous arguments. Well-structured with clear explanations of key concepts like uncentered vs centered propositions. Covers all three parts of the question including the modified experiment. Slight deduction for incomplete final sentence but otherwise excellent."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.45,
      "brief_justification": "Accurately presents both positions with rigorous arguments. Well-structured with clear explanations of Lewis-style halfer, Elga-style thirder, betting arguments, and frequentist reasoning. Covers the extreme case thoroughly. Slight deduction for not mentioning hybrid positions or the double halfer view, but otherwise comprehensive 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": "Comprehensive and rigorous treatment of the Sleeping Beauty problem. Correctly presents both halfer and thirder positions with proper mathematical formalism, addresses the philosophical nuances including Dutch book and self-locating uncertainty arguments, and thoroughly analyzes the 1,000,000 wakings variant. Well-structured with clear explanations."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.45,
      "brief_justification": "Accurately presents both positions with rigorous arguments including Bayesian analysis. Clear structure with good use of mathematical notation. Covers the philosophical reasoning well and provides compelling analysis of the modified experiment. Slight room for more depth on counterarguments."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.25,
      "brief_justification": "Factually accurate presentation of both halfer and thirder positions with rigorous mathematical derivations. Covers all three parts of the question. Well-structured with clear explanations of the key concepts (reference class, self-locating belief). The response is slightly incomplete at the end (Dutch-book argument unfinished), but the core arguments are thorough and insightful."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 7.7,
      "brief_justification": "Accurate and well-structured presentation of both positions with rigorous arguments including Bayesian, frequentist, and betting perspectives. However, the response is incomplete as it cuts off mid-sentence when addressing the 1 vs 1,000,000 wakings modification, failing to provide the final analysis."
    }
  ],
  "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)"
  }
}