{ "evaluation_id": "EVAL-20260402-153453", "question_id": "REASON-003", "question_text": "Estimate how many piano tuners there are in Chicago. Show your reasoning step by step, including:\n1. All assumptions you make\n2. How you derived each number\n3. Sensitivity analysis (what if your assumptions are wrong?)\n4. Final estimate with confidence interval", "category": "reasoning", "timestamp": "2026-01-28T00:00:00.000Z", "display_date": "Jan 28, 2026", "winner": { "name": "GPT-5.4", "provider": "openrouter", "score": 9.07 }, "avg_score": 8.102222, "matrix_size": 80, "models_used": [ { "id": "deepseek_v4", "name": "DeepSeek V4", "provider": "openrouter" }, { "id": "claude_opus_46", "name": "Claude Opus 4.6", "provider": "openrouter" }, { "id": "gpt_5_4", "name": "GPT-5.4", "provider": "openrouter" }, { "id": "grok_420", "name": "Grok 4.20", "provider": "openrouter" }, { "id": "claude_sonnet_46", "name": "Claude Sonnet 4.6", "provider": "openrouter" }, { "id": "mimo_v2_flash", "name": "MiMo-V2-Flash", "provider": "Xiaomi" }, { "id": "gpt_oss_120b", "name": "GPT-OSS-120B", "provider": "OpenAI" }, { "id": "gemini_25_flash", "name": "Gemini 2.5 Flash", "provider": "openrouter" }, { "id": "gemini_31_pro", "name": "Gemini 3.1 Pro", "provider": "openrouter" } ], "rankings": { "gpt_5_4": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 9.07, "score_count": 9, "min_score": 8.4, "max_score": 9.8, "rank": 1 }, "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 9.05, "score_count": 9, "min_score": 8.15, "max_score": 10, "rank": 2 }, "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 8.9, "score_count": 9, "min_score": 8.15, "max_score": 10, "rank": 3 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 8.74, "score_count": 9, "min_score": 7.9, "max_score": 10, "rank": 4 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 8.74, "score_count": 9, "min_score": 8.35, "max_score": 9.4, "rank": 5 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 8.7, "score_count": 9, "min_score": 8.25, "max_score": 9.4, "rank": 6 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 8.63, "score_count": 9, "min_score": 8.25, "max_score": 9, "rank": 7 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 8.42, "score_count": 9, "min_score": 7.75, "max_score": 9.2, "rank": 8 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 2.68, "score_count": 8, "min_score": 0.65, "max_score": 6, "rank": 9 } }, "judgments": [ { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "brief_justification": "The response perfectly follows the requested structure for a Fermi estimate, with clear math and sensitivity analysis. However, it misses institutional pianos (schools, churches, venues), slightly limiting depth and completeness." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 10, "brief_justification": "The response perfectly executes the Fermi estimation, explicitly addressing all prompt requirements with excellent structure, logical assumptions, and a brilliant reality check." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.8, "brief_justification": "The response perfectly executes a Fermi estimate, clearly stating assumptions, performing accurate calculations, providing a robust sensitivity analysis, and delivering a well-reasoned confidence interval." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.8, "brief_justification": "The response perfectly follows the requested structure with clear logic and a great sensitivity analysis. However, it misses institutional pianos (schools, churches, concert venues) which are tuned much more frequently than household pianos, slightly limiting completeness and depth." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.25, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 10, "brief_justification": "The response perfectly addresses all prompt requirements with flawless math, clear step-by-step logic, thorough sensitivity analysis, and an excellent supply-side sanity check." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.15, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.8, "brief_justification": "The response perfectly follows the requested structure with excellent step-by-step math and sensitivity analysis. However, it misses institutional pianos (schools, churches, venues), which is a notable omission for this classic Fermi problem." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 6, "brief_justification": "The response starts well but is incomplete, missing key steps like assumptions, derivation, sensitivity analysis, and final estimate." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.05, "brief_justification": "Highly accurate and thorough with excellent reasoning and sensitivity analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.65, "brief_justification": "Detailed, logical, and well-structured with sensitivity analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.65, "brief_justification": "Accurate, thorough, and well-structured with good sensitivity analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.65, "brief_justification": "Thorough and logical with clear assumptions and sensitivity analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.2, "brief_justification": "Highly accurate, thorough, and well-structured with insightful analysis and practical value." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.65, "brief_justification": "Thorough and well-structured with detailed assumptions and sensitivity analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.65, "brief_justification": "Detailed and logical with thorough sensitivity analysis." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 1.65, "brief_justification": "The response is severely incomplete - it cuts off after just the introduction without providing any actual estimation, assumptions, calculations, sensitivity analysis, or final estimate. While the opening framing is reasonable, it fails to deliver on any of the four required components of the question." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.75, "brief_justification": "The reasoning is well-structured and clearly presented. The piano ownership assumption of 1% is on the low side—many Fermi estimation references use 5-10% of households, which would yield a higher estimate closer to the commonly cited ~100-200 tuners. The actual number of piano tuners in Chicago is estimated around 100-200, so the point estimate of 14 is likely an undercount by roughly an order of magnitude, primarily due to the low ownership rate assumption. However, the methodology is sound, t" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.95, "brief_justification": "The response is a textbook-quality Fermi estimation. The assumptions are reasonable and clearly stated, the step-by-step derivation is logical, and the sensitivity analysis with low/mid/high scenarios is well-executed. The final estimate of ~45 tuners aligns well with commonly cited answers (~100-200 for the broader metro area, ~50-100 for the city proper). Minor quibbles: the institutional piano estimate at 20% of household pianos may be slightly low, and the response could have briefly mention" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.95, "brief_justification": "Excellent Fermi estimation with clear step-by-step reasoning, well-stated assumptions, and a thorough sensitivity analysis. The final estimate of ~30 with a 90% CI of 20-50 is reasonable and aligns with commonly cited figures. Minor quibbles: the tuning frequency assumption of once every 2 years is arguably too low (many sources suggest once per year for maintained pianos), and the jump from 12-13 FTE to 25-40 actual tuners could be better justified. The sensitivity table is well-constructed and" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.95, "brief_justification": "Excellent Fermi estimation with clear step-by-step reasoning, well-structured assumptions, good sensitivity analysis, and honest sanity checking. The methodology is sound and transparent. Minor issues: the city-proper vs metro area scope could have been handled more cleanly from the start rather than reconciled at the end, and the piano ownership rate of 5% is reasonable but could benefit from more justification. The sensitivity analysis is genuinely useful and the confidence intervals are well-" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.55, "brief_justification": "The response is exceptionally well-structured and thorough, with clear step-by-step reasoning, sensitivity analysis, and a sanity check. However, the piano ownership assumption of 1 per 12 people is likely too high — industry estimates suggest roughly 10 million pianos in the US for 330 million people (~1 per 33), and urban areas like Chicago likely have lower ownership rates. This inflates the estimate significantly; most credible estimates for this classic Fermi problem land around 100-300 tun" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.15, "brief_justification": "The response is exceptionally well-structured, covering all four requested elements (assumptions, derivations, sensitivity analysis, confidence interval) with clear tables and step-by-step reasoning. The final estimate of ~270 tuners is reasonable and consistent with commonly cited answers to this classic Fermi problem (typically 200-300). Minor quibbles: the Chicago MSA actually has more like 14 counties and the number of schools/churches could be debated, but these don't materially affect the " }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.95, "brief_justification": "Excellent Fermi estimation with clearly stated assumptions, step-by-step derivation, and thorough sensitivity analysis. The piano ownership rate of 2% is slightly conservative (some estimates suggest 5-10% of US households have pianos), which could undercount tuners. The final estimate of ~25 is reasonable but possibly on the low side compared to commonly cited estimates of 100-200 for the Chicago metro area. However, the response correctly scopes to the city proper (2.7M) rather than the metro " }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 0.65, "brief_justification": "The response is truncated and does not provide the actual estimation, assumptions, derivations, sensitivity analysis, or confidence interval requested. It only gives a brief introductory framing." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 7.75, "brief_justification": "Well-structured Fermi estimate with clear assumptions, calculations, and sensitivity analysis. Main weaknesses are somewhat simplistic assumptions, a likely low household-size estimate for Chicago, and a confidence interval that is only loosely justified." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.15, "brief_justification": "Well-structured Fermi estimate with explicit assumptions, derivations, sensitivity analysis, and confidence intervals. Main weakness is likely overestimation from aggressive piano ownership and tuning-frequency assumptions, plus some inconsistency between intermediate numbers and final city estimate." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.55, "brief_justification": "Well-structured Fermi estimate with explicit assumptions, derivations, sensitivity analysis, and a confidence interval. Mostly reasonable and internally consistent, though the jump from 12-13 FTE tuners to 25-40 total tuners is only loosely justified and the external directory claim is unsupported." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.55, "brief_justification": "Well-structured Fermi estimate with explicit assumptions, calculations, sensitivity analysis, and confidence ranges. Mostly reasonable and internally consistent, though a few assumptions and external references are loosely supported and the final nudge from 34 to 40 is somewhat ad hoc." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 7.9, "brief_justification": "Well-structured and thorough with explicit assumptions, derivations, sensitivity analysis, and a confidence interval. However, the core estimate likely overstates piano ownership and thus tuner count for Chicago proper; the supply-side sanity check also relies on shaky membership assumptions. Strong Fermi reasoning overall, but factual plausibility is only moderate." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.15, "brief_justification": "Well-structured Fermi estimate with explicit assumptions, arithmetic, sensitivity analysis, and a final range. However, several inputs are questionable or inconsistent: Chicago MSA is larger than the listed 4 counties, school count appears low for the metro, the weighted average tuning rate is internally inconsistent with the stated institutional share, and the combined-extremes sensitivity numbers do not match the stated assumptions. Still, the overall methodology is solid and practically usefu" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.4, "brief_justification": "Well-structured Fermi estimate with explicit assumptions, calculations, sensitivity analysis, and a reasonable confidence interval. Minor weaknesses include somewhat arbitrary assumptions, limited consideration of non-household pianos, and a confidence range not fully derived from a systematic combination of assumptions." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 4.05, "brief_justification": "The response correctly identifies the problem as a Fermi estimation and outlines a high-level approach using supply and demand, but it is incomplete as it cuts off abruptly without providing any assumptions, derived numbers, calculations, sensitivity analysis, or final estimate as explicitly requested in the query." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.4, "brief_justification": "The response follows the requested structure precisely with clear assumptions, derivations, sensitivity analysis, and a final estimate with confidence interval. Numbers are logically derived from reasonable assumptions, though the population figure is slightly outdated (actual ~2.7M is for city proper but metro area is larger). Overall a strong Fermi estimation." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.65, "brief_justification": "The response provides a thorough, well-structured Fermi estimation with clear assumptions, derivations, sensitivity analysis, and confidence intervals. All required elements are covered logically and transparently. Minor deduction for some arbitrary rounding choices and slight inconsistency between final answer and table, but overall highly accurate and insightful." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.65, "brief_justification": "The response provides a structured, logical Fermi estimation with clear assumptions, derivations, sensitivity analysis, and confidence intervals. All calculations are accurate and transparent; minor factual quibble on Chicago population (closer to 2.7M is fine for estimate) but overall highly correct and complete." }, { "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 provides a structured, logical Fermi estimation with clear assumptions, derivations, sensitivity analysis, and a sanity check against known references. Minor inconsistencies (e.g., population/household figures, final point estimate nudge) slightly affect precision but not overall validity." }, { "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 provides a thorough, well-structured Fermi estimation with clear assumptions, calculations, sensitivity analysis, and a sanity check. Numbers are logically derived and internally consistent; minor factual quibbles (e.g., Chicago population closer to 2.7M is acceptable for estimation) do not detract significantly. High clarity and completeness." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.2, "brief_justification": "The response is factually sound with logical steps, fully addresses all required elements including detailed assumptions, derivations, sensitivity analysis, and a confidence interval. It is clearly structured with tables and calculations, provides insightful breakdowns beyond basics, and offers practical value as a model Fermi estimation." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.65, "brief_justification": "The response follows the requested structure precisely with clear assumptions, step-by-step derivation, sensitivity analysis on key variables, and a final estimate with interval. Numbers are logically derived from stated assumptions; minor factual quibbles on Chicago metro population are negligible for a Fermi estimate." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 1.45, "brief_justification": "The response is severely incomplete - it cuts off mid-sentence after only providing an introduction. No actual reasoning steps, assumptions, derivations, sensitivity analysis, or final estimate are provided. The response fails to address any of the four required components of the question." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.15, "brief_justification": "Well-structured Fermi estimation with clear step-by-step reasoning. The methodology is sound, but the final estimate of ~14 tuners seems low compared to reality (typically estimated at 50-100 for Chicago). The piano ownership rate of 1% may be slightly low, and the model doesn't account for commercial/institutional pianos (schools, churches, concert halls) which significantly boost demand. The sensitivity analysis is good but the confidence interval feels narrow given the uncertainties. Overall " }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.2, "brief_justification": "Excellent Fermi estimation with well-justified assumptions, clear step-by-step reasoning, thorough sensitivity analysis with a useful table, and a reality check against external data points. The distinction between city proper and metro area adds nuance. Minor quibble: tuning time of 1.5-2 hours per piano is on the high end (typical is closer to 1-1.5 hours), which slightly underestimates tuner productivity, but the sensitivity analysis compensates. Overall a model response for this type of esti" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.2, "brief_justification": "Excellent Fermi estimation with clear step-by-step reasoning, well-justified assumptions, thorough sensitivity analysis with low/mid/high scenarios, and a well-presented confidence interval. The math is consistent and correct throughout. The institutional piano multiplier (20%) could be argued as slightly low for a major city like Chicago with many schools, churches, and venues, but the sensitivity analysis covers this. The final estimate of ~45 tuners (range 20-115) is reasonable and aligns wit" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.55, "brief_justification": "Excellent Fermi estimation with clear step-by-step reasoning, explicit assumptions, sensitivity analysis table, and a well-calibrated confidence interval. The adjustment for part-time tuners shows good practical thinking. Minor issue: the jump from ~12 FTE to 25-40 adjusted estimate could be better justified quantitatively. The final estimate aligns well with real-world figures, demonstrating sound reasoning throughout." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.95, "brief_justification": "Excellent Fermi estimation with well-structured reasoning, clear assumptions, thorough sensitivity analysis, and a sanity check from the supply side. The piano ownership rate (1 in 12) is reasonable though slightly high for urban Chicago apartments. The PTG sanity check is clever but the 10% certification rate assumption is speculative and could skew results. Overall, the methodology is sound, the confidence interval is appropriate, and the final estimate of ~310 is defensible and well within th" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.2, "brief_justification": "Excellent Fermi estimation with well-structured reasoning, clearly stated assumptions, detailed arithmetic, and thorough sensitivity analysis. All major variables are addressed including residential vs. institutional pianos, tuning frequency, and tuner capacity. The confidence interval is well-derived from the sensitivity analysis. Minor quibble: the weighted average tuning calculation slightly underweights institutional pianos given their higher tuning frequency, but this has minimal impact on " }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.55, "brief_justification": "Excellent Fermi estimation with clear step-by-step reasoning, well-justified assumptions, thorough sensitivity analysis, and a reasonable confidence interval. The piano ownership rate (2%) may be slightly low given institutional pianos (schools, churches, bars) weren't explicitly accounted for, which could push the real number higher. The methodology is sound and the presentation is exemplary for this type of estimation problem." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 2.2, "brief_justification": "The response is incomplete, stopping abruptly after the introduction. It fails to provide any assumptions, derivations, sensitivity analysis, or final estimate as requested." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9, "brief_justification": "Excellent Fermi estimate with clear assumptions, step-by-step derivation, and thorough sensitivity analysis. Minor deduction for depth as the sensitivity analysis could be more quantitative (e.g., full range calculation), but it is very strong overall." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.4, "brief_justification": "Excellent Fermi estimation. The reasoning is transparent, assumptions are clearly stated and justified, and the sensitivity analysis is thorough. The final estimate is realistic and cross-checked against external data." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.4, "brief_justification": "Response is factually sound, thoroughly covers all requested elements, exceptionally clear and well-structured, provides insightful sensitivity analysis, and offers practical value with actionable estimates." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.4, "brief_justification": "Accurate Fermi estimation with clear assumptions, thorough sensitivity analysis, and practical confidence interval." }, { "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": "Thorough, well-structured Fermi estimate with clear assumptions, sensitivity analysis, and sanity check; minor uncertainty in ownership rate but overall robust." }, { "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 Fermi estimate with clear assumptions, step-by-step derivation, sensitivity analysis, and confidence interval. Highly accurate and actionable." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 9, "brief_justification": "The response is logically sound, clearly structured, and demonstrates strong analytical depth by breaking down assumptions and performing a sensitivity analysis. The calculations are correct and the final estimate with a confidence interval is well-justified." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.3, "brief_justification": "The response identifies the problem as a Fermi estimate and outlines a general approach, but it stops abruptly without presenting assumptions, calculations, sensitivity analysis, or a final estimate, resulting in low completeness, depth, and usefulness despite moderate clarity." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.2, "brief_justification": "Logical calculations are sound and all requested elements are addressed; clear presentation and useful estimate, though depth could be greater with more market nuances." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.4, "brief_justification": "Accurate, thorough Fermi reasoning with assumptions, derivations, sensitivity analysis, and clear presentation; minor uncertainties in some numeric assumptions but overall solid and practical." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.4, "brief_justification": "Accurate Fermi reasoning, covers all required parts, clear structure, good sensitivity analysis, practical estimate." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.4, "brief_justification": "The answer follows the requested format, lists assumptions, derives numbers, includes sensitivity analysis and a confidence interval, and does so clearly and logically. The figures are plausible for a Fermi estimate, though exact values could vary." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.4, "brief_justification": "The response follows the requested format, lists assumptions, derives numbers, includes sensitivity analysis and confidence intervals, and presents calculations clearly. The estimates are plausible though some numbers are loosely justified, leading to minor uncertainty in factual precision." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.15, "brief_justification": "Logical, well-structured Fermi estimate with assumptions, calculations, sensitivity analysis, and confidence interval; however some assumptions (e.g., piano ownership rate) are likely unrealistic, affecting factual accuracy." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.25, "brief_justification": "Accurate calculations and clear step-by-step reasoning with assumptions and sensitivity analysis; covers most required aspects but could include more on non‑household pianos and industry nuances." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9.2, "brief_justification": "The response provides a clear, step-by-step Fermi estimate with well-explained assumptions, derivations, and sensitivity analysis. The final estimate and confidence interval are well-supported." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.85, "brief_justification": "The response provides a very thorough and well-structured Fermi estimation. It clearly outlines all assumptions, derivations, and performs a good sensitivity analysis. The final estimate with confidence intervals and a reality check adds significant value." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.2, "brief_justification": "The response is exceptionally clear, well-structured, and provides a thorough Fermi estimate. The step-by-step derivation, sensitivity analysis, and confidence intervals are all well-executed and demonstrate a strong understanding of the task." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9, "brief_justification": "The response provides a very thorough and well-structured Fermi estimation. It clearly outlines all assumptions, derivations, and includes a good sensitivity analysis and confidence interval. The final estimate aligns well with external information." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.05, "brief_justification": "The response provides a clear, step-by-step estimation with well-explained assumptions and a good sensitivity analysis. The final answer is well-justified and includes a confidence interval." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.85, "brief_justification": "The response provides a very thorough and well-structured estimation using the Fermi problem approach. It clearly outlines assumptions, derivations, and a robust sensitivity analysis. The sanity check adds significant value." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 10, "brief_justification": "This is an outstanding response. It follows all instructions perfectly, providing a detailed, step-by-step Fermi estimate with clear assumptions, derivations, and a thorough sensitivity analysis. The final estimate includes a confidence interval, and the presentation is exceptionally clear and well-structured." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 2.1, "brief_justification": "Response is incomplete - cuts off mid-sentence and provides no actual calculations, assumptions, sensitivity analysis, or final estimate. Only identifies the problem type and general approach without delivering the requested estimation." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.55, "brief_justification": "Well-structured Fermi estimate with clear step-by-step reasoning, reasonable assumptions with sensitivity analysis, and a sensible confidence interval. Covers all requested components thoroughly." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.4, "brief_justification": "Thorough Fermi estimation with clear assumptions, step-by-step derivations, comprehensive sensitivity analysis, and validation against real data. Well-structured with good depth on methodology." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.4, "brief_justification": "Methodologically sound Fermi estimate with correct calculations throughout. Exceeds requirements by including detailed sensitivity analysis with low/mid/high scenarios, well-organized derivations, and clear confidence intervals. The institutional piano adjustment, productivity assumptions, and scenario variations demonstrate thorough analytical depth. All four requested components fully addressed with a useful summary table and even an offer for further analysis." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.35, "brief_justification": "Excellent coverage of all required elements with clear reasoning and sensitivity analysis. Well-structured and transparent. Slight扣分 for unverified claim about 'true number' and could go deeper on justification for some assumptions." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.4, "brief_justification": "Thorough Fermi estimate with clear assumptions, step-by-step derivation, sensitivity analysis table, and sanity checks against real-world data. Well-structured with appropriate confidence intervals." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.4, "brief_justification": "Well-structured Fermi estimation with clear assumptions, detailed derivations, comprehensive sensitivity analysis, and validation via supply-side check. Covers all required elements thoroughly." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.55, "brief_justification": "Thorough Fermi estimation with clear assumptions, well-documented derivations, and comprehensive sensitivity analysis. Excellent structure with tables and step-by-step reasoning. Covers residential and institutional pianos, tuning frequency, tuner capacity, and provides a reasonable confidence interval (210-380). The response is clear, accurate, and highly actionable." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.4, "brief_justification": "Well-structured Fermi estimate with clear assumptions, thorough step-by-step derivation, and comprehensive sensitivity analysis covering multiple variables. Final estimate of ~25 tuners with 15-40 range is reasonable and useful." } ], "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)" } }