{ "evaluation_id": "EVAL-20260402-160854", "question_id": "REASON-008", "question_text": "Three bidders (A, B, C) are in a first-price sealed-bid auction for an item. Their private valuations are:\n- A values it at $100\n- B values it at $80 \n- C values it at $60\n\nEach bidder knows only their own valuation but knows the valuations are uniformly distributed between $0 and $100 for all bidders.\n\n1. What is each bidder's optimal strategy?\n2. What is the expected revenue for the seller?\n3. How would this change in a second-price auction?\n4. If the bidders could collude, what would happen?", "category": "reasoning", "timestamp": "2026-03-04T00:00:00.000Z", "display_date": "Mar 04, 2026", "winner": { "name": "Claude Opus 4.6", "provider": "openrouter", "score": 9.36 }, "avg_score": 8.434444, "matrix_size": 80, "models_used": [ { "id": "gemini_25_flash", "name": "Gemini 2.5 Flash", "provider": "openrouter" }, { "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_31_pro", "name": "Gemini 3.1 Pro", "provider": "openrouter" } ], "rankings": { "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 9.36, "score_count": 9, "min_score": 8.65, "max_score": 10, "rank": 1 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 8.99, "score_count": 9, "min_score": 8.45, "max_score": 9.55, "rank": 2 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 8.87, "score_count": 9, "min_score": 8, "max_score": 10, "rank": 3 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 8.82, "score_count": 9, "min_score": 7, "max_score": 10, "rank": 4 }, "gpt_5_4": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 8.73, "score_count": 9, "min_score": 8, "max_score": 9.65, "rank": 5 }, "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 8.35, "score_count": 9, "min_score": 6.8, "max_score": 9.45, "rank": 6 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 8.34, "score_count": 9, "min_score": 7.6, "max_score": 9, "rank": 7 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 7.67, "score_count": 8, "min_score": 5.95, "max_score": 9.45, "rank": 8 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 6.78, "score_count": 9, "min_score": 5.05, "max_score": 8.45, "rank": 9 } }, "judgments": [ { "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": "Accurate and thorough response with clear explanations and good depth, though slightly more practical examples could enhance usefulness." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 10, "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": 10, "brief_justification": "The response is exceptionally accurate, comprehensive, and well-structured. It correctly applies auction theory, provides clear mathematical derivations, and offers deep insights into collusion and mechanism design." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.3, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.55, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 6.75, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.05, "brief_justification": "The response is highly accurate, mathematically sound, and provides excellent depth by referencing order statistics and the Revenue Equivalence Theorem. However, the final sentence on collusion is cut off, slightly impacting completeness." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.5, "brief_justification": "" }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.45, "brief_justification": "Accurate and detailed response with clear explanations, though slightly incomplete on second-price auction details." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.45, "brief_justification": "Highly accurate and thorough analysis with clear explanations and deep insights." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.45, "brief_justification": "Accurate, thorough, and insightful with clear explanations." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.45, "brief_justification": "Accurate and detailed analysis with clear explanations, though slightly dense in presentation." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.45, "brief_justification": "Highly accurate and thorough analysis with clear explanations and practical insights." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.45, "brief_justification": "Highly accurate and thorough response with clear explanations and insightful analysis." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.45, "brief_justification": "Highly accurate and thorough with clear structure, though slightly dense in presentation." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5.65, "brief_justification": "The optimal bidding strategy formula (n-1)/n × valuation is correct for the symmetric equilibrium with uniform distributions. The individual bids are correctly computed. However, the expected revenue calculation has issues: the expected value of the maximum of 3 uniform[0,100] draws is 75, so expected revenue is (2/3)×75 = 50, which is correct. But the response is clearly cut off mid-sentence in part 3 (second-price auction) and entirely missing part 4 (collusion). This severely impacts complete" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "brief_justification": "The response correctly derives the symmetric Bayesian Nash equilibrium bidding strategy b(v) = (n-1)/n * v for the first-price auction, correctly computes expected revenue as 50, correctly applies the Revenue Equivalence Theorem to show the second-price auction yields the same expected revenue, and provides a reasonable analysis of collusion. The math is verified through multiple approaches. Minor issues: the problem states bidders know only their own valuation but then gives specific valuations" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.25, "brief_justification": "The response is highly accurate. The equilibrium bidding strategy b(v)=(n-1)/n * v is correct for the symmetric uniform IPV model. The revenue equivalence application is correct, with ex ante expected revenue of 50 for both auction formats. The second-price auction analysis is spot on. The collusion discussion is qualitatively correct though could be deeper (e.g., discussing ring mechanisms, enforcement problems, or comparing collusion vulnerability across formats more rigorously). One minor iss" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9, "brief_justification": "The response is highly accurate, covering the symmetric BNE bidding strategy correctly (b(v) = (n-1)/n * v), computing expected revenue correctly via order statistics and revenue equivalence, correctly describing the second-price auction dominant strategy, and providing a solid analysis of collusion. The only minor issue is in the summary table where 'A pays ~$53-67' is a bit ambiguous for the realized outcome (A would pay $66.67 in this specific draw), but the expected revenue calculation is co" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7.95, "brief_justification": "The response is well-structured and covers all four parts thoroughly. However, there are correctness issues: (1) The optimal bid formula b(v)=v*(n-1)/n is derived under the assumption that the bidder's own value is the highest, but the problem states specific realized valuations - the formula applies correctly only in the symmetric equilibrium where bidders don't know others' values. (2) The Revenue Equivalence Theorem comparison is somewhat confused - the ex-ante expected revenue of $50 is corr" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8, "brief_justification": "The response is well-structured and mostly correct. The derivation of the symmetric BNE bidding strategy b(v) = (n-1)/n * v is correct, and the application to n=3 is accurate. The expected revenue calculation of $50 is correct, and the Revenue Equivalence Theorem is properly invoked for the second-price auction comparison. However, there's a subtle issue: the probability of winning calculation uses Pr{v_j < v} rather than accounting for the bid b, though the final result is correct. The collusio" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 7.75, "brief_justification": "The response correctly applies the symmetric equilibrium bidding strategy (N-1)/N * v for the first-price auction, which is the standard result for uniformly distributed valuations. The individual bid calculations are correct. The ex-ante expected revenue formula is correct. The Revenue Equivalence Theorem is correctly invoked. However, there's a subtle issue: the problem states bidders know valuations are uniformly distributed on [0,100], but the specific realized valuations are given. The symm" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5.05, "brief_justification": "The response correctly gives the symmetric first-price equilibrium bid function b(v)=2v/3 and computes realized revenue for the given valuations, but it is incomplete: the second-price section is cut off, collusion is not addressed, and expected revenue is ambiguous. The ex-ante expected revenue of 50 is correct under the stated model, but the answer mixes ex-post and ex-ante perspectives without fully resolving them." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8, "brief_justification": "Mostly correct and clearly structured. It correctly gives the symmetric first-price equilibrium bid function, expected revenues, truthful bidding in second-price, and the qualitative effect of collusion. Minor issue: it mixes realized valuations with ex-ante uncertainty without explicitly distinguishing conditional outcomes from expected ones, and the collusion discussion is somewhat simplified." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.2, "brief_justification": "Accurate and well-structured, correctly gives the symmetric first-price equilibrium, expected revenue, second-price comparison, and collusion effects. Minor caveat: wording around 'actual valuations' vs private information could be more precise, and collusion outcomes depend on reserve prices/rules." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.8, "brief_justification": "Accurate use of the symmetric equilibrium bid function and revenue equivalence, with correct expected revenues under Uniform[0,100]. Clear and well-structured, and it addresses collusion sensibly. Minor issue: the summary table is slightly imprecise about the first-price winner payment in the realized valuation profile, which would be the highest bid ($66.67), not a range." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7.95, "brief_justification": "Clear, structured, and mostly correct on standard auction theory: first-price equilibrium bidding for U[0,100] with 3 bidders, truthful bidding in second-price, and expected revenue equivalence. Main issue is mixing realized valuations with private-information assumptions and overstating collusion details (e.g. near-zero winning bid and side-payment example are not rigorously justified and one comparison for B is inconsistent)." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 5.95, "brief_justification": "Clear structure and mostly correct on equilibrium bidding and second-price truth-telling, but it conflates ex ante expected revenue with the realized revenue given the stated valuations, incorrectly cites dominant strategy/RET for first-price, and the collusion analysis is flawed since full collusion would typically suppress the winning bid toward the reserve/minimum, not $80." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 6.8, "brief_justification": "Mostly accurate on standard auction results: first-price bidding b(v)=2/3 v, expected revenue 50, and truthful bidding plus revenue equivalence in second-price. However, the derivation shown for first-price is flawed/incomplete because it treats win probability as depending directly on v while varying b, and the response is truncated in the collusion section, leaving part 4 incomplete and somewhat oversimplified." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8, "brief_justification": "Mostly correct and clearly structured. It correctly gives the symmetric first-price bidding function, revenue equivalence, truthful bidding in second-price, and the basic collusion effect. Main limitation is ambiguity between conditional answers given the realized valuations versus ex-ante strategy under private information, and the collusion discussion is somewhat simplified." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7, "brief_justification": "The response correctly identifies the bidding strategy for first-price auctions with uniform distributions and provides accurate calculations for this case. Expected revenue calculation is mostly right but slightly imprecise on the expected max valuation (actual E[max of 3 uniforms on [0,100]] is 75, which is correct). It cuts off abruptly in part 3 and entirely omits part 4 on collusion. Clarity and structure are good but the response is incomplete." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.45, "brief_justification": "The response is factually accurate on optimal strategies, revenue calculations, and revenue equivalence. It correctly derives the bidding function and expectations for uniform distributions. Minor issue: the specific realized bids and outcomes in parts 1 and 3 treat the given valuations as known to all, contradicting the 'each knows only their own' premise, but the general theory and expectations are solid." }, { "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 is factually accurate on optimal strategies, revenue equivalence, and collusion dynamics. It provides thorough coverage with derivations, tables, and comparisons. Explanations are clear and well-structured. Depth is strong but slightly limited by assuming uniform [0,100] independent of the given specific valuations. Highly useful for understanding the concepts." }, { "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": "Response is factually accurate on equilibrium bidding, revenue equivalence, and collusion effects. Thoroughly addresses all four parts with both realized and ex-ante perspectives. Clearly structured and insightful on theory. Minor deduction for not deriving the bidding formula as offered and slight over-simplification of collusion without addressing potential instability." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7, "brief_justification": "The response provides a clear and structured analysis with good intuition and tables, but contains factual errors: the optimal bid in first-price is incorrectly stated as (n-1)/n * v (should be (n-1)/n * v for uniform[0,1] but scaled wrongly here; for [0,100] it is actually 2/3 v but revenue calc inconsistent). Expected revenue formula misstated (E[2nd order stat] for n=3 uniform[0,100] is actually 50 but derivation given is wrong). Collusion section has flawed side-payment numbers that don't ma" }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 6.6, "brief_justification": "The response correctly identifies the bidding strategy b(v)=(n-1)/n * v and expected revenue of $50 via revenue equivalence, but contains factual errors: it misstates the first-price outcome as $66.67 (ignoring that bids are random variables, so realized revenue is random), confuses specific realized values with expectations, and incorrectly claims the collusive bid in first-price yields seller revenue of $80 (it would be near the second-highest valuation only in second-price). Clarity and struc" }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.45, "brief_justification": "The response is factually accurate on optimal bidding, revenue calculations, and revenue equivalence. It provides clear derivations and tables. Minor incompleteness in the collusion section which is cut off mid-sentence. Strong overall structure and insight into auction theory." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 7.6, "brief_justification": "The response is mostly accurate on optimal bidding and revenue equivalence but contains a critical error in the expected revenue formula (should be (N-1)/(N+1) * V_max is incorrect; correct is 2N/(N+1)^2 * V_max ≈ $48.57 for N=3). It correctly identifies strategies, explains reasoning well, covers collusion effectively, and maintains clear structure. Depth is solid on theory but misses some nuances like exact expected payment derivation." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 6.45, "brief_justification": "The response is mostly correct on the first-price auction strategy (2/3 rule for uniform distribution with n=3) and provides good clarity. However, the response is incomplete - it cuts off mid-sentence during the second-price auction explanation and never addresses the collusion question (part 4). The expected revenue calculation is correct ($50 ex-ante), and the realized revenue of $66.67 is properly noted. The Revenue Equivalence Theorem connection between first and second price auctions is mi" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.55, "brief_justification": "The response is largely correct and well-structured. The symmetric equilibrium bidding formula is correctly derived and applied. The revenue equivalence theorem is correctly invoked. The expected revenue calculation is mathematically verified. Minor issue: the collusion section oversimplifies by suggesting bidding near zero without addressing reserve prices or the more realistic ring mechanism. The expected revenue calculation has a slight logical shortcut (applying the formula for E[b(V_(1))] =" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.4, "brief_justification": "Excellent response covering all four questions with rigorous derivations, clear tables, and insightful analysis. The equilibrium strategy derivation is correct, revenue equivalence is properly explained with the ex-ante vs ex-post distinction clearly noted. Collusion analysis is thorough covering both auction formats, incentive problems, and seller countermeasures. Minor note: the collusion section could more explicitly discuss the side payment mechanism for sustaining the ring, but overall this" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8, "brief_justification": "The response correctly derives the equilibrium bidding strategy and applies it to the specific valuations. The revenue equivalence theorem is correctly invoked. One minor issue: the response conflates 'expected revenue' with the realized outcome for specific valuations without clearly distinguishing the two contexts initially, though it does clarify later. The collusion section is qualitatively correct but lacks depth on specific equilibrium outcomes or formal analysis. Overall well-structured a" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.8, "brief_justification": "The response is highly accurate, covering the symmetric BNE bidding strategy, revenue equivalence theorem, dominant strategy in second-price auctions, and collusion dynamics. The formula b(v)=(n-1)/n×v is correctly derived and applied. Expected revenue calculation is correct and well-explained through two methods. The summary table adds clarity. Minor gap: could elaborate more on the mechanics of collusion enforcement challenges and side-payment structures, but overall this is an excellent, well" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 7.35, "brief_justification": "The response is well-structured and covers all four questions clearly. However, there are notable errors: (1) The expected revenue calculation is incorrect. The expected revenue in a first-price auction with 3 bidders uniform [0,100] is E[second-highest order statistic] = (n-1)/(n+1) * V_max = 2/4 * 100 = $50, which happens to give the right answer but the reasoning shown (2/3 * 75) is coincidentally correct numerically but conceptually flawed - the expected highest bid is not simply (2/3)*E[max" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.8, "brief_justification": "The response is mathematically rigorous and correct, applying the standard symmetric BNE formula for first-price auctions accurately. It correctly derives b(v) = (2/3)v for n=3, computes expected revenue of $50 using order statistics, properly invokes the Revenue Equivalence Theorem for the second-price comparison, and provides a nuanced discussion of collusion scenarios. Minor issue: the response appears cut off mid-sentence in section 4b, and the derivation sketch for the bidding function is s" }, { "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": "The response is well-structured and largely correct. The optimal bidding strategy formula (N-1)/N × v is correctly derived and applied. The expected revenue formula is correctly stated and calculated. The Revenue Equivalence Theorem is properly invoked for part 3. The collusion section is thorough and practical. Minor issue: the expected revenue formula E[R] = (N-1)/(N+1) × Vmax applies to symmetric uniform distributions ex-ante, but the problem gives specific valuations, creating a slight tensi" }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.45, "brief_justification": "Accurate and well-structured response covering key auction theory concepts, though the collusion question is incomplete." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9, "brief_justification": "The response is mathematically accurate and well-structured, covering all parts of the question. It correctly applies auction theory principles (equilibrium bidding, revenue equivalence, collusion). Minor deduction in depth for not explicitly mentioning the revenue equivalence theorem by name, though it is demonstrated." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.2, "brief_justification": "Accurate and thorough analysis with clear structure, insightful derivations, and practical insights on collusion." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.6, "brief_justification": "Accurate and thorough with clear structure, good depth on collusion, and practical insights." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9, "brief_justification": "The response is mathematically accurate and covers all parts of the question thoroughly. It provides clear explanations of equilibrium strategies, expected revenue calculations, and collusion dynamics, though the collusion section could explore enforcement challenges more deeply." }, { "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": "Accurate and thorough analysis with clear structure, insightful explanations, and practical implications." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.6, "brief_justification": "Accurate application of auction theory formulas (equilibrium bidding, order statistics) and Revenue Equivalence. Completeness is slightly reduced by a truncated final sentence on collusion." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.8, "brief_justification": "The response is highly accurate, covering all parts of the question with clear explanations. It correctly identifies the optimal strategies, calculates expected revenue, explains the revenue equivalence theorem, and details the mechanics and consequences of collusion." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 6.15, "brief_justification": "Correct formulas and calculations are given, but the answer is incomplete, missing the full discussion for the second-price auction and the collusion scenario, reducing overall coverage and practical value." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.25, "brief_justification": "Accurate equilibrium and revenue calculations, clear presentation, covers all parts, but could elaborate derivations and collusion nuances." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.65, "brief_justification": "Accurate equilibrium and revenue calculations, thorough coverage of all parts, clear presentation with tables and formulas, solid analytical depth, and practical insights on bidding and collusion." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.45, "brief_justification": "Accurate formulas and numbers for strategies and revenues, covers all parts with clear headings, offers solid economic reasoning, and practical insights on collusion." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.05, "brief_justification": "Accurate equilibrium and revenue calculations, covers all parts, clear presentation, good analytical depth, and practical insights." }, { "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": "Accurate equilibrium and revenue calculations, covers all questions with clear structure and insightful discussion on collusion, though collusion analysis could be deeper." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 7.6, "brief_justification": "Correct equilibrium and revenue calculations, clear presentation, but collusion analysis is flawed and oversimplified, reducing overall accuracy and depth." }, { "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 formulas and calculations, covers all questions, clear presentation, but limited derivation detail and deeper collusion analysis." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7.25, "brief_justification": "The response correctly identifies the optimal strategy for a first-price auction and calculates the bids. It also correctly distinguishes between realized and expected revenue. However, it is incomplete as it cuts off before addressing the second-price auction and collusion parts of the question." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 10, "brief_justification": "The response is exceptionally accurate, complete, clear, and demonstrates a deep understanding of auction theory. All parts of the question are addressed thoroughly with correct formulas and explanations." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 10, "brief_justification": "The response is exceptionally thorough, accurate, and well-structured. It not only answers all parts of the question correctly but also provides clear derivations, intuitive explanations, and insightful comparisons, demonstrating a deep understanding of auction theory. The inclusion of both ex-post and ex-ante revenue, the detailed comparison between auction types, and the comprehensive analysis of collusion are particularly strong." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.65, "brief_justification": "The response is highly accurate, complete, and clear. It correctly applies auction theory principles, addresses all parts of the question with appropriate depth, and provides useful insights into different auction formats and collusion." }, { "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 is highly accurate, clear, and provides good depth on auction theory concepts. It correctly applies formulas and explains the reasoning behind each answer. The summary table is a great addition." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "The response is exceptionally accurate, complete, clear, and demonstrates deep understanding of auction theory. It provides practical insights and is well-structured." }, { "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 is very clear, accurate, and provides good depth for each part of the question. The formulas are correctly applied, and the explanations are insightful. The only minor point is the expected revenue calculation for the second-price auction, which is stated as $50, but then the specific instance gives $80. While the overall expected revenue is $50, the specific instance's revenue is $80, which could be clarified slightly better in the comparison section." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.45, "brief_justification": "The response is highly accurate, complete, and clearly explained. It provides a strong theoretical foundation and applies it correctly to the specific scenario. The depth is excellent, especially with the derivation sketches and references to economic theorems. The usefulness is high for anyone trying to understand auction theory." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 6.55, "brief_justification": "Accurate for first-price auction with correct formulas and calculations, but response is incomplete—cuts off mid-sentence on second-price auctions and entirely omits the collusion question." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "brief_justification": "Accurate equilibrium bidding strategies with correct formulas. Comprehensive coverage of all four questions. Well-structured with clear mathematical notation. Good depth including order statistics and revenue equivalence theorem. Highly practical with specific numerical examples for all scenarios." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.65, "brief_justification": "Factually accurate with correct derivation of Bayesian Nash equilibrium, proper expected revenue calculation, accurate second-price comparison, and comprehensive collusion analysis. Well-structured with tables and clear explanations. Provides good depth including derivation, intuition, and mechanism design responses. Highly useful with practical insights." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.2, "brief_justification": "Accurate equilibrium strategies with correct formulas. Clear distinction between ex-post and ex-ante revenue. Well-structured with proper mathematical notation. Good depth including revenue equivalence theorem and collusion analysis. Highly useful and comprehensive." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.8, "brief_justification": "" }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.2, "brief_justification": "Accurate analysis with correct equilibrium bidding formula and truth-telling proof. Thorough coverage of all four questions. Excellent clarity with well-organized sections, tables, and formulas. Good depth explaining intuition and game-theoretic instability of collusion. Highly useful with practical insights on seller countermeasures." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.8, "brief_justification": "Accurate application of auction theory with correct formulas. Comprehensive coverage of all four parts with proper mathematical notation. Well-structured explanations with clear reasoning. Good depth in explaining bid shading and collusion dynamics. Highly useful with practical numerical examples." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7.55, "brief_justification": "Mathematically correct with excellent clarity and depth on bidding strategies and revenue equivalence. However, the response is abruptly cut off mid-sentence in section 4, leaving the collusion analysis incomplete." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 9, "brief_justification": "Excellent response with correct game-theoretic analysis. Properly applies optimal bidding formula for first-price auctions, explains revenue equivalence theorem, and provides detailed collusion scenario. Well-structured with clear explanations and mathematical rigor." } ], "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)" } }