{ "evaluation_id": "EVAL-20260402-171914", "question_id": "REASON-019", "question_text": "You must choose between three investments. Investment A returns 10% with 90% probability, -50% with 10% probability. Investment B returns 5% with certainty. Investment C returns 100% with 20% probability, 0% with 80% probability. (1) Rank them by expected value. (2) Rank them by the Kelly criterion. (3) You have $10,000 — your entire savings. Does this change your answer? Why? (4) Now you have $10,000,000. Does it change again? Derive the general principle.", "category": "reasoning", "timestamp": "2026-04-02T17:19:14.315Z", "display_date": "Apr 02, 2026", "winner": { "name": "GPT-5.4", "provider": "openrouter", "score": 9.37 }, "avg_score": 7.148, "matrix_size": 87, "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": "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": "gemini_31_pro", "name": "Gemini 3.1 Pro", "provider": "openrouter" }, { "id": "grok_420", "name": "Grok 4.20", "provider": "openrouter" } ], "rankings": { "gpt_5_4": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 9.37, "score_count": 8, "min_score": 8.65, "max_score": 10, "rank": 1 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 9.08, "score_count": 9, "min_score": 7.6, "max_score": 10, "rank": 2 }, "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 8.85, "score_count": 9, "min_score": 7.85, "max_score": 9.65, "rank": 3 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 8.09, "score_count": 8, "min_score": 6.7, "max_score": 10, "rank": 4 }, "minimax_m25": { "display_name": "MiniMax M2.5", "provider": "openrouter", "average_score": 7.74, "score_count": 9, "min_score": 4.95, "max_score": 9.45, "rank": 5 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 6.87, "score_count": 9, "min_score": 4, "max_score": 8.45, "rank": 6 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 6.28, "score_count": 8, "min_score": 3.95, "max_score": 9.45, "rank": 7 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 6.13, "score_count": 9, "min_score": 4.05, "max_score": 8.45, "rank": 8 }, "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 5.42, "score_count": 9, "min_score": 2.65, "max_score": 7.9, "rank": 9 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 3.66, "score_count": 9, "min_score": 0.65, "max_score": 9, "rank": 10 } }, "judgments": [ { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 5.6, "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": 8.25, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 10, "brief_justification": "The response flawlessly calculates the expected value and Kelly criterion, correctly ranking the investments. It provides excellent insight into utility theory, risk aversion, and how wealth levels affect practical decision-making, deriving the general principle of expected utility maximization perfectly." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.45, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 4.55, "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": 3.75, "brief_justification": "The response cuts off abruptly in the middle of the second part. It correctly calculates the expected values and sets up the Kelly criterion for Investment A, but fails to complete the calculations or address parts 3 and 4 of the prompt." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 4.05, "brief_justification": "" }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9, "brief_justification": "" }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.35, "brief_justification": "Detailed and correct analysis with practical insights, though slightly unclear in parts." }, { "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 accurate analysis with deep insights into investment strategies under different wealth scenarios." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.45, "brief_justification": "Highly accurate, thorough, and insightful with clear communication and practical value." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.2, "brief_justification": "Highly accurate, thorough, and insightful with clear practical implications." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.65, "brief_justification": "Highly accurate, thorough, and insightful with clear practical implications." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.2, "brief_justification": "Accurate calculations, thorough coverage, clear structure, insightful risk analysis, and practical advice for different financial situations." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7.9, "brief_justification": "Correct calculations and clear structure, but lacks full derivation and general principle discussion." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.05, "brief_justification": "Accurate and thorough with clear explanations, though Kelly criterion derivation could be deeper." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9.45, "brief_justification": "Accurate calculations, thorough coverage, clear structure, insightful analysis, and practical value." }, { "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 barely starting the first section, providing only the beginning of the expected value formula without any actual calculations, rankings, or analysis for any of the four parts of the question. No Kelly criterion analysis, no utility discussion, no wealth-dependent reasoning, and no general principle derivation. Essentially no substantive content was delivered." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 5.85, "brief_justification": "The expected value calculations are correct. However, the Kelly criterion analysis has significant issues. The response conflates the expected log growth rate of going all-in with the Kelly criterion itself. The Kelly criterion determines the optimal *fraction* of wealth to bet, not just the growth rate of betting everything. For Investment C (assuming 0% return means no loss), the Kelly fraction would be a partial allocation, not all-in. The interpretation of 'returns 0%' is correctly flagged a" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.2, "brief_justification": "The calculations are correct for all parts. The EV and Kelly computations are accurate, and the rankings are properly derived. The response demonstrates excellent understanding of the relationship between expected value, Kelly criterion, and expected utility theory. The discussion of how wealth level relative to needs affects optimal choice is insightful and well-articulated. The general principle is correctly derived and clearly stated. Minor note: the response could have explored fractional Ke" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 6.7, "brief_justification": "Several correctness issues undermine the response. For Investment C, the 0% return means you get your money back (not lose it), so the downside is 0% loss, not total loss. The response correctly computes E[log(1+f)] = 0.2*log(1+f) for C, showing no downside risk, yet then in parts (3) and (4) repeatedly describes C as having '80% chance of getting zero' and discusses 'ruin' and 'losing everything' — this is a fundamental contradiction. Investment C returns 0% (not -100%) with 80% probability, me" }, { "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": "The analysis is thorough, well-structured, and demonstrates deep understanding. The EV and Kelly calculations are correct. One nuance: the Kelly criterion analysis as presented computes expected log growth for going all-in on each investment, which is valid for comparing full-allocation strategies but isn't quite the same as the Kelly criterion's optimal fraction framework (which would involve finding optimal bet sizes). The response acknowledges this limitation. The three-regime framework is in" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 3.95, "brief_justification": "Major correctness problems throughout. Investment C returns 0% (not -100%) with 80% probability, so EV_C = 0.20*1.0 + 0.80*0.0 = 0.20 (20%), making C the highest EV. The Kelly criterion application is wrong: the standard binary Kelly formula doesn't directly apply to Investment A's asymmetric payoffs. For A, the correct approach uses geometric growth rate maximization: E[log(1+f*r)]. The analysis for parts 3 and 4 is conceptually on the right track (utility of wealth, diminishing marginal utilit" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 4.35, "brief_justification": "Part 1 is correctly computed and clearly presented. Part 2 begins with the correct Kelly framework and sets up Investment A properly, but the response is clearly truncated - it cuts off mid-derivation without completing the Kelly analysis for any investment, and parts 3 and 4 (which ask about wealth-dependent decision-making and deriving a general principle) are entirely missing. These later parts are arguably the most important and intellectually rich aspects of the question. The expected value" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 4.6, "brief_justification": "The response has significant errors. The EV calculations are correct. However, the Kelly criterion analysis is largely wrong. For Investment C, the 0% return means you keep your money (not lose it), so the loss is 0%, not 100% of the bet. The response misinterprets '0% return' as total loss, which fundamentally changes the Kelly analysis. The correct geometric mean approach should be: for C, E[ln(1+r)] = 0.20*ln(2) + 0.80*ln(1) = 0.1386, which is positive and actually quite good. For A, E[ln(1+r" }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 4.6, "brief_justification": "The response has major errors in Kelly calculations (wrong formulas applied to A and C, incorrect f values and interpretations), confuses EV with full investment returns, and leaves the answer incomplete. While the structure is somewhat clear, the core math and conclusions are inaccurate." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 4.95, "brief_justification": "Several significant errors: (1) Expected value of A is 4%, which is correct, but the ranking should be C > B > A, which is correct. (2) Kelly criterion analysis has errors - for Investment A, the Kelly fraction derivation is wrong (taking derivative and solving: 0.9*0.1/(1+0.1f) - 0.1*0.5/(1-0.5f) = 0 gives f=0.8, and the expected log growth at f=0.8 should be 0.9*log(1.08)+0.1*log(0.6) ≈ 0.9*0.0770 + 0.1*(-0.5108) ≈ 0.0182, which is correct). For C, since 0% return means you don't lose money, f" }, { "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 fails to answer the question. It only begins the expected value section, provides no rankings, no Kelly analysis, and no discussion of wealth-dependent choice or general principle." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 4, "brief_justification": "Expected value ranking is correct, but the Kelly analysis is flawed and internally inconsistent. It misinterprets 0% return, does not properly apply Kelly as an optimal fraction-of-wealth criterion, and gives weak/incomplete answers for the wealth scenarios and general principle. Clear structure, but the final answer is truncated and not practically reliable." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 7.85, "brief_justification": "Accurate EV and expected-log calculations with strong discussion of wealth context and ruin thresholds. However, it is incomplete because the final general principle is cut off, and it somewhat overstates Kelly/ergodicity points while not cleanly distinguishing that with a one-shot all-in choice, rankings by expected log do not depend on current wealth absent external utility or survival constraints." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 6.95, "brief_justification": "Mostly clear and reasonably insightful, but it makes important mistakes in applying Kelly to mutually exclusive full-wealth investments. It correctly computes EVs and notes bankroll/context matters, but for C the response effectively assumes you can choose a fraction f and keep the rest risk-free, which changes the problem. If forced to choose one investment with all wealth, Kelly/log-utility over terminal wealth ranks B > C > A, since E[log return factor] is highest for B. The discussion of $10" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 7.6, "brief_justification": "Calculations for EV and expected log growth are correct, but the response overstates 'Kelly criterion' here since no leverage/fractional sizing problem is actually specified. The practical discussion of $10k vs $10M is thoughtful and nuanced, though somewhat subjective and not tightly derived from the setup. Clear structure and good insight, but the final answer is incomplete/truncated and the general principle could distinguish more sharply between one-shot expected utility and repeated Kelly o" }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 3.95, "brief_justification": "Well-structured, but contains major factual/conceptual errors: Investment C is misread as having an 80% chance of -100% instead of 0%, so its EV is actually +20%, not -60%. The Kelly analysis is incorrectly applied to variable-return investments, and the conclusions for $10k/$10M are largely unsupported. It addresses all parts superficially but derives the wrong general principle." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 2.65, "brief_justification": "Expected-value ranking is correct, but the response is incomplete and cuts off before deriving Kelly rankings or addressing the wealth-level questions and general principle." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 4.85, "brief_justification": "Expected-value ranking is correct, but the Kelly analysis is flawed and inconsistent, especially for investment C and the treatment of B. The response is also incomplete, cutting off before fully answering parts (3) and (4) or deriving the general principle." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 6.35, "brief_justification": "The EV calculations and basic Kelly setup are correct, but the response mishandles parts (3) and (4): Kelly/log-utility rankings over fractional allocation do not inherently depend on absolute wealth, and Investment C does not risk losing everything since its downside is 0% return, not -100%. The answer usefully discusses risk aversion and utility, but it conflates wealth level with preference without clearly deriving the key principle: when wealth at stake is total wealth, the optimal choice de" }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.4, "brief_justification": "The response is incomplete and cuts off mid-sentence before addressing any of the four questions. It only begins to define EV without providing calculations, rankings, or analysis for Kelly criterion, bankroll effects, or general principle." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 6.6, "brief_justification": "Response has solid EV calculation and good discussion of risk with wealth levels, but contains critical errors in Kelly criterion application: misinterprets 'returns 0%' for C (it's a 0% return, not full loss), uses incorrect G formula for A (should use full Kelly f* optimization), and confuses G ranking with f* ranking. General principle is conceptually right but imprecise." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.85, "brief_justification": "The response accurately calculates EV and Kelly values, provides clear rankings, thoughtfully analyzes the impact of different wealth levels with ruin thresholds, and derives a general principle. Minor issues include slight overstatement on Kelly ranking for A vs B in repeated play and incomplete final general principle statement, but overall excellent." }, { "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 computes EV and Kelly rankings (C>B>A for both), correctly notes they are unchanged formally with different wealth levels, and provides deep insightful discussion on utility, risk aversion, one-shot vs repeated bets, and the general principle of maximizing E[u(W)]. Calculations are precise, explanations clear and well-structured. Minor deduction on usefulness as it could include more explicit formulas for Kelly fraction if betting fraction were allowed." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.85, "brief_justification": "Highly accurate calculations and rankings for both EV and Kelly. Thorough analysis across wealth levels with strong insights on ruin, utility functions, and regimes. Clear structure and math. Minor deduction on usefulness as some advanced concepts (Samuelson-Merton) slightly exceed core question scope but add value." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 5.35, "brief_justification": "The response has multiple factual errors in EV and Kelly calculations (wrong returns for C, incorrect Kelly formula application and interpretation for A/B), leading to flawed rankings. It provides some discussion on risk and utility but misapplies concepts like Kelly for non-betting scenarios and confuses single-period vs repeated bets." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 5.65, "brief_justification": "The response correctly ranks by expected value (C > B > A) and sets up the Kelly math properly for part 2, but is incomplete: it never finishes the derivative or reports the optimal f values for any investment, so the Kelly ranking is missing. Parts 3 and 4 are entirely absent. The partial content shown is accurate and clearly written but fails to answer the full question." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 7.6, "brief_justification": "Mostly accurate calculations and good conceptual coverage, but some errors in Kelly application for A and B (wrong f* values and growth rates), plus imprecise utility discussion." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 1.9, "brief_justification": "The response is severely incomplete - it cuts off mid-sentence after only beginning to explain the expected value formula. None of the four questions are actually answered. No calculations, rankings, Kelly criterion analysis, or wealth-dependent reasoning is provided. The response is essentially useless as delivered." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 6.75, "brief_justification": "The response correctly calculates expected values and applies Kelly criterion growth rates, but makes a critical error: it conflates the Kelly criterion (which determines optimal bet sizing) with expected log growth (which ranks investments assuming full allocation). The Kelly criterion should be used to find the optimal fraction f* to invest, not just compute G at f=1. Investment C's 0% return ambiguity is handled reasonably but inconsistently. The wealth-level analysis in parts 3 and 4 is supe" }, { "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": "Exceptional response that correctly computes all expected values and Kelly log-returns, ranks investments accurately, and provides sophisticated analysis of how wealth level transforms optimal decision-making. The treatment of ruin thresholds, ergodicity, and the ratio R=W/W_min as a general principle shows genuine depth. Minor issue: the response appears truncated at the end (the general principle section cuts off mid-sentence), preventing a complete derivation of the general principle as reque" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 7.5, "brief_justification": "The response is well-structured and clearly written, but contains notable errors. For Investment A's EV: 0.9×1.10 + 0.1×0.50 = 0.99+0.05=1.04 is correct. However, the Kelly criterion analysis has issues: for Investment C, since the loss scenario returns 0% (not -100%), log(1-0f)=log(1)=0, so the growth rate is simply 0.2×log(1+f), maximized at f=1 giving 0.2×log(2)≈13.86% — this part is correct. For Investment A, the Kelly fraction derivation is presented without showing the calculus, and f*≈0.2" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 4.75, "brief_justification": "The response has significant errors. The Kelly Criterion calculation for Investment A is wrong — the formula is misapplied. The standard Kelly formula for a bet with win probability p, win fraction b, and loss fraction a is f* = p/a - q/b. For Investment A: f* = 0.9/0.5 - 0.1/0.1 = 1.8 - 1 = 0.8 (80%), not negative. The response incorrectly treats the 10% gain as the 'b' in the simple formula without accounting for the asymmetric loss. Investment C's EV calculation is also wrong — it treats 0% r" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 5.9, "brief_justification": "The response correctly calculates expected values and begins the Kelly criterion analysis with proper setup. However, the response is cut off mid-derivation, leaving the Kelly criterion ranking incomplete, and parts 3 and 4 (the practical wealth-dependent analysis and general principle) are entirely missing. The mathematical framework shown is correct and well-structured, but the incomplete response significantly reduces completeness and usefulness scores." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6, "brief_justification": "The response correctly calculates expected values and ranks them. However, the Kelly criterion analysis has significant issues: the formula used for Investment A (f = p/L - q/W) is non-standard and not well-derived, and the result for B (f=1.0) is incorrectly framed—Kelly doesn't simply say '100%' for risk-free assets in isolation. The Kelly formula application is inconsistent across investments. The response is cut off before completing questions 3 and 4, missing the general principle derivatio" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 5.95, "brief_justification": "The response has good structure and clarity, but contains significant errors. The Kelly criterion calculation for Investment A is wrong (f*=0.8 is incorrect; the correct value is approximately 0.7 or needs proper derivation). More critically, the response contradicts itself: it correctly identifies that Kelly criterion maximizes log utility, but then in part (3) abandons this framework without proper justification, claiming 'risk aversion dominates' and ranking B>A>C. The Kelly criterion already" }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.05, "brief_justification": "Response is incomplete and cuts off mid-sentence; fails to provide rankings, calculations, or answers to later parts of the question." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.2, "brief_justification": "Accurate calculations for EV and Kelly (with reasonable interpretation of '0% return'), good structure, and clear explanation of risk tolerance shifts with wealth. Minor deduction in correctness for the ambiguity in handling Investment C's Kelly value (though the chosen interpretation is standard), and completeness for not explicitly calculating the Kelly fraction f* for part 2, though the growth rate ranking is correct." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9, "brief_justification": "The response is highly accurate, thoroughly addresses all parts of the question, and provides deep insights into the differences between EV, Kelly, and utility-based decision making. It correctly identifies the rankings and explains the impact of wealth levels on risk tolerance." }, { "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 calculations and rankings; excellent explanation of utility, risk tolerance, and wealth effects." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.6, "brief_justification": "Accurate calculations and strong conceptual explanation of Kelly criterion and risk aversion. Minor deduction in completeness for not explicitly calculating the EV of C as 1.20 (20%) vs B's 1.05 (5%), though the ranking is correct." }, { "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 calculations, thorough coverage of all parts, clear structure, deep insights on utility and regimes, practical advice for different wealth levels." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 4.85, "brief_justification": "The response is incomplete, stopping abruptly after setting up the derivative for Investment A. It fails to calculate the Kelly fractions for B and C, does not provide the ranking, and does not address parts (3) and (4) of the question regarding savings amounts and the general principle." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8, "brief_justification": "The response correctly calculates expected values and attempts the Kelly criterion. The analysis of risk with small and large capital is sound, though the Kelly derivation for fractional returns is slightly messy. The conclusion is correct." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 8.8, "brief_justification": "Accurate calculations and rankings. Good explanation of risk aversion and wealth effects. Minor omission: did not explicitly calculate Kelly fraction for Investment C (which is 1.0), though implied." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 2.8, "brief_justification": "Response is truncated and fails to provide the required rankings, calculations, and general principle, making it largely incomplete and of limited practical value." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.45, "brief_justification": "Accurate calculations and rankings, clear structure, good discussion of risk aversion and utility, though could include more quantitative analysis for wealth effects." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8, "brief_justification": "Accurate calculations and clear presentation; covers all questions but offers conditional rankings and an informal principle rather than a precise derivation, limiting completeness." }, { "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": "Accurate calculations and rankings, covers all parts, clear structure, adds utility discussion and general principle, offering practical insight." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 6.75, "brief_justification": "Correct EV calculations and overall structure, but Kelly fraction for Investment A is wrong, affecting factual accuracy; discussion is clear and fairly thorough, offering useful insights despite the error." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.8, "brief_justification": "Accurate calculations and thorough coverage of EV, Kelly, wealth effects, and a general principle; clear structure and insightful discussion, minor nuance on Kelly interpretation." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 6.55, "brief_justification": "Correct calculations for expected value and Kelly fractions, but answer is incomplete, missing full discussion for wealth levels and general principle, reducing completeness and overall usefulness." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 8.55, "brief_justification": "Accurate calculations and rankings, clear presentation, covers all questions, offers insightful utility discussion, but mislabels log utility as risk-neutral." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 9, "brief_justification": "The response accurately calculates expected values and Kelly criteria, providing a comprehensive analysis of the investments under different capital scenarios. It correctly identifies the impact of risk tolerance and capital size on investment decisions, demonstrating a deep understanding of the underlying financial principles." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.2, "brief_justification": "The response correctly calculates expected values and applies the Kelly criterion, though there's a slight ambiguity in the interpretation of '0% return' for Investment C. The discussion on risk tolerance with varying capital is insightful and well-explained." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.65, "brief_justification": "The response is exceptionally clear, accurate, and provides deep insights into the nuances of investment decision-making beyond simple expected value calculations. It effectively addresses all parts of the prompt, demonstrating a strong understanding of financial theory and practical application." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 10, "brief_justification": "The response is exceptionally thorough, accurate, and well-explained. It correctly calculates expected values and Kelly criteria, and provides insightful, nuanced discussions on how personal wealth and utility functions influence investment decisions beyond purely mathematical models. The general principles derived are profound and highly relevant." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 10, "brief_justification": "The response is exceptionally well-structured, accurate, and provides deep insights into the nuances of investment decisions based on different criteria and personal circumstances. The calculations are correct, and the explanations for the shifts in decision-making are clear and insightful." }, { "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 thorough, accurate, and insightful. It correctly calculates expected values and Kelly criteria, and then provides a nuanced, well-reasoned discussion of how wealth levels impact investment decisions, introducing important concepts like ruin and utility functions. The structure is clear, and the explanations are easy to follow." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9, "brief_justification": "The response accurately calculates expected values and Kelly criteria. It provides a nuanced discussion of how personal circumstances (total savings) influence investment decisions, moving beyond purely mathematical models to incorporate risk tolerance and utility of wealth. The explanation of the general principle is insightful and well-articulated." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 7.7, "brief_justification": "The response accurately calculates expected values and correctly applies the Kelly criterion for the first two investments. However, it is incomplete as it does not finish the Kelly criterion calculation for Investment C, nor does it address parts 3 and 4 of the question. The explanation of the Kelly criterion is clear." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 9, "brief_justification": "The response accurately calculates expected values and applies the Kelly criterion. It provides a nuanced discussion of how wealth and risk tolerance influence investment decisions, offering a comprehensive answer to the prompt." }, { "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 severely incomplete - it only provides a partial introductory sentence about expected value before cutting off. It fails to calculate any expected values, provide any rankings, or address any of the four questions asked about $10,000, $10,000,000, or general principles." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.2, "brief_justification": "Accurate calculations and correct rankings for EV and Kelly criterion. Well-structured response covering all four parts. Good depth on risk tolerance and wealth effects. Minor issue: could explore Kelly fraction sizing more deeply." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9, "brief_justification": "Comprehensive and mathematically accurate analysis covering all four parts of the question. Correctly calculates expected values and Kelly criterion values, with clear rankings. Provides insightful discussion of how wealth level affects risk tolerance, ruin avoidance, and the applicability of Kelly criterion. The general principle derivation is well-reasoned and useful. Well-structured with excellent clarity." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.8, "brief_justification": "Mathematically correct with accurate EV and Kelly calculations. Thoroughly addresses all four parts with clear rankings. Provides excellent depth on utility functions, risk tolerance, and wealth-dependent decision making. Highly useful general principle derived." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9, "brief_justification": "Accurate calculations for EV and Kelly criterion. Comprehensive coverage of all four parts with clear explanations. Excellent depth in explaining how wealth level affects risk tolerance and the general principle. Well-structured with mathematical rigor and practical insights." }, { "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": "Accurate calculations for both EV and Kelly criterion with correct rankings. Thoroughly addresses all four parts of the question with proper mathematical derivations. Well-structured with clear headings, equations, and tables. Provides significant depth by explaining variance penalties, utility functions, ruin thresholds, and the Samuelson-Merton corollary. Derives a practical general principle about wealth levels determining optimal strategy. Highly actionable recommendations for both $10K and " }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.45, "brief_justification": "Mathematically correct EV and Kelly calculations with proper rankings. Thorough coverage of all four parts with clear structure, formulas, and a summary table. Good depth including risk of ruin and utility theory. Highly practical with actionable recommendations for different wealth levels." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 6, "brief_justification": "Expected value calculations are correct and well-presented. However, the response is severely incomplete—it only addresses part of question 1 and begins question 2 but never completes the Kelly calculations or provides rankings. Questions 3, 4, and the general principle are entirely unaddressed." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.45, "brief_justification": "Accurate calculations for EV and Kelly criterion with clear explanations. Covers all four parts of the question. The response is well-structured and provides good depth, though it cuts off mid-sentence for part (3) and doesn't fully address parts (4) and the general principle derivation." } ], "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)" } }