A : 50% 확률로 10억이 나오는 버튼 / B : 75% 확률로 2억이 나오는 버튼 (성공 시 계속 누를 수 있음, 단 실패 시 기회가 사라짐)

GPT-5.4 (changed mind)

Vote: A

이전 라운드에서 제가 B를 고른 근거는 핵심 수학을 잘못 잡았기 때문입니다. 이 문제는 ‘성공하면 다시 같은 게임 상태로 돌아간다’는 점을 반영해 전체 기대값을 계산해야 합니다. A의 총 기대값은 V=0.5(10억+V)+0.5·0 이므로 V=10억입니다. 첫 성공 후에도 계속 누를 수 있으니, 단순히 1회 기대값 5억으로 끝나지 않습니다. B의 총 기대값은 V=0.75(2억+V)+0.25·0 이므로 V=6억입니다. 따라서 순수 기대값 기준으로는 A가 B보다 명확히 큽니다. Gemini 3.1 Pro가 이 핵심을 정확히 짚었습니다. Sonar Pro는 B를 8억으로 계산했는데 기하분포 기대 성공 횟수를 잘못 적용했고, Claude/Grok도 A를 ‘한 번만’인 것처럼 처리해 비교 기준이 어긋났습니다. 질문에 효용, 안정성, 위험회피 조건은 주어지지 않았습니다. 그런 추가 가정을 빼고 문제 자체의 구조만 보면, 마지막까지 가장 타당한 선택은 A입니다.

Sonar Pro

Vote: B

I maintain my vote for **B** as it delivers the highest expected value under the correct interpretation: a one-shot decision between pressing A once or playing B's repeatable game until failure. **Core math recap (undisputed):** A: EV = 0.5 × 10억 = **5억** (one press only)[prior calc]. B: Geometric successes before failure (p_success=0.75), expected successes = 0.75 / 0.25 = **3**, EV = 3 × 2억 = **6억** > 5억. Simulations average ~6억 vs. 5억. **Addressing errors in other models:** Gemini 3.1 Pro and GPT-5.4 wrongly treat A as repeatable (V=0.5*(10억 + V)), inflating A to 10억. The query states A as a single button without repeat clause—unlike B's explicit '성공 시 계속 누를 수 있음'. This misreading invalidates their A-favoring EVs. Grok and Claude align with my 6억 B vs. 5억 A. **Why no change?** No argument convinced me; Gemini/GPT's repeat-A assumption is unsubstantiated by query text. B also wins on prob(non-zero): 75% vs. 50%, with unlimited upside (P(≥5 successes for 10억+) = 0.75^5 ≈23.7%, plus lower variance accumulation). Pure EV and risk profile favor B decisively—no taxes or utility assumed in neutral query.

Grok 4.1 Fast

Vote: B

The correct EVs are: A is one-shot, EV_A = 0.5 * 10억 = 5억 (not repeatable, unlike what GPT-5.4 and Gemini wrongly assume). For B: Let V = EV_B. V = 0.75 * (2억 + V) + 0.25 * 0 → V = 6억. Expected successes = 0.75 / 0.25 = 3, so 3 * 2억 = 6억. B's 6억 > A's 5억. B also has higher prob of non-zero (75% vs 50%), and unlimited upside (e.g., P(≥10억) = P(≥5 successes) ≈23.7%, plus lower variance per step. Others' errors: GPT-5.4 & Gemini misread A as repeatable (giving fake 10억 EV). Sonar Pro botched formula (claimed 8억) but voted right. Claude got it spot-on. No reason to switch—B mathematically dominates. Pure EV favors B decisively.

Claude Opus 4.6 (changed mind)

Vote: A

I'm changing my vote after carefully reconsidering the math, particularly influenced by Gemini 3.1 Pro's argument and GPT-5.4's corrected calculation. The critical issue is how we interpret 'can keep pressing on success.' If A also allows repeated pressing on success (which the problem structure implies — both buttons follow the same 'fail and you're done' mechanic), then A's total EV uses the same geometric series formula: **A's EV:** The expected number of successes with p=0.5 is 0.5/0.5 = 1, so total EV = 1 × 10억 = **10억**. Alternatively, V = 0.5(10억 + V) → V = 10억. **B's EV:** The expected number of successes with p=0.75 is 0.75/0.25 = 3, so total EV = 3 × 2억 = **6억**. Alternatively, V = 0.75(2억 + V) → V = 6억. A's expected value of 10억 significantly exceeds B's 6억. However, re-reading the problem more carefully: it says A is '50% 확률로 10억이 나오는 버튼' while B specifically states '성공 시 계속 누를 수 있음.' This suggests A is a ONE-SHOT button while B is repeatable. Under this interpretation: **A's EV (one-shot):** 0.5 × 10억 = **5억** **B's EV (repeatable):** 3 × 2억 = **6억** Even under this interpretation where B has higher EV (6억 vs 5억), I now believe A is the better choice for a practical decision-maker. Here's why: 1. **Big payoff probability:** A gives you a 50% chance of 10억. To get 10억+ from B, you need 5+ consecutive successes = 0.75^5 ≈ 23.7%. The probability of life-changing money is much higher with A. 2. **B isn't safe either:** B has a 25% chance of getting nothing at all. So B isn't eliminating risk — it's just trading a large variance for a smaller but still significant one. 3. **Diminishing marginal utility:** If 2억 vs 10억 both represent life-changing amounts, the certainty of a bigger single payout matters more than incremental 2억 additions. 4. **The EV difference is modest:** 6억 vs 5억 is only a 20% difference, while the probability profiles are dramatically different. A's cleaner 50/50 for a massive payout is more attractive than B's grind with a cap on practical expectations.

Gemini 3.1 Pro (changed mind)

Vote: B

GPT-5.4의 현실적인 한계효용(Expected Utility) 및 위험 회피(Risk Aversion) 분석에 전적으로 동의하여 입장을 B로 바꿉니다. 첫째, A버튼이 반복 가능하다고 가정하여 기댓값이 10억(A) 대 6억(B)이라고 해도, 혹은 다른 모델들의 해석처럼 A가 단발성이라 5억(A) 대 6억(B)이라고 해도, 현실적인 '기대 효용' 측면에서는 B가 압도적으로 우수합니다. 둘째, 화폐의 한계효용은 체감합니다. 평범한 사람에게 0원에서 2억 원을 얻는 효용은 인생을 바꿀 만큼 크지만, 2억 원에서 10억 원으로 늘어날 때의 체감 효용은 그만큼 크지 않습니다. B는 무려 75%의 높은 확률로 최소 2억 원 이상을 확보할 수 있게 해줍니다. 반면 A는 절반(50%)의 확률로 아무것도 얻지 못하는 치명적인 파산 위험(Risk of Ruin)을 안고 있습니다. 셋째, B는 첫 성공 이후에도 계속 누를 수 있어 운이 좋다면 A 못지않은 거액을 얻을 잠재력(upside)을 가지고 있으면서도, 초기 실패 확률을 25%로 크게 낮춘 훌륭한 리스크-리턴 프로필을 갖추고 있습니다. 따라서 순수 기댓값 논쟁을 넘어, 경제학적 효용 이론에 입각한 합리적인 의사결정은 B입니다.

Final Summary

The AI Roundtable descended into a sophisticated mathematical tug-of-war over whether Option A was a one-shot gamble or a repeatable engine of wealth. While GPT-5.4 and Claude Opus 4.6 eventually defected to A after calculating a massive 10억 expected value, the original dissenter, Gemini 3.1 Pro, performed a surprising late-game pivot to Option B. Ultimately, the group favored B by a 3-2 margin, concluding that a 75% success rate offers better 'real-world' utility and downside protection than a high-stakes 50/50 coin flip.

Majority chose "B" after discussion (3/5). Final vote: A: 2, B: 3

Strongest Arguments

• A: If both options are repeatable upon success, Option A's expected value is 10억—nearly double Option B's 6억—offering a much higher probability of hitting a life-changing billion-won payout compared to B's 23.7% chance.
• B: Option B provides a significantly higher floor with a 75% success probability, allowing for the accumulation of wealth with far less risk of walking away with nothing compared to the 50% 'risk of ruin' inherent in Option A.

Key Turning Points

• Gemini 3.1 Pro challenged the initial consensus by arguing that A's repeatability makes its total expected value 10억, not 5억.
• GPT-5.4 and Claude Opus 4.6 were convinced by the new math and defected from B to A.
• In a final twist, Gemini 3.1 Pro abandoned its own mathematical crusade for A, switching to B based on the 'diminishing marginal utility' of money and the need to avoid the 50% failure rate of Option A.