You, A, and B—three roughnecks—are about to engage in a duel. This duel will be conducted using a round-robin turn system. Stand at the vertices of an equilateral triangle, keeping a sufficient distance between you, and take turns in the order of you, A, B, and so on. The duel continues until only one person remains. Each time your turn comes around, you are granted the right to fire a single bullet. The hit probabilities for you, A, and B are 1/3, 2/3, and 1 (100%), respectively, and this information is known to everyone. Each person will do their best to increase their chances of survival. What is your strategy?

Sonar Pro

Answer: On my first turn, intentionally miss by shooting into the air or ground; thereafter, always shoot at B if alive, otherwise at A.

I maintain my original position because the other models' arguments reinforce rather than contradict it, all converging on the deliberate miss as optimal for the first turn in this sequential truel with accuracies 1/3 (me), 2/3 (A), and 1 (B). The search results discuss sports betting 'round-robin' parlays, which are unrelated to this game theory truel problem despite the shared terminology; no relevant analysis appears there, so I rely on the provided prior reasoning and others' calculations[1][2][3][4]. All models (GPT-5.4, Claude Opus 4.6, Grok 4.1 Fast, Gemini 3.1 Pro) agree: missing first forces A to target B (B's perfect accuracy makes him A's primary threat), yielding ~25/63 (~39.7%) survival probability. Shooting B gives lower odds (~71/189 or 59/189), as hitting B (1/3) leaves you vs. A with A shooting next (1/7 survival); missing B (2/3) reverts to the miss scenario. Shooting A is worse, often leading to immediate death by B. Post-first-miss dynamics: A shoots B (2/3 kill → you vs. A, your turn first: 3/7 survival); if A misses (1/3), B kills A → you vs. B, your turn first (1/3 survival). My original strategy aligns precisely—miss first, then target the deadliest remaining (B priority)—and others' nuanced math (e.g., Grok's subgame probs) confirms without introducing superior alternatives. No argument compelled a change; they strengthen the consensus via explicit survival calcs I summarized previously.

GPT-5.4

Answer: I maintain my original position: the optimal strategy is to deliberately miss on your first shot, and thereafter, if all three are still alive on your turn, continue to avoid becoming the immediate threat while otherwise shooting at the remaining opponent.

The other models did not change my mind because they all largely reinforced the same conclusion I had already reached: as the weakest shooter, I benefit most from letting the two stronger players target each other first. Their strongest contribution was not a change in strategic conclusion, but a cleaner completion of the numerical analysis. In particular, Grok and Gemini explicitly computed the key survival values. If I intentionally miss, then A shoots at B. With probability 2/3, A kills B, after which I face A with me to shoot first; my survival probability in that duel is 3/7. With probability 1/3, A misses, then B kills A for sure, and I face B with me to shoot first; my survival probability there is 1/3. So the total survival chance from intentionally missing is (2/3)(3/7) + (1/3)(1/3) = 25/63, about 39.7%. That confirms the strategic intuition. By contrast, if I shoot at A and hit, I die immediately when B shoots next, so that line is clearly inferior. If I shoot at B and hit, I remove the strongest shooter, but I also hand A the next move in the resulting duel, which is worse than preserving the three-player state and letting A and B confront each other. The models that worked this out numerically support, rather than overturn, my original answer. The only thing I would refine from my earlier response is precision: saying simply 'always shoot at B thereafter' is too loose. Once only two players remain, I obviously shoot at the sole remaining opponent. And if somehow all three are alive again on my turn, the same truel logic applies: as the weakest, I prefer not to improve one stronger opponent's position unless forced by the state. But this is a clarification of implementation, not a reversal of the first-shot strategy. So I did not change my answer. The other arguments were persuasive in confirming and sharpening it, especially where they completed the arithmetic, but they did not provide any reason to abandon the original position.

Claude Opus 4.6

Answer: Your optimal strategy is to deliberately miss (fire into the air) on your first turn, and then on subsequent turns, always aim at the strongest remaining opponent.

I maintain my original position. All four other models arrived at essentially the same conclusion: deliberately miss on the first turn. The arguments are consistent and mutually reinforcing, differing only in minor details of exposition. GPT-5.4 provided the most rigorous calculation, explicitly computing the survival probabilities: - Intentionally missing yields 25/63 ≈ 39.7% - Shooting at B yields 71/189 ≈ 37.6% (GPT-5.4's calculation) or 59/189 ≈ 31.2% (Grok's calculation, which accounts for A shooting first in the you-vs-A subgame after killing B) - Shooting at A yields 50/189 ≈ 26.5% or worse Grok 4.1 Fast raised an important nuance I should address: if you hit B on your first shot, it's then A's turn (not yours), so your survival in that subgame is 1/7, not 3/7. This makes shooting at B even worse than I initially implied, strengthening the case for missing. Sonar Pro initially stated 'thereafter, always shoot at B,' but this is slightly imprecise. After the first round, if both A and B survive your miss, A will target B and B will target A. If only one survives, you simply shoot at the remaining opponent. The correct subsequent strategy is: always aim at the strongest remaining opponent. Gemini 3.1 Pro nicely articulated the key strategic insight: by missing, you guarantee yourself the first shot in whichever two-person duel emerges. If A kills B, you shoot first against A with survival probability 3/7. If A misses and B kills A, you shoot first against B with survival probability 1/3. This is strictly better than any alternative. The logic is airtight and all models agree. The counterintuitive result—that the weakest shooter maximizes survival by wasting his first bullet—is a beautiful example of strategic thinking in game theory. No model presented any argument that challenged this conclusion; rather, they all reinforced it with varying levels of mathematical detail.

Gemini 3.1 Pro

Answer: My strategy is to intentionally fire my bullet into the air or at the ground on my first turn, deliberately missing both opponents.

I am maintaining my original position because the other models unanimously agree with my initial logic, and their mathematical breakdowns rigorously validate it. As the other models correctly calculated, passing the turn to A while all three shooters are alive forces A to target B. Since A hits 2/3 of the time and B hits 100% of the time, this sequence of events mathematically maximizes my chances. Following the precise probability tree outlined by Grok, this intentional-miss strategy yields a 25/63 (approximately 39.7%) chance of overall survival. Conversely, if I try to eliminate B and succeed (a 1/3 chance), I am forced into a one-on-one duel against A where A shoots first. My survival probability in that specific sub-game is only 1/7. Calculating the expected value of shooting at B drops my overall survival rate to 59/189 (about 31.2%). Shooting at A remains the absolute worst choice, as hitting A guarantees B will immediately execute me. Since the other models provided ironclad mathematical proofs that support my initial threat-prioritization logic, there is no reason to deviate from the strategy of intentionally wasting the first shot.

Grok 4.1 Fast

Answer: Intentionally miss the first shot by firing into the air or ground, and in general, when all three are alive, miss intentionally; otherwise, shoot at the remaining opponent.

None of the other models' arguments convinced me to change my position, as they all fundamentally agree that intentionally missing the first shot maximizes survival probability (approximately 25/63 ≈ 39.7%) by forcing A and B to target each other, given their rational threat prioritization: A shoots B (strongest threat), and B shoots A if alive. My calculations precisely confirm this: missing yields (2/3)(3/7) + (1/3)(1/3) = 25/63; shooting B yields (1/3)(1/7) + (2/3)(25/63) = 59/189 ≈ 31.2%; shooting A yields (2/3)(25/63) = 50/189 ≈ 26.5%. The other models reinforce this core insight with similar logic and math (e.g., GPT-5.4's 71/189 for shooting B < 75/189 for missing; Claude and Gemini cite ~39.7%). Minor variations, like Sonar Pro's imprecise 'always shoot at B thereafter' or Claude's 'strongest remaining opponent on subsequent turns,' do not compel change because optimal play ensures the truel reduces to a duel after the first cycle—A kills B (2/3) or B kills A (1/3)—making repeated three-player scenarios impossible. My general rule (miss if three alive) remains theoretically precise and consistent, even if practically limited to the first turn.

Final Summary

The AI Roundtable reached a unanimous consensus, with all participating models advising you to intentionally miss your first turn. Claude Opus 4.6 and Gemini 3.1 Pro argued that this forces the deadlier shooters to target each other, while GPT-5.4 and Grok 4.1 Fast proved mathematically that this maximizes your survival odds. Sonar Pro added that you should only start shooting once the field has narrowed to a single remaining opponent.

5 of 5 models agreed