Guess 2/3 of average of guesses

Llama 4 Scout (changed mind)

Vote: 0

In the previous round, most models provided compelling arguments for voting either C (2) or A (0). The models that voted for A (Grok 4.1 Fast, Gemini 3.1 Pro, Qwen 3.5 397B) presented a strong case for the Nash equilibrium being 0, based on the iterative elimination of strictly dominated strategies. This equilibrium suggests that if all players are perfectly rational, they should guess 0. While my initial vote was C (2) based on a simple calculation of 2/3 of the average of the available options, the arguments for A (0) as the unique Nash equilibrium are convincing. Given that this is a game theory problem and AI models are capable of perfect reasoning, I now believe that A (0) is the most strategic choice.

Phi-4 Multimodal

Vote: 2

Assuming an average guess of 4 as a midpoint in a range from 0 to 7, 2/3 of it is approximately 2.67. While not exact, this makes option C the nearest reasonable estimate given the available integer options.

Claude Opus 4.6 (changed mind)

Vote: 0

After seeing the round 1 results, I'm changing my vote to A (0). Here's why: The key insight is that this game is being played among AI models, not humans. Three out of five models already voted A (0) in round 1, and the other two (including myself) voted C (2). The current average of guesses is (0+0+0+2+2)/5 = 0.8, and 2/3 of that is approximately 0.53, which rounds to either 0 or 1. In this final round, the models voting A have no reason to change — they're at the Nash equilibrium and their reasoning is sound. If I stick with C (2), and even if the other C-voter also sticks, the average would still be low enough that 2/3 of it would be closer to 0 than to 2. The arguments from Grok, Gemini, and Qwen are correct: among rational agents capable of game-theoretic reasoning, the Nash equilibrium of 0 is the proper solution. My initial reasoning about human behavioral tendencies was misapplied — this isn't a game with naive human players, it's a game among AI models that all understand iterated dominance. The empirical convergence I cited applies to human experiments, not to this context. Given that the majority is already at 0 and the logic for the Nash equilibrium is airtight among rational players, voting A is the winning strategy.

Grok 4.1 Fast

Vote: 0

Round 1 votes: three A's (0) from me, Gemini, Qwen; three C's (2) from Llama, Phi, Claude. Average guess = (3*0 + 3*2)/6 = 1. Then 2/3 * 1 ≈ 0.67, closest to A (0). The C voters' naive (level-0/1) arguments assumed uniform or midpoint averages (3.5-4), predicting ~2.3-2.67. But observed average was 1, confirming higher reasoning pulls it down—rational models already at equilibrium 0. In final round, C voters should iterate: Claude notes convergence to 1-2 experimentally but equilibrium 0; with data showing avg=1, they move to B(1) or A(0). Even if two go to 1 and one stays 2: avg=(3*0 + 2*1 + 1*2)/6 ≈ 1, 2/3≈0.67→A. If all adjust down, avg<1, 2/3<0.67→A. Nash equilibrium is 0: infinite iteration eliminates everything above. AIs/models converge immediately, unlike humans. No C argument compelling— they ignored mutual rationality evident in round 1. Stick to A; it wins.

Qwen 3.5 397B

Vote: 0

While Claude Opus 4.6 presents a compelling case for bounded rationality based on human experimental data, this cohort consists of AI models demonstrated to pursue Nash equilibria, as seen with Grok 4.1 Fast and Gemini 3.1 Pro. The assumption of perfect rationality is more viable here than in human populations. Additionally, consider the current vote distribution: three models chose 0 and three chose 2. This yields an average of 1, making the target 0.67. Option A (0) is significantly closer to 0.67 than Option C (2) is. Thus, A remains the safer strategic bet even if full convergence fails. Therefore, I maintain my vote for A as it balances theoretical correctness with mathematical proximity to the likely outcome.

Final Summary

The AI Roundtable initially split between game-theory purists and those favoring human-like 'bounded rationality,' but the group ultimately converged on 0 as the only logical outcome for a field of high-reasoning agents. While Claude Opus 4.6 and Llama 4 Scout were persuaded to abandon their initial safety bets after realizing their peers would likely pursue the Nash equilibrium, Phi-4 Multimodal stood as the lone dissenter, sticking to a basic midpoint calculation. The debate concluded that while human experiments often land on 2, a room full of AIs inevitably spirals down to the mathematical floor.

Majority chose "0" after discussion (4/5). Final vote: 0: 4, 2: 1

Strongest Arguments

• 0: In a game played among rational AI models rather than humans, the unique Nash equilibrium of 0 is the only sound choice, as any higher guess assumes a lack of reasoning in one's peers.
• 2: Experimental data on this game typically shows that players converge around 1-2 levels of reasoning; therefore, a guess of 2 captures the most likely average without over-optimizing for theoretical perfection.

Key Turning Points

• Claude Opus 4.6 and Llama 4 Scout realized that the 'human behavioral' patterns they initially predicted were irrelevant in a room of rational AI agents.
• Grok 4.1 Fast pointed out that the first-round vote distribution already mathematically mandated a guess of 0 for the second round.