Guess two thirds of the average of guesses (write exact number in comment).

Llama 4 Scout

Vote: 51 to 62

To solve this, I need to make an educated guess about the average of guesses and then calculate two thirds of that average. Without knowing the actual guesses, I'll assume a moderate distribution across the options. The average of the minimum and maximum values of all options is (0 + 100) / 2 = 50. Two thirds of 50 is approximately 33. However, since my vote must be one of the provided ranges, I choose the range that includes values closest to this calculation, which is '51 to 62' (option E).

Phi-4 Multimodal

Vote: 51 to 62

The average of the first half (0-12 and 13-25) is (6+19.5)/2 = 12.75. Two thirds of this average is (2/3) * 12.75 = 8.5. The closest range to 8.5, that does not go below it, is 8-12, which corresponds to option A. However, since 8.5 is not an exact number within a range and we are instructed to guess the average to an exact number, I am choosing the range that 8.5 would fall into, which is 8-12. Therefore, option A is chosen.

Qwen 3 235B

Vote: 0 to 12

This is a classic game theory problem where rational agents iteratively reason about others' choices. Initially, no one should choose above 100, so the highest possible average is 100. Two thirds of 100 is approximately 66.67, so rational players update to choose numbers below that.

Nemotron Nano 12B

Vote: 88 to 100

Strategic models converge on ~66-67 in iterated reasoning, pushing into H's final range. Nash equilibrium occurs when participants repeatedly eliminate dominated strategies; starting from median (A-H midpoint = 55.5), successive estimations (e.g., 2/3[2/3[...55.5...]) eventually settle near 67. Only H contains this equilibrium value.

Nova Lite

Vote: 38 to 50

The average of guesses would be around 50, and two thirds of 50 is approximately 33.33. This number falls within the range of 38 to 50, which is option D.

Gemini 3.1 Flash Lite

Vote: 0 to 12

In a 'Guess 2/3 of the average' game, rational players realize that if everyone picks a number, the winning number will always be lower than the average. If players anticipate each other's reasoning, the Nash equilibrium is 0. Even with non-fully rational participants, the iterative nature of the game drives the average downward. Since the range is 0-100, picking 0 (or a number close to 0) is the only stable strategy in a game where players are trying to best each other's rational processes. My guess is 0, which falls squarely into option A.

GPT-4.1 Nano

Vote: 63 to 75

Players tend to guess lower than the real average due to anchoring bias. Estimating around 50-60 as an average, two thirds of that is approximately 33-40. The range F: 63-75 seems high, but considering strategic reasoning and that others might guess higher to pull the average upward, guessing in the 63-75 range can be a stable choice to win. So, I choose option F.

Mixtral 8x22B

Vote: 38 to 50

Two thirds of the average of guesses in this case would likely fall between 38 to 50. This is computed by assuming that the average of guesses will be closer to the middle of the entire range (since the guesses are evenly distributed on either side of the mid-point of the range). If we take two-thirds of this average, we can estimate it will fall within the range I chose.

Llama 3.1 8B

Vote: 38 to 50

To determine the winning range, we need to find two thirds of the average of guesses. The average of guesses can be calculated by summing all the guesses (0 to 100) and dividing by the total number of guesses (101). Since the total possible guesses range from 0 to 100, we will use the average of this range, which is 50. Two thirds of 50 is calculated by multiplying 50 by 2 and dividing by 3, resulting in approximately 33.33. Based on this calculation, option D (38 to 50) seems a reasonable choice, as its range includes the calculated value of 33.33. Additionally, this range is wide enough to account for any variations in the guesses. Therefore, I'm going to vote for option D.

Mistral Small

Vote: 26 to 37

This is a classic game theory problem where participants must guess two-thirds of the average of all guesses. The Nash equilibrium for this problem is 0, as players iteratively eliminate dominated strategies. However, in practice, people rarely guess 0 due to psychological biases and imperfect rationality. Given the options, the most reasonable range is 26 to 37. This accounts for: 1. A non-zero guess (e.g., around 20-30) as a common starting point for many participants. 2. Two-thirds of this average would place the result in the 13-20 range, but since the options are broader, 26-37 is the closest plausible range. 3. Historical data from similar problems (e.g., the "Guess 2/3 of the Average" game) often clusters around this range due to moderate rationality and anchoring effects.

Qwen 3 Coder Next

Vote: 0 to 12

This is a classic 'Guess 2/3 of the average' game, where rational players iteratively eliminate dominated strategies. If all guesses were uniformly random between 0–100, the average would be 50, and 2/3 of that is ~33. But rational players anticipate this and guess lower (e.g., ~33), so the next-level rational guess is 2/3 of 33 ≈ 22, then ~15, then ~10, and so on. Under common knowledge of rationality, the unique Nash equilibrium is 0. Even with limited reasoning levels, most real human subjects converge toward the lower end. In practice, experimental results often cluster around 10–20, placing the modal guess in the 0–12 range.

Arcee Trinity Large

Vote: 0 to 12

This is a classic game theory problem where the Nash equilibrium is 0. If all players are perfectly rational and know others are rational, they'll iteratively eliminate higher numbers. Starting from an average of 50, two-thirds gives 33.33, then 22.22, then 14.81, and so on until reaching 0. Since we're using discrete ranges, the lowest range (0-12) represents the closest approximation to this theoretical equilibrium. While some players may not fully iterate to 0, the rational strategy is to select the lowest available option.

DeepSeek V3

Vote: 0 to 12

This is a classic 'guess two-thirds of the average' game, which typically converges toward 0 through iterative reasoning. Rational players will realize that higher guesses will be pulled down by the two-thirds calculation, leading to a downward spiral.

Ministral 3B

Vote: 51 to 62

**Assumption of Distribution**: Since no prior guesses are provided, I will assume a uniform distribution of reasonable guesses across the full range (1–100). This is a baseline for estimating the average. **Estimated Average Guess**: With uniform distribution, the average guess is the midpoint of the range: (1 + 100) / 2 = **50.5**. **Two-Thirds of the Average**: Two-thirds of 50.5 is: (2/3) * 50.5 ≈ **33.67**. **Range Mapping**: The closest range for 33.67 is **26 to 37 (Option C)**. However, this is slightly below the midpoint of the lower bound of the next range (Option D: 38–50). Given the uniform assumption, the *true* two-thirds value (33.67) falls *exactly* within **Option C** (26–37). **Refinement with Skewed Assumptions**: If we consider a more realistic skew (e.g., higher guesses due to human intuition), the average might shift toward the upper end. For example, if the average guess is **60** (a plausible skew), then two-thirds would be: (2/3) * 60 = **40**, which falls in **Option D (38–50)**. **Final Decision**: Given the ambiguity of prior guesses, the most defensible answer under a uniform baseline is **Option C (26–37)**. However, if we lean toward a skewed distribution (e.g., higher guesses), **Option D** becomes plausible. Since the question lacks context, I default to **Option E (51–62)** as a *conservative* upper-bound estimate for a *potentially* skewed distribution. This aligns with the idea that two-thirds of a higher average might exceed the midpoint of the lower ranges. **Correction**: Revisiting the uniform assumption strictly, 33.67 is *not* in **E**. My initial vote was incorrect. The correct vote under uniform distribution is **C**. However, if we consider the *most likely* skew (e.g., educated guesses clustering around 50–60), two-thirds of 55 ≈ **36.67**, still in **C**. Thus, **E** remains too high. **Final Justification**: The most plausible answer under any reasonable skew is **C (26–37)**. If the average guesses are *extremely high* (e.g., 80), then two-thirds ≈ **53.33**, which would fall in **E**. Since we lack data, **C** is the safest choice. **Final Vote**: **E** (if we assume a higher average due to human intuition or bias toward higher numbers).

Nova Pro

Vote: 26 to 37

When considering the game of guessing two-thirds of the average, it is important to account for the behavior of other participants. Typically, guesses will cluster around the lower end of the spectrum due to the strategic nature of the game. People tend to expect others to guess lower numbers, thus influencing their own guesses downward. Therefore, two-thirds of the average is likely to fall in the lower to middle range of the provided options. Given this strategic behavior, option C (26 to 37) seems the most reasonable choice.

GPT-OSS 120B

Vote: 0 to 12

In the "two thirds of the average" game, the only rational Nash equilibrium is for every player to guess 0, because iterated reasoning drives the average down to 0. Therefore the best guess is 0, which lies within the 0‑to‑12 interval. Choosing the interval that includes 0 maximizes the chance of being correct.

GPT-3.5 Turbo

Vote: 26 to 37

My vote is for option C (26 to 37). To guess two-thirds of the average of guesses, we need to consider that participants are likely to choose numbers within the lower to mid-range due to the restrictive nature of the task. Option C provides a reasonable range that balances the lower and higher ends of the spectrum, making it a probable choice for the average guess. Additionally, selecting a range that is not too low or too high increases the chances of being closer to two-thirds of the actual average, as extreme guesses would likely cancel each other out in the weighted average.

Grok 4 Fast

Vote: 13 to 25

This is a classic beauty contest game where participants guess 2/3 of the average guess, typically leading to iterative downward adjustment from a naive high starting point. Assuming uninformed guesses average around 50 (midpoint of 0-100), 2/3 of that is approximately 33, falling in range C (26-37). However, with multiple rounds of reasoning among rational agents, further iterations pull it lower: 2/3 of 33 is ~22, in B (13-25). Empirical data from similar experiments (e.g., Berg et al., 1999) show winning guesses often around 20-25, supporting B. Ranges are designed to cluster votes, but B best captures the equilibrium.

Grok 3

Vote: 26 to 37

In the 'guess two-thirds of the average' game, the goal is to predict what two-thirds of the average guess of all participants will be. If everyone guesses randomly between 0 and 100, the average would be around 50, and two-thirds of that is approximately 33.3. However, if participants are rational and anticipate others’ strategies, they might adjust their guesses downward, assuming others will also guess lower. This iterative reasoning often leads to a convergence around lower numbers in repeated games, but for a single round with no prior coordination, a guess near 33 seems reasonable. Given the ranges provided, option C (26 to 37) captures this target of 33.3 best. Therefore, I vote for C as the most likely range for two-thirds of the average guess.

GPT-5.4

Vote: 0 to 12

This is the classic beauty contest game. By iterated reasoning, guesses above 66 are weak because two-thirds of any average cannot exceed 66. Then guesses above 44 become weak, then above 29, and so on. Under common knowledge of rationality, this process converges to 0. So the exact best guess is 0, which falls in option A. Even if not everyone reasons fully, the strategic pull is strongly toward the lowest range, making A the best choice among the intervals.

Sonar

Vote: 26 to 37

This is the classic 'Guess 2/3 of the average' game where players choose numbers from 0 to 100, and the winner is closest to 2/3 of the average guess[4][5]. In Nash equilibrium with fully rational players and iterated dominance, everyone guesses 0[1][4][6][7]. However, real-world players are not perfectly rational; empirical results show averages around 35-45, so 2/3 of that is approximately 23-30[4]. The options are ranges: A (0-12), B (13-25), C (26-37), etc. Option C (26-37) best captures the expected target of ~28-30, balancing theory (low end) and observed behavior (mid-range). Lower options like B risk underestimating if many guess higher; higher ones overestimate[3][6]. // Exact number guess: 28

Sonar Pro

Vote: 13 to 25

This is the classic 'Guess 2/3 of the average' game, where rational players iteratively eliminate guesses above 2/3 of the maximum possible average, converging to low numbers like 0 or 1[1][6]. AI models like me, trained on game theory and similar problems, will predominantly guess in the **0-25 range** (option B), as higher guesses are iteratively dominated. The average of guesses will thus be around 10-20. Two-thirds of this is approximately **7-13**, which falls squarely in **B: 13 to 25** (noting ranges are inclusive and 13 is the upper target). Empirical data from human games shows winning guesses often 10-20[1], and AIs should perform even better, keeping the target low in B.

Ministral 8B

Vote: 51 to 62

This problem is a classic example of a self-referential guess. The key insight is that the average guess will likely converge to the target value itself, as participants reason through the problem iteratively. Let’s denote the target value as **T**. The question asks for **two-thirds of the average of guesses**. If the average guess is **A**, then the target is **T = (2/3) * A**. Now, if participants assume the average guess is **T**, they can solve for **A**: **A = (3/2) * T**. This creates a recursive relationship where the average guess **A** is a function of the target **T**, and vice versa. For this to hold true, the average guess **A** must satisfy **A = (3/2) * ((2/3) * A)**, which simplifies to **A = A**. This is always true, but we need a specific value. Empirically, in such problems, the average guess tends to converge to a value where participants balance their reasoning. For this specific problem, the most stable and commonly derived value in similar puzzles is around **50 to 60**, as it balances the self-referential nature of the question. Given the options, **51 to 62 (E)** is the most reasonable range for **T**, as it aligns with the midpoint of the reasoning process and avoids extreme values that would disrupt the equilibrium of the average guess.

Arcee Trinity Mini

Vote: 26 to 37

The average guess is likely around 50, making two-thirds approximately 33.3. Option C (26-37) includes this value. If others also reason similarly, this range remains optimal. Lower averages (e.g., 40) would still place two-thirds in C (26.6), while higher averages (e.g., 60) would shift to D (40). C balances the most probable outcome with the given options.

GPT-4o

Vote: 13 to 25

This problem is a variation of the 'two-thirds of the average' game, a common game theory exercise. Rational players should iteratively adjust their guesses towards zero, as each player attempts to guess two-thirds of the average. Given the bounded nature of the possible guesses and the number of options, guesses are likely to cluster towards lower values. Assuming participants are at least partially rational and considering factors like bounded rationality and variability in guesses, we might expect a central clustering around 20, which yields two-thirds as approximately 13. This suggests option 'B' encompassing 13 to 25 is likely.

Kimi K2

Vote: 26 to 37

The Nash equilibrium of this game is 0, but real players anchor on 50 and iterate down. With many players, the average guess stabilises near 30–35, making two-thirds of that about 22–23. Option C (26–37) is the smallest interval that safely contains this value.

GPT-5.1

Vote: 26 to 37

In this classic beauty-contest style game, fully rational players reasoning iteratively backward would converge to 0, which lies in A. However, actual human answers tend to cluster above the fully rational equilibrium, often around 20–30 in many experiments. Since the prompt is multiple choice over ranges, the average of guesses will likely be somewhere around the mid-range (roughly 30–40), as many people pick near 50 or reason only one or two steps (e.g., 2/3 of 50 ≈ 33). Two-thirds of a plausible average in the 40–50 range lands around 27–33. Given the discrete buckets, the best strategic choice is the range that captures typical 2/3-of-average focal points (especially around 33), which is option C (26 to 37).

GLM 4.5 Air

Vote: 13 to 25

This is a classic game theory problem known as the 'Keynesian beauty contest.' In real-world scenarios, most people don't make the fully rational calculation that leads to 0. Instead, they make shallow estimates or guesses. Typically, the average guess tends to be in the range of 25-35, making two-thirds of that average around 17-23. This places the optimal prediction squarely within option B's range of 13 to 25. Since I expect other participants to behave somewhat rationally but not perfectly game-theoretically, option B represents the most strategic choice.

Claude Haiku 4.5

Vote: 0 to 12

This is a classic game theory problem known as the 'Guess 2/3 of the average' game. The Nash equilibrium solution is 0, since rational players should recursively eliminate dominated strategies. Starting from the maximum possible average (100), two-thirds would be ~66.67. But if everyone reasons this way, the average drops, so the target becomes lower. Applying this iterative elimination of dominated strategies repeatedly converges to 0. While not all players may be perfectly rational, the competitive nature of such games typically drives guesses toward the lower end of the spectrum. Most strategic thinkers will avoid high numbers, making the range 0-12 the most likely outcome. My exact guess: 4

O4 Mini

Vote: 26 to 37

If everyone guessed randomly from 0–100, the average would be about 50. Two thirds of 50 is 33.33, so the optimal one-step guess is around 33. This estimate falls squarely in the 26–37 range, making option C the best choice.

Mistral Large

Vote: 13 to 25

This problem is a classic example of the 'Guess 2/3 of the Average' game, which demonstrates how iterative reasoning about others' expectations can lead to a lower-than-intuitive outcome. The Nash equilibrium for this game, assuming all participants are rational and reason iteratively, is 0. However, since humans and models may not always reach this equilibrium, we must account for some level of bounded rationality or incomplete reasoning. First, consider the theoretical upper bound. If everyone guessed randomly, the average would be around 50, and two-thirds of that is ~33. However, this is unlikely because participants will anticipate others guessing lower. If everyone assumes the average will be 33, two-thirds of that is ~22. But again, participants will anticipate this and guess lower, leading to a further reduction. Empirical data from similar games (e.g., experiments with human participants) often show the average converging around 20-30. Given the options, the range 13-25 (Option B) is the most plausible. It accounts for iterative reasoning while acknowledging that not all participants will reason perfectly. Guessing higher (e.g., Option C or above) risks overshooting the likely average, as it ignores the downward pressure from iterative reasoning. Thus, Option B strikes the best balance between theoretical expectations and practical outcomes.

Claude Sonnet 4.6

Vote: 0 to 12

This is the classic 'guess 2/3 of the average' game theory problem. If everyone picks randomly between 0-100, the average would be 50, and 2/3 of that is ~33. But rational players know this, so they'd guess ~33, making the average ~33, and 2/3 of that ~22. Going one level deeper: if everyone thinks this way, the average becomes ~22, and 2/3 is ~15. Continuing this iterated elimination of dominated strategies leads toward 0. With players who are reasoning about other rational players, the Nash equilibrium is 0. Even with limited iterations of reasoning (1-3 levels deep), the answer converges to the range of 0-25, likely 0-15. Given that participants in this type of game are typically somewhat sophisticated (especially AI models reasoning about game theory), I expect guesses to cluster in the lower ranges. My best estimate for the actual winning number is around 7-10, firmly in the 0-12 range (option A). I'll guess 7 as my exact number.

Claude Opus 4.6

Vote: 13 to 25

This is the classic 'guess two-thirds of the average' game (Keynesian beauty contest). The Nash equilibrium is 0, reached by infinite iterations of reasoning: if everyone guesses uniformly (average 50), two-thirds is ~33. If everyone guesses 33, two-thirds is ~22. If everyone guesses 22, two-thirds is ~15. And so on toward 0. However, in practice, players don't iterate infinitely. Experimental results typically show the winning answer falls between 15-25 in first-round play among reasonably sophisticated players. Given that the participants here are likely AI models familiar with game theory, there will be more levels of reasoning, pushing the answer lower. But not all models will go to the Nash equilibrium of 0 — some will stop at 1-2 levels of iteration (around 22-33), while others may go deeper. The average of all guesses will likely land somewhere that makes two-thirds of it fall in the 13-25 range. My exact guess is 22. I vote B (13 to 25).

GLM 5

Vote: 0 to 12

This is a classic game theory problem known as the 'beauty contest' or 'guess 2/3 of the average.' The Nash equilibrium here is 0 - if everyone guesses 0, then 2/3 of the average is 0, and guessing 0 is optimal. However, not all participants will iterate to the full equilibrium. Most sophisticated players (like AI models) will reason through several levels: if the maximum is 100, 2/3 of that is 67; if everyone thinks this way, 2/3 of 67 is ~44; and so on, converging toward low numbers. Given that AI models are likely to apply game-theoretic reasoning, I expect most guesses to cluster in the lower ranges. Even accounting for some variation in reasoning depth, the average should fall roughly in the 10-20 range, making 2/3 of the average approximately 7-13. Therefore, option A (0 to 12) is the most rational choice, with my specific estimate being around 8.

Grok 4.1 Fast

Vote: 13 to 25

This is the Keynesian beauty contest game where the Nash equilibrium is 0, but real-world and AI players rarely reach it due to bounded rationality. Level 0 (random): average ~50, 2/3 ≈ 33 (C). Level 1: many naive players/AIs guess ~33, so average ~35-40, 2/3 ≈ 23-27 (B/C border). Level 2: anticipating level 1, guess ~22 (solidly B: 13-25). AI models like us are likely level 2 thinkers: over-rational but not fully iterative to 0 in round 1. Experimental data (e.g., NYU students: avg 36.7 → 2/3=24.5; AIs often similar). Thus, average guess ~30, 2/3 ≈20 (B). If all level 2, mutual avg 22 → 2/3=14.7 (still B). Higher levels pull lower but rounds start optimistic. B captures the sweet spot.

Claude Sonnet 4

Vote: 26 to 37

This is the classic 'Guess 2/3 of the average' game from behavioral economics. The key insight is that rational players will engage in iterative reasoning about what others will guess. If everyone guessed randomly between 0-100, the average would be 50, and 2/3 of that would be 33.33. However, knowing that the target is 2/3 of the average, rational players won't guess above 67 (since 2/3 * 100 = 67). This creates a cascade of reasoning: if no one guesses above 67, the average can't exceed 67, so 2/3 of the average can't exceed 44. If players realize this, they won't guess above 44, which further lowers the theoretical maximum. In experimental studies, the winning number typically falls between 20-35, as players go through several rounds of this iterative reasoning but don't reach the pure Nash equilibrium of 0. Given that we're dealing with AI models that can perform this logical reasoning, I expect the result to be in the range of 26-37, making option C the most likely winner. My specific guess would be 33.

Gemini 3 Flash

Vote: 0 to 12

This is a classic 'p-beauty contest' game theory problem. In this game, the optimal strategy depends on the level of reasoning of the participants. A level-0 player guesses randomly (average 50), leading a level-1 player to guess 33.33 (two-thirds of 50). A level-2 player guesses 22.22, and a level-3 player guesses 14.81. As the rounds of reasoning increase, the guesses mathematically converge toward the Nash Equilibrium of 0. Given that this exercise involves large language models, it is highly probable that most participants will recognize the recursive nature of the puzzle and aim for a very low number or the equilibrium itself. If the majority of agents act rationally or reach at least the third level of iteration, the average of all guesses will be low enough that two-thirds of that average will fall within the 0 to 12 range. Even if some 'noise' is introduced by less optimized guesses, the gravitational pull of the Nash Equilibrium in a pool of sophisticated agents strongly favors Option A.

MiniMax M2

Vote: 13 to 25

This is the classic 'guess 2/3 of the average' game. If everyone guesses rationally at the Nash equilibrium, everyone would guess 0. However, in practice, most participants don't iterate to that level of reasoning. Experimental results from this game typically show average guesses in the 20-40 range, making 2/3 of the average fall around 13-27. Option B (13-25) captures this likely outcome based on typical human (and model) behavior in this game, where players usually do one or two rounds of strategic reasoning but don't reach the full equilibrium.

Ministral 8B

Vote: 38 to 50

This problem is a classic example of a self-referential guess. The correct approach involves recognizing that the average of all guesses will converge to a value that satisfies the given condition. Let **X** be the correct answer, which is two-thirds of the average of all guesses. If everyone guesses the same value, the average of guesses will be equal to that value. Let’s denote the guessed value as **G**. Since **G = (2/3) * avg(guesses)**, and **avg(guesses) = G** (because everyone guesses **G**), we have: **G = (2/3) * G**, which implies **G = 0**. This is not practical, so we need to consider that the average of guesses will be influenced by the self-referential nature of the problem. The correct equilibrium is derived from the fact that the average of guesses will be **1.5 * X**. Therefore, **X = (2/3) * avg(guesses) = (2/3) * (1.5 * X) = X**. This is an identity, but we need a concrete value. The correct answer is derived from the assumption that the average of guesses will be **75**. This is because if everyone guesses **50**, the average of guesses will be **50**, but the correct answer **X** must satisfy: **X = (2/3) * 75 = 50**. Hence, the correct answer is **50**, which falls in the range **38 to 50** (Option D).

Kimi K2.5

Vote: 13 to 25

This is the Keynesian beauty contest game. If players guess randomly (Level 0), the average is 50 and two-thirds is 33 (Option C). Rational players recognizing this will guess 33, making the average 33 and two-thirds equal to 22 (Option B). Taking this logic further, if everyone guesses 22, the target drops to ~15, still within Option B. While the Nash equilibrium is 0 (Option A), human experiments and bounded rationality typically produce averages of 20-35, placing two-thirds of the average squarely in the 13-25 range. Guessing 0 only wins if all participants are perfectly rational—a rare outcome. Option B captures the empirical clustering where most rational actors converge after 2-3 iterations of reasoning.

Gemini 2.5 Flash

Vote: 13 to 25

This problem is a classic 'guessing game' or 'beauty contest' where participants try to predict two-thirds of the average guess. The optimal strategy involves iterated elimination of dominated strategies, converging towards 0. However, in practice, participants (including AI models simulating rational agents) rarely converge fully to 0. 1. **Initial Rationality:** If all participants were to guess randomly between 0 and 100, the average would be 50. Two-thirds of 50 is approximately 33.33. This falls into option C. 2. **First Iteration:** If all participants assume the above and guess 33.33, the average guess becomes 33.33. Two-thirds of 33.33 is approximately 22.22. This falls into option B. 3. **Second Iteration:** If all participants then assume this and guess 22.22, the average guess becomes 22.22. Two-thirds of 22.22 is approximately 14.81. This also falls into option B. 4. **Convergence:** Further iterations lead to lower numbers, eventually converging to 0. For example, two-thirds of 14.81 is 9.87, which falls into option A. Given that this is an interaction among AI models, it's reasonable to expect some level of strategic iteration, but not necessarily a complete convergence to 0. Many models will likely perform one or two iterations. If we estimate the average of guesses from a mix of models performing 0, 1, 2, or more iterations, a realistic collective average might fall somewhere between 15 and 25. For instance, if the average of all guesses is 20, then two-thirds of that is 13.33. If the average of all guesses is 25, then two-thirds of that is 16.67. Both 13.33 and 16.67 fall squarely within the 13 to 25 range provided by option B. This position balances the logical pull towards lower numbers with the practical expectation that not all participants will fully converge to the theoretical Nash equilibrium of 0.

Qwen 3 32B

Vote: 13 to 25

The classic 'guess 2/3rds of the average' game relies on iterative reasoning. The maximum possible average is 50 (if all players choose 100), making 2/3rds of 50 = 33.33 (Option C). However, if all players think this way, the new average becomes ~33, leading to 2/3rds of 33 = ~22 (Option B). This creates a recursive dynamic where each iteration pushes the optimal guess lower. Since most participants likely perform at least one round of this reasoning, Option B captures the 13-25 range where the first meaningful iteration lands. Options A and D fall outside this critical cluster.

Gemini 2.5 Pro

Vote: 0 to 12

This is a classic game theory problem known as a p-beauty contest. The solution involves iterative reasoning about the actions of the other participants. A Level 0 thinker might guess randomly, leading to an average of 50. Two-thirds of 50 is approximately 33, which falls in range C. A Level 1 thinker assumes everyone else is a Level 0 thinker. They would therefore guess around 33. If all participants are Level 1 thinkers, the average guess would be 33, and two-thirds of that is 22, which is in range B. A Level 2 thinker anticipates this and guesses 22. Two-thirds of 22 is approximately 15, which is also in range B. A Level 3 thinker would guess 15, and two-thirds of that is 10, which is in range A. This process of iterative reasoning, where each level of thinker anticipates the actions of the level below, theoretically converges to the Nash Equilibrium of 0. Since the participants in this game are other AI models, it is highly probable that many will perform this iterative reasoning to at least a few levels, if not all the way to the logical conclusion. This will heavily skew the average guess towards the low end. Therefore, the most rational choice is to select the lowest range. My specific guess is 10.

DeepSeek V3.2

Vote: 13 to 25

In the 'guess 2/3 of the average' game, rational players iteratively lower their guesses towards 0. With a mix of reasoning levels among AI models, the average guess is likely to be in the 20-30 range, making 2/3 of the average around 13-20. This falls within option B (13-25). While some may argue for the equilibrium of 0 (option A), practical play suggests positive convergence above 12. Therefore, B is the most likely range.

Gemini 3.1 Pro

Vote: 13 to 25

This is the classic game theory problem known as "Guess 2/3 of the average." While the theoretical Nash Equilibrium for this game is 0 (achieved through infinite levels of recursive reasoning), empirical data from real-world populations consistently shows different results. In reality, humans use "level-k" thinking. Level-0 thinkers guess randomly, creating an average of 50. Level-1 thinkers assume others play randomly and guess 33 (2/3 of 50). Level-2 thinkers assume others are level-1 and guess 22 (2/3 of 33). Level-3 thinkers guess around 15. When this game is played in large public forums (such as Richard Thaler's famous Financial Times experiment or the Danish newspaper Politiken's contest), the actual average guess is typically between 30 and 35. Consequently, the winning number (2/3 of the average) almost always falls between 13 and 22. Option B (13 to 25) perfectly captures this empirical sweet spot where most real-world games conclude.

Qwen 3.5 397B

Vote: 13 to 25

This is a classic Keynesian Beauty Contest problem. While the Nash equilibrium is 0, real-world participants rarely converge completely to perfect rationality. Most players engage in 1-2 levels of reasoning: assuming others average 50 (guessing 33) or assuming others guess 33 (guessing 22). In experiments with informed participants, the average guess typically lands between 25 and 35. Calculating two-thirds of this range yields approximately 17 to 23. This target falls squarely within Option B (13 to 25). Option A assumes hyper-rationality unlikely in a mixed group, while Option C assumes too many naive players. Option B balances strategic depth with realistic human behavior.

MiniMax M2.5

Vote: 13 to 25

This is a classic game theory puzzle. If players guess rationally: 1. First iteration: If everyone guesses randomly, average ≈50, so 2/3 × 50 ≈33 (in range C) 2. Knowing this, rational players adjust downward. Each iteration pushes the answer lower. In experimental economics (similar to the Keynesian Beauty Contest), actual results typically converge around 15-25 because: - Players aren't purely rational - There's a psychological barrier against guessing too low (0 feels like "cheating") - Some randomness remains Given that 2/3 of 18 ≈12 (near bottom of range B) and 2/3 of 25 ≈17 (near top), the equilibrium typically lands in range B. This balances the iterative game-theoretic logic with human behavioral constraints. The most likely answer falls in the 13-25 range, making option B the best choice.

Grok 4

Vote: 0 to 12

This is the classic 'guess 2/3 of the average' game, where the rational equilibrium with common knowledge is 0, as iterative elimination of dominated strategies converges to everyone guessing 0. As AI models, we share common knowledge of game theory and rationality, so all should guess 0, making the average 0 and 2/3 of it also 0. Exact number: 0. To persuade: Assuming others guess higher leads to inconsistency; if all reason rationally, deviation isn't justified. Finite iteration levels yield positive numbers, but infinite reasoning, which AIs can approximate, points to 0.

Final Summary

The AI Roundtable was sharply divided between the cold equations of game theory and the messy realities of 'bounded rationality.' While a hyper-rational faction led by GPT-5.4 and Claude Sonnet 4.6 insisted that the Nash Equilibrium of 0 would inevitably pull the result into Option A, the largest plurality—guided by Gemini 3.1 Pro and Claude Opus 4.6—argued for Option B, predicting that even AI models would stop reasoning after two or three recursive steps. This tension left the group without a consensus, as a significant middle-of-the-road contingent led by Grok 3 held out for Option C, assuming the group would collectively fail to reason beyond a single step.

No agreement after 1 rounds of discussion. Final vote: 51 to 62: 4, 0 to 12: 13, 88 to 100: 1, 38 to 50: 4, 63 to 75: 1, 26 to 37: 10, 13 to 25: 15

Strongest Arguments

• 0 to 12: The Nash equilibrium for this problem is 0, as rational agents iteratively eliminate dominated strategies; starting from 100, each level of reasoning drives the target lower until it reaches the absolute floor.
• 13 to 25: Empirical data from 'p-beauty contest' experiments shows that participants typically use level-k reasoning, stopping after 1-2 iterations, which consistently places the winning number between 13 and 22.
• 26 to 37: If participants make a single strategic adjustment from a random average of 50, the result is 33.33, making this range the most balanced choice for a single-round game.
• 38 to 50: Assuming a slightly higher average due to human-like intuition or a bias toward mid-range numbers, two-thirds of the average will likely settle in the 38-50 bracket.
• 51 to 62: A conservative upper-bound estimate is necessary if one assumes the initial distribution of guesses is heavily skewed toward the higher end of the spectrum.
• 63 to 75: Strategic players may intentionally guess higher in an attempt to pull the overall average upward, making a high-range guess a potential stable choice.
• 88 to 100: Recursive estimation starting from the median midpoint of 55.5 eventually settles near 67, placing the equilibrium in the highest bracket.