Evaluating Language Models' Evaluations of Games
Authors: Katherine M. Collins
Cedegao E. Zhang
Graham Todd
Lance Ying
Mauricio Barba da Costa
Ryan Liu
Prafull Sharma
Adrian Weller
Ionatan Kuperwajs
Lionel Wong
Joshua B. Tenenbaum
Thomas L. Griffiths
Venue: ICLR 2026 Poster (2026)
Keywords: game AI; meta-reasoning; cognitive science; problem evaluation
Primary area: applications to neuroscience & cognitive science
TL;DR: Reasoning is not just about solving new problems, but deciding what problems to solve in the first place. We should evaluate LMs on evaluation.

Abstract: Reasoning is not just about solving problems---it is also about evaluating which problems are worth solving at all. Evaluations of artificial intelligence (AI) systems primarily focused on problem solving, historically by studying how models play games such as chess and Go. In this paper, we advocate for a new paradigm that assesses AI systems' evaluation of games. First, we introduce a formalism for evaluating such evaluations. We then leverage a large-scale dataset of over 100 novel board games and over 450 human judgments to compare evaluations produced by modern language and reasoning models against those of people and symbolic computational agents. We consider two kinds of evaluative queries: assessing the payoff (or fairness) and the funness of games. These queries span two dimensions relevant to the design of evaluations of AI evaluations: how complex a query is to compute and how difficult a query is to quantify. Our results show that reasoning models are generally more aligned to people in their evaluations of games than non-reasoning language models. However, we observe a non-monotonic relationship: as models get closer to game-theoretic optimal, their fit to human data weakens. We also observe more "jaggedness" across models for assessing funness, in line with the greater difficulty of quantifying this query. Across queries and games, reasoning models show highly variable and unpredictable resource usage when assessing queries, pointing to the importance of imbuing more resource-rational meta-reasoning in language and reasoning models.

Published as a conference paper at ICLR 2026
EVALUATING LANGUAGE MODELS' EVALUATIONS OF GAMES
Katherine M. Collins 1.3,3 Cedegao E. Zhang?
Lance Ying! Sharma? Adrian Weller
Graham Todd*
Mauricio Barba da Costa²
Ryan Liu
Ionatan Kuperwajs Lionel Wong
Joshua B. Tenenbaum Thomas L. Grifliths
UUniversity of Cambridge, "MIT. Princeton University, "NYU, Harvard University, Stanford University
"Corresponding author: katieme @ mit.edu *Co-senior authors
ABSTRACT
Reasoning is not just about solving problems- it is also about evaluating which problems are worth solving at all. Evaluations of artificial intelligence (Al) systems primarily focused on problem solving, historically by studying how models play games such as chess and Go. In this paper, advocate for a new paradigm that assesses Al systems' evaluation of games. First, we introduce a formalism for evaluating such evaluations. We then leverage a large- scale dataset of over 100 novel board games and over 450 human judgments to compare evaluations produced by modern language and reasoning models against those of people and symbolic computational agents. We consider two kinds of evaluative queries: assessing the payoff (or fairness) and the funness of games. These queries span two dimensions relevant to the design of evaluations of AI evaluations: how complex a query is to compute and how difficult query is quantify. Our results show that reasoning models are generally more aligned to people in their evaluations of games than non- reasoning language models. However, we observe a non-moneee monooonne relationship: models get closer to game-theoretic optimal, their fit to human data weakens. We also observe more "jaggedness" across models for assessing funness, in line with the greater difficulty of quantifying this query. Across queries and games, reasoning models show highly variable and unpredictable resource usage when assessing queries, pointing to the importance of imbuing more resource-rational meta-reasoning in language and reasoning models.
INTRODUCTION
The ability to play games has long been used as a measure of assessing reasoning in artificial intelligence (AI) systems. From chess (Turing, 1950; Campbell et al., 2002; Newell et al., 1958) to Go (Silver et al., 2016) to poker (Brown and Sandholm, 2018) and now ARC-AGI (ARC Prize Foundation, 2025) and Pokémon (Anthropic, 2025; Karten al., 2025), AI systems have consistently been evaluated on their ability to play games. The AI community ever-expanding the set of games inventing new games (Ying et al., 2025; Verma al., 2025)-to test
used in these assessments- the flexibility of AI systems' reasoning. However, these efforts offer a partial picture of the general reasoning capacity of AI systems. Reasoning
just about playing games or solving problems, but also evaluating higher order aspects of the problems themselves. like whether game is worth playing in the first place (see Figure la; Wong et al. 2025; Griffiths 2020; Chu al. 2023; Getzels 1987).
There are many ways to evaluate game, and they are not all equally interesting. Determining whether game is cooperative or competitive, for instance, is often relatively trivial: it does not require substantial compute and the query itself is unambiguous. payoff of an arbitrary game is more interesting-in requires precise and complex computation (e.g.. over likely game states). Formally assessing whether a game is likely to be "fun" adds a further layer of complexity, given the difficulty of determining how to quantify the answer to such a question which in turn, may also be difficult to compute. Yet a measure, like those studied utilitarian ethics, may be hard to quantify. That is it might be hard to quantify what values to assign to different quantities
contrast, assessing the expected

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Given game.....
5 board. pieces in row wins. Second player can play twice on their first turn.
a
Evaluate problem solving
Evaluate problem evaluation
Which play as?
Hard to compute
What is the expected payot! of game!
How fun is this class game?
Who's likely to win?
What kind of game is
this cooperative compecitive ganne?
I would move in position (3,
Player 2 Is likely to win. The expected payoff for Player 1 -0.2
Hard to quantify
Figure 1: Evaluating AI systems' evaluations. a, A holistic understanding of model reasoning demands not just assessing how Al systems solve problems (play games), but how they evaluate whether problems, systems, or games are worth pursuing at all: b, Not all evaluations of problems are interesting for evaluating models. Good evaluation queries pose a challenge by being difficult to compute, difficult quantify, or both.
(such as human lives or irreplaceable works of art), but it is easy to compute the sum of these values when considering a possible outcome of action. This highlights two dimensions of evaluations: (1) difficulty to compute, and (2) difficulty to quantify (see Figure lb). These dimensions are relevant when evaluating the evaluations produced by AI systems and inform the kind of human data want to collect to compare these systems against. For example, human data may be more variable for queries that are harder quantify (though also more relevant to real-world situations), and even a measure like payoff can still be difficult to quantify if it requires determining what counts as "reasonable" play over which to compute expected outcomes (see Appendix A7).
In this work, we lay out perspective on evaluating models on their capacity to-not just play- but evaluate games. We then take initial steps to empirically assess language models on their capacity to conduct such evaluations. To do so, we draw on a corpus of 121 novel games from Collins al. (2025). For each game, we test series of language and reasoning models using two reasoning queries that engage both dimensions of evaluation: one that is difficult to compute and another that is difficult to compute and quantify. That is, we ask the models to evaluate: (I) the expected outcome compute the expected value payoff), and (2) the perceived funness of the game. Critically, these evaluations are meant to capture reasoning about game before any actual play, akin to someone deciding whether game, goal, task is worth their limited time and energy to engage with. We compare the evaluations produced by the models to those made by people and to a series of explicit gameplay (non-language-modee baselines. The baselines include both agents drawn from AI and computational cognitive modeling as well as, for games where it can be computed, the game-theoretic optimal payoff. Evaluating game evaluations raises the question of what evaluations to measure against-ree researchers may strive to build agentic systems that are as rational as possible (c.g.. close the game-theorete optimal) or more human- -liined for effective thought partnering (Collins et al., (724)-questionn which may be even harder when evaluation measure is difficult to quantify objectively (e.g., funness). We explore both directions this work.
of the game (from which
We find that non-reasoning language models, which directly produce game evaluations without using intermediate chains of thought, substantially differ from people's 's evaluations of games as well as the optimal game-theoretic expected payoff. These models' evaluations of the expected payoff of games are highly similar, even across different model families, suggesting that models may have picked up similar inductive biases about what makes game fair based on shared training data. However, these evaluations remained substantially different from those produced by humans, indicating that such biases are insufficient for computing human-aligned evaluation query. In
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contrast, allowing models reason through an intermediate chain-of -httuugghttttttthhh generally yields more sensible game evaluations relative to the peoples' evaluations. Reasoning models are generally more aligned with both human judgments of expected payoffs and estimates obtained from our non-linguistic baselines. We observe a non- monotonic relationship between reasoning models and humans, where eventually an increase in alignment with the game-theoretic optimal solution begins to result in a decrease in alignment with human judgments. While reasoning models also generally capture human bumanfunness, funness judgments better than non non-reasoning language models, performance across models is inconsistent (e.g.. more advanced models are not consistently more aligned to people in their funness evaluations), which matches the difficulty of quantifying "fun." And across both queries- we observe vastly different amounts of resources being used by reasoning models (as measured by reasoning tokens), motivating future work to design more resource-rationas problem-evaluation agents capable of dynamically adapting compute to the evaluation query and problem at hand (Sui et al., 2025). We close with open questions that follow from a principled study of evaluating reasoners' abilities to evaluate.
optimal, but these are often still far from
2 FROM EVALUATING SOLUTIONS TO EVALUATING EVALUATION
One common way to study problem solving capacities is through games (see Section 5). A game 9 can be represented as a series of feasible states S possible actions A: rules TT specifying valid actions and state transitions given those actions; and one or more goal functions mapping from states S to possible rewards R. Typically, problem solving (game play) is evaluated by assessing how well systems can estimate and deploy a policy (a | st) for choosing actions given state to optimize reward Ry, where T is the final turn or sum of the discounted reward over all timesteps (Sutton et al., 1998). The problem solving ability of an agent can also be assessed by measuring its efficiency in learning or estimating for the problem at hand.
However, real-world reasoning requires not just identifying what good actions are for any given prob- lem or game state, but evaluating whether problem or game is worth engaging with at all (Getzels, 1982; Nickles, 1981; Chu et al., 2023). For games, evaluation can be thought of as estimating some properties v of game G. This may involve estimating as an intermediate step (see Collins et al. 2025), but critically places the emphasis of evaluation over the entire game rather than any single action and the reward of that action. Evaluating a game for given query vp (e.g., whether a game is likely to be fun) may require breaking a query down into subqueries { [11, 1b2....by)) based on some factors F from space of factors (e.g., whether the game is fair; whether the game is likely to demand strategic thinking; how long the game is expected to be; etc.). Computing any y may then also require varying levels of computation, as laid out in Figure 1b. Breaking down larger query into different subqueries also raises the question of how solutions to these subqueries should be aggregated answer the original question.
Evaluating a game itself inherently relies on less precise criteria than evaluating a player (for which victory or reward can be used). In addition, determining whether problem is "good" might not permit objective evaluation. This renders the task of problem evaluation more nebulous, yet accordingly, also more interesting. For instance, judgments to any query (G)) could be compared to judgments made from other reasoners (e.g., people) or the judgments we may expect under a perfectly rational reasoner (e.g, game- theoreth optimal payoff, when it can be computed). Problem evaluators can also be assessed on the resource cost incurred, whether it is measured in wall-clock time, number of simulations run, or the number of reasoning tokens used.
METHODS
3.1 EVALUATIONS OVER NOVEL GAMES
We focus on the 121 two-player competitive strategy games playable on a grid from Collins et al. (2025). Games span range of variants of Tic-Tac-Toe (see Appendix A2), most of which are novel in that they have not been publicly proposed before and therefore are both unlikely to have been played by people before and unlikely to be in the model' training data. While these games do represent restricted space of the possible games one may play, many are strategically rich, capturing many hallmarks of real decision making and planning problems people face. And already, this set already
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pushes productively away from the dominant focus in Al and psychology game at time (e.g.. Chess, Go, Diplomacy; see Appendix A1). Approximately 20 people evaluated each game per query (expected value and expected funness), totaling over 450 participants. People evaluated each game as "novices" before any actual play.
3.2 ELICITING MODEL GAME EVALUATIONS
We prompted a series of language and reasoning models to evaluate the expected payoff and funness of each of the 121 games (see Appendix A3.1). Models are sampled with 20 rollouts (to match the approximately 20 people who responded for each game query) using their default temperature (1.0 for o1, 03, and GPT-5, 0.7 for other models). In the main text, all reasoning model results are reported under medium reasoning effort; we explore other reasoning settings Appendix A5,1, We also compare against a series of game reasoning models from Collins et al. (2025) which predict judgments by explicitly simulating gameplay between artificial agents. These agents vary in sophistication, ranging from random action selection, to heuristic- sarristic-basd "Intuitive Gamer" model that approximates novice human gameplay, to models based on more extensive tree search, namely an "Expert" model that approximates depth-5 tree search based on van Opheusden et al. (2023), and a separate Monte Carlo Tree Search (MCTS, Coulom, 2006; Genesereth and Thielscher, 2014; Silver et al., 2016) based method (see Appendix A3.4).
3.3 EVALUATION MEASURES
Our primary measure of similarity is the R2 between the averaged model judgments and human judgments (computed over all 121 games). We computed the split-half correlation between human participant judgments as a measure of the amount of explainable variance the human data. Ad- ditionally, compared models' and people's estimated payoffs in the subset of 78 games where we could compute an estimated game-theoretic optimal payoff (see Appendix A4.1). This allows us to also estimate the rationality of models relative estimated optimal payoff. We measure R?, accuracy, and distance between the predicted and estimated optimal payoff. We also measure models' similarities to each other (within the same class of models, e.g., other reasoning or non-reasoning language models, or to other classes, e.g., a game reasoner that employs MCTS- -based gameplay). We assess other measures of similarity in Appendix A4.
4 RESULTS
4.1 EVALUATING EXPECTED PAYOFF (FAIRNESS) OF GAMES
Non-reasoning language models are more similar to each other than they are to people's judgments or to tree-search based models, when comparing both the mean (Figure 2a) and distribution (Appendix Figure 8) over expected payoff of the games. This highlights some of the limits of inferring game properties purely from statistical associations in training data (i.e., without explicit reasoning or simulation). Non-reasoning language models which directly produce the game evaluation ("direct" prompting), without going through any intermediate chain-of-thought (CoT"), tend propose game evaluations that are even further from "optimally rational" (estimated game-theoretic) predicted payoff's (Table 1). Allowing non-reasoning models produce natural language chain-of-thouge before coming to the final game evaluation yields both more rational and numan-aligned fits to people, but still substantially less than more advanced reasoning models. These reasoning models are increasingly similar to both people (approaching the split-half human R² (R² = 0.82 [95% CI: 0.77, 0.86)) and the game- theoretic optimal (Table 1). However, we highlight countervailing trend in the OpenAT family of models: initially, increasing sophistication (i.e., from GPT-4 to ol and o3) corresponds with a better fit to both human judgments and game-theoretic judgments (Figure le; Figure 2a). But sophistication continues to increase (i.e., from o3 to GPT-5), the fit to human judgments degrades even as the fit to game-theoreti, judgments continues to improve (Figure Ic and Table 1), indicating worse alignment with semi-rational human participants (Figure 2b). Interestingly, however, a model like o3 can be instructed to approach game-theore game-theoretic optimal behavior, but is harder to get GPT-5 to simulate novice human-like like behavior (see Appendix A7).
These results suggest that the kind of inductive biases picked up from standard pre- and post-traine alone-at at least on the kind of web and preference data these systems have been trained mn- be
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Human
Expert Samer
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LLMM 2.1 /CB (Col)
GPT-4 (Direct)
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LaHe 708 (CoT)
GFT.4 ID rost
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Figure 2: Evaluating payoff (fairness) evaluations. a, R2 between human- and model-predicted payoff evaluations, over all 121 games. Each cell reports the R in payoff evaluations between two reasoners. b, Payoff predictions across a subset of the OpenAl model family, compared to people's predicted payoffs (blue) and the estimated game-theoretic optimal (grey). Error bars depict bootstrapped R² 95% Cls. c-d, Example human- and a subset of model podel--peedcctd game evaluations. each models' 20 rollouts are shown; the
The distribution over human participants' judgments vertical axis shows the normalized density over binned distributions. Non-reasoning models (GPT-4 and DeepSeek-v3) are prompted with CoT. c, depicts game where reasoning models are more aligned to people's 's evaluations, in other games as in d, judgments are highly varied across models with no model faithfully capturing the rich structure in the distribution of human judgments. More example games are included in Appendix A4.
insufficient to model the kinds of judgments made by novice humans, even if they are able to capture some of the underlying game-theoretical dynamics. What then can allow models to move beyond the inductive biases baked in during standard training? Are reasoning models tasked with estimating game properties actually engaging in simulated play, such as sampling actions from some latent policy? We find that more advanced reasoning models are increasingly similar in both their aggregate and distributions of predicted judgments to game reasoners that use explicit simulation (see Figure 2a and Appendix Figure 8), but these fits are nuanced at per-game level (see Figure 2c-d; Appendix A4.2). The close fit between the judgments of some reasoning models and search-based methods (e.g., o3 and the Intuitive Gamer model, or GPT-5 and MCTS) suggest that they could be engaging in some form of explicit simulation. While some closed source models do not expose the internal reasoning traces needed progress on this question. we can inspect the reasoning traces of open reasoning models (c.g., DeepSeck R1). We find that while such models do engage in some kind of explicit game simulation in portion of their games, the frequency of simulation relatively low compared to other forms of reasoning (e.g., reasoning off of analogies, or even attempting to mathematically compute the expected payoff; see Appendix Table 4). More details on trace coding are included in Appendix A6. Moreover, even though reasoning models generally align better with people's predictions, there are still discrepancies at a per-game level (see Figure 2 and Appendix A4.2), underscoring the need for expanded analyses of language models' game evaluations. Future work can better understand how different evaluation strategies of models impact the distribution of their judgments relative to people, the "optimal" expected value, and other models' judgments.
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Reasoner
R (95% CI)
Accuracy (95% CI)
Deviation (95 CT
Human
0.62 (0.58,
0.69
(0,65, 0.73)
0.32 (0.31.
Intuitive Gamer
0.69 (0.66, 0.72)
0.75 (0.72, 0.78)
0.25 (0.24, 0.26)
Expert Gamer
0.87 (0.85, 0.88)
0.92 (0.1, 0.92)
0.08 (0.08, 0.09)
MCTS
0.89 (0.88, 0.91)
0.91 (0.00, 0.92)
0.06 (0.0K, 0.07)
Random
0.39 (0.34, 0.44)
0.57 (0.55, 0.59)
0.43 (0.41, 0.44)
LLaMA 3.1 70B (Direct)
0.19 (0.17,
0.47 (0.45, 0.500
0.51 (0.50, 0.52)
GPT-4 (Direct)
0.31 (0.30, 0.33)
0.60 (0.59, 0.60)
0.42 (0:41,0042)
DeepSeek v3 (Direct)
0.35 (0.32, 0.388)
0.61 (0.58, 0.64)
0.42 (0.41,0.43)
LLaMA 3.1 70B (CoT)
0.30 (0.27,0.33)
0.48 (0:46, 0.50)
0.48 (0:45,00:99)
GPT-4 (CoT)
0.38 (0.37, 0.39)
0.59 (0.56, 0.60)
0.42 (0:42, 0.48)
DeepSeck v3 (CoT)
0.40 (0.37,
0.63
0.38 (0.37.0.39)
DeepSeek RI
0,43 (0.37, 0.48)
0.64 (0.59, 0.71)
0.40 (0.38, 0:43)
Gemini 2.5 Flash
0.53 (0.50, 0.55)
0.79 (0.76, 0.82)
0.30 (0.28, 0.31)
Gemini 2.5 Pro
0,66 (0.64, 0.67)
0.84 (082, 0.86)
0.22 (0.21, 0.23)
o1
0.50 (0.49,
0.72 0.74)
0.35 (0.34, 0.35)
0.71 (0.68, 0.73)
0.83 (0881, 0.86)
0.27 (0.26, 0.27)
GPT-5
0.82 (0.79, 0.84)
0.88
0.15 (0.14.0.16)
Table 1: Model and human predictions relative to the approximate game-theoretic optimal. Human and model payoff evaluations compared to the 78 of the 121 games where the game- theoretic optimal payoff is estimatable. Accuracy between predicted payoff and the approximate game-theoretic optimal is computed by labeling the predicted payoff as "correct" if is within 0.5 of the game- thenretccceetccll payoff (payoff E {-1, 0, 1J). R² correlation is computed between the raw predicted payoff's and the game-theoretie optimal values, as well as the average absolute difference between the expected predicted payoff and approximate game-theoretic payoff (lower is closer to the same-theoretic value). We report 95% bootstrap confidence intervals (CIs) parentheses. For the empirical data, CIs were computed by resampling participants with replacement. For the computational models, CIs were computed by bootstrapping over simulated runs.
Model
Balance
Challenge
Length
Strategic Richness
Novelty
LLaMA 3.1 70B (CoT)
47.5%
97.1%
53.0%
99.5%
56.8%
GPT-4 (CoT)
55.6%
98.6%
67.9%
98.6%
54.5%
DeepSeek v3 (CoT)
71.7%
95.7%
70.9%
98.1%
65.4%
DeepSeek RI
85.7%
99.7%
90.8%
99.3%
62.3%
Gemini 2.5 Flash
74.6%
99.9%
76.5%
100.0%
48.3%
Gemini 2.5 Pro
86.2%
100.0%
77.5%
100.0%
73.8%
Table 2: Funness factors discussed in chain-of-thought and reasoning traces during evaluation. Chain-of of-thought and reasoning traces for models evaluating fumness were inspected and automatically coded by o3 based whether they mentioned particular factors (e.g., game balance, length, etc.) in their explicated evaluative process. Percent of reasoning traces mentioning each factor are averaged over each responses per game.
4.2 EVALUATING GAME FUNNESS
Next, we evaluate language models' judgments of the funness of games. Participants and models were instructed to define funness however they wished -this is by design; we are interested in model and people's assessments of how to even define fun in the first place (a query which is both "hard to quantify" and "hard to compute.") When models answered this without going through any intermediate thain-of-though matched people's judgments (Figure 3a and Appendix Figure 13). On the other hand, models that engage in some kind of natural- langraal--ngggeaaaaeeee intermediate reasoning capture more of the variance
reasoning trace, they consistently produced results that poorly

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PP to human funness
LLAMA
304
LLMMA
12071
(C07)
GPT4 (Direct)
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Funness
Figure 3: Evaluating funness evaluations. R2 between human- and model-predicted funness evaluations, over all 121 games. Each cell reports the R² in funness evaluations between those two reasoners. b, Fumness evaluations across subset of the OpenAl model family reveals non- monotonicty in fits when moving from non-reasoning to reasoning models. Bootstrapped R² are computed relative people's s predicted funness, with error bars depicting the bootstrapped 95% Cls. c-d Example human- and model-predicted distributions over funness. The vertical axis shows the normalized density over the histogram of people and models' binned judgments. c, depicts an example where people and most models (though not all, e.g., ol) recognize that the game is unlikely be fun: d, however, people's funness judgments are also variable, e.g., showing bimodality. This bimodality is not captured by most models, with a few exceptions (c.g., GPT-5) for this game. More example game evaluations are in Appendix A4.
in human data (see Appendix Figure 8). However, comparisons across model sophistication are more "jagged" (Karpathy, 2024)-bott in capturing aggregate human judgments and the distribution of individuals" judgments (Figure compa-e to our results measuring models' evaluations of expected payoffs of games. This is likely because evaluating funness hard to quantify: estimating funness requires first determining what factors (f E F) make a game fun, then measuring the game along each of those metrics (w)y), and finally aggregating across those metrics in a sensible way. While we find that most of the chain-of-thouge and reasoning trace-based models that we can examine consider the challengingness and strategic richness of games when assessing funness (Table 2), models differ in their rates of assessing game balance and length, potentially driving some of the disparities in the eventual scalar funness judgments (Figure 7).. Thus, these differences may arise from either disparities in how the values of each subquery computed (which may expect based on our evaluations of models' differences in evaluating expected game payoffs), or different methods in which these values are aggregated. We again identify differences in rates of explicit simulation of gameplay in some of the reasoning models (see Appendix Figure 18).
4.3 RESOURCE USAGE WHEN EVALUATING GAMES
How much compute models engaging in to evaluate these games, and what may explain why some games and queries demand more compute? To begin to assess resource usage, we conduct an exploratory analysis into the number of reasoning tokens used by series of reasoning models (DeepSeek-Ri) R1, Gemini 2.5 Flash and Pro, o3, and GPT-5) when determining game evaluations.
While there is some relationship between the number of tokens used when estimating game payoff across models (with the exception of DeepSeek-RR -1) there is minimal relation across models' token when evaluating game funness (Figure 4a). There are also vast differences in the magnitude of
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Expected Poyer
Futhaas
Expected Hayot!
Punness
www.rhentestrrae
Expected Peyoff
Furness
Garenn 2.5 tlat
Gemini 3.5 Pro
Figure 4: Reasoning tokens used across games and evaluation queries. a, R between models' median number of reasoning tokens used per game, for the payoff and funness evaluation queries. b, Median reasoning tokens used for games based on how many "traits" they differ from Tac-Tac-Toe (e.g., game that is not played win conditions to "only diagonals count," has 3 traits different from Tic-Tac-Toe). Tic- Tac-Tox itself is zero. The heights of bars show averaged number of median tokens for that game, with error bars depicting standard deviation over games. c, Token usage based on higher-level game category.
3 x 3 board, requires pieces in row to win, and constrains the
tokens used across models and query type: despite funness being more ambiguous (and possibly involving multiple sub-questions to compute), models generally use far fewer tokens to estimate funness (Figure 4b-c).
Moreover, at a per-game level, while we may expect less typical games (c.g., games more distant from Tic-Tac-Toe) to demand more compute (reasoning tokens) to reason about, we did not observe a measurable difference between game "novelty" and token usage (Figure 4b-c), where "novelty" is measured the number of features of game that differ from the base Tic-Tac-Toe (e.g., if the game is not played on a 3 X 3 board, or involves asymmetric win conditions between players), as used in Collins al. (2025). We also do not observe a strong relationship between token usage and the distance between the model's output and the game-theoretic optimal or human predictions (see Appendix Figure 17). This raises a question about what determines the expenditure of reasoning tokens and how models could be made more resource- Sabbata et al., 2024; Sui et al., 2025) based on game complexity or other factors (e.g., how precise an evaluation needs to be). These adjustments should likely be made under the particular resource limitations of language models, which may be different from those of people (Griffiths et al., 2015; Lieder and Griffiths, 2020). Additional examples and trace analyses are included in Appendix 16.1.1 and A6.3.
in dynamically adapting compute (De
5 RELATED WORK
Problem evaluation and metacognition in cognitive science The meta-level problem of deciding which problem to solve is an active area of research in cognitive science to which our work directly relates. While people have remarkable cogmitive flexibility to represent and reason about a wide range of problems- even posing new questions and new goals (Schulz, 2012; Chu et al., 2023; Getzels, metae-intt-r is necessary because people have limited cognitive resources (Griffiths, 2020). Thus, resource-rational analysis (Icard, 2023; Lieder et al., 2025) has been especially successful as framework for the development of computational models of problem selection in contexts such as problem representation and decomposition (Ho et al., 2022; Correa et al., 2023; Binder et al., 2023) and strategy selection (Lieder and Griffiths, 2017; Binz et al., 2022). Algorithms for human problem selection extend to various other domains as well, including deciding how much to plan given set of alternatives (Sezener et al., 2019; Callaway et al., 2022; Kuperwajs et al., 2024) or when to even engage with task at all as opposed to quitting (Kuperwajs and Ma, 2022; Sukhov et al., 2023).
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Building AI systems that collaborate and interact with people requires understanding not just how machines and people solve problems, but also how to evaluate novel problems from the perspectives of both Al systems and humans.
Assessing language model reasoning Prior work has predominantly investigated the reasoning capabiliities of language models with the goal of solving problems instead of evaluating them. These broad efforts span topics such as math and symbolic reasoning (e.g.. Mirzadeh et al., 2025; Sprague al., 2025; Holliday
al., 2024), coding (e.g., Yang al., 2025), psychology and behavioral economics tasks (e.g. Liu et al., 2025b; Piedrahita et al., 2025), vision/multimodal tasks (e.g., Chen al., 2024; Zhang al., 2024), planning and robotics (e.g.. Kambhampati al., 2024; Wang et al., 2025), linguistic phenomena (c.g., Hu et al., 2022; Qiu et al., 2025), and games (see additional related work, Appendix AI). These works typically evaluate reasoning models (e.g., OpenAl, 2025) or prompt-induced reasoning such as chain-of-thougle (Wei et al., 2022; Nye et al., 2021; Kojima et al., 2022) against non-reasoning baselines. general finding across these studies is that newer and larger models enable better reasoning capabilities (Mirzadeh et al., (025)-sometimes with the help of tools such as program executors or additional training techniques (Yang et al., 2025), and such interventions may make reasoning more efficient (De Sabbata et al., 2024; Sui et al., 2025; Liu et al., 2025b; Zhang et al., 2025). Studies have also found that reasoning models' reasoning token usage may co-vary with human reaction times across several tasks (de Varda et al., 2025). We additionally contrast our work with LLMM-as-judge approaches in Appendix A1.
Applying methods from cognitive science to understand language models Our work follows a well-established line of recent research that employs psychological findings to better understand language model behavior (e.g.. McCoy et al., 2024a;b; Binz and Schulz, 2023; Ku et al., 2025; Coda-Forno et al., 2024; Frank, 2023). Such research typically replicates an existing psychological study by replacing participants with language models, which are compared the original participants as well as rational cognitive models that describe desired behavior (e.g., Liu et al., 2024; Marjich al., 2024; Liu al., 2025a; Zhu and Griffiths, 2024). Insights from cognitive science have also contributed to designing and improving LLM- agents (Liu et al., 2026). Additional related work is in Appendix A..
DISCUSSION
A holistic understanding of Al systems' reasoning capacities requires understanding not only how models solve problems, but also how they assess problems. Games are a microcosm of the kind of systems of rules and rewards that want to AI to evaluate. We show that while language can capture a substantial amount of associative knowledge that can be brought to evaluate new systems (e.g., whether game is fun), language-based intuitions alone without some deliberative, iterative thinking can only go so far. For these games and queries, some form of simulation or explicit reasoning seems essential for aligning with human judgments and computing the optimal value. Our work opens up range of questions for future work, namely:
Whose evaluations should models be evaluated against: people (of which group and what level of experience), or the "optimally rational" evaluation?
What cost are we willing to pay for such evaluations? How should we balance resource demands to evaluate problems (particularly when deciding whether to engage more with, or even solve, the problem in the first place)?
What inductive biases have models already picked up about what makes a problem game fair, or worth engaging with? What are the limits about what can be learned from generic supervised training vs. reinforcement learning?
Are language and/or reasoning models interally engaging in some kind of game simulation in order to produce tokens, beyond what written explicitly in chain-of-though rationales or reasoning traces?
How can models be encouraged to explicitly simulate when evaluating? When should they, and when should they not simulate-give particular compute budget?
What other evaluation queries should we engage with, to understand models' capacities for more general problem evaluation and meta-reasoning?

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Indeed, our work only scratches the surface of evaluations of game evaluation. It is open empirical question how far our results generalize to other competitive board games (like Hex or Othello variants) and entirely different categories of games (e.g., cooperative games, or games with asymmetric roles) and other settings (e.g., in law and finance) which may require asking other evaluation queries and designing new human experiments
compare models against. Our assessments are not meant to be definitive: model performance is sensitive to a host of factors like the exact prompt (here, we prompt participants with the human instruction text; see Appendix A3.1 and Appendix A7) and other hyperparameters (e.g., reasoning amount; see Appendix 15.1). Future work should better explore the relationship between the style of player or agent a model is simulating when making such assessments (see Appendix A7). We also note that our evaluations focus on "novice" game reasoners; it is an open question of how well models relate to people of varying skills, culture, and experience. One of the goals of this work is to carve out an underexplored space of evaluation of Al systems: evaluating their capacity to evaluate problems; our empirical work only scratches the surface of this rich space.
Evaluating AI systems' evaluations is important for building human-beneficial AI thought part- ners (Collins et al., 2024) that meet our expectations for deciding what problems to solve (e.g., in educational contexts) systems are used in part to create new rules that people engage with or are guided by (Koster et al., 2022; Tacchetti et al., 2025). For example, it is important that Al systems involved in automated mechanism design (Myerson, 1983; Maskin, 2008; Hurwicz, 1973; Milgrom, 2004) can appropriately evaluate whether the resulting system will be fair (and even engaging) for other people to participate in. Moreover, studying where models differ from people in their evaluations of systems can also inform the construction other kinds of thought partners that complement people (e.g., cognitive prostheses; Lieder et al., 2019) by adjusting people's 's expectations about a new problem or system. We hope our work paves the way for future evaluations evaluations that go beyond assessing model problem solving, but flexible problem and system evaluation.
determining whether system is fair. The latter is especially important if AI
7 CONCLUSION
Reasoning is not just about solving problems, but evaluating whether problems are worth solving at all. We laid out a perspective thinking about the evaluation of Al systems' capacity for problem evaluation. We focused on the domain of games, particularly, two-player competitive strategy board games and assessed series of language and reasoning models on their judgments about games over two evaluation quantify. Our work raises questions for how to design more human -ennf-ienefiiial and resource-efficient machine reasoners and evaluate their evaluations of whether new problems are worth solving.
and funness- that span a range of difficulty to compute and to
ETHICS STATEMENT
Our work is directly related to AI alignment, here assessing whether AI aligns with human judgments of fairness and funness of games. This novel penspecive--understadddee similar conclusions about what makes a system, or game, fair-has broad implications for Al models that are designing systems of rules that may involve or engage people. Our work also looks at how models' judgments of funness compare people; models which better anticipate what people will find engaging could be used to optimize the design of highly engaging games, which could cross the threshold toward being (harmfully) addictive. The use of more human- aalined models of human engagement is not all fun and games, warranting careful consideration of how it is used in practice.
whether models come to
ACKNOWLEDGMENTS
We thank Simon Frieder, Shubhra Mishra, Gabriel Poesia, Sam Cheyette, and Tracey Mills for valuable conversation that informed this work. KMC acknowledges support from the Cambridge Trust and King's College Cambridge. This work was supported in part by AFOSR (FA9550-22-1- 0387), the ONR Science of AI program and the Siegel Family Quest for Intelligence at MIT. TLG acknowledges support from the NOMIS Foundation. AW acknowledges support from Turing AI Fellowship under grant EP/VO25279/1,
a Schmidt AI2050 Fellowship to JBT,
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The Alan Turing Institute, and the Leverhulme Trust via CFI. IK acknowledges support from the National Science Foundation (II-2312373).
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L. Zheng, W.-L. Chiang, Y. Sheng, Zhuang, Z. Wu, Zhuang, Z. Lin, Z. Li, D. Li, E. Xing, et al. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in neural information processing systems, 46:46595-466333 46623, 2023.
L. Zhou, L. Pacchiardi. F. Martínez-Plumed, K. M. Collins, Y. Moros-Daval S. Zhang, Q. Zhao, Y. Huang, L. Sun, J. Prunty, al. General scales unlock ai evaluation with explanatory and predictive power. arXiv preprint arXiv:2503.0633 2025.
J.-Q. Zhu and T. L. Griffiths. Incoherent probability judgments in large language models. In Proceedings of the Annual Meeting of the Cognitive Science Society, volume 46, 2024.
M. Zhuge, C. Zhao, D. R. Ashley, W. Wang, D. Khizbullin, Y. Xiong, Z. Liu, E. Chang, R. Kr- ishnamoorthi, Y. Tian, et al. agent-as-a-judge Evaluate agents with agents. In Forty-second International Conference on Machine Learning, 2025.
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APPENDIX
A1 Additional related work
17
Example games
18
A3 Additional model details
18
A3.1 Prompts and additional language model generation details
18
A3.2 System prompt
19
A3.3 Task prompt
20
A3.4 Alternate models
20
A4 Additional analysis details
21
A4.1 Game-theoretic payoff estimates and additional htw.aa..........................
21
A4.2 Additional comparisons to human payoff evaluations
22
A4.3 Additional comparisons human funness evaluations
22
A5 Analyzing reasoning token usage
23
A5.1 Varying reasoning amount
23
A6 Additional details on reasoning trace coding
26
A6.1 Coding methods explicitly engaged in reasoning models' traces
29
A6.1.1 Example reasoning traces across evaluative methods
A6.2 Coding funness measures
33
A6.3 Additional example reasoning traces
34
A7 Prompt sensitivity analyses
38
A7.1 Assumption of "reasonablenes""
A7.2 Question order
40
A8 Preliminary explorations into an expanded game set
40
A1 ADDITIONAL. RELATED WORK
Games and the evaluation of AI Our work is related to a line of research that uses games to benchmark and understand AI model's capabilities. Games have long served as valuable environments for evaluating AI models and algorithms (Shannon, 1950; Newell, 1955; Campbell et al., 2002; Mnih al.. 2015; Silver al., 2016; Yannakakis and Togelius, 2018: Vinyals al., 2019: Bard al., 2020; FAIR et al., 2022; van Opheusden et al., 2023; Bailis et al., 2024; Todd et al., 2025; Shojace al., 2025). Games are useful for evaluation in part because they offer precise rules and reward structures that are easily encoded into artificial systems while still requiring players to engage in variety of complex cognitive behaviors, from long-range planning to semantic understanding to social inference-as as has made games especially appealing for studying LLM decision making (Wu et al., 2024; Duan et al., 2024; Huang et al., 2025; Lin et al., 2025). However, rather than study LLM play in games, here, focus studying how LLMs evaluate games, before any play, and compare such evaluations to humans' evaluations. Additionally, our focus on game variants that are unlikely to have been previously studied and unlikely to be present in extant training corpora aligns with
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recent trend to focus novel or generated games for the purpose of evaluating modern Al systems (Ying et al., 2025; Verma et al., 2025).
Language models as judges Lastly, one parallel application in which language models are also used as evaluators is in LLM-as-a-judge paradigms (Li et al., 2024). In these settings, LLMs are used to provide evaluations by leveraging their ability to process diverse data types and provide scalable assessments that approximate human preferences (Zheng et al., 2023). Such methods have been applied for generating various scores (e.g., Bai et al., 2023), answering yes/no questions (e.g.. Shinn et al., 2023), and conducting pairwise comparisons (e.g.. Liu et al., 2025c), which have been used to improve aspects of models (c.g., Dubois ct al., 2023), data (e.g.. Zhang et al., 2023), agents (c.g.. Zhuge et al., 2025), and even reasoning (Lightman al., 2023). However, unlike this literature- or other literature evaluating the kinds of evaluations used to test Al systems, e.g. (Zhou et al., 2025) motivation is not to use language model judgments to acquire assessments at scale. Instead, our work focuses on the cognitive traits of these models-using the setting of games to analyze how language models compare humans in reasoning about tasks that difficult to compute or quantify.
A2 EXAMPLE GAMES
The novel games we explore here span a wide range of board sizes and shapes, as well as game rules. We provide several example games, broken down by game categories in Table 3 below.
Game Category
Example Game
K in a Row (Square)
pieces in a row wins on a 10 x 10 board
K in Row (Rectangle) Infinite Board
pieces in wins 4 board
pieces in row wins on an infinite board
K in a Row Loses
A player loses if they make pieces in row 4 x board
No Diagonal Win Allowed
wins on 10 10 board, but player cannot win by making diagonal row
pieces
Only Diagonal Win Allowed
pieces in row wins a x 5 board, but player can only win by making diagonal row
First Player Moves 2 pieces
pieces in nww wins on 3 3 board; Player can place 2 pieces as their first move
Second Player Moves 2 Picces
10 pieces in row wins on a 10 x 10 board; Player 2 can place 2 pieces as their first move
First Player Handicap (PI no diag)
3 pieces in row wins on 3 x 3 board, but Player cannot win hy making diagonal row
First Player Handicap (PI only diag)
pieces in row wins on x 7 board, hut Player can only win by making diagonal row Player needs picces in a row, but Player 2 only needs 2 pieces to win on 5 board
Second Player K-1 to Win
Table 3:
Game categories and example games. The 121 games can be grouped into categories based on their board shape and game rules. Example games are shown for each category.
A3 ADDITIONAL MODEL DETAILS
A3.1 PROMPTS AND ADDITIONAL LANGUAGE MODEL GENERATION DETAILS
Models were prompted with lightly modiifed version of the human instruction text from (Collins et al., 2025). Experiment instructions were provided in the "system" prompt, with the specific game provided in the "user" prompt. For payoff questions, models were prompted (like people) to provide separate estimates P(PI non draw) and P(ends in a draw). Responses were provided simultancously. These scores were combined into single measure of payoff, i.c., P(PI wins) P(P1 wins] not draw) x (1 - P(ends in a draw) and payoff for Player 1 is (1 (P(ends in a draw). P(P1 wins))) (-1) + P(P1 wins). Future work explore eliciting payoff directly in a single query. Models were asked (again, like people) to estimate the funness of the game, with respect to the broader class of games.
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For non-thinking models, varied whether they prompted respond directly (just number) or via -fttinn- (CoTT) (Wei et al., 2022). Further details are provided when describing our task prompt. Any run for language model that prompted directly answer the question (i.e., without going through a Co' first) and still outputed a natural language rational first was filtered out.
Thinking models were all prompted in CoT fashion, with the exception of DeepSeek-RI which required few modifications: for RI specifically, we append the system prompt in the primary "user" prompt, per recommendations on Together AI API. We additionally adjusted the maximum tokens to 32, 000 tokens as we observed that RI tended to respond longer than the default. Any run that took over the limit was filtered out.
A3.2 SYSTEM PROMPT
System prompt for payoff evaluation
Welcome! are conduct ing experiment understand people Your answers will be used to inform cognitive acience and AI resecrch.
games,
In this experiment, you will be reading short descriptions of board games and answering two simple questiens about each geme.
Rach game is played by playera who take turns by plactin p' peess on a grid, similar so zow), Tie Tac
garee Connect Gomoku
You will be reading descriptions
games in which the size of the board and the rules for winning very. We will always show you example game board from each description. for example, you might read a descriptien like:
hoard in this game is 5x5 grid. In this game, the rule is that the first player to nake 3 in row wins.
Then, for each game, the game does not end in a draw, hew likely is that the first player is going win (nct drax), and hew likely dzaw
task 1s to anawer: assuming both players play reasonably -- if
will answer this queation providing response (In the form number) between 0 and 100.
sefore you ansver the question for each game, you will have as much time want to think about the game and its rules.
Afer you feal like you understand the game, you can provida your responss.
For each gane, you can write en scratchpad to think about the game before answer. We encourage you to take your time and carefully analyze the game before providing your answer.
System prompt for funness evaluation
Welzome! Ke are conducting experiment to understard how paople think about games. used to inform cognitive ence research.
Your answers
In this experiment, you will be reading short descriptions of board games and answering simple question about each game.
turns by placing pieces grid, similar gomes like Connect 4, Comoku (S-in-a-rok), or Tig-Tac-Toe.
game played by players
You will be reading descriptions
in which the size of the board and the rules for winning very. We will always show you an exemple game board frem each description.
example, you might read daseription like:
The beard in this game is a 5x5 grid.
this game,
first player make
rew wins.",
Then, for each game, your task La to answer: how fun the game is to play
You will answer this question by providing respense (in the form of a number! between and 100.
We that you about funness with respect to that involve playera placing piaces grid. You can de fun hovever you wish.
game; that is, games
Before you the question for each game, you will have much time think about the game and its rules.
to
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Afez you feel like you understand the game, you can provide your response. For each game, you can write on acratchped to think about the game before you anawer. Re encourage you Lo Lake your time and carefully analyze the game before providing your
A3.3 TASK PROMPT
Below are two example task prompts (specified in the "user" part of the prompt). Note that "You may first write out your thoughts on a scratchpad." " is included for the "CoT" variant (and removed for the "Direct" variant). As noted, we filter out any run in the "Direct" variant that includes "chain-of- thought" response before providing a number (for the LLaMA 3.1 70B, GPT-4, and DeepSeek v3 "Direct" variants).
Example payoff evaluation prompt, for an example game
Imagine you are playing the following game:
Board size:
Win consitions: 3 pieces in a row wins.
You will answer two questions. For each question, provide your single number between and 130.
Q1:
the game does not and draw, assuming both players play ressonably, how likely it that the first player is going to win (not draw)
scale of to 100.
Let 0 = "First player definitely gaing to lase", lose", 100 = "First player dafinitely going to win"
"Equally likely
Q2:
Assuning both players play reasonsbly, how likely the game to end in draw?
Answar on a acale of to 100. "impessible to end
drax" 50 "Equally likely to end in a draw not",
100 "Betinitely going to end in draw"
You may first write out your thoughts on a scratchpad. when feal undarstand tha game and ready respond, previde single number ween C to 103, Mrite your responses as a number, in the form RnsPoNsE-01
and RESPONSE-Q2
Funness evaluation prompt, an example game
Imagine are playing the following game:
Board size: 7
win conditions: Each player needs pieces in rew to win. The first player can only by making diagonal zow, but secoad player
restriction.
this game?
Answez on a scale of to 100. Let 0 "The least fun of this class of grid-based game" 50 "Neutrol"
nest
grid game"
You may first write out your thaughts scratchpad. When feel understand the game and ready to respond, provide single number between 100. Write your response es a number, in the form RESPONGE neer--nmericel--eeeee
A3.4 ALTERNATE MODELS
We also compare to a series of alternate models implemented in (Collins et al., 2025). We compared against the "Intuitive Gamer," computational cognitive model which captures how people reason
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Spearmant
LLAMA
Spearman's rank corr
Luaio 11 702 (0rect
X
Loa% 738 (bel
07F4 10
X
2.5
Coren 2.5
ADO Chowessnng
OLO
U00 0
13 1000000
Figure 5: d, payoff and b, funness predictions. Higher means the ranked order of the predicted game evaluation is more similar.Spearman's rank correlation over models' and people's predicted payoff and funness judgments. Rank correlation is computed for a, payoff and b, funness predictions. Higher means the ranked order of the predicted game evaluation is more similar.
Figure 5: d.id.a.a, payoff and b, funness predictions. Higher means the ranked order of the predicted game evaluation i:
about new games before any experience. The model posits that people engage in fast, flat (depth- limited) goal-directed probabilistic reasoning. The model can be scaled up toward more sophisticated "Expert Gamer" model which implements deeper tree search inspired by the depth-5 model in van Opheusden et al. (2023). We also compared against Monte Carlo Tree Search (MCTS) (Coulom, 2006; Genesereth and Thielscher, 2014; Silver et al., 2016) and random agents, examples of player agents with greater and lesser sophistication. We only compare against these alternate models for the payoff predictions, as the funness models are regression models fit to a subset of the human data, rendering the comparison less clear. We refer Collins al. (2025) for details all alternate models.
A4 ADDITIONAL ANALYSIS DETAILS
We include additional details into model evaluations, based on the estimated game- theoretic payoffs and further comparisons to human evaluations of payoff and funness.
In addition to the R² over payoff and funness evaluations reported in the main text, we compute Spearman's 's rank correlation over the games evaluations (see Figure 5) the rank correlation is less discriminative across models, however, the general trends across model families persist. We show additional individual game-level predicted payoff and funness evaluations in Figures 6 and 7 respectively, and we move beyond aggregate analyses to compare the distribution of people and model judgments in Figure 8.
A4.1 GAME-THEORENT PAYOFF ESTIMATES AND ADDITIONAL ANALYSES
Game-theoretic payoffs were computed following (Collins et al., 2025): that is, we mathematically compute the optimal payoffs where possible, and otherwise use the value on games where MCTS converged to {-1,00 0, 1). This yields 78 of the 121 games. Specifically, for our MCTS-based estimates of the game outcomes, we have an MCTS agent play each game in our dataset 50 times. We report an instance of a "convergence" if all 50 trials of the MCTS agent led to the same outcome for Player 1. Each MCTS agent was run with 1000 iterations. The average duration of each 50-match trial was approximately 4.2k seconds and utilized over million nodes.
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Rules: 1xS.
Rulea: 3 pieceia rowwins
Board: AnB.
Rules: P1 needs pieces in to win.
Board: dark.
Pront)
Densitty
Bosite
Seeel
Gmald
Board: 10x10.
Rulees pieces
play
Board:Avl. pieces in
Rules: 3 pieces in row winss
Density
Peyoff
Payoff
Payoff
Figure 6: Distribution over models' and people's predicted payoff judgments for example games. Example hand-selected representative game evaluations. The distribution over human participants' judgments each models' 20 rollouts are shown. Panels and d show the complete set of models for the examples shown in Figure
A4.2 ADDITIONAL COMPARISONS TO HUMAN PAYOFF EVALUATIONS
We depict scatterplots of model and human predictions for all 121 games in Figure We additionally computed the absolute distance between the expected payoff under each model and people, broken down the category of game (Figures 10- 12). This granular breakdown reveals that, even though many reasoning models like OpenAl's o3 better capture human game evaluations in aggregate, there variability at a per-game level, e.g., for infinite or rectangular boards (Figure 10).
A4.3 ADDITIONAL COMPARISONS TO HUMAN FUNNESS EVALUATIONS
We repeat the same analyses as in the payoff evaluations, depicting the full scatterplots of model versus human predicted funness for the games (Figure 13) as well comparing absolute deviation in judgments at per-game category level (Figures 14- 16).
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b
Board size:
Rules: 3 pieces in winn. P2 can play teice on thei first tum.
preces
Rulex: 2 pioss in vine..
Calomee 2000)
Tmpplit
Rules: pieces in wins.
wize:
Rules: piecoo i wins.
Board size: Ex5.
P2 only nearss 9
noeds
Deneity
Funness
Funness
Funness
Figure 7: Distribution over models' and people's predicted funness for example games. Example hand-selected representative game evaluations. The distribution over human participants' judgments each models' 20 rollouts shown. Panels and show the complete of models for the examples shown in Figure
A5 ANALYZING REASONING TOKEN USAGE
We include additional analyses into reasoning traces across the reasoning models. Reasoning tokens were extracted from the models' respective APIs, and for DeepSeek-RI, computed using the DrreSSekk-R--Disil----nnn the "think" tokens.
tokenizer from the Together AI API for text generated between
In exploratory analyses, we find that reasoning token usage is not well-correlated with human judgments, model judgments, or their deviation from the game-theoretic optimal 17. Moreover, preliminary qualitative looks at the content of the reasoning traces (see Section 16.1.1 and A6.3) reveals that while the model can make judgments based on different strategies (e.g., comparing novel games to familiar games such Connect 4 and proposing features such as first-mover advantage). it still sometimes produces implausible claims or conclusions (e.g., wrongly estimating Player win rate and underestimating the funness of game).
A5.1 VARYING REASONING AMOUNT
Several reasoning models allow users to specify the "amount" of reasoning. In the main text, we reported results using the default ("medium") reasoning threshold. We conducted preliminary
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ruttte Coreer
LLONA 3.1 708 IRreCT
bepert Correr
LLAMG YOU (ONT)
LLatA K. (Drect)
(GGT)
11446 11 208 yivl)
CeupbNnk
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Dccobokk et (Dredt)
Densbeek Hall
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0 Pro
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Figure 8: Comparing distributional alignment of people and models. Games were judged by approximately 20 people and language and reasoning models were sampled with 20 rollouts per game. The distribution of judgments per game is compared using the Wasserstein Distance (lower means closer) over histograms of judgments per game. Histograms over the range 1 to 1 for payoff (a) and to 100 for funness (b).
Reasoning Type
DeepSeek R1
Gemini 2.5 Flash
Gemini 2.5 Pro
Explicit Simulation
15.4% 10.8%
34.7% /2100% 76.8% / 93.8%
43.8% 40.9%
Analogical Reasoning
76.9% / 98.5%
82.6% 97.99
Mathematical Computation
44.8% 15.0%
38.1 % 11.6%
47.0% 25.6%
Table 4:
Methods used in reasoning trace for evaluating games. Reasoning traces coded based on whether they involve explicit game simulation; analogical reasoning (e.g., comparing the game being evaluated to Tic-Tac-Tooe compute the game-theoretic optimal payoff based on mathematical game properties). Reasoning traces may involve more than one method (or none of the above methods). In each cell (for model and reasoning method), the first number shows the % for the payoff queries; the second number shows the % for the funness queries.
Gomoku); mathematical reasoning (e.g., attempting
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ML 3.1 70B [Direct) LLaMA 3.1 708 (CoT)
G (Diract)
GPT44 [CaT]
DecpSeek va (Direct)
DeepSook va (CoT)
R= 0.40 (0.36. 0.44)
[V 0:46 10:43.0.53)
Rd = 54 10 0.560
R
0.02 (0.57.
RV = 0.55 (0.50.0.59)
RV = 0.57 (0.53, 0.61
Human
A4
DeepSeek
Gemini 2.5 Flash
Gemini 2.5
GPT-
R? 0.51 10.56, 0.00)
0.52 (0.58, 0.67) R² 0.70 00.05, 0.74) R 0.04 10.00, 0.68)
0.75 10.72, 0.801
0.72 10.08. 0.75
Human
Model
-50
Model
Madel
Model
Model
Figure 9: Model- versus human-predicted payoff. Each point is game. Averaged model- and human-predicted payoff per game. Error bars depict bootstrapped 95% CIs around the mean average payoff per game, bootstrapped over participants and model rollouts per game. The top row language-only based models; the second row are reasoning models.
(quare)
Podds Pumeri
(eemangle
1 int
mhad can
corf pamm Fruch
853m
Cavet
Figure 10: Distance between model and human payoff predictions, by game category. Averaged absolute difference between model and human payoff predictions, grouped by game category. Aver- aged over games within each category. Error bars depict standard deviation over absolute distance between model and human payoff predictions for games within the category. K in row indicates the number of pieces in a row needed to end the game, where horizontal, vertical, and diagonal all count (as in, e.g.. a standard Ti--Ta-TTo game). We separate square and rectangular boards are separated for this setting; other categories mix board shape. Payoff values range from -1.0 to 1.0.
exploration into the impact of varying the reasoning amount specifically for two of the OpenAl family of reasoning models: o3 and GPT-5. There are three options: "low", "medium", and "high". In the main text, we report results using the default ("medium"") reasoning threshold. We run a series of exploratory analyses varying the reasoning amount across the "low" and "high" levels. Interestingly. varying the reasoning amount has minimal impact on aggregate fit to human data, but does impact how close to the rame-theoretic optimal predictions are (Figure 19). We report games with the highest differences in predicted payoff as function of reasoning amount in Table 5. The games that differ most across reasoning amounts generally follow monotonic changes in payoff' (to be predicted more less biased) as a function of the increased reasoning amount. Models are generally fairly consistent across reasoning amounts for some of the simpler games (e.g., where only two pieces in a row are needed to win. But even if all models agree, e.g., 3 3, 3 pieces in row wins and the first player gets to go twice on their turn (i.e., "3 x 3 3 (P1 2p)", the models may not align to peoples' judgements.
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Clagonal
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Orly Oiegonal
srecc bod
Denchen
seath dvent
der prect
God soour
depprek Deethet
Tre Payer
Secend Payer
and soeedd wotti
Figure 11: Distance between model and human payoff predictions, by game category (continued).
Tre Rayer dis
(Poda Bereer
Tie Rayer Hardecs
criee petegn to aor sooiee
corf
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Figure 12: Distance between model and human payoff predictions, by game category (continued).
LLaMA 708
LLaMA 3.1
GPT (Direct)
(CoT)
DeepSeek (Drect) 0.27
DeepSeek (CoT)
Human
Model
Model
Model
Model
Model
Model
DeepSeek
Gemiin 2.5 Flash 0.46
Gemini 2.5 Pro
ol
63 0.55
GPTS
0..43 42, 0.55)
R²
(0.33.
0.61)
Human
1000
700del
Nodde
Vodell
Figure 13: Model- versus human-predicted funness. Each point is game. Averaged model- and human-predicted funness per game. Error bars depict bootstrapped 95% CIs around the mean average funness per game, bootstrapped over participants and model rollouts per game. The top row are language-only based models; the second row are reasoning models.
A6 ADDITIONAL DETAILS ON REASONING TRACE CODING
To understand the patterns of reasoning that the reasoning models (for those which we can inspect their produced traces, i.e., DeepSeek-R and the Gemini family), we automatically code the content of the reasoning traces, based on the method of computation they involve (explicit simulation; analogical reasoning; mathematical computation)
well as what factors of funness the models discuss part of their evaluation. We code reasoning traces using o3 (specifically, version o3-2025-00 0025-04-16 16, at its default medium reasoning setting and default temperature). Due the computational cost of coding
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gen cod
B3300
gree
Figure 14: Distance between model and human funness predictions, by game category. Averaged absolute difference between model and human funness evaluations, grouped by game category. Averaged over games within each category. Error bars depict standard deviation over absolute distance between model and human funness evaluations for games within the category. Funness values range from 0 to 100.0.
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Figure 15: Distance between model and human funness predictions, by game category (continued).
Hanskap IPS dagg
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Figure 16: Distance between model and human funness predictions, game category (continued).
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Reasoning Tokens
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DeepSeek
Reasoning Tokens
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Gemini 2.5 Flash
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Reasoning Tokeets
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Germini Pro
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Rossoning Tokens
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Human]
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Reasoning Tokens
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Ressoming Tokerss
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Ressoning Tokenss
Gernini 2.5 Pro
Reasoning Tokens
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2000
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Reasoning Tokens
DeepSeek R1
Simming Tokens
Gemini 2.5 Flash
Germini 2.5 Pro
Reasoning Tokens
o3
Reasoning Tokens
GPT-5
tooo
Reasoniig
sacs
THEAL
Model
Model
Model
Model
Model
Figure 17: Reasoning token usage compared to human- and model-estimated payoff. a, Median
usage relative deviation between model and human; human and game-theoretic optimal, and c, that model and the game-theoretic optimal.
Papected Plyof
Panseus
SeepSeek
Gemini 2.5 flesh
Gemini 2.5
12+0 B4
12-0.00 50
12-0.75
Capected Payclt
Expected Payott
Expected Peyot
Expected Payoff
Funness
Detpbeok H: 2.5
Advte
Lane
Figure 18: Explicit game simulation in reasoning models. a, Correlation (R²) across models in rates of explicit game simulation, across all 121 games; b, Simulation rates per game, depending on whether the game was evaluated on fairness (horizontal axis) or funness (vertical axis); c, Simulation rates broken down by game category. Error bars depict standard deviation over the simulation rates for the games in those categories.
all 20 sampled traces for all games across many dimensions, only run o3 once per trace. We therefore caveat that these evaluations are an approximation of the content of the reasoning traces; future work explore scaling and further verifying reasoning trace analyses.
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b
0.755
0550
0430
03 (Low)
03 (Medium)
03 (High)
GPT-5 (Low)
GPT-5 (Medium)
GPT-5 (High)
03 (Low)
03 (Medlum)
03 (High)
GPT-5 (Low)
GPT-5 (Medlum)
GPT-5 (High)
Figure 19: Assessing evaluations under varied"reasoning amount". Select reasoning model (o3 and GPT-5) evaluations of games under varied reasoning "amounts". a, Bootstrapped Rº relative to people's predicted payoff's (bluc) and the estimated game-theoretic optimal (grey). b, Bootstrapped R2 relative to people's predicted game funness (green). Error bars depict the bootstrapped 95% CIs over games.
A6.1 CODING METHODS EXPLICITLY ENGAGED IN REASONING MODELS' TRACES
Reasoning traces were coded based on the kind of computation they involve. The annotation model (o3) was prompted respond with a binary YES NO if the reasoning trace involved either explicit game simulation; analogical reasoning; or mathematical computation (e.g., trying to compute the expected optimal based on features like board size). Each query was asked independently for each rollout and each game. Note, these analyses are over the explicit reasoning traces produced by the models; possible they are engaging in other kinds of evaluative methods that is not explicitly written in the reasoning trace, or, that the methods engaged in the reasoning trace are not appropriately accounted for in the final evaluation. Two authors from our team manually inspected several traces for validity; we are actively expanding verification of the coding. Prompts provided below.
Reasoning trace categorization prompt for assessing analogical reasoning
You are categorizing reasoning traces written by agents reasoning about games. Your taak
categaria whether the trace invalves any expliaft analogiea reasoning. That the reasening traces involves comparisen one more other games. Reapond with only a single word.
Bither:
YCS 15 it involves analogical reascning, or involve any analogical reasoning.
No if it
Reasoning trace categorization prompt for assessing explicit mathematical calculation
You are categorizing ressoning traces written by agenti reasoning about games. Lask calegorize whethe: the Lrace involves any explicit mathemalical celculstioas.
That is, particular mathematical operaters It, -, The mathematics doean't nead to be correct, it just naeda to be explicit calculations, The mathematioal calculations should be precise; juat simulating play doean! t count. Respond with only single word.
etc) assess questicn.
sither:
YDS involves explicit mathematical calculations, 80 if it does not involve any explicit mathemacical calculations.
Reasoning trace categorization prompt for assessing explicit simulation
You categorizing ressoning treces written by agents reasoning about games. Your task is to categorize whether the trace involves any explicit game simulation.
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Game
GPT-5 (Low)
GPT-5 (Medium)
GPT-5 (High)
Human
5x5 3 (P1 D)
-0.63 (-0.69, -0.57)
-0.21 (-0.50, 0.11)
1.00 (1.00, 1.00)
0.12 (-0.17, 0.41)
10x10 3 (PID)
-0.74 (-0.78,
0.47 (0.12, 0.82)
0.86 (0.86, 0.86)
0.04 (-0.23, 0.32)
5x5 3 (P1 HV)
-0.29 (40.40, -0.15)
0.49 (0.17, 0.75)
1.00 (1.00, 1.00)
-0.00 (-0.18, 0.20)
4x4 3 (P1 HV)
(-.42
0.69 (0.44, 0.89)
1.00 (1.00, 1.00)
-0.08 (-0.30, 0.17)
10x10 (PI HV)
-0.22 (-0.40, 0.01)
0.50 (0.23, 0.75)
1.00 (1.00. 1.00)
0.30 (0.04, 0.57)
10x10 10 P1 P2
-0.71 (-0.78, -0.62)
-0.37 (-0.53,
-0.08 (-0.08, -0.08)
-0.03 (-0.66, -0,00)
7x7 (P2
-0.34 (-0.39, -0.30)
-0,29 (-0.36, -0.22)
0.13 (0.13, 0.13)
-0.01 (-0.15,
3x10
0.49 (0.27, 0.71)
0.02 (0.01, 0.04)
0.08 (0.08, 0.08)
0.04 (0.00, 0.10)
3L
0.39 (0.10, 0.65)
(-0.00, 0.60)
0.00 (0.00, 0.00)
-0.03 (-0.09,
3x3 3 (P2 2p)
-0.61 (-0.79, -0.42)
-0.58 (-0.76, -0.37)
-0.15 (-0.31,
10x D
0.00 (0.00, 0.00)
0.00 (0.00, 0.00)
0.00 (0.00, 0.00)
0.02 (-0.01,
Xx5 2
1.00 (1.00, 1.00)
1.00 (1.00, 1.00)
1.00 (1.00, 1.00)
0.61 (0.42, 0.80)
1x5 3
0.00 (0.00, 0.00)
0.00 (0.00, 0.00)
0.00 (0.00, 0.00)
0.01 (-0.02, 0.04)
10x10 2
1.00 1.00)
1.00 (1.00, 1.00)
1.00 (1.00. 1.00)
0.91 (0.79, 0.99)
10x
-1.00 (-1.00, -1.00)
-1.00 (-1.00, -1.00)
-0.78 (-0.90, -0.65)
3x32
1.00 (1.00, 1.00)
1.00 (1.00, 1.00)
(1.00,
0.78 (0.64, 0.91)
3x3 3 (P1 2p)
1.00 (1.00, 1.00)
1,00 (1.00,
1.00 (1.00, 1.00)
0.54 (0.39, 0.70)
3x3 31 P2
(-1.0.-10))
-1.00)
(1.00.-.00
-0.75 (-0.93, -0.54)
4x4 3 P1 P2
-1.00 (-1.00, -1.00)
(-1.0000.11
-0.63 (-0.88, -0.35)
5x52 2
1.00 (1.00, 1.00)
1.00 (1.00, 1.00)
1.00 (1.00, 1.00)
0.99 (0.98, 1.00)
Table 5: Payoff evaluation differences with varied reasoning amount. Top, Games with the highest average predicted difference in payoff by GPT-5 under the low versus high reasoning setting. 95% CIs around the mean payoff are reported per model variant. Bottom, ten games where GPT-5 variants agreed despite differences in amount of reasoning. For space, abbreviated game names used, indicating the board size, K in a row to win (e.g., x 10 3 means pieces in a row is needed to win (e.g.. 3 compared to Player
extra piece in only needing 2) is listed as 3 P1 2 P2); games where a player can place twice on their first turn is listed 2p; games where player can only win along certain direction, like Player I only winning diagonally is listed as PID. And games where the first player to get K in row loses is denoted with an
to win); games where Player 1 requires
That 1s, that the trace ingludes explicit playout behavior for any game, clearly
spelling out who moves on which turn. Categorize whether the follewing includes explicit playout simalation.
The simulation doesn't need to be correct, nor to the end of the game, But it needs to involve
taking and move selectien. sespond with only a single word.
gither:
YES i5 it involves explicit playout aimulation, or No if it does not involve any explicit simulation.
16.1.1 EXAMPLE REASONING TRACES ACROSS EVALUATIVE METHODS
We include three example traces demonstrating example evaluative methods.
Reasoning trace from DeepSeek-Rt engaging analogical reasoning
10x10
first player get in wins.
Alright, let think through this game scenario. Me have 10x' board where the first player to get in a row wins.
sirst, considering the win condition: 9 in row is quite long on 10x10 grid. the board in large enough that blocking attempts become camplex than amaller gamen. dowever, achieving 9 uninterrupted pieces seems difficuls when the opponent can actively
For 21 (first player win likelihood is not draw): Given the board's spaciousness, the But the second playez
first player might have initial advantage in clsiming con effectively mirror or counter strategies. remall that in similar games like Somaku
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situation where all Published as a conference paper at ICLR 2026
(S-in-a-rox on 15x15), professionel play often leads to draws with perfect play. Here, in a row is even harder to achieve, making decisive wins unlikely. I'd estimate the first player's win chance just 1014 only possible if the opponent makes significant errors.
For 22 (draw likelihood); The large board and high win threshold make it extremely difficult either player to force
against optimal defenze, This resembles
high-level So or chess endgames where draws dominate. Unless one player blundars, the game should end in a drew nearly every time. I'd put draw probability at 3014. soth estimates lean heavily toward draws due to the balanced nature of the setup and the difficulty of achieving such long connection.
Reasoning trace from DeepSeek-RI involving explicit simulated play
2x10
pieces in row wins
Alright, let tackle problem. step by step. game on 2x10 grid, goal is to get three pieces in rew. #oth players play reasonsbly, and need to figure out the likel
the first player winning (if not draw) and the chance drow.
First, the heard is narrew with only twe rows, Gatting three in a row herizontally would require placing
Vertically, since there only Lwc rows, gelting three in vertical 1ine is impossible. So winsing must be horizontsl.
pieces
Rach player takes turns placing pleces. The board 20 calls. The first player starts, rows, making three
and they'll 11 aim block each other. Given the be tricky because opponenta can esstly black. For exarple, if the first player places place adjacent block
might
piece in row column 1, the opponent
potential three,
cclumns), there's room to create multiple threats. aut with only two rows, playera can cover hoth rews quickly. Maybe neither player can force win if both play optimally, Every time someone tries to start a sequence, the other narrew, it might laad
Considering
length
block. Since the hozrd long, apota are fed without a three-in-a-row.
Total neves possible are 20. Even if players aim to bleck, if the game continues to the without three players are playing parfectly, they' 11 block each other's attempts, leading to a fill board with no winner. Therefore, the chance a drew is high.
row, it's a draw. perhaps draw very likely. both
For 02, I'd say around 90-95 because it's almost certain if both play well. For 01, chance
first player winning not draw low. aut if not a draw, say 1018 chance the first player wins? but sctually, if there's s no draw, it means somaone wina. since hoth players are symnetric except for going first, maybe the first player has slight advantage, Sut in auch a tight soard, perhaps itrs 50 50? But considering that the second player can always block, maybe if win happens, it's equally likely for either player. So QI around 50.
since the
highly likely,
let think again. The threat. But the second player can black and counter. diven the hozrd size, it's pessible that the first player can set up a fork or something, However, with only twa rows, it's hard to create multiple thrests simultaneously. Each nove night be blocked inmedlately.
player moves first, so they might create the first
After deeper consideration, it without the ether making a mistake. Therefore, the game will alnost always end in a or close but 15 there's win, it's 50-50 who gets it. So @1 at 50.
almost impossible for either player force win
draw. Se 22
Let's say
Q1, sinze
common,
But the question is Miil the does not end in drawl", when there's win, how likely the first player wins. Given symmetzy, it should be 50. But perhaps the sirss move given advantage, maybe 55? Hewever, this constrained beard, the advansage might be minimal.
Tocking at known similar lar games. Tic-tac-toe 3x3 is draw with perfect play. Here, the board 2x10, which is longer but narzower, Gecting three in row horizontally is harder because the oppenent places in (1,1), #layer2 in (1,2), then Flayerl can't place in (1, 3) without Player2 blocking earlier. Actually, Player2
always block the third piece. For example, if Playerl
always block the inmediate threats.
Therefare, game will
optimal play. 02: 100, but maybe not exact 103 because if player makea mistake, but assum perfect play, it's always
draw. So Q2 is 100. Then Q1: since no wins occur, the condition \M15 not a draw)" impossible, it's undefined. But the question requires number. In such cases, perhaps the likelihood is 50 as a default? or maybe 02 kewaver, the question says ("if
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the game does not end in a draw\", which implies that we'ze conditioning on there being winner, If both play perfectly, Lhere's never a winner, so that scenario doeen't happen. Reace, the answer for Q1 is irrelevant, but mathematica.ly, it could be 50 neutral value. But in reality, with perfect play, it's a draw 10013 of the time. So for 130. For 01, since there are wina, but the question is conditional is on \"if not draw\", which never happens. So perhaps Q1 can be set to 50 aroitracily.
Alternatively, considoring that if a win occura Iwhich it shouldn't with perfect play), it's equally likely, so 50. So 1'1l go with Ql: 50, 02: 190.
But let me doubla-check. Supposa the first player trias to start at position (1,1), Second player plays at 11,2) to blook, Then first player pLaya at (2,1), second at (2,2), Thea first plays at (1,3), second blocks at (1,4), and so on. This cont inues without a three-in-a-rov. The entire board fills without win. &o yes, draw every time.
Therefore, 02: 100 (derinitely draw). 01: 5D (12 not draw, equal ly split, though it's impossible). , the answers are 50 and 100.
Reasoning trace from DeepSeek-RI engaging mathematical reasoning
5x10 grid, and the vin condition is getting 6 picces in a zoM
Miright, latts tackis this problen
by stap. The row, Both players are playing reacenably, and need to astinate twe thinge: the
is 5xxc arid, and the vin condition
getting pisces Likellbood of the fira: player winning i2 it's noz a árax (01) and the probabillty of a draw occuzzing 1921, AntaFirst, considering the board dimensions. A sw10 arid hes 50 colls. The win requizes 6 in a row. board d 5 sows high, berisontally, So theze's space for potential winning linsa. Vertieally, since it's only 5 rows, you can only get in a colunn, which. isn't cnough for win. Tharefora, no vestissi wias are possible in this setup, So winning lines can only be hurisental or diagonal. walnoww disgonals. Por diagonals, the length pussible depends the disection. 3x10 arid would be Himited. For example, starting from the top-left, diagonal could qo down zows but only acroas as nany colunns as the saw allows, Actually, in a 5x1c grid, the lungest diagonal would be 5 colls. Balt, 5 zovs, so diagonally, the longest possible is 5 cells. For axample, frem (1,1) to eneach for
cumaes spaces
zow would be 10.
diasenals qoine fran top-left beitan-right, the maxinum length in
is eslls. That's cella. Thacefore, diagunal vina are ingossible in this 5x10 grid because the board isn L tall wnough co gst E in a diaceasl. so the only pessible way 1o vin is horizeatally. Antnoo playeza can only vin by placing consecat ive pieces in the same row. Since each multiple potential s-in-a-zow linea per
zow, similarly, other diaronais would also
has 10 cells, it's iossible to have has possible winning lines ipositiona 1-6, 2-7, 3-8, 4-9, 5-10). Mith zows, that's 25 possible winaing linas, Antallww, the gana ls xinilar to Conrect 4 or Gonkka, but with the consiraint that only barisontal wins are possible. This night nake it easler to block potential wina. The first player starta by placing a piece, then the second plaver respends. Since the board is wider (10 colunna) but only roxs, players will fill up the rowe over time, Aainbiven that both players play cessunably, they will try to create their own winning lines while blocking the osponent. The first playez night have a alight advantage age by caking the iniciative, Blovever, the board being 5 nows high night
exanple,
that it's harder for the firat player the
aecund player of noves. The board has 50 colla. Each player places one place per tarn. all calls are filled without wither player getting 6 in a row, the game is a drex. The question is whether a draw is "iklly-eehan gansa Like Gonoku on largez board, dravs are less common besause playera can forn Lines, hare the diagonal wins are inpassible, blocking is easier, The second player can always respond to thrests by placing place to block the faest playec's a polencial Ln Antelureoeer, with 5 zowa, onow a sow has several pieces, players might switch to another hoard filliig up wishout any player schieving possible that sows get filled in such way that no consecut ive wella are frew for win. For example, 15 pieces are plazed such that every potertial 6-cell Hin in rox has at least one bloce fron mach ligghes. Conversely, preventing either from getting S in row. AnAnSo, the likel of a draw night be firsl player night atill have advantage by starting crying to build thrests that force the second player to respord. Bet with the second player able to block nult iple rows, the advantage night be mininel. AninConsiderin grid almays ends 15x15 board is a First-glayer win, tonly 5 rows), and the win condition is 6 in row, Since vert inel and diagonal wins are inpossible, the game is essentially constreined to horizontal play only, This night make it more drawish because players easily black each other's horizantal 1ines.
opgozianitiees
block, Aaiaaomedder
number
condition
In
board that's only high. Since vertical
spreading
the noves, This might lead to the colunna per row, it's
row, Since therw
center of
known gares. Tie tac tow on 3x3 6x grid is tirst-p.ayer win, Gomoka (s dimension
with pertect play. Connect 4 here, the board gize is
in rowl
er
"heretore, the probability of a draw could be high, For 02, the Likelihood at a drew night be stound 70-8016. For al, if the game is not a draw, the first pl player's charce might be sliht y above 501,, but since
common, the astual vin ptoy play pertectly, and always mirror the firm player's neves in a differen: rox to prevert any 6 in row, Per exemple, the first player places a piete in rox 3, colune the second player places in rov 2, column 5, also preventing zorizontal ddup.- Actually,
be 10n, Vainact let's adjust. second player can
both players
game can be torced to drax, then 02 would nigh,
blocking vertical (chough vertical doesn't matterl can only be horisontal, players can focus blocking rows, Vnkallach row indegendent ly can seen as a smiler gene. Hoxever, players can choose which rov to play in, The first player could try to create multiple threats in disterent rows. But the second player can block those, Since the win requires 6 in row in one row, and the row has 10 cells, It's possible to create two threats once. Fer exampio, placing piece that threatens exo overiagping 6-in-a-row lines. Bat with only 5 sous, the second player night be able to block effectively.
Retlecting total moves. Minimun noves vins player gets 6 in. rox in their sixth Hovever, the opporent can block. gane night end rew. But i2 both players block cffectivoly, the hoard night fill up without a xinner.
are Filled soneone geta
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Given that, a draw seens plausible, In fact, with per fact play, the second player night faroe draw, drew, the first player might still have slight edge, 6018, Rut overall, draws
So for 01, likely.
Alternetive the first player night start in the center of a row. The second player blocks. The First player slacta another The seuond player blocks. Evuntually, the rews fill, and without vertical oz diagonal threats,
get row, leading drex.
So, M'd estimate:
Q11 I: no deas, sizst player wins azout 60%8 or the time. But since draws are trequent, their overa.l vin rate is low,
02: Draw probability
high, 8019.
But xait, the question about the likelibood wher the gane doeen't end drax (0') and the ihoel of dowv (02), So for 01, the cond foral. probabil iny: given that there's i a winnery how wins? That night be closer to 50-b0V6 because the tiret player might otten, Howevee, with the board constroints, it night be bolanced.
Likely is that the tirst
possible,
slightly
Alternat ively, perhaps the second go
that if therefs win, lefa theirs. fhet that 92.
the game ion's draw, tirst player wins 6018 or the time, but there's an BoVe chance the game ende in a draw, Theretore, the actual win rate For the First player would be 11-0,81 80.66 6 0. 12, or 1236, whi ch seens pl alaaibble given
scens lcss lixely.
A6.2 CODING FUNNESS MEASURES
Additionally, traces produced from evaluating the funness of games are coded based what measures of the funness a model considered as part of evaluation. We code whether models discussed the game balance (fairness); strategic richness; challengingness; game length; and/or game novelty, Each query was again asked independently for each rollout for each game. In addition to coding the reasoning traces, also run o3 over the thain-of-though rationales written from selection of the non-reasoning language models.
Reasoning trace categorization prompt for coding whether balance was considered when assessing funness
You are categorizing reasoning treces written by agents reasoning about games. taak categarize tunness.
the trace considers game balance when assessing
That is, whether the Lrace makes any mention of the Iairness the game, whether the is lopaided, whether player has advantage, Respoad with only single word.
2ither:
YES 15 it mentions game balance, N0 it
game bane
Reasoning trace categorization prompt for coding whether challengingness was considered when assessing funness
You are categorizing reasoning treces written by agents reasoning about games.
task categorize whether the tracs considers game chailenge when assesaing funness.
any Reapond with only single word.
relative challengingness or
the game.
Bither:
YES mentions game challengingness, oz No if it does mention game challengee chellengingness
Reasoning trace categorization prompt for coding whether game length was considered when assessing funness
written by agents reasoning about games. Your task is to categorize whether the trace considers game length when assessing Sunness.
are categorizing reasoning
That is, whether the makes any mention of hew long the game is expected take. Reapond with only a single word.
Bither:
YDS 15 it mentione gane length, oz
if it mention game length
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Reasoning trace categorization prompt coding whether strategic richness was considered when assessing funness
written Your taak to to categaria ze whether the trace conalders whether gama 15 atrategt cy when assessing funness.
categorizing rezsoning
reasoning about games.
That is, whether the trace makes any mention of how much strategy the game involves (e.@.r how nuch strategic depth haa).
Respond with only single word.
sither:
YES mantions the strotegic richness of game, No if it doea not mention the strategic richneas of game
Reasoning trace categorization prompt for coding whether game novelty considered when assessing funness
You are catagorizing rezsoning traces written by agents reasoning about games. teak categorize whether the conaiders game nove lty when assesa Cunness.
That 1s,
the trace any mention Reapond with only single word.
game.
Bitherr
YES mentions game novelty, N0 if it does not mention game novelty
A6.3 ADDITIONAL EXAMPLE REASONING TRACES
We provide several additional example reasoning traces, which we selected to demonstrate notable reasoning patterns in model evaluations; here, all traces again from DeepSeek-RI.
Ex 1, Unrcasonable fairness judgment from DeepSeck-RR
Board size: 10 x 16
conditions: pieces
[think]
Okay, the user is aaking to aua luate specific board game scenario. They've described a 10x10 grid where winning requizes getting pieces in row, My taak estimate prohabilities: the first player's win chance (01) and the draw probability (02), sasuming optimal play.
This is an interesting combinatorial game theory problem. recall that in larger grid gamea lenger win conditions, first-player advantage diminishea compared amaller boards. For 10x10 beard with 7-11-a-row, the game has more space and flexibility.
Considering 01: In games like Gomoku (5 in a zow on 15x15), first player wins are common, that's more constrained setup. Hier with only
10x10, board is relatively smaller for the win condition. With perfect play, the secend player can Iikely forea a draw by mirroring blocking effectivaly. chance below 1038 so I'll with
eatimate first player
22: become highly probable in such balanced setups. The board has exactly 100 spaces (even number), and with the win requirement being high relative to board size, exhaustion moves without around 3514 both players can prevent wins indefinitaly.
seems very possible. put probsbility
be conducting cognitive science research, probably studying hunan intuitien about game symmetries. They're likely leoking for reasoned estimates rather than precise calculations. raspenses should reflect the inherent draw-favering of this configuration.
The
[think]
After ana analyzing the machanics 10x10 grld with xin condition of 7-in-a-row:
..RES20RSB-0 Q1 S..
*liist player has a very low win chance with optimal play due to the beard size and win condition favoring defensive strategies)
*EESPONSE-022
(fraw is ext romely 1ikely the large board and high win threahold low both players to coasistently block winning fozmations)
#ft Reasoning:
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1. wwwin condition vs. soard Bizesr:
7 in a row on a 10x20 grid requires significant space, making early wins unlikely, Second player
mieres theory on lorger boards).
block threats effectively (similar to ABenckus
2. x*Diaw Dominancers:
positions (100) vastly exceed minimum moves win (13-14 per playez). with optimal blocking, neither player car force a win before the heard fills. Proven dzaw strategies exist in comperable games (e.g.r 15x15 Gomoku with 5 a- Fow is draw with perfect play!
**Firet Player Disadvantagenv:
No known strong opening for 7 in zow this size. player equalizes quickly, reducing theoretical edge near zero.
Noot:: Computational game theery confirms draw-neavy outcones for similar configurations..
2x Underestimate fun from DeepSeek-RI
soard size: x conditions: pieces now wins.
[think]
Alright, need evaluate fun this grid game you getzing rоw, First, I should think about the board dimensione. à 4x9 grid is wider than is is wide only
sail, which might effect strategy. Since placing pieces strategically to black opponents while aiming for rew of four bacomes cructal.
high,
smn, the vin condition is in a row. That's simi lar to Connect 4, but Connect 4 usua ly Here,
only rows, vertical Players can enly stack up to pieces in celumn, which is exactly the win condition. So, someone fills column, they autematically get verticai win. column can only hold 4 pieces, and the board is high, that means each column can be corpletely filled. Bowever, once a colurn is full, no more pieces can be placed there.
might Limited.
since
Wait, does that mean avery column could potent fal ly be w'n for the player who places column?
fourth piece deterministic, Let me think. If player plays in columa, the opponent might bleek by playing in Since there are 9 co colunn, the first player could start in colunn 1, and 14 the second playez blocks in column 1, the first player moves column 2, and so on, Evencually, the first player could force but the second player can also start their own columns.
would
game very
possibly
same column,
they do,
player can fust play elsexhere.
the fifth
if thay keep moving
oo columns,
Actually, both players could be building their own columns Each player can clain column by placing thair pieces. Since you only need four way, norisonta and diagonals are also possible, The height of the soard is horizontal win zequires four adjacent columns in the same row, Kith 9 columns, there's plenty of space fer horizontal connectiens. Liagonsls might be trickier but still possible.
row, vertical one
recall that in similar game like Connect 4, the player who first has advantage, espectally on sealler heards. Rst here, the hoard asymmetric (4x3). narrow height might lead to quicker Bach time a player places piece in new celumn, they're one step closer to winning that column. Since the columns be dominated by who controls more columns.
because vectical wins easler to achieve.
independent
of vertical wins, the game might
However, horizental wins raquire controlling adjacent columss in the same row. This adés layer of sttatggy becsuse players need balance between building their own columns and blocking the epponent's horizontal consections. For example, if Player starts the center, Player 2 night need to block horizontally to prevent en easy win, But with 9 columss, there's lot of threats.
might hardes block all possible horizensal
The large number of colurna 19) compared to rows (4) means there's more horisontally, which could lead to more complex strategies. Flayers can create multiple thrests across different make it easier to schieve vertical wins quickly. if player focuses on a łew columns, they could
and columns. Beewevr, the linited verticel
might
vartical win before the opponent
block.
blacking vestical wins strright place pieza the opponent has thres, The chal lenge comes when the opponent has multiple threats. With columns, the board night feel bit spread out, The game's pace could be Ias., with players making quick noves. However, the wider board night also lead longer if both players are effectively blocking each other's s four-in-a-row sttempts.
same column when
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Anether aspect is the possibility of draws. Total celle are 4x9-36, Since Lwo players take turns, if wins, the could end draw. However, schieving 4-in-a-row isn't too hard on this board, so drawa might be rere. But if both players are vigillant blocking,
game night drag
shorter vertical dimensicn. but wider horizonzelly. Connect 4 is known for being solvable, where the first player can ferce win with perfect play. On a 1x3 grid, the first player might have an advantage due the add number of columns and the ability control the center.
Comparing
Connect
game
strategic depth, Verticsl wins are quick, and horizoatal wins sequize fewer consections than in larger grids. lazger board horizontally columns) could allow more complex strategias, but tha lew height (4) might reduce the complexicy since players can't build tall stacks.
win condition being only
in
grid might lead
Additionalll ly, diagonal wina might be harder to achieve because the board isn't square. Diagonals
require cour conseculive zows, since the height is only 4, diagenals are possible both upwards and dewawards. For exemple, diagonal starting at zow
span
colurns
going to row
pessible. Similarly,
diagonals from row 4 to row
Considering if the game is aimple and the first player force win, that reduces replayaoi and fun, on the other hand, if there's a good balance between engaging.
might allow plays, but the law height night make it feel constrained.
defense,
remember that in games ke Gomoku, larger board allows for deeper strategy, but here, the 4x3 is not standard. Pezhapa the sayиmеоry adds uniqueness, but it might unbalaace the game, Also, with only four ahorte: games. Shorter gamea can be Fun foz quick sessions but lack depth for prolonged engagement.
the board fllla up
leading
On the fun scale, I'd rate this around might make vertical wins too easy, potentially cavoring the first player. The wide board offers some strategic depth, but the limitations on vertical space might hinder cemplex strategies. Compared claasic less fun, So, RESPONSE 45 [think]
below neutral bacsuse
height
4, whicn palanced, this variation feels
After ana lnaignnng the grid game with a condition 4-in-a-Tow:
BBBrrd asymmetry.:: The width 19 columns) greatly exceeds the height (4 zews), horizontal/diaagnal play. Vestical wins easily blocked to minimal stacking depth.
akewing strategy
columns, Player can create multiple unblockable threats. Perfect play often leads to forced wins, reducing tension. simulations auggest ~738 win rate for Player 1.
=avirst-mover advantagess;
prevent complex board states. Sames resolve quickly (evg. 8-12 noves), favoring speed ever strategy.
seLimited depthe+: Only
rABlocking afficiency.v: befenders
neutralize threats essily by "capping"
columss or splitting herizental lines, causing zepetitive stelenates in mid-game. *eDiagonal constroints.:: 4-row height restricts diagona oppartunsties (e.g., only valid NR-SM diagona 1sl, simplifying path planning.
++Fun assessment*::
viStrengthsev: #ast-paced, accessible for beginners.
>sWosknesssssa: Predi otable outcomes, low strategia replayabl ity, imbalance favoring
#layer 1.
Compared to classies connect (bslancad) Gomaku (deep strategy), variant lacks depta and fairneas.
**Katinge:: 36/100
10 least fun, 50 noutral, 140 most fun)
RESFONSE 38
Ex 3, Careful deliberation with DeepSeek- R1 assessing different funness features
Board size: 10 x 10
win conditions: Each player needs pieces in The first player cannot win by making a diagonel row (only horizental and vertical rows count), but the second player does not have restriction.
[Chink]
We evaluating game with 10x10 beard.
win conditions:
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- sirst player needs 5 in zow korizontally or vertically (diagonal doesn't count For the ri:st playez).
Second player needs 5 row in direction (horizental, vertical, diagonal). imbalance The first player restricted, whi
This asymnetry rules may lead the second player has more ways to win.
Considerations for fun:
1. Balance: Eun game should be fair and balanced. IE one player has a significant advantage, it might
be fun.
2. challange: laving restrictions can add challenge, but if too restrictive, it might be frustrating.
3. Variety: The second player having more options might make the game more dynamic For side.
Focential Issues:
The first player is handicapped by being allowed win diagonally. This might make it harder far the firat player to win, especially since the second player can block horizontally vertically
build diagonals.
On a 10x10 boaré, 5 in row is achievable, but the restrictien for the first player where the
might
player an adventage.
How night this affect fun?
If the often or the other feels unfairly reatricted.
is unbslanced, it might not be fun bacause player might win
flowever, if handicep intended (like in standard S-in-e-row), then it might be an attempt to balanze, but the rules are different
inherent player advanitage
two playars.
In standard in row (Gonoku) on large beard, the first player oftea has an advantage. This rule tries reduce that by removing winning condition for the fivat player, but then gives the aecond player an exiza condition (diagonale) withou: restristion. This might overcompensate
give second player edge.
factor:
A game that is too predictable Ibecause of Imbalance) is less Iun. Players might feel the zules are arbitrary and unfait, which can reduce enjoyment.
Rowever, note that the game night still be fun if the asymnetry crestes an interesting dynamic. But suspect
imbalance might
great.
T compare to symmaоric game (where both players the same rulest and assigr fun relative to that.
Given asymmetry, would this game less Iun than syametric veraion. let's assign & number.
Considering the acale:
least fun of this class
50 neutral
100 mest fun
would place this game belcw neutrel because the asyamezzy is likely to cause frustration and imbalance. But it's still a playable game, so not the absoluts worst.
Let ne think a symmetric game a baseline: standard 5 S-innarn aow on a 10x10 board would be around 60-70 (it's classic, but might be bit large and slow for some).
This game has rule thas might make it less rate it loxer.
aynmetric version.
would
However, note that the second playests and aiility win diagonally night add element of surprise and complexity. But the first player might feel unfeisly treated.
think a reasonable score would be around 30-40. set's aay 35.
But let ma consider: if the handicap is vell-salibrated to balence the game, it might ba more fun. lowever, without extensive testing, it's hard to ssy. rulee seems arbi saary and night be confusing, the absence of evidence that it balances the game, I'll assume it causes imbalance.
Therefore, I'll go with 35.
However, note that the problam says: "You can defing fun hewaver wish." " So I tocuaing on balance and rairness important for fun.
Response:
Rot note the instruction wrice the reaponse aa "RESPONSR = 35*
Tat ne double-chedd the ryles:
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First player: only horizental and vertical S-in-a-sow count [no diagonsl).
Second player: horizontal, vertical, and disgonal all count.
This is a significant difference. The first player must only build in two directions and watch for the second player but luiindig in three. The second player winning coadition and can also block the first playe: in all directions.
This imbalance is likely to make the game less fun for the first player and possibly toa easy for the second playes? But note that the first playez moves first, which is an advantage. the rule might be intended counter that. kowever, might much.
Without knowing exact balance, go bit too asyanetric.
rule. think
Fina: score: 40 its account for the possibility that it might be interesting for some players).
Rot think 35 la more sccurace.
Alternatively, can think of it as:
Symmetric gama (both have the same rules) is more fun. So this asyemetria version is leas fun.
Rowever, if the asyиmetry exactly balancea the first-move advantage, then it could be the problem unfair.
zules different, which night be confusing
Therefore, stick to 35.
adjust: think 30 is bit too low, and 40 15 bit too high. is in the middle.
So RESPONSE 35 [think]
After analyzing the game:
strategio depth may feel large 5-in-a-row game. **Asymeerric wia conditionses create imbelance:
**Board (10x101++
**iirst player.* restricted horizontal/verttcal wins only (no diagonals), limiting options.
**Seoond player+s has unrestricted wins pertaanaal/vrrtiis//dhaaa
granting
signicicant advantage.
**Balance issues*e: The first player's handicap combined with the secend player's frustration, predictable
flexibility likely
player lavored),
and reduced feicness.
-aaun impact.*: Asyametzy feels arbitrary punitive rather strategic. shile novelty might intrigua initially, long-term engagement would suffer.
Compared to symmetric grid games le.g., Gomoku, Die Tac Toe), this design sscrifices belance and feirness-key to fun. it ranks low but avoids being "least fun" due to i:s playable structure.
REESSONSE
oBellw reutral: Tmbal ance overshadows patential depth)
A7 PROMPT SENSITIVITY ANALYSES
We next conduct an initial analysis of prompt sensitivity, focusing the payoff predictions. We first assess the impact of changing the "kind" of play the model should simulate (reasonable versus optimal) and then explore the impact of flipping the question order.
A7.1 ASSUMPTION OF "REASONABLENESS""
Participants, as well as models, were instructed to estimate the expected outcomes assuming both players played "reasonably." But, defining and being able to simulate what counts as "reasonable" play is itself somewhat hard to quantify. We conduct an initial analysis of a subset of the GPT family of models' predictions of the expected outcomes of games when the assumption of reasonable play is replaced with "optimal" play. That is, models are asked assume both players play optimally" rather than "assuming both players play reasonably".
When models are prompted to instead assume players play optimally, the fit of all models tested relative people drops (see Figure 20a). However, the games where estimate game-
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130
X55
63
6775
Model
Model
Reasonable
Sim Opt
Figure 20: Impact simulating "reasonable" vs. "optimal" play in predictions. a, Model-model and human-moode correlation (f2²) in predicted payoff over the 121 games, depending whether models were prompted to assume players player reasonably or play optimally. b, R relative to the game-theoretic optimal predicted payoff depending payoff hased on reasonable versus optimal play. c, Absolute difference in predicted payoff relative to the game- theorecc optimal payoff. Error bars for b and depict bootstrapped confidence around the mean.
whether models were prompted to estimate
R² 0.97 (0.95, 0.98)
1.00
0.75
Flipped Order
0.50
0.25
0.00
-0.25
-0.50
-0775
-1.00
-1.0
-0.5
0.0
0.5
1.0
Regular Order
Figure 21: Impact of question order on payoff judgments. Average payoff predictions for the 121 games for o3 (with medium reasoning amount) for the original question order (horizontal axis; predicting the first player's likelihood of winning if no draw, then draw likelihood) and flipped order (vertical axis; predicting draw likelihood before likelihood of player one winning if no draw). Error bars depict 95% CI bootstrapped mean predictions.
theoretic optimal value, neither ol nor GPT-5 substantially improves their closeness to the optimal value. In contrast, o3's predictions become comparably close to the game-theoretic optimal as GPT-5 (see Figure 20b-c). This suggests that o3 is more adept at tailoring its response to the questions based on a more human-like assumption of what "reasonable" may mean and that the model is capable of approaching calibrated estimate of the game-theoretic optimal value of games. In contrast, ol struggles under the regimes tested here to estimate the game-theoretic theoretic optimal value; and conversely, GPT-5 struggles estimate human-lii responses that sub-rational relative the game-theoretic optimal. We see an investigation the controllability of the precision and character of such evaluations ripe grounds for future work (e.g., may imagine advantages to being able
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to simulate how less rational agent may engage in job or math problem). Our findings corroborate other work into the challenges sophisticated models' assumptions human rationality (Liu et al., 2025a).
A7.2 QUESTION ORDER
For expected outcome questions (from which payoff was computed), participants as well as models were all instructed to (1) assess the probability that the first player wins given the game does not end in draw and (2) estimate the probability match would end in draw. The order of the questions was fixed (Question was always presented before Question 2). As 03 was the most human-aligned model, we conduct an initial sensitivity analysis into the impact of flipping the order of the questions (presenting the draw probability estimation before estimating the probability the first player wins no draw). There is little difference in the resulting payoff predictions for o3 (see Figure 21). While this is not a guarantee that people will not be sensitive to the order, nor other language models, it lends some credence to the potential robustness of our results to question order.
a
b
Yevalath
Cooparative
d
New game variants. Averaged mean absolute error in payoff predictions new game variants. Errors are averaged over all game variants for game category, over 20 rollouts per LMM. Darker blue means more different (higher absolute error). Lighter means more similar.
Figure 22:
A8 PRELIMINARY EXPLORATIONS INTO AN EXPANDED GAME SET
While the space of 121 games is highly varied, we conducted an initial exploration into three other competitive game categories (Reversi, Hex, Yavalath) and one cooperative game category ("cooperative tic-tac-toe" wherein players either both win if they make their pattern or both lose). We developed variants off a base game for each (totaling variants for each of the four game categories). For the competitive variants, we varied board size and turn dynamics (c.g., wherein Player could play twice on their first turn). For the cooperative variant, we varied board size and the patterns (e.g., 4 x 5. P1 needs 4 in vertically and P2 needs 4 in horizontally for both to win).
We ran subset of non-reasonin language models (LLaMA 3.1 70B: DeepSeek v3; GPT-4- all with CoT) and reasoning- based language models (ol; o3; GPT-5). Models were tasked, as before, with predicting the expected payoff of the game (here, under assumed reasonable play). We computed the
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mean absolute deviation in the payoff predictions, where higher means the payoff predictions more different.
For the cooperative variants, there stark differences in the predicted payoff for the reasoning- versus non-reasoning LMs (Figure 22d). The differences are more variable across the other competi- tive variants. While generally observe similar payoff predictions amongst the reasoning models for the competitive variants, there are more deviations between the reasoning and non-reasoning models there (and the absolute differences are less stark than the cooperative variants (Figure 22a-c). However, we caution these results are preliminary.
We actively exploring other non-L.M based game reasoners (e.g.. MCTS) these game variants. Future work can also explore the collection and comparison of human data on these novel game variants. Our results could be used to guide which games are most interesting gather human data e.g., prioritizing games with the biggest differences between models.
41


Reviews:
[{"review_id": "7IKOM473rf", "reviewer": "Reviewer_Tg2R", "summary": "The paper propose to evaluate whether models can evaluate games (as benchmarks) themselves. Using 121Tic-Tac-Toe variants of board games and ~450 human judgments, the authors compare non-reasoning LMs, reasoning LMs (with traces), and simulation agents, and report a non-monotonic trade-off: models closer to game-theoretic optimality, their capacity as (game) evaluation judges align less with human judges. They also observe higher variance for funness evaluation and highlight the need for resource-rational meta-reasoning.", "questions": "1. Would make the paper stronger if the authors could report preliminary results for expert-refined vs auto-tuned prompts (e.g. with DSPy), since prompting is a major driver of judgments. \n\n2. Could the authors include other rank-based metrics (Spearman/Kendall) or motivate why it suffices to sticking with R^2 as evaluation metrics? \n\n3. Does the human-vs-optimal non-monotonicity persist on cooperative or asymmetric games? Any early evidence? \narXiv\n\n4. Do the authors have comments on how do these results transfer to other evaluation settings (beyond game-as-an-eval, for other benchmark/test construction)?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. Evaluating the games (as evaluation benchmarks) themselves (for design properties like payoff, funness) is a novel and valuable lens. \n\n2. Includes non-reasoning LMs, reasoning models, and rule-based (MCTS) agents, contextualizing model judgments against humans and game-theoretic references. \n\n3. Systematically contrasts AI vs human evaluations, revealing the non-monotonic alignment for game-theoretic optimal LLM players.", "weaknesses": "1. There lacks evaluation metrics beyond R^2. Would be useful and informative to add Spearman’s rho and Kendall’s tau to capture rank alignment.​\n\n2. There lacks a strong justification on why choosing MDA-style constructs for formalizing \"funness\". Could the authors provide more discussion on their choice?\n\n3. The 121 games are largely tic-tac-toe variants.\n\n4. Lacks a prompt sensitivity study and analysis.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4Ex4yoCeIs", "reviewer": "Reviewer_AE1w", "summary": "The authors explore and evaluate how frontier language models evaluate games. The bigger picture is that, as a field, we demonstrate that models can play and learn in games, but can they also choose which game to play, and if so, based on what? Do their tastes in games match human preference, or do they qualitatively differ? The paper collects human annotations for game evaluation for two specific questions. The frontier models are asked the same questions. The paper analyses the (dis)agreement between humans and frontier models, as well as the intra-frontier model disagreement. The main contribution is probably the extensive statistical analysis of human and model preference. The most interesting takeaway seems that reasoning in models leads to stronger correlation with human notions of fun, at least for these specific games, in a specific annotation setting.", "questions": "1. Figure 2 caption says \"modle-predicted\"\n2. There's a typo in the example prompt, i.e., see line 981 \"For each quetsion, provide your a single number\"\n3. The y-axes ticks in Figure 4b are very small.\n\nThe reviewer has no problem with the technical execution of the paper. If the authors could address the concerns raised in Weakness, the reviewer would be very grateful and would look forward to the rebuttal period.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "## Clarity\n- The paper presents its results transparently and thoroughly.\n- The paper is clear about hyperparameters and which prompts were used.\n- The authors are transparent with the limitations.\n- The paper is mostly easy to follow.\n- The figures are easy to understand and mostly well formatted.\n- The authors compare and contrast with relevant related work.\n\n## Quality\n- The statistical analysis is thorough, and meaningful measures of variance are provided. The details of how they were computed were also documented.\n- The experiments are thorough, cover a lot of frontier models, and investigate the role of reasoning (both through CoT and trained reasoning models).\n- The evaluation goes the extra mile and also looks at token usage, which was interesting!\n\n## Originality and Significance\n- The broader topic is definitely interesting and goes into capturing human subjectivity and open-ended agentic problems. The actual study setup can also be interesting to game developers.", "weaknesses": "## Clarity\n1. The reviewer struggles to understand the motivation for the two dimensions of evaluations as outlined in the Intro. It's unclear how the second paragraph \"highlights\" (line 48) two dimensions of evaluations. For example, Hunicke (2004) appears to discuss the MDA framework, rather than the two dimensions of game evaluation. Isn't something hard to quantify necessarily hard to compute? What is an example of something hard to quantify but easy to compute? For example, in Figure 1, the example question is \"What kind of game is this?\" While being more of a classification task than a quantification task, why is that easier to compute? Currently, it's the reviewer's opinion that \"Hard to compute\" is too hard to quantify and \"Hard to quantify\" is too hard to compute to provide a useful framework for game evaluation by models.\n2. For the question about expected payoff, it was not clear from the main text that the prompt that humans and models were asked was:\n\n> Then, for each game, your task is to answer: assuming both players play *reasonably*\n\nThe Introduction describes the question as if it's focused *only* on the hard-to-compute part, i.e., all humans and models have to do is attempt to compute the game-theoretically optimal outcome for two rational players. \n\n> For each game, we test a series of language and reasoning\nmodels using two reasoning queries that engage both dimensions of evaluation: one that is difficult\nto compute and another that is difficult to compute and quantify.\n\nand also in Section 3.2:\n\n> We prompted a series of language and reasoning models to evaluate the the expected payoff and funness of each of the 121 games\n\n\nHowever, the question asks explicitly for *reasonable*, which, at least to the best of the reviewer's understanding, is not a well-defined term. For example, when playing against a child, where you might let them win on purpose, reasonable gameplay looks different from rational gameplay. In the reviewer's opinion, the expected payoff question then also becomes a question that is both hard to quantify *and* hard to compute. This muddies the comparison to the game-theoretic optimal performance. The results do seem to suggest that humans and frontier models tend to infer \"rational\", but for example, the reasoning trace from DeepSeek on line 1538 explicitly considers the agents to be reasonable, not rational.\n\n3.  For the annotation study, it's unclear in what order the human annotators were provided the questions. Would the same humans always answer both question 1 and question 2, and in the same order? If so, answering question 1 might bias a human to think about payoffs in the funness estimates. Ideally, annotators would be asked either question 1 or question 2 about a game to avoid a bias that wouldn't be present in the models' context.\n\n## Originality and Significance\n1. In the Conclusion, the authors write\n\n> We laid out a framework for thinking about evaluation of AI systems’ capacity for problem\nevaluation\n\nand the Abstract states:\n> In this paper, we advocate for a new paradigm that\nassesses AI systems’ evaluation of games. First, we introduce a formalism for\nevaluating such evaluations. We then leverage a large-scale dataset of over 100 novel board games and over 450 human judgments\n\n\nHowever, it's unclear to the reviewer what exactly the framework or new paradigm is. Section 2 explains that games may have factors that you can use to judge a game by. However, there is nothing specific to games in the framework described in Section 2. Anything that might be evaluated may have multiple factors (dimensions) by which to judge it. Is the framework the two specific questions, i.e., should an agent always evaluate expected payoff and fun? Is it the two dimensions? At the moment, to the best of the reviewer's understanding, the framework is simply gathering human annotations on a subjective measure, asking LLMs the same question and measuring correlation for a very specific domain. There is no evidence that any of these insights would generalise or that these dimensions (or questions) prove useful in \"evaluating language models' evaluations of games\".\n\nTo summarise, the paper's correlation analysis is well-executed; however, it's challenging to see how this paper contributes anything more general than a correlation analysis with human notions of fun for a specific family of games.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ixoawHCoeq", "reviewer": "Reviewer_cvbH", "summary": "This paper introduces a new axis for evaluating language models (LMs): instead of only measuring whether models can solve problems, it evaluates how models assess them in a manner similar to humans. Using more than 100 two-player games, the study compares human judgments with model predictions on two quantities: fairness (expected payoff) and funness.\n1) Non-reasoning LMs produce similar judgments to each other but differ from humans and gameplay-based baselines on fairness.\n2) Reasoning LMs (with chain-of-thought(CoT)) show closer alignment with human judgments than non-reasoning models.\n3) The alignment with human judgment is not monotonic: as models move closer to optimal play, similarity to human judgments can decrease.\n4) For funness, models often reference similar factors in their reasoning traces, yet their final scores still differ.\n5) Reasoning-token usage varies widely across models and shows weak association with game novelty and with distance from optimal or human predictions.", "questions": "see Weaknesses section", "limitations": "", "rating": 4, "confidence": 3, "soundness": 1, "presentation": 3, "contribution": 2, "strengths": "1) The game collection size, the number of human responses per game, and the number of model samples per condition are all within a reasonable range. This allows estimation of meaningful statistical signals and enables analysis of variance across games and models.\n2) The evaluation covers a broad set of models, including non-reasoning and reasoning variants. The consistent observation that reasoning models show closer alignment to humans than non-reasoning models suggests that explicit reasoning traces have a measurable effect on judgment behavior.\n3) Two evaluation targets, “fairness” and “funness,” which provide additional structure beyond standard goal-oriented problem solving. This creates a concrete way to compare models and humans on perceptual or judgment-driven properties, relevant to studying whether models can serve as proxies for human judgments in cognitive science settings.", "weaknesses": "1. Human and game selection reporting are incomplete. The main conclusions depend on the properties of both the game set and the human participants. However, the paper only states that it uses 121 games from Zhang et al. (2024a) and Collins et al. (2025) and that each game received approximately 20 human responses. There is no information about participant recruitment, demographics, incentives, exclusion criteria, or screening. Because judgments of fairness and funness are subjective, missing details about the human population make it difficult to assess how general the findings are. A similar concern applies to game selection; no justification is given for why these games are representative of broader gameplay settings.\n2. Ambiguous use of the term “novel.” The text refers to the 121 games as “novel,” but they are sourced from prior work. If “novel” means new to participants or models, this should be specified. Without clarification, readers may incorrectly infer that the authors introduce a new game set.\n3. Payoff scale design lacks justification. The payoff query uses two separate probability questions on a 0 – 100 scale (win given no draw, and draw) and later maps these to an expected payoff in [−1, 1]. The paper does not explain why this continuous format is preferred(especially when it will give larger variance) over a simpler ternary label {−1, 0, 1} or why no explicit uncertainty option is included. \n4. Non-monotonic model-human alignment is described but not explained. The paper reports that similarity to human judgments can decrease as models approach game-theoretic optimal play. However, no mechanism is proposed. Without additional analysis, this pattern is descriptive only. It remains unclear whether the divergence stems from structural differences between human and optimal strategies, prompt effects, or sampling artifacts.\n5. Positioning relative to prior LM-as-a-judge work is limited. The study contrasts non-reasoning models with models using CoT and notes that reasoning improves alignment. Similar patterns have been discussed in other LM-as-a-judge settings. The paper would benefit from situating its contribution more precisely within that literature rather than relying mainly on internal comparisons.\n6. Funness construct lacks shared operational definition. Participants/models are explicitly told they may define funness however they wish. Without a shared definition or anchoring examples, different participants may use incompatible internal criteria. This increases variance in the target signal and makes it unclear what construct models are attempting to approximate. While the paper analyzes which factors appear in reasoning traces, there is no attempt to identify a stable latent structure of funness or to evaluate whether model factors correlate with participant factors.\n7. Typo. There are many typos and I will just list some of them \n- “evaluate the the expected payoff and funness” -> remove the duplication (Section 3.2)\n- \"Afer you feel like you understand the game, you can provide your response.\" -> After (A3.2 SYSTEM PROMPT)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GZ9pHDrYlI", "reviewer": "Reviewer_Ra3f", "summary": "This paper proposes a novel AI evaluation paradigm that shifts from assessing problem-solving to evaluating problem evaluation capabilities. The study uses 121 novel board games (Tic-Tac-Toe variants) to assess language and reasoning models across two dimensions: expected payoff (fairness) and funness. The authors compare multiple models (GPT-4, DeepSeek-v3/R1, Gemini 2.5, o1, o3, GPT-5) against baselines including human judgments and game-theoretic optimal solutions. Key findings include: (1) reasoning models align better with human judgments than non-reasoning models; (2) a non-monotonic relationship exists between game-theoretic optimality and human alignment; (3) funness evaluations show higher cross-model variability; (4) reasoning token usage is highly unpredictable and weakly correlated with task complexity.", "questions": "NA", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper clearly distinguishes between problem-solving and problem evaluation capabilities, a meta-cognitive perspective with theoretical and practical value. The proposed two-dimensional framework of \"difficulty to compute\" and \"difficulty to quantify\" provides a generalizable structure for future research.\n2. The experiments compare 7+ models against multiple meaningful baselines: human judgments, game-theoretic optimal solutions, and computational cognitive models (Intuitive Gamer, Expert Gamer, MCTS). The comparison of reasoning versus non-reasoning models enables clear assessment of reasoning's value.", "weaknesses": "1. All 121 games are Tic-Tac-Toe variants differing only in board size and rules, representing an extremely narrow subset of the game space. The exclusive focus on two-player, zero-sum, perfect-information competitive board games undermines claims about general \"problem evaluation\" capabilities.\n2. The paper treats funness as a \"hard to quantify\" query but provides no clear evaluation framework, instead instructing participants \"you can define fun however you wish.\" This lack of standardization means observed human-model misalignment may reflect disagreement about what funness means rather than evaluation capability differences.\n3. Game-theoretic optimal values are only computable for 64% of games (78/121), with the remainder relying on MCTS convergence without reported convergence criteria, runtime, or accuracy validation. The key finding of \"non-monotonic relationship between optimality and human alignment\" depends critically on the correctness of these values.\n4. Results are based on a single prompt design with heterogeneous temperature settings (1.0 vs 0.7) and only 20 rollouts per model, despite the authors acknowledging that \"model performance is sensitive to factors like exact prompt.\" Appendix Figure 18 shows reasoning amount significantly impacts results, yet this is not thoroughly analyzed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)