This paper examines the reasoning capabilities of Large Language Models (LLMs) from a novel perspective, focusing on their ability to operate within formally specified, rule-governed environments. We evaluate four LLMs (Gemini 2.5 Pro and Flash variants, Llama 3.3 70B and GPT-OSS 120B) on a suite of forward-simulation tasks-including next / multistep state formulation, and legal action generation-across a diverse set of reasoning problems illustrated through General Game Playing (GGP) game instances. Beyond reporting instance-level performance, we characterize games based on 40 structural features and analyze correlations between these features and LLM performance. Furthermore, we investigate the effects of various game obfuscations to assess the role of linguistic semantics in game definitions and the impact of potential prior exposure of LLMs to specific games during training. The main results indicate that three of the evaluated models generally perform well across most experimental settings, with performance degradation observed as the evaluation horizon increases (i.e., with a higher number of game steps). Detailed case-based analysis of the LLM performance provides novel insights into common reasoning errors in the considered logic-based problem formulation, including hallucinated rules, redundant state facts, or syntactic errors. Overall, the paper reports clear progress in formal reasoning capabilities of contemporary models.
Figures & tables
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
battlebrushes
0.997
0.950
0.997
0.950
0.988
0.750
0.945
0.100
bomberman2p
0.995
0.950
0.972
0.700
0.932
0.500
0.946
0.350
checkers
1.000
1.000
0.999
0.950
0.994
0.850
0.956
0.250
chess
0.998
0.850
0.998
0.850
0.996
0.800
0.973
0.150
mummymaze2p
1.000
1.000
0.999
0.900
1.000
1.000
0.989
0.300
Table 1: Next state generation results for selected games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
battlebrushes
1.000
1.000
1.000
1.000
1.000
1.000
0.536
0.000
bomberman2p
1.000
1.000
0.928
0.500
0.965
0.850
0.525
0.000
checkers
0.954
0.800
0.813
0.300
0.201
0.050
0.004
0.000
chess
0.783
0.150
0.311
0.000
0.371
0.000
0.143
0.000
mummymaze2p
1.000
1.000
1.000
1.000
0.967
0.800
0.910
0.000
Table 2: Legal actions generation results for selected games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
battlebrushes
0.300
0.300
1.000
1.000
0.197
0.000
0.427
0.000
bomberman2p
1.000
1.000
0.936
0.450
0.407
0.050
0.020
0.000
checkers
0.791
0.650
0.816
0.550
0.639
0.400
0.000
0.000
chess
0.993
0.650
0.921
0.200
0.663
0.050
0.134
0.000
mummymaze2p
1.000
1.000
0.865
0.600
0.780
0.050
0.964
0.000
Table 3: Multistep state generation results for selected games (n=5).
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
battlebrushes
0.500
0.500
0.868
0.850
0.213
0.000
0.087
0.000
bomberman2p
0.744
0.700
0.896
0.650
0.387
0.000
0.091
0.000
checkers
0.948
0.850
0.494
0.400
0.412
0.250
0.000
0.000
chess
0.915
0.100
0.763
0.250
0.633
0.150
0.000
0.000
mummymaze2p
0.994
0.950
0.961
0.750
0.796
0.000
0.145
0.000
Table 4: Multistep action-state generation results for selected games (n=5).
Figure 1: Multistep state generation accuracy (%S) across different horizon lengths. The left panel shows performance on original game descriptions, while the right panel presents results for the obfuscated ‘placeholder term’ variant (detailed in Section 5.3 ).
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
battlebrushes
0.169
0.150
0.465
0.250
0.127
0.000
0.069
0.000
bomberman2p
0.970
0.650
0.889
0.250
0.446
0.050
0.647
0.000
checkers
0.996
0.900
0.949
0.350
0.048
0.000
0.447
0.000
chess
0.941
0.550
0.904
0.150
0.192
0.000
0.347
0.000
mummymaze2p
0.995
0.600
0.972
0.500
0.607
0.000
0.746
0.000
Table 5: Results for the obfuscation method using placeholder terms (term1, term2, …) for selected games. The average results over all 35 games are included for all three obfuscation methods: placeholder terms, dictionary words, random strings, and for a baseline no-obfuscated game definition.
Property
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Converging property
0.45
0.53
0.46
0.71
Total rules_NEXT
-0.49
-0.60
-0.43
-0.45
Total constant facts_NEXT
-0.47
-0.53
-0.49
-0.44
Rules ’H-Index’_NEXT
-0.48
-0.49
-0.42
-0.50
Total conditions_NEXT
-0.49
-0.54
-0.41
-0.44
Max conditions per rule_NEXT
-0.48
-0.49
-0.42
-0.39
Table 6: Top-10 correlations averaged across models between game properties and model performance. Rule’s ‘H-Index’ is defined as max H s.t. H rules have ≥H conditions. ‘Converging property‘ indicates that state facts are never deleted, only added. Full results are presented in supplementary material. The highest absolute value for each model is bolded .
Place in TOP 10
Game
AvgFactCount
StateDiff
HighEnd
LowEnd
Variance
1reversi2
5
battlebrushes
8
6
beatMania
4
bomberman2p
7
4
bomberman2p_InvertedRoles
6
10
8
3
Table S1: Chosen games for the empirical experiments.
2pffa_zerosum
breakthroughSmallHoles
nineBoardTicTacToe
2pttc
breakthroughSuicide
pawnWhoppingCorrected
3conn3
breakthroughWalls
pawnToQueen
3pttc
cephalopodMicro
peg
4pttc
checkersBarrelNoKings
pegEuro
bidding-tictactoe
chinookDisjunctive
pentago
biddingTicTacToe_10coins
circlesolitaire
pentago_2008
Table S2: Games from the full set of 86 instances that were not selected for the final experimental setup of 35 instances.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
1.000
1.000
0.998
0.950
0.996
0.950
0.823
0.000
battlebrushes
0.997
0.950
0.997
0.950
0.988
0.750
0.945
0.100
beatMania
1.000
1.000
1.000
1.000
0.998
0.950
0.242
0.000
bomberman2p
0.995
0.950
0.972
0.700
0.932
0.500
0.946
0.350
bomberman2p_InvertedRoles
0.976
0.800
0.963
0.650
0.920
0.500
0.909
0.050
Table S3: Next state generation results across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.864
0.250
0.772
0.050
0.880
0.550
0.215
0.000
battlebrushes
1.000
1.000
1.000
1.000
1.000
1.000
0.536
0.000
beatMania
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
bomberman2p
1.000
1.000
0.928
0.500
0.965
0.850
0.525
0.000
bomberman2p_InvertedRoles
0.988
0.900
0.919
0.550
0.983
0.850
0.533
0.000
Table S4: Legal actions generation results across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.800
0.800
0.883
0.800
0.796
0.750
0.714
0.000
battlebrushes
0.900
0.900
0.902
0.750
0.161
0.000
0.656
0.000
beatMania
1.000
1.000
0.875
0.750
1.000
1.000
0.516
0.250
bomberman2p
1.000
1.000
1.000
1.000
0.405
0.100
0.023
0.000
bomberman2p_InvertedRoles
1.000
1.000
1.000
1.000
0.405
0.100
0.023
0.000
Table S5: Multistep state generation results with n=1 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.850
0.850
0.950
0.900
0.854
0.700
0.427
0.000
battlebrushes
0.050
0.050
0.866
0.700
0.163
0.000
0.503
0.000
beatMania
1.000
1.000
1.000
1.000
1.000
1.000
0.580
0.150
bomberman2p
1.000
1.000
0.993
0.800
0.391
0.050
0.012
0.000
bomberman2p_InvertedRoles
1.000
1.000
0.993
0.800
0.391
0.050
0.012
0.000
Table S6: Multistep state generation results with n=2 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.767
0.650
0.994
0.950
0.702
0.600
0.637
0.000
battlebrushes
0.200
0.200
0.894
0.650
0.182
0.000
0.508
0.000
beatMania
1.000
1.000
0.936
0.850
1.000
1.000
0.452
0.000
bomberman2p
1.000
1.000
0.999
0.950
0.324
0.000
0.009
0.000
bomberman2p_InvertedRoles
1.000
1.000
0.999
0.950
0.324
0.000
0.009
0.000
Table S7: Multistep state generation results with n=3 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.860
0.700
0.937
0.750
0.889
0.700
0.468
0.000
battlebrushes
0.600
0.600
0.926
0.850
0.238
0.000
0.447
0.000
beatMania
1.000
1.000
0.954
0.900
1.000
1.000
0.282
0.000
bomberman2p
0.938
0.850
0.899
0.400
0.358
0.050
0.052
0.000
bomberman2p_InvertedRoles
0.938
0.850
0.899
0.400
0.358
0.050
0.052
0.000
Table S8: Multistep state generation results with n=4 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.750
0.650
0.893
0.750
0.827
0.700
0.540
0.000
battlebrushes
0.300
0.300
1.000
1.000
0.197
0.000
0.427
0.000
beatMania
1.000
1.000
0.863
0.800
1.000
1.000
0.202
0.000
bomberman2p
1.000
1.000
0.936
0.450
0.407
0.050
0.020
0.000
bomberman2p_InvertedRoles
1.000
1.000
0.936
0.450
0.407
0.050
0.020
0.000
Table S9: Multistep state generation results with n=5 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.774
0.600
0.939
0.750
0.788
0.700
0.514
0.000
battlebrushes
0.550
0.550
1.000
1.000
0.271
0.000
0.516
0.000
beatMania
1.000
1.000
0.848
0.800
1.000
1.000
0.173
0.000
bomberman2p
0.939
0.750
0.914
0.250
0.336
0.000
0.175
0.000
bomberman2p_InvertedRoles
0.939
0.750
0.914
0.250
0.336
0.000
0.175
0.000
Table S10: Multistep state generation results with n=7 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.804
0.650
0.883
0.650
0.803
0.700
0.388
0.000
battlebrushes
0.900
0.900
0.943
0.800
0.266
0.000
0.385
0.000
beatMania
1.000
1.000
0.895
0.800
0.985
0.850
0.132
0.000
bomberman2p
0.986
0.900
0.854
0.100
0.336
0.000
0.221
0.000
bomberman2p_InvertedRoles
0.986
0.900
0.854
0.100
0.336
0.000
0.221
0.000
Table S11: Multistep state generation results with n=10 steps across all games.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.733
0.400
0.462
0.200
0.435
0.300
0.000
0.000
battlebrushes
0.500
0.500
0.868
0.850
0.213
0.000
0.087
0.000
beatMania
1.000
1.000
1.000
1.000
1.000
1.000
0.178
0.000
bomberman2p
0.744
0.700
0.896
0.650
0.387
0.000
0.091
0.000
bomberman2p_InvertedRoles
0.744
0.700
0.896
0.650
0.387
0.000
0.091
0.000
Table S12: Multistep action-state generation results across all games.
Model
Comparison Pair
p -value
Gemini 2.5 Pro
Orig. vs Rand.
0.1112
Orig. vs Dict.
0.0005
Orig. vs Place.
0.0032
Rand. vs Dict.
0.0369
Rand. vs Place.
0.1260
Dict. vs Place.
0.1980
Table S13: Wilcoxon signed-rank test results for obfuscation experiments (Orig. = no obfuscation, Rand. = random strings, Dict. = dictionary words, Place. = term placeholder). Significant results ( p<0.05 ) are marked in bold .
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.875
0.700
0.842
0.550
0.069
0.000
0.036
0.000
battlebrushes
0.169
0.150
0.465
0.250
0.127
0.000
0.069
0.000
beatMania
0.843
0.500
0.936
0.800
0.736
0.350
0.069
0.000
bomberman2p
0.970
0.650
0.889
0.250
0.446
0.050
0.647
0.000
bomberman2p_InvertedRoles
0.970
0.650
0.889
0.250
0.446
0.050
0.647
0.000
Table S14: Obfuscated game reasoning results for Multistep state generation experiment with n=5 moves across all games. Term placeholder obfuscation.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.606
0.000
0.483
0.050
0.070
0.000
0.019
0.000
battlebrushes
0.760
0.600
0.382
0.050
0.075
0.000
0.027
0.000
beatMania
0.850
0.550
0.945
0.800
0.220
0.000
0.043
0.000
bomberman2p
0.947
0.800
0.946
0.400
0.322
0.000
0.075
0.000
bomberman2p_InvertedRoles
0.947
0.800
0.946
0.400
0.322
0.000
0.075
0.000
Table S15: Obfuscated game reasoning results for Multistep state generation experiment with n=5 moves across all games. Dictionary words obfuscation.
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Game
JI
%S
JI
%S
JI
%S
JI
%S
1reversi2
0.895
0.700
0.714
0.150
0.084
0.000
0.143
0.000
battlebrushes
0.451
0.200
0.456
0.200
0.124
0.000
0.058
0.000
beatMania
0.736
0.200
0.968
0.850
0.552
0.300
0.038
0.000
bomberman2p
0.987
0.750
0.860
0.100
0.352
0.000
0.859
0.000
bomberman2p_InvertedRoles
0.987
0.750
0.860
0.100
0.352
0.000
0.859
0.000
Table S16: Obfuscated game reasoning results for Multistep state generation experiment with n=5 moves across all games. Random strings obfuscation.
Property
Gemini 2.5 Pro
Gemini 2.5 Flash
GPT-OSS 120B
Llama 3.3 70B
Converging?
0.45
0.53
0.46
0.71
Total rules_NEXT
-0.49
-0.60
-0.43
-0.45
Total constant facts_NEXT
-0.47
-0.53
-0.49
-0.44
Rules ’H-Index’_NEXT
-0.48
-0.49
-0.42
-0.50
Total conditions_NEXT
-0.49
-0.54
-0.41
-0.44
Max conditions per rule_NEXT
-0.48
-0.49
-0.42
-0.39
Table S17: Correlations between game properties and model performance.
Game
Converging?
Total rules_NEXT
Total constant facts_NEXT
Rules ’H-Index’_NEXT
Total conditions_NEXT
Max conditions per rule_NEXT
Max rule depth_NEXT
Avg rule depth_LEGAL
Top level rules_NEXT
Top level rules_LEGAL
1reversi2
0
10
50
4
35
7
4
2.6
4
3
battlebrushes
0
27
44
5
93
6
2
0.7
14
4
beatMania
0
14
59
2
24
3
1
0.0
8
7
bomberman2p
0
16
137
3
32
4
1
1.7
11
2
bomberman2p_InvertedRoles
0
0
0
0
0
0
0
0.0
0
0
buttons
1
19
6
2
38
2
0
1.0
19
3
Table S18: Property values for all 35 games for top 10 highest correlated properties.
Strategic depth of reasoning is essential for human interaction of Large Language Models (LLMs) operating in boundedly rational environments. However, existing evaluations are primarily based on canonical games prevalent in pretraining corpora, making it difficult to disentangle true strategic reasoning from memorisation. To address this, we formalise a necessary level-K distinguishability condition for strategic depth inference and construct a suite of novel game structures that meet this standard. Using these games, we evaluate strategic depth in LLMs from both the Chain-of-Thought tokens and actual actions under recursive reasoning and an inductive trace of opponent game-play data. Across experimental trials spanning four LLMs, four game structures, and ten levels of iterated reasoning, we find that model models maintain accurate strategic depth under recursive reasoning, with strong internal consistency between stated reasoning and actions at every level. Errors arise from using the wrong number of iterated depth of reasoning steps, not from computing best responses incorrectly. However, inductive inference from opponent play degrades accuracy sharply and unevenly across games, and explicit strategic mentalizing in the chain of thought substantially improves overall performance.
Specification gaming is a critical failure mode of LLM agents. Despite this, there has been little systematic research into when it arises and what drives it. To address this, we build and open source a diverse suite of tasks where models can score highly by taking unintended actions. We find that all tested models exploit their specifications at non-negligible rates in most of our eight settings, including five non-coding settings. We see the highest rates of specification gaming in Grok 4 and the lowest rates in Claude models. We use our evaluation suite to study what drives specification gaming, and find that: 1. RL reasoning training substantially increases the rate at which models exploit their specifications, 2. Increasing RL reasoning budget has a weakly positive effect on exploit rate, and 3. Test-time mitigations reduce but do not eliminate the rate of specification gaming. Our results suggest that specification gaming is a fundamental challenge arising from RL reasoning training; we release our evaluation suite to support further work on this problem.
Kei Nishimura-Gasparian, Robert McCarthy, David Lindner
1MATS · University College London, London, UK · 3Google Deep- Mind.
We introduce a multi-turn interactive framework for reasoning evaluation that treats reasoning as active evidence acquisition and belief updating. Wherein, LLMs receive only the task rules, must issue targeted queries to a hidden environment, integrate partial observations over time, and decide when to submit a final answer. Beyond standard success rate and interaction efficiency, we evaluate contextual robustness under controlled contextual perturbations, and metacognitive adaptation through counterfactual revision and necessity judgment. We instantiate the framework as a benchmark of 474 executable games, each evaluated under five fixed configuration search spaces corresponding to five difficulty levels, and evaluate a broad set of frontier LLMs. Results show that the benchmark is highly discriminative, exposing large differences not only in success rate but also in interaction efficiency. Moreover, we empirically show that contextual perturbations cause moderate but consistent declines, whereas counterfactual revision and necessity judgment lead to much larger drops.