The Internal Anatomy of Strategic Choice in Large Language Models

Motivation

• The introduction identifies an unresolved gap: models may represent strategic information without reliably using it, so similar choices do not establish similar reasoning.Prior work found weak and inconsistent effects of internally tracked hidden-information beliefs on action, motivating direct measurement of strategic computation.Source: §1 · pp. 1–2

Study design

• The study tests four open-weight models on all 144 strict ordinal 2 × 2 games, comparing their choices with human data and mapped cardinal-payoff data.The sample spans a matched base–instruction-tuned Qwen2.5 pair, Llama-3.1-Instruct, and GPT-OSS, covering dense and mixture-of-experts architectures.Source: §1 · p. 2

Analytical framework

• A prespecified canonical action and payoff incentive against a uniform opponent provide one quantity that can be followed from each game prompt through internal processing to choice.The incentive is defined for every game using a hierarchical behavioural model and a logistic quantal-best-response function.Source: §1 · pp. 2–3

Measurement strategy

• Linear probes test incentive and choice availability, incentive recruitment into decisions, layer- and prompt-position alignment, and susceptibility to intervention along the incentive signal.The same analyses test whether fixed risk, loss, inequity, selfishness, or maximin cues are represented, recruited, and able to shift choices.Source: §1 · pp. 3–4

Contribution

• Across all four models, choices were systematic and incentive, choice, and cue identities were decodable, yet models differed in how encoded incentives reached decisions.These differences persisted even between matched base and instruction-tuned Qwen2.5 checkpoints that chose almost identically.Source: §1 · p. 4

2.1 Strategic behaviour across the complete game space

• Canonical-choice rates ranged from 0.714 to 0.889, declining with strategic complexity as in humans; GPT-OSS stayed highest, while only Llama retained a controlled decline.Dense-model complexity correlations were ρ = −0.40 to −0.42, versus −0.44 in humans and −0.18 for GPT-OSS; controlling for incentive left only Llama’s interval excluding zero.Source: §2.1 · pp. 4, 6, Figure 2 · p. 5

2.2 Internal encoding of strategic information and its coupling to choice

• Own incentive and canonical choice were decodable in every model, but incentive–choice alignment emerged late in dense models and only Qwen2.5-Instruct retained a partial bridge beyond objective incentive.Choice decodability was AUC = 0.79–0.87; the partial bridge was +0.047 for Qwen2.5-Instruct versus −0.012 for base Qwen2.5, while GPT-OSS’s association disappeared after separating capture sites.Source: §2.2 · pp. 6, 8, 9

2.3 Causal steering along the encoded incentive direction

• Steering shifted the output policy toward incentive-favoured actions in all nine double-dominance games for both Qwen models, but not systematically for Llama or non-dominance families.The effect concerned graded output policy rather than regenerated choices; Qwen2.5 and Qwen2.5-Instruct showed +0.055 [.035, .073] and +0.421 [.363, .475] against permuted directions, respectively, whereas regenerated-choice intervals included zero.Source: Figure 5 · p. 10, §2.3 · pp. 11–12

2.4 Answer-axis expression across prompt regions

• Despite similar Qwen baseline behaviour, answer-axis incentive expression differed sharply: λlens was 1.43 for Qwen2.5-Instruct, 0.79 for Llama, and 0.17 for Qwen2.5.Behavioural λ was 2.04 versus 1.92, with matching choices in 555 of 576 presentations; the behavioural bridge was stronger in the instructed model by +0.059, and only it showed a reliable opponent-payoff increment of −0.65.Source: §2.4 · p. 12, Figure 6 · p. 13
• Incentive information was recoverable at own-payoff tokens before becoming expressed along the answer axis near commitment, while cue-target projections likewise strengthened toward the final position.At own-payoff tokens, probe AUC was 0.60–0.67 with lens performance near chance; at the final token, lens performance reached 0.98–0.99 in instructed models and 0.70 in Qwen2.5 base.Source: §2.4 · pp. 13–14, §2.5 · p. 14

2.5 Prompt-induced dispositions: from neural encoding to choice

• Cue effects were selective: inequity aversion shifted choices most strongly, while risk, loss, and maximin had intermediate effects and selfish maximisation produced little additional movement.Raw mean absolute cue–baseline shifts were 0.643 across 17 inequity-aversion games, 0.311, 0.309, and 0.331 across 48 risk, loss, and maximin games, and 0.093 across 96 selfish-maximisation games.Source: §2.5 · p. 14
• Game structure drove the largest held-out prediction gain, incentive gap contributed most, cue identity added ∆AUC ≈0.04, and cue–structure interactions improved prediction further.Strong neural cue encoding did not imply uniform behavioral expression: fixed wording perturbed an existing strategic process constrained by the underlying game.Source: §2.5 · p. 14

Core findings

• Strategic information was decodable in every model, yet incentive–choice linkage and externally induced susceptibility differed, establishing a three-way dissociation between availability, recruitment and susceptibility.Dense models reproduced broad human-like declines in canonical choice as game complexity increased, but similar choices did not imply similar computations.Source: §3 · p. 14

Decision cues

• Fixed decision cues were nearly perfectly identifiable internally, but their choice effects depended on cue, model and game; game structure predicted choices better than cue identity.The cues modified an existing strategic process rather than imposing a uniform decision rule across games.Source: §3 · pp. 15–16, Figure 7 · p. 15

Internal processing

• In dense models, incentive and choice representations aligned mainly in deeper blocks and near-choice processing, while GPT-OSS probes recovered incentive more accurately from router gates than selected routes.These activation comparisons are descriptive because recordings followed reasoning and were not manipulated.Source: §3 · p. 16

Matched Qwen comparison

• The base and instruction-tuned Qwen2.5 models agreed on 96.4% of baseline decisions, but post-training strengthened how available incentive information reached choice.The models represented payoff incentive similarly, while the instruction-tuned model showed stronger incentive–choice linkage and preference reflection.Source: §3 · p. 16

Causal intervention

• Causal strengthening of the incentive-associated activation shifted graded preference toward the incentive-favoured action in both Qwen models, especially double-dominance games, but not selectively in Llama.Qwen base showed that a weak observed incentive–choice link can coexist with a signal that remains causally influential.Source: §3 · p. 16

Implications and limitations

• The evidentiary framework separates what models choose, what information is available, what ordinary processing recruits and what intervention can influence, while generalisation remains limited to one-shot strict ordinal 2 × 2 games and four checkpoints.The causal tests covered one prespecified incentive, one answer position and limited layers, and human similarity was behavioural rather than computational.Source: §3 · p. 17

4.1 Experimental design and model sample

• The study evaluates four large language models on the complete set of 144 distinct strict ordinal 2 × 2 games, scoring responses relative to a fixed canonical action.The models comprise three dense transformers and one mixture-of-experts model.Source: §4.1 · pp. 17–18
• Choices are measured from generated answers, with cue effects compared against matched player-1 baselines and unusable responses treated as missing rather than imputed.There were 18,326 usable choices out of 18,432 responses; five cues and a neutral control were applied to player 1, while player 2 was measured at baseline.Source: §4.1 · pp. 19–20

4.2 Behavioural analysis and human calibration

• Behavioural analysis compares model and human choices using equilibrium conformity, payoff efficiency, coordination, incentive sensitivity, revealed reasoning depth, and complexity-adjusted canonical-choice associations.Candidate behavioural models are evaluated by ten-fold held-out-game binomial log loss, while cue uptake uses cue-specific identifiability sets and structure-versus-cue attribution uses grouped cross-validation and held-out AUC.Source: §4.2 · pp. 20–22

4.3 Internal representations and causal intervention

• Internal analyses capture dense-model residual states through the answer position and GPT-OSS commitment states, then decode choice, both players’ incentive signs, and a raw-payoff control on held-out games.GPT-OSS analyses distinguish residual and router signals, and router decodability is treated as descriptive rather than causal.Source: §4.3 · p. 23
• Representation geometry, recruitment bridges, token-level logit lenses, and causal interventions test how incentive signals relate to decisions while preserving objective-gap controls and explicitly distinguishing association from causation.The recruitment bridge residualizes internal projections and choice rates on the objective incentive; GPT-OSS geometry uses a separate empirical-belief sensitivity analysis, while the main incentive gap remains defined at q = 0.5.Source: §4.3 · pp. 23–24
• The analysis compared an output lens with a game-held-out linear probe, while interpreting token and cue projections as descriptive answer-axis signals rather than causal attribution or compliance.GPT-OSS was excluded from matched-position lens comparisons because its activations were captured after generated reasoning.Source: §4.3 · pp. 24–25
• Causal steering used an incentive direction fitted on all 144 games, orthogonalized to the answer-letter axis, and injected at dense-model answer positions with norm-scaled positive and negative doses.Interventions targeted layers 65 and 79 with α ∈ {0, ±0.05, ±0.1, ±0.25}; the answer-axis removal was intended to prevent directly writing the answer token.Source: §4.3 · p. 25
• Intervention effects were estimated from 54 deterministically selected games using dose-zero subtraction, form averaging, and slopes over six non-zero doses, with graded preference primary and regenerated choices secondary.Family estimates averaged nine game slopes, while pooled summaries weighted six families equally rather than the unequal 144-game catalogue.Source: §4.3 · p. 25
• Controls tested dose-zero identity, random directions, layer choice, orthogonalization, family resolution, and permuted incentive directions, while GPT-OSS was not intervened on because its routed, post-reasoning activations were not comparable or modifiable.The three permutations supported structured comparison rather than a permutation p-value.Source: §4.3 · pp. 25–26

4.4 Statistical inference and reproducibility

• Uncertainty used game-level bootstrap intervals that preserve repeated forms, with participant clustering for human analyses; intervals were 95%, and no multiplicity correction was applied to exploratory geometry or descriptive cross-model comparisons.For models, the 144 games are treated as a catalogue, so bootstrap intervals quantify sensitivity to game composition rather than ordinary sampling error.Source: §4.4 · p. 26
• Collection and decoding were deterministic, with fixed prompts, disabled sampling, counterbalanced forms, complete run records, prompt hashes, and consolidated estimator-specific settings and validation checks.Analyses used Python, PyTorch, Transformers, and scikit-learn, with further implementation details in the supplementary methods.Source: §4.4 · p. 26

• ChatGPT and Claude assisted with language editing, Claude Code assisted with analysis-code development, and the authors reviewed all assisted work and took responsibility for the final content.Source: Use of generative AI tools · p. 26

• The supplementary methods contain the full experimental specification.Source: A Supplementary Methods · p. 27
• These sections document the experimental details supporting the main text.Source: A Supplementary Methods · p. 27
• The supplementary material establishes the specification underlying the study’s main-text analyses.Source: A Supplementary Methods · p. 27

Game space

• The game space comprises six ordinal structural families, with equilibria, dominance profiles, iterated-dominance depth, and pure-equilibrium counts computed from one prespecified feature table.Supplementary Table 1 defines the six families and their equilibrium structure.Source: §A.1 · p. 27, Table 1 · p. 27

Prompt lineage and scope

• The single-shot prompt preserves sentence-level game elicitation while omitting repeated-game framing, history, rounds, and cumulative payoffs; wrappers vary by model without changing user-level game text.The dataset therefore does not estimate repeated-game learning, adaptation, or history effects.Source: §A.1 · pp. 27–28

Readout and capture conventions

• Dense models use answer-prefix residuals from L0–L80 and graded policy probabilities, whereas GPT-OSS uses post-reasoning residual, router, and expert state at commitment or analysis-to-final transitions.GPT-OSS router measurements are sampled at 14 prespecified layers, while its residual capture remains full-depth across L0–L36.Source: §A.1 · p. 30

Classification of GPT-OSS commitments

• GPT-OSS responses are classified as pure, mixed, or uncommitted; across 4,608 responses, 4,415 were pure, 190 mixed, and three uncommitted.Mixed responses retain stated probabilities or use a uniform mixture when no numerical probability is given, with one fixed-seed Bernoulli resolution for analyses requiring a realized choice.Source: §A.1 · p. 30

Collection factorial

• The complete collection contains 144 × 8 × 4 = 4,608 observations per model and 18,432 observations across four models before excluding unparseable or uncommitted responses.Each cue applies only to the player-1 prompt, while player 2 is always baseline; readout depth is not an experimental condition.Source: §A.1 · p. 31

Human reference data

• Human reference data retain 451 complete ordinal sessions and 832 cardinal games after rank-based eligibility filtering, with canonical reorientation preserving structural properties and a stricter sensitivity universe of 279 games.The cardinal panel contributes 2,416 oriented roles from 1,208 original games; the ordinal panel retains one participant’s repeated session as a separate analysis unit.Source: §A.1 · pp. 31–32

Canonical incentive and likelihood units

• Canonical incentives use qg = 0.5 for behavior, decoding, recruitment, and steering, whereas GPT-OSS sensitivity additionally recomputes gaps from its elicited opponent behavior.The reduced-form likelihood uses canonical choices across four LLM forms or 64,944 session–game choices, with uncertainty clustered over 451 sessions; QRE instead mutually couples both players’ logistic responses.Source: §A.2 · pp. 32–33

Conformity, efficiency and coordination: aggregation rules

• Aggregation preserves the estimand by matching LLM coordination within game and presentation form, pooling ordinal role marginals, and evaluating cardinal functionals once per original game.The section also defines quantal level-k gaps from uniform, level-1, and level-2 opponents, with binomial maximum-likelihood fitting across the same response counts.Source: §A.2 · p. 33

Probe protocols and targets

• Probe analyses fit logistic or ridge models inside training folds, score equilibrium and stimulus targets with macro one-versus-rest AUC, and report held-out results in Supplementary Table 9.GPT-OSS geometry was additionally standardized at a shared analysis-to-final transition, reproducing all 576 outputs exactly; mixed responses without numerical probabilities were excluded, yielding n = 541.Source: §A.2 · pp. 33–34
• Final-layer AUCs were 0.802/0.832 for canonical choice, 0.916/0.894 for incentive sign, and 0.601/0.567 for stimulus control in pure/all responses.Dense analyses used 576 baseline observations; choice sensitivities used 525 pure commitments, while GPT-OSS objective-incentive and stimulus probes used all 576 responses.Source: §A.2 · p. 34

Cue-identity decoding

• Cue differences achieved 0.97–1.00 game-held-out classification accuracy, showing distinguishable fixed-wording shifts without identifying general psychological constructs.Cued and baseline activations were differenced within game and presentation form, with a procedural control providing the reference level.Source: §A.2 · p. 34

Token-level conventions

• Token-level decompositions are descriptive and order-dependent, compressed probes can discard low-variance directions, GPT-OSS is excluded from homologous lens comparisons, and stopping-rule precision remains unverifiable.Causal interventions use fixed layer-index offsets; layer-65 injected and fitted axes have cosine 0.91–0.98, while archived records lack wavewise precision calculations.Source: §A.2 · pp. 34–35

Implementation

• Deterministic generation ran on one NVIDIA DGX Spark, using four presentation forms instead of seed-varying repeats.The hardware had a GB10 processor and 128 GB of unified memory.Source: §A.3 · p. 35
• Observation and run records captured model, game, role, condition, presentation, prompt, action mapping, decoding, stopping, layers, code revision, and integrity metadata.Unparseable or uncommitted responses remained missing; GPT-OSS records additionally retained response-specific mixture and fixed-seed information.Source: §A.3 · p. 35

Estimator settings

• Estimator settings specified validation, dimensionality reduction, uncertainty quantification, and random seeds, with game bootstraps preserving all four presentation forms together.Supplementary Table 6 documents these settings and identifies components unused by particular estimators.Source: Table 6 · p. 36

Reproducibility limitations

• The archived records lack immutable checkpoint and tokenizer hashes, a complete environment specification, and an accessible archival tag for token-level collection.The collection environment was not fully locked, so exact versions should be recovered from collection hosts or logs and documented with final code revisions and persistent identifiers in the public release.Source: §A.3 · p. 36

• Supplementary analyses add comparisons and diagnostics that support the main-text results.Source: B Supplementary Analyses · p. 36

Checkpoint agreement

• At baseline, the Qwen checkpoints agreed on 555 of 576 presentations (96.4%) and had identical four-form policies in 124 of 144 games.Across all eight conditions, agreement fell to 3,959 of 4,505 presentations (87.9%), indicating that decision cues produced most checkpoint divergence.Source: §B.1 · p. 37

Behavioural estimates

• Supplementary tables report per-agent behavioural parameters and cue-uptake estimates, including incentive sensitivity, revealed depth, decision sharpness, and control-adjusted cue shifts.These estimates include structural-complexity correlations and cue-specific magnitude, directed shift, and aim measures with bootstrap intervals.Source: §B.1 · pp. 36–37, Table 7 · p. 37, Table 8 · p. 37

Behavioural-model comparison

• QLk had the lowest out-of-fold log loss for the three dense models and humans, while CH was marginally lower for GPT-OSS.For GPT-OSS, CH achieved 0.328 versus 0.334 for QLk, with overlapping intervals.Source: Figure 1 · p. 38

Conformity prediction

• Predicted conformity increased monotonically with signed incentive gaps, while game structure produced the largest held-out discrimination gain.Cue identity and its interaction with structure further improved prediction, supporting situation-dependent cue effects rather than fixed action biases.Source: §B.1 · p. 38

Activation analyses

• The decodability analyses use parsed generated choices for realised choice and 576 player-1 baseline captures per model for other targets, with GPT-OSS read post-reasoning.The GPT-OSS realised-choice row contains 525 pure commitments.Source: §B.2 · p. 38
• Supplementary Table 9 reports held-out decodability for every prespecified probe target using final-residual-layer AUC with 95% game-bootstrap intervals.Equilibrium type and three-valued stimulus control use macro one-versus-rest AUC; dense-model dominance decodability is near ceiling, reflecting recovery of an explicit prompt feature rather than demonstrated use.Source: Table 9 · p. 38, §B.2 · p. 38
• Behavioural attribution checks find that the canonical-signed incentive gap contributes monotonically to predicted canonical conformity, while nested held-out prediction adds model, game, cue, and interaction features.A logistic mixed-model cross-check evaluates cue family, game family, and incentive clarity using generated dense-model decisions and GPT-OSS final-channel commitments.Source: Figure 2 · p. 39

Process link

• Stronger gate-level strategic evidence predicted shorter final-channel commitments after objective controls, with partial Pearson ρ = −0.351 (p = 1.6 × 10−5).Adding gate evidence increased objective-controlled R2 for log token length by 0.050, compared with 0.010 for the top-k set; this process link is supplementary and non-causal.Source: §B.2.1 · p. 39, Figure 3 · p. 40

GPT-OSS gate-versus-route readout

• Gate logits reached AUC 0.891 for incentive-sign decoding, versus 0.640 for top-k weights and 0.609 for top-k expert sets at layer 18.Across layers, the best AUCs were 0.891 for gate logits, 0.714 for top-k weights, 0.660 for top-k sets, and 0.915 for residual state; the dense-versus-sparse comparison is descriptive, not evidence that routing discards strategic information.Source: §B.2.1 · p. 39, Figure 3 · p. 40

Geometry onsets and capture-position sensitivity

• Under the objective incentive, GPT-OSS showed no three-layer geometry onset: the final-layer angle was 39.0° with p = 0.28, while router gate-logit geometry had no below-null layer.Empirical-policy geometry fell below the null through layer 17 but ended at 33.6° with p = 0.16; alternative capture-site estimates were confounded or endogenously selected and were not interpreted.Source: §B.2.1 · p. 40

Recruitment bridge

• The recruitment bridge was near zero on pure responses: slope 0.006 [−0.029, 0.037] for 525 responses and 0.003 [−0.029, 0.034] for 108 all-pure games.The all-response slope 0.072 mixed capture sites and was excluded; the bridge was not re-estimated on the uniform-site capture.Source: §B.2.1 · p. 40

Layer-specific empirical null

• Fusion required a layer-specific empirical null because null width tracked activation geometry, with rank correlations from −0.78 to −0.98 across models.A fixed 90° reference is unreliable, especially in anisotropic late layers; the GPT-OSS pattern was descriptive and capture-site sensitive.Source: §B.2.1 · pp. 40–41, Figure 4 · p. 41

Token-level analyses

• Qwen2.5 mean net signal was +0.44, but recruitment magnitude is measured by λlens rather than this annotated average.The annotated mean net signal captures the map’s level component, whereas λlens provides the recruitment magnitude.Source: §B.3 · p. 42
• Answer-prefix commitment produced the largest mean positive projection toward the canonical action in every dense model.Supplementary Figure 5 localizes this movement at the decision boundary and is descriptive rather than a recruitment metric.Source: Figure 5 · p. 42, §B.3 · p. 42
• Because the incentive gap was approximately zero-mean across games, signed payoff-token contributions averaged toward zero and the map reflected intercept or level effects.For Llama, chat-scaffolding tokens further diluted the average.Source: §B.3 · p. 42

Saturated-dose control

• At |α| ≥ 0.5, injections moved slot preference 0.84–1.00, flipped 78–99% of regenerated decisions, and produced letter-coherent but strategy-incoherent effects that counterbalanced to a null.Saturation was complete by |α| = 0.5, outside the validated |α| ≤ 0.25 linear regime; the large-dose sweep therefore does not test objective-incentive steering.Source: §B.4 · p. 43

Apparatus validity

• Dose-zero generations matched baseline in 0/432 cases, while direction, layer, channel, and structural controls formed a hierarchy validating the intervention apparatus.The incentive axis was non-degenerate, random directions were orthogonal, and layer-specific claims should rely on layer 65 because alignment was stronger there.Source: §B.4 · p. 43

Permutation control

• The true incentive direction outperformed permutation-seed means in both Qwen models, but Llama’s interval included zero and three seeds cannot establish unique potency among structured directions.Against the seed mean, paired double-dominance differences were +0.145 [+0.118, +0.170] and +0.431 [+0.352, +0.506]; against the most null-favourable seed, they were +0.055 [+0.035, +0.073] and +0.421 [+0.363, +0.475].Source: §B.4 · pp. 43–44

Dose–slope tables

• Supplementary Tables 10 and 11 report canonical-action incentive dose–slopes by cell, family, and model using game-cluster bootstrap confidence intervals and positive-slope counts.Table 10 defines DD and OD2 and reports cell-mean slopes; Table 11 organizes the corresponding family-by-model results.Source: Table 10 · p. 44, Table 11 · p. 44

Regenerated choices

• The regenerated-choice analysis estimated pooled incentive slope +0.068 [−.002, .140] versus random +0.027 [−.013, .069], with both overlapping confidence intervals including zero.Supplementary Table 12 reports family-balanced regenerated-decision canonical dose–slopes at layer 65.Source: Table 12 · p. 44