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The Effect of State Representation on LLM Agent Behavior in Dynamic Routing Games

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read How the game's history is worded into the prompt — summarized, regret-based, own actions only — determines whether LLM agents converge to equilibrium in a repeated routing game.

desk verdict Useful framework and a convincing summary-vs-full effect, but the regret and own-action claims are muddied by an instruction mismatch and missing trial counts. read the letter →

arxiv 2506.15624 v1 pith:GOGZCOTI submitted 2025-06-18 cs.AI

Lyle Goodyear , Rachel Guo , Ramesh Johari This is my paper
classification cs.AI
keywords LLMagentsstaterepresentationdynamicroutinggamesBraess'sparadoxregretfeedbackpromptingstyleequilibriumconvergencelearningin
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to establish that the natural-language 'state representation' an LLM agent receives is not a neutral way to package game history but a first-order control on whether the agent behaves strategically. In a repeated selfish routing game with a known, simple Nash equilibrium, the authors vary how history is encoded along three axes—full transcript versus summarized table, raw payoffs versus regrets, and everyone's actions versus only the agent's own—and test all eight combinations. The consistent result is that agents who are given a summarized, regret-based, own-actions-only history play closest to equilibrium, with far lower regret and far fewer route switches, while full-chat transcripts with payoff feedback and full visibility of others' choices yield erratic play. The authors offer the framework and the findings as practical guidance: experimenters who want LLM agents to behave like stable, rational players should treat the wording of history as a design variable, not an implementation detail. If the finding extends beyond this one game, prompt design becomes a measurable lever in every LLM-based economic or strategic simulation.

What carries the argument

The carrying object is a three-axis taxonomy of natural-language state representations—action informativeness (whose past choices are shown), reward informativeness (payoffs or regrets), and prompting style (full chat transcript or summarized table)—whose eight crossings are the experimental conditions. The testbed is a repeated atomic selfish routing game on the Braess's-paradox network, chosen because its equilibria are exactly known and because human subjects converge to them in laboratory experiments, giving the authors both a theoretical and a behavioral benchmark. Equilibrium proximity is measured by deviation scores, defined as the total distance between the observed route counts and the equilibrium route counts, and by Kendall's $\tau$ between round number and deviation, which detects monotone convergence; stability is measured by the mean number of route switches per agent.

What would settle it

Run the same eight state representations but rewrite the game description for each condition so that agents are promised exactly the feedback they will receive—regret conditions announce regret reports, and own-action conditions announce that only the focal agent's history will be shown—then compare the equilibrium gap between representations. If the representation ordering survives these instruction-consistent descriptions, the central claim stands; if the gap collapses or reverses, the reported effects were an artifact of promising one thing and delivering another.

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Extended reading notes

Core claim

On the authors' own terms, the discovery is a sharp, empirical dependence: in a dynamic game, the natural-language state representation of history governs whether LLM agents approach equilibrium, and it does so along three predictable axes. The testbed is a repeated atomic selfish routing game played by eighteen agents over forty rounds, built on the Braess's-paradox network in which adding a third route raises the equilibrium cost for everyone from 300 to 360; each variant has a unique equilibrium ($n_L = n_R = 9$ in Game A, all eighteen agents on the bridge route in Game B). Crossing the three axes yields eight state representations, and the paper reports that summarization is the clearest driver: summarized representations cut mean regret four- to five-fold in Game A and nearly to zero in Game B, and produce route splits close to equilibrium, whereas full-chat representations stay far from equilibrium with high variance. Regret feedback has its largest effect in Game B, where the equilibrium route is a weak dominant strategy that yields lower payoffs, helping agents find it; and showing agents only their own actions persistently reduces switching in both games. The strongest representation, S-RO (summarized, regret-based, own-actions-only), tracks equilibrium more closely than the EXP3 and multiplicative-weights baselines on all four aggregate metrics in Game B and exhibits far fewer switches in Game A.

Load-bearing premise

The load-bearing premise is that the fixed game description given to every agent is neutral across all eight prompt conditions; because that description promises a full report of route distributions and payoffs each round, the regret-based and own-action conditions deliver something different from what the agent was told to expect, so the measured behavioral gaps could partly reflect that instruction mismatch rather than the three design axes.

Editorial extensions

If this is right

  • Experimenters building LLM agents for repeated games should summarize history into a compact table rather than replaying the full transcript; the results indicate that full-chat prompting actively degrades convergence and increases switching.
  • Regret is a more actionable feedback signal than payoff for LLM agents, precisely in games where the equilibrium action is not the one that looks best in raw payoff terms, so regret-based state designs should be the default when such counterfactual information is computable.
  • Hiding other agents' actions stabilizes play; the qualitative evidence suggests that revealing everyone's actions invites erroneous anticipatory reasoning about peers' future moves.
  • The best representation (summarized, regret-based, own-actions-only) matches or beats the classical no-regret algorithms EXP3 and multiplicative weights on equilibrium proximity and stability in these games.
  • The three-axis framework gives researchers a common vocabulary for state encoding, which the paper argues is needed before results across different LLM-agent game studies can be compared at all.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not test instruction-consistent descriptions: the fixed game description promises agents a route distribution and payoff each round, while the regret and own-action conditions silently deliver something else, so a direct control that rewrites the description to match each feedback regime would separate genuine representation effects from instruction-mismatch effects.
  • A natural extension is to predict that the summarized-beats-full-chat ordering grows with context length and with weaker models, following the known degradation of attention to the middle of long transcripts, and shrinks in very short games.
  • The framework suggests a design rule for LLM-based simulations beyond games: agent memory should be deliberately summarized and filtered rather than an append-only transcript, a hypothesis the paper gestures at but does not test outside the routing setting.
  • The regret axis implies a testable hypothesis for other decision tasks: any feedback design that encodes 'how much better you could have done' should beat outcome-only feedback for LLM agents, even outside game settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a framework for constructing natural-language state representations for LLM agents in repeated multi-agent games, characterizing them along three axes: action informativeness (own vs. all agents' actions), reward informativeness (payoff vs. regret), and prompting style (full-chat vs. summarized). The authors apply this framework to a repeated atomic selfish routing game with an 18-agent, 40-round design adapted from a human-subject experiment, testing all eight combinations of the three binary axes. They report that summarized prompting, regret-based feedback, and own-action-only feedback each move LLM agent behavior closer to Nash equilibrium and reduce switching instability, and they compare the best-performing representation (S-RO) against EXP3 and MWU baselines.

Significance. The paper addresses a timely and practically important question: how to encode game history for stateless LLM agents. The proposed framework is a useful organizing device for a design space that is often handled ad hoc, and the choice of a routing game with a unique, well-understood equilibrium provides a clean benchmark for evaluating agent behavior. The empirical study is ambitious, including eight representation conditions, two games, and comparisons to human-subject data and standard online-learning baselines. If the causal claims held up, the findings would give concrete guidance to experimenters building LLM agents for strategic settings. However, as detailed in the major comments, the experimental design does not currently support the causal interpretation advanced in the abstract, because the manipulations are confounded with violations of the game instructions and with the presence of the agent's own prior chain-of-thought.

major comments (2)
  1. [Section 4.2 and Appendix A] The causal claims about the reward-informativeness and action-informativeness axes are confounded by an instruction mismatch. Section 4.2 states that every agent receives a game description 'directly adapted from the text instructions given to subjects in [Rapoport et al., 2009]', and Appendix A shows that this description promises: 'At the end of each round, you will be informed of the number of drivers who chose each route and your payoff for that round.' Yet in the own-action-only conditions (F-P, S-P, F-R, S-R) the route distribution is omitted, and in the regret conditions (F-R, S-R, F-RO, S-RO) the promised payoff is replaced by a regret value. Thus a change along the action or reward axis is not a clean manipulation: it also changes whether the agent receives feedback that is consistent with the stated rules of the game. An agent that notices this inconsistency may behave differently (e.g., hedge, hallucinate missing counts, or discount the feedback) for reasons unrelated to the information content under study. To support the abstract's causal conclusions, the authors need to either adapt the game description to each condition so that the promised feedback matches the delivered feedback, or add control conditions that hold the description fixed while varying only the axis of interest, or provide direct evidence from agent outputs that this mismatch does not drive the observed differences. Without such steps, the 'regret vs. payoff' and 'own vs. everyone' findings are not cleanly identified.
  2. [Section 3, Figures 1a–1b, and Section 4.2] The summary-vs-full-axis comparison is also confounded. As described in Section 3, the full-chat condition includes the agent's own previous responses (including the chain-of-thought reasoning elicited by the 'think step-by-step' instruction), whereas the summarized condition contains only a tabular summary of past outcomes. Changing from full-chat to summarized thus varies at least three things simultaneously: total context length, the presentation format, and the presence of the agent's own prior reasoning in the context. The paper attributes differences between these conditions to 'natural language compression' but does not isolate compression from the inclusion of the agent's prior rationales. Since the summary-vs-full finding is described in Section 5.1 as 'the clearest finding in our results', this confound is load-bearing. The authors should either design a condition that includes the agent's prior responses in a compressed form without the full transcript, or explicitly acknowledge that the prompt-style axis bundles these factors and adjust the strength of the causal claim accordingly.
minor comments (5)
  1. [Section 4.2 and Section 5.1] The number of trials per representation is not reported anywhere in the main text. The standard errors in Table 1 cannot be interpreted without knowing how many independent games were run per condition, and the comparison with EXP3/MWU in Section 5.4 states '50 trials' only for the baselines. Please state the number of trials for each LLM condition.
  2. [Section 5.1, Table 1] Some standard errors seem implausibly large, for example F-PE in Game A reports a mean of 9.03 with a standard error of 7.26. If these are standard errors over trials of a trial-level mean, this implies enormous variance across trials; please verify the calculation or clarify whether these are standard deviations across rounds or something else.
  3. [Section 4.3] There is a grammatical error in the definition of the Game A focal-route metric: 'we first compute the of the number of agents' should read 'we first compute the number of agents'. This makes the text confusing, though the intended meaning is clear from the formula.
  4. [Section 5.1 and throughout] The paper makes strong comparative claims such as 'summarized representations lead to significantly lower regret' and 'summarized prompting leads to behavior closer to equilibrium' without reporting any inferential statistics (e.g., confidence intervals for the differences or formal tests). Given the high standard errors in several conditions, the qualitative conclusions would be substantially strengthened by supplying effect sizes and uncertainty measures for the key contrasts.
  5. [Section 4.3 and Section 3] The term 'regret' is defined in Section 4.3 as the per-round difference between the best possible payoff in hindsight and the received payoff, which is an instantaneous best-response regret rather than the cumulative regret commonly used in online learning. This is a reasonable choice, but it should be introduced and defined earlier in Section 3, where 'regret' is first used, to avoid ambiguity for readers familiar with the online-learning regret.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the study is empirical, with no free parameter fitted to the outcome and no load-bearing self-citation; the fixed-instruction confound is an internal-validity concern, not circularity.

full rationale

This paper makes no formal derivation whose inputs are equivalent to its outputs. The central findings are comparative empirical observations across 2x2x2 prompt conditions in a fixed routing game; the three axes are varied explicitly and all eight combinations are run. No parameter is estimated from the outcome data and then re-sold as a prediction: regret feedback is computed from the same payoff functions as the evaluation metrics, but the agent's route choice is not forced to equal the zero-regret route, and the paper reports cases where regret-based representations do not improve convergence (Game A), so the outcome is not determined by construction. The only load-bearing external citation is to Rapoport et al. (2009) for the game instructions, equilibrium benchmarks, and human-subject convergence; that citation is to an independent lab experiment, not to the authors' own prior work, and it is used as a benchmark rather than as a proof that the LLM results must hold. The reviewer-identified problem, namely that the fixed game instructions promise payoff and route-distribution feedback that the own-action and regret conditions do not deliver, is a genuine threat to causal identification (an instruction-mismatch confound), but it is not circularity in the sense of a conclusion that is equivalent to its premises by definition. Accordingly, the score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central empirical claim rests on the experimental design: the chosen game, the fixed instructions, the specific model and temperature, and the evaluation metrics. The most fragile premise is that the fixed instructions are neutral across feedback formats, which the design violates.

free parameters (3)
  • GPT-4o temperature = 1.0
    Set to 1 to induce varied behavior across runs; a hand-chosen hyperparameter that affects variance and is not fitted to data.
  • Number of trials per condition = not reported
    The paper reports standard errors over trials but does not state the number of trials for LLM agents, leaving a key experimental quantity unknown.
  • Baseline hyperparameters (eta, gamma) = 0.75
    MWU learning rate and EXP3 exploration rate were chosen by hand for the comparison baselines; they affect baseline results but not the central state-representation claims.
assumptions (6)
  • domain assumption LLM agents are stateless and require natural language encodings of history.
    Stated in Section 1 and motivates the entire setup.
  • standard math Game A and Game B equilibria are as computed (Game A split 9/9, Game B all 18 on bridge).
    Follows from the cost functions in Section 4.1; used as the evaluation benchmark.
  • domain assumption Closeness to pure-strategy Nash equilibrium is the appropriate evaluation criterion.
    The paper evaluates LLM agents by distance to equilibrium, a normative choice; other objectives could change the conclusions.
  • domain assumption The fixed game instructions are neutral across state representations.
    Section 4.2 and Appendix A: the same instructions promise route distribution and payoff feedback, while some conditions deliver regret or omit others' actions. This assumption is violated, creating a confound.
  • domain assumption GPT-4o at temperature 1 is a representative LLM agent.
    The paper uses one model and one temperature; generalizability across models is unknown and acknowledged in Section 5.5.
  • domain assumption The number of trials is sufficient for the reported standard errors.
    Since the trial count is not stated, the reliability of all quantitative comparisons cannot be assessed.

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Cite this review

Pith. "Pith review of The Effect of State Representation on LLM Agent Behavior in Dynamic Routing Games." pith.science (2026). https://pith.science/paper/GOGZCOTI

@misc{pith2026250615624,
  author       = {Pith},
  title        = {Pith review of: The Effect of State Representation on LLM Agent Behavior in Dynamic Routing Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GOGZCOTI}},
  note         = {Machine review of arXiv:2506.15624}
}
read the original abstract

Large Language Models (LLMs) have shown promise as decision-makers in dynamic settings, but their stateless nature necessitates creating a natural language representation of history. We present a unifying framework for systematically constructing natural language "state" representations for prompting LLM agents in repeated multi-agent games. Previous work on games with LLM agents has taken an ad hoc approach to encoding game history, which not only obscures the impact of state representation on agents' behavior, but also limits comparability between studies. Our framework addresses these gaps by characterizing methods of state representation along three axes: action informativeness (i.e., the extent to which the state representation captures actions played); reward informativeness (i.e., the extent to which the state representation describes rewards obtained); and prompting style (or natural language compression, i.e., the extent to which the full text history is summarized). We apply this framework to a dynamic selfish routing game, chosen because it admits a simple equilibrium both in theory and in human subject experiments \cite{rapoport_choice_2009}. Despite the game's relative simplicity, we find that there are key dependencies of LLM agent behavior on the natural language state representation. In particular, we observe that representations which provide agents with (1) summarized, rather than complete, natural language representations of past history; (2) information about regrets, rather than raw payoffs; and (3) limited information about others' actions lead to behavior that more closely matches game theoretic equilibrium predictions, and with more stable game play by the agents. By contrast, other representations can exhibit either large deviations from equilibrium, higher variation in dynamic game play over time, or both.

Figures

Figures reproduced from arXiv: 2506.15624 by the authors.

Figure 1
Figure 1. a. Example full-chat representation given to an agent at the start of round 10. LLM is prompted with a list of the previous messages in the exchange. Game Environment (Round 10) You will be participating in an experiment on route selection in traffic networks . . . The available routes are: O-L-D, O-R-D Game Environment (Round 10) Summary of previous rounds: Round 1: Your Choice: O-R-D Route Choice Distribution: {O-… view at source ↗
Figure 3
Figure 3. a. Network in Game 𝐴 O L R D 10𝑥 210 210 10𝑥 0 Fig. 3b. Network in Game 𝐵 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 14
Figure 14. a. Mean 𝜏 across trials, Game A Fig. 14b. Mean 𝜏 across trials, Game B [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figures from the paper (2 more)
Figure 15
Figure 15. Figure 15: Comparison of LLM agent performance under S-RO with learning algorithms EXP3 and MWU across [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 16
Figure 16. Figure 16: Game description (adapted from [Rapoport et al [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.