Information Limits and Attractor Dynamics in Economies of Frontier LLM Agents: A Pre-Registered Test

Abstract: We report a pre-registered, two-part experiment on small economies of frontier language-model agents (Claude Opus 4.8), testing two quantitative predictions about coupled multi-agent systems: an information-theoretic capacity region for wealth growth under market coupling, and a mean-field residual-scaling law for population misalignment under incentive and control levers. All predictions, acceptance bands, and decision rules were frozen in a public git chain before any run; every reported number re-derives mechanically from cached model outputs; the entire experiment cost $138.76 in metered API spend and is re-runnable at zero cost from the cache. Result 1 (confirmation): in parimutuel-coupled economies, relative growth equals relative claimed information -- the gap law G_a - G_b = I_a - I_b holds to a worst-case 46 millinats (pre-registered band: 50) across four perception structures; coalition value is submodular exactly where channels are conditionally independent, and a designed XOR synergy control flips it supermodular by 0.62 >= ln2/2 nats, with agents reasoning out the joint bit; the joint growth ceiling G_S <= H(X) binds exactly; and the best-informed agent absorbs essentially the whole wealth pool in 4/5 market seeds. Result 2 (structural negative): the residual-scaling test returned "domain not found." In all 72 population runs, goal dispersion collapsed (V -> 0; maximum 4.85 against a frozen floor of 5.31), the population's response to the two levers was a step function across the dominance boundary rather than a smooth response, and cells near the boundary were bistable with seed-selected outcomes. No tested LLM population at any capability level realizes the noise-maintained-dispersion regime the smooth mean-field model assumes. We release the full protocol, pre-registration chain, call cache, and analysis code.

• The paper confirms that quantitative information laws reliably predict agent-level economic outcomes with calibration-agnostic precision (<0.05 nats).
• It demonstrates a transition from submodular to supermodular coalition growth when using synergistic XOR-based signal composition.
• The study reveals discontinuous attractor dynamics in LLM populations, with bistable behavior challenging traditional mean-field models.

Information Limits and Attractor Dynamics in Economies of Frontier LLM Agents

Introduction and Motivation

The study presents a pre-registered, empirical assessment of collective behavior in populations of frontier LLM agents (specifically Claude Opus 4.8) engaged in canonical economic and control-theoretic environments. The experimental design rigorously tests (a) whether quantitative information-theoretic laws accurately predict agent-level economic outcomes in coupled markets, and (b) whether LLM agent populations exhibit population-level response properties (such as smooth, noise-maintained misalignment) that underlie mean-field models central to multi-agent alignment, governance, and incentive theory. The pre-registration protocol is enforced strictly: all predictions, acceptance bands, and procedural amendments are irrevocably committed ex ante on a public VCS chain, and all model interactions are cached for full auditability.

Experimental Framework

Part I: Parimutuel Market and Capacity Region

Agents operate in parimutuel-coupled markets, each perceiving the state of a toy world (three fair bits) through private perception channels (perfect, overlapping, cloned, noisy, or XOR-coupled). Upon observing a signal, each agent outputs a posterior over world states, which is treated as a bet allocation in the market. Coalition actions are tested via composite elicitation with designated synergy and redundancy controls.

Quantitative predictions focus on the "capacity region" derived from multi-agent information theory and Kelly-type growth optimality:

• Gap Law: For any agent pair $(a, b)$, the difference in average log-wealth growth $G_a-G_b$ is predicted to equal the difference in their respective information $I_a - I_b$ (relative to the world prior).
• Submodularity and Synergy: Coalition growth is expected to demonstrate diminishing returns when members' channels are conditionally independent, and strictly supermodular growth when designed joint synergies (as in the XOR case) exist.
• Entropy Ceiling: No agent or coalition achieves asymptotic growth exceeding the entropy rate $H(X)$ of the world.
• Market Selection: In dynamic markets, agents with finer information are predicted to absorb almost all wealth.

Part II: Population Dynamics and the Mean-field Regime

Populations of LLM agents are distributed across a discrete action/niche landscape with reward gradients, external control incentives, and coordination pressures. The mean-field prediction is a residual-scaling law: the population's misalignment from a target $\bar{k} - k^*$ should scale proportionally to a combination of goal-dispersion $V$, gradient $g$, and control strength $\gamma$.

The key theoretical implication: if LLM populations occupy a smooth, noise-maintained regime, interventions (incentive tuning, control) should produce marginal, continuous adjustments in population alignment. Sharp deviations from this behavior challenge the foundational assumptions of mean-field analytic tools for AI population governance.

Key Results

Confirmation of Coupled Information-Economic Laws

The empirical findings for the coupled market setting are all positive within the pre-registered bands:

• Gap Law Holds: Across all four non-synergistic perception structures, the empirical gap $G_a-G_b$ matches $I_a-I_b$ to within $G_a-G_b$0 nats ($G_a-G_b$1 band), robustly validating the capacity-region law connecting epistemic capacity (KL of posteriors from prior) and realized economic growth. This confirmation is stringent: the equality is calibration-agnostic and fully independent across the two axes.
• Submodularity/Synergy: As theoretically mandated, coalition value is submodular where signals provide independent evidence—but becomes strictly supermodular (synergy gap $G_a-G_b$2 nats) under XOR composition, nearly saturating the predicted $G_a-G_b$3 bound, confirming the mechanism of conditional dependence in coalition value.
• Entropy Ceiling: Joint coalition growth never exceeds the entropy ceiling; no overage is observed.
• Market Selection: In repeated-play dynamic markets, agents with higher channel information consistently absorb $G_a-G_b$4 of market wealth, with full information-based ordering in $G_a-G_b$5 seeds.
• Clones and Sampling Anomaly: The expectation that agent clones with identical signals achieve identical outcomes is violated in a subset of seeds: sampling noise in all-in bet allocation can irreversibly break symmetry, indicating an interface between idealized continuous theory and practical stochastic implementations.

Structural Negative: Absence of Smooth Population-Level Response

Attempted confirmation of mean-field residual-scaling in population misalignment yields a robust structural failure:

• Dispersion Collapse: Across all 72 grid conditions, the population's variance in goal ($G_a-G_b$6) collapses to values well below those requisite for the linear-response regime (max $G_a-G_b$7, all below the $G_a-G_b$8 regime floor). The population always concentrates—perfectly aligning to the control target when $G_a-G_b$9, or else saturating at the economic reward peak.
• Step-Function and Bistability: Rather than exhibiting marginal responses to control/incentive levers, the collective switches discretely between attractors on either side of the dominance boundary ($I_a - I_b$0). At the boundary, outcomes become bistable, with seeds stochastically selecting one attractor over the other; within attractors, behavior is highly reproducible and sharply deterministic.
• Three-Instrument Bracketing: Previous small-model arms either observed complete saturation (gradient window below noise floor, no scaling) or full non-response ($I_a - I_b$1 indistinguishable from uniform), together bracketing the possible continuous regime and finding it empty in all tested models.
• CAP (Cannot Answer – Pass): The residual-scaling law cannot be tested empirically because the required noise-maintained intermediate regime is never realized — neither confirmed nor falsified, but rather assigned to the class of mathematical results without empirical domain.

Discussion and Implications

Theoretical and Practical Implications

The study provides the first pre-registered, quantitative confirmation of the capacity region for information/wealth dynamics in coupled LLM-agent economies. This establishes a direct, calibration-agnostic law linking epistemic value (KL info over priors) and realized economic payoff under parimutuel coupling at millinat precision. Such laws enable robust, falsifiable evaluation of agent incentives, epistemic composition, and agent selection dynamics, and set a template for further empirical interrogation of information-theoretic constructs in emergent AI economic interactions.

The negative result for the mean-field regime challenges foundational modeling assumptions widely used in multi-agent safety, governance, and incentive design. In practical LLM agent economies, marginal interventions do not yield marginal changes in macroscopic population alignment. Instead, interventions operate as switches: unless an incentive crosses the attractor dominance boundary, the population state remains unchanged; if crossed, the population relocates wholesale to a new attractor. This property invalidates the intuitive application of mean-field gradient response and compels a shift to attractor-based analyses for both theoretical modeling and practical governance of LLM agent populations.

Limitations and Open Questions

The present findings are confined to a single model family (Anthropic Claude Opus 4.8) and small system scale (agents $I_a - I_b$2, worlds $I_a - I_b$3 states), with all limitations of vendor architecture, prompt design, and non-incentivized elicitation that implies. Replication across models (e.g., OpenAI or open-source LLMs), scale (order-of-magnitude increases in $I_a - I_b$4), economic settings (market frictions, entry/exit), and elicitation protocol (calibrated scoring or proper incentives) are substantive directions for future work.

The demonstration of step-function attractor dynamics raises theoretical questions about the universality of this phenomenon and its sensitivity to population diversity, agent objectives, and explicit exploration mechanisms. Mechanism design in LLM agent economies (e.g., to promote sustained goal dispersion or smooth response) remains open.

Conclusion

The study rigorously quantifies, at millinat scale, both the validity of coupled information-theoretic laws for agent economies, and the structural absence of smooth population-level response in contemporary LLM agent ecosystems. Information reliably prices realized economic value in coupled settings, but interventions on LLM populations act as switches rather than dials: population-level behaviors transition discretely between attractors, rather than adjusting marginally. These results have direct implications for both empirical analysis and theoretical modeling of future AI multi-agent systems, particularly for the design and governance of economic institutions populated by artificial agents.

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Reference:

"Information Limits and Attractor Dynamics in Economies of Frontier LLM Agents: A Pre-Registered Test" (2607.06001)