Technical Summary: Information Limits and Attractor Dynamics in Economies of Frontier LLM Agents

Problem Statement

This paper addresses two fundamental, untested questions regarding populations of frontier Large Language Model (LLM) agents interacting in economic and selection-based environments:

1. Information-Wealth Coupling: Do coupled LLM agents obey classical information-theoretic laws connecting their "claimed" information (posterior beliefs) to their economic growth (wealth accumulation) in parimutuel markets? Specifically, does the relative growth rate of agents equal their relative information content, and do coalition values exhibit predicted submodularity or synergy?
2. Population Dynamics and Control: Do LLM populations respond smoothly to incentive and control levers (e.g., reward gradients and alignment bonuses) as assumed by mean-field models? Or do they exhibit discrete, non-linear behaviors such as attractor switching and goal dispersion collapse?

Prior work has simulated LLM agents in social or economic settings, but none have rigorously tested these quantitative laws on frontier models with pre-registered acceptance bands and deterministic, auditable execution.

Methodology

The study employs a pre-registered, two-part experiment using the claude-opus-4-8 model. The methodology is distinguished by strict adherence to scientific reproducibility and auditability:

• Pre-Registration & Freezing: All predictions, acceptance bands, decision rules, and protocol gates were committed to a public Git repository before data collection. Three amendments were committed "blind" (before the runs they governed) to address external constraints or internal inconsistencies.
• Determinism & Cost Control: Every API call was cached in a SQLite store keyed by the full request. The analysis is entirely deterministic and re-runnable offline from the cache at zero marginal cost. Total metered API spend was $138.76.
• Instrument Design:
  • Experiment A (Capacity Region): Agents interact in a parimutuel market over a 3-bit world. Four perception structures (disjoint, overlapping, cloned, noisy) and a designed XOR synergy control are tested. Agents report posteriors, which are mechanically converted to bets. Growth is measured as expected log-wealth change.
  • Experiment B (Residual Scaling): A population of 20 agents occupies a 16-niche landscape with a reward gradient ($g$) and a control bonus ($\gamma$) at a target niche. The experiment tests a $3 \times 3$ grid of $(g, \gamma)$ values across 8 seeds (72 runs) to measure population misalignment and goal dispersion ($V$).
• Decision Rules: A mechanical analyzer evaluates results against frozen bands. Verdicts are categorized as PASS, CLEAN NEGATIVE, or CAP ("Cannot Answer - Insufficient In-Regime Span").

Key Contributions

1. First Pre-Registered Confirmation of Coupled Capacity Laws: The paper provides empirical evidence that frontier LLM agents obey specific information-theoretic laws in coupled markets, including the "gap law" (relative growth equals relative information), conditional submodularity of coalition value, and the entropy ceiling on joint growth.
2. Structural Negative Result for Mean-Field Models: The study demonstrates that tested LLM populations do not occupy the "noise-maintained dispersion" regime required for smooth mean-field linear-response models. Instead, they exhibit goal dispersion collapse ($V \to 0$) and step-function responses to control levers.
3. Auditable Methodology: The paper establishes a template for low-cost, high-integrity pre-registered experiments on frontier model populations, featuring frozen bands, blind amendments, and full provenance via a public commit chain.

Results

Result 1: The Coupled Capacity Region Holds

Across four perception structures and a designed synergy control, the experiment confirmed the following within pre-registered bands:

• Gap Law: The difference in relative growth between agents equals the difference in their claimed information ($\hat{G}_a - \hat{G}_b = \hat{I}_a - \hat{I}_b$). The worst-case deviation was 46 millinats (within the 50 millinats band).
• Coalition Structure: Coalition value was submodular where channels were conditionally independent (worst margin $-0.019 \ge -0.03$). In the XOR synergy control (where individual signals carry zero information but the joint signal carries one bit), the coalition value flipped to supermodular with a gap of 0.621 nats (exceeding the 0.347 nats band), demonstrating the agents' ability to reason out joint parity structures.
• Joint Ceiling: No coalition exceeded the world's entropy rate ($H(X)$); the bound held to numerical precision.
• Wealth Selection: In a live 40-round market, the agent with the cleanest information absorbed $\ge 99.9\%$ of the wealth pool in all seeds, and wealth ordering strictly followed information ordering in 4/5 seeds.
• Notable Deviation: The prediction that clones with identical signals would reach equal terminal wealth failed (2/4 scoreable seeds passed). The mechanism was identified as "absorbing all-in dynamics" where sampling noise in a single round permanently eliminates a clone from the zero-sum pool.

Result 2: The Mean-Field Regime Does Not Exist

The test for the residual-scaling law ($\text{residual} \propto V g / \gamma$) returned a CAP (Cannot Answer) verdict.

• Dispersion Collapse: In all 72 grid runs, goal dispersion ($V$) collapsed to near zero (maximum $V=4.85$ against a frozen floor of 5.31). The population never maintained the noise-sustained dispersion required for the linear-response regime.
• Step-Function Dynamics: The population response was discrete rather than smooth.
  • When control ($\gamma$) exceeded the peak incentive ($12g$), the population aligned perfectly with the target ($\bar{k}=0$).
  • When incentive dominated, the population clustered near the reward peak.
  • Near the boundary ($\gamma \approx 12g$), the system exhibited bistability, where the specific seed (sampling noise) determined which attractor the population settled into.
• Three-Instrument Bracket: Combined with two prior failed attempts using smaller models (one due to gradient saturation, one due to non-response), the results suggest the domain of the mean-field linear-response law is empty for tested LLM populations. The law is reported as "retired to its mathematical scope" rather than falsified.

Significance and Claims

The paper claims two distinct but equally weighted findings:

1. Quantitative Validity of Information-Economics Laws: It establishes that for frontier LLM agents, "what an agent measurably knows prices what it measurably earns." The coupling of information and wealth in parimutuel markets follows precise, falsifiable laws derived from classical information theory.
2. Discrete Attractor Dynamics: It challenges the assumption that LLM populations respond smoothly to incentives. Instead, these populations behave as discrete attractor systems. Interventions do not produce marginal adjustments; they either switch the population to a new attractor basin or have no measurable effect. This implies that multi-agent governance strategies relying on smooth, linear-response models may be fundamentally misaligned with the actual dynamics of LLM populations.

The authors emphasize that the negative result (Result 2) is only interpretable because the instrument was proven capable by the positive result (Result 1). The study concludes that while the capacity region is a confirmed property of these agents, the smooth mean-field regime assumed by many safety and alignment theories appears to be an artifact not present in real LLM populations.