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arxiv: 2605.06525 · v1 · submitted 2026-05-07 · 💻 cs.GT · cs.MA· econ.TH

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Sustaining Cooperation in Populations Guided by AI: A Folk Theorem for LLMs

Jonathan Shaki , Eden Hartman , Sarit Kraus , Yonatan Aumann
Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:02 UTC · model grok-4.3

classification 💻 cs.GT cs.MAecon.TH
keywords folk theoremlarge language modelsrepeated gamescooperationNash equilibriummulti-agent systemsAI advice
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The pith

Shared LLM guidance sustains all feasible and individually rational outcomes as ε-equilibria in repeated games despite indirect observation and hidden advisor identities.

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

The paper studies how large language models advising multiple agents create coupling that affects cooperation when underlying incentives are misaligned. In one-shot interactions, shared instructions change equilibrium behavior only if an LLM influences more than one role in the same game, and the impact of each LLM's client share can be beneficial, harmful, or non-monotone. The main contribution proves a folk theorem for the repeated setting: all feasible and individually rational outcomes can be sustained as ε-equilibria. This holds even though clients cannot identify which LLM advised their opponents and observations of actions are indirect, requiring new proof techniques beyond the standard repeated-game folk theorem.

Core claim

In the repeated setting where multiple LLMs each advise populations of clients playing instances of an underlying game, all feasible and individually rational outcomes of that game can be sustained as ε-equilibria in the induced meta-game among the LLMs. This result holds despite indirect observation of actions and clients' inability to identify the specific LLM advising their opponents. The construction does not follow from the classical folk theorem and relies on new equilibrium strategies that operate at the level of LLM advice.

What carries the argument

The meta-game among the LLMs, created when each model advises a population of clients who interact in the underlying repeated game.

If this is right

  • In one-shot games, cooperation emerges only when an LLM can advise multiple roles within the same interaction.
  • Varying the share of clients per LLM can increase, decrease, or non-monotonically affect equilibrium cooperation depending on the base game.
  • Repeated play allows any rational outcome to be sustained approximately even without direct identification of advisors.
  • Shared LLM guidance couples agents who appear independent, expanding the set of sustainable cooperative outcomes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Different LLM providers may effectively compete or coordinate through the client populations they serve, creating a new layer of strategic interaction.
  • Design choices in how LLMs are deployed across agent populations could be used to steer long-run outcomes toward cooperation.
  • The result suggests testing whether similar ε-equilibrium constructions survive when LLMs have memory limits or when client populations overlap in more complex ways.

Load-bearing premise

The repeated structure must allow construction of equilibria that overcome indirect observation and the clients' inability to identify which LLM advised their opponents.

What would settle it

A concrete feasible and individually rational payoff profile that cannot be approximated as an equilibrium when clients cannot distinguish which LLM advised their opponents.

read the original abstract

Large language models (LLMs) are increasingly used to provide instructions to many agents who interact with one another. Such shared reliance couples agents who appear to act independently: they may in fact be guided by a common model. This coupling can change the prospects for cooperation among agents with misaligned incentives. We study settings in which multiple LLMs each advise a population of clients who participate in instances of an underlying game, creating strategic interaction at the level of the LLMs themselves. This induces a meta-game among the LLMs, mediated through clients. We first analyze the one-shot setting, where shared instructions can change equilibrium behavior only when an LLM may influence more than one role in the same interaction; in such cases, cooperation may emerge, and the effect of client share can be beneficial, harmful, or non-monotone, depending on the base game. Our main result concerns the repeated setting. We prove a folk theorem for LLMs: despite indirect observation and the clients' inability to identify which LLM advised their opponents, all feasible and individually rational outcomes can be sustained as $\varepsilon$-equilibria. The result does not follow from the standard folk theorem and requires new proof techniques. Together, these results show that shared LLM guidance can sustain cooperation among populations of agents even when the underlying incentives are misaligned.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper examines how shared LLMs advising populations of clients in underlying games induce a meta-game among the LLMs. In the one-shot setting, shared instructions alter equilibrium behavior only when an LLM influences multiple roles in the same interaction, with client-share effects that can be beneficial, harmful, or non-monotone. The central result is a folk theorem for the repeated setting: despite indirect observation and clients' inability to identify which LLM advised their opponents, all feasible and individually rational outcomes can be sustained as ε-equilibria. The proof requires new techniques beyond the standard repeated-game folk theorem.

Significance. If the folk theorem holds, the result provides a theoretical basis for how LLM-mediated guidance can sustain cooperation in populations facing misaligned incentives, even under realistic constraints like indirect observation. The explicit development of new proof techniques to handle the meta-game induced by shared advisors is a notable strength, as it directly addresses a setting where classical folk theorems do not apply.

major comments (1)
  1. [Main result (folk theorem proof)] The central claim rests on a folk theorem whose proof uses new techniques to construct equilibria under indirect observation and non-identifiability of advisors. The manuscript states that the result does not follow from the standard folk theorem, yet the provided abstract and high-level description do not include the full derivation, the explicit equilibrium strategies, or verification that the construction succeeds in the induced meta-game. This gap is load-bearing because the viability of the ε-equilibrium construction is the sole support for the claim that all feasible IR outcomes are attainable.
minor comments (2)
  1. [One-shot setting] The one-shot analysis mentions non-monotone effects of client share but does not illustrate them with a concrete base game or payoff matrix; adding a small example would clarify the claim.
  2. [Introduction] Notation for the meta-game (e.g., how client populations map to LLM influence) could be introduced earlier to improve readability before the repeated-game section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for identifying the need to strengthen the presentation of our main result. We address the concern about the folk theorem proof below and commit to revisions that improve clarity while preserving the manuscript's contributions.

read point-by-point responses

  1. Referee: [Main result (folk theorem proof)] The central claim rests on a folk theorem whose proof uses new techniques to construct equilibria under indirect observation and non-identifiability of advisors. The manuscript states that the result does not follow from the standard folk theorem, yet the provided abstract and high-level description do not include the full derivation, the explicit equilibrium strategies, or verification that the construction succeeds in the induced meta-game. This gap is load-bearing because the viability of the ε-equilibrium construction is the sole support for the claim that all feasible IR outcomes are attainable.

    Authors: We appreciate this feedback on the presentation of the central result. The full proof appears in Section 4, which develops new techniques for the induced meta-game: we explicitly construct LLM strategies that sustain any feasible and individually rational payoff vector as an ε-equilibrium despite clients' inability to identify which LLM advised opponents. The construction uses a block structure with coordinated reward and punishment phases, where deviations are detected via aggregate client play (leveraging the shared-advisor coupling) rather than direct identification; we then verify incentive compatibility by bounding the one-shot deviation gain by ε, accounting for the indirect observation. This does not reduce to the standard folk theorem because the meta-game payoffs are not directly observed by the LLMs. That said, we agree the introduction and abstract provide only a high-level sketch. In revision we will expand the main-text outline to include the explicit strategy form, the key detection mechanism, and a step-by-step verification that the construction works in the meta-game, moving only the most technical lemmas to the appendix. This addresses the load-bearing concern without changing the result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a novel proof extension

full rationale

The paper claims to prove a folk theorem for LLM meta-games in repeated settings that does not follow from the classical folk theorem and requires new proof techniques to handle indirect observation and non-attribution of advisors. No equations, parameters, or self-citations are presented that reduce the central result to a fit, definition, or prior author work by construction. The one-shot analysis and repeated-game construction are described as independent extensions of standard repeated-game theory, with the result explicitly positioned as non-derivable from prior theorems. This is a self-contained mathematical argument with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard repeated-game assumptions plus new proof techniques for the LLM meta-game; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard assumptions of repeated games (discounting or infinite horizon) allow equilibrium constructions
    Folk theorems in repeated games typically invoke these; the abstract invokes the repeated setting for the main result.

pith-pipeline@v0.9.0 · 5543 in / 1169 out tokens · 82795 ms · 2026-05-08T04:02:16.769357+00:00 · methodology

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    Call a time perioddiscrepantifH l h,s ‰ ¯H h, and letd l :“ 1 T řn h“1 řTh s“1 1tH l h,s‰ ¯H hu be the discrepancy fraction in the phase-lblock

    LetH l h,s denote the realized public aggregate at thes-th time period of blockh. Call a time perioddiscrepantifH l h,s ‰ ¯H h, and letd l :“ 1 T řn h“1 řTh s“1 1tH l h,s‰ ¯H hu be the discrepancy fraction in the phase-lblock. At the end of the block: - If there existh, s, i, asuch thatH l h,spi, aq ąΓ h,lpi, aq,it is anexcess deviationand the protocol mo...