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arxiv: 2503.01976 · v3 · submitted 2025-03-03 · 💻 cs.GT

Learning a Game by Paying the Agents

Brian Hu Zhang , Tao Lin , Yiling Chen , Tuomas Sandholm This is my paper

Pith reviewed 2026-05-23 01:13 UTC · model grok-4.3

classification 💻 cs.GT
keywords no-regret learningutility elicitationrepeated gamesprincipal-agentequilibrium steeringpayment mechanismsgame theory
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The pith

A principal learns every agent's utility function to any precision ε in polynomially many rounds by choosing payments, for any no-regret algorithm.

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

The paper shows that a principal who observes play and issues payments based on actions can recover the payoff matrices of all agents up to arbitrary accuracy. The number of rounds needed grows polynomially with the number of actions and with 1/ε. Recovery succeeds even when the principal does not know which no-regret algorithm each agent is using. The same technique yields the first method for steering agents toward a chosen equilibrium without knowing their utilities in advance.

Core claim

The principal constructs, for each agent, a zero-sum game whose pure strategies for the principal are all possible payment functions over action profiles. The value of this game encodes the agent's utility function. By playing a mixed strategy over payments that solves the zero-sum game, the principal forces the observed sequence of agent actions to reveal the payoffs to precision ε. The procedure runs in time polynomial in the game size for any no-regret dynamics whose regret bound is known.

What carries the argument

A zero-sum game between the principal and each agent whose strategies for the principal are payment functions over observed actions.

Load-bearing premise

The agents follow no-regret algorithms whose total regret is bounded by some known function of the number of rounds.

What would settle it

Apply the payment procedure to a small game whose utilities are known in advance, recover candidate utilities, and verify whether fresh play by the agents matches the behavior predicted by those utilities within ε.

read the original abstract

We study the problem of learning the utility functions of no-regret learning agents in a repeated normal-form game. Differing from most prior literature, we introduce a principal with the power to observe the agents playing the game, send agents signals, and give agents payments as a function of their actions. We show that the principal can, using a number of rounds polynomial in the size of the game, learn the utility functions of all agents to any desired precision $\epsilon > 0$, for any no-regret learning algorithms of the agents. Our main technique is to formulate a zero-sum game between the principal and the agents, where the principal chooses strategies among the set of all payment functions to minimize the agent's payoff. Finally, we discuss implications for the problem of steering agents. We introduce, using our utility-learning algorithm as a subroutine, the first algorithm for steering arbitrary no-regret learning agents to a desired equilibrium without prior knowledge of their utility functions.

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

2 major / 1 minor

Summary. The paper studies learning utility functions of agents playing a repeated normal-form game who use arbitrary no-regret algorithms. A principal observes play, sends signals, and makes action-dependent payments. The central claim is that the principal can learn all agents' utilities to precision ε > 0 using a number of rounds polynomial in the game size, via a zero-sum game reduction in which the principal chooses payment functions to minimize agent payoffs. The paper also gives a steering algorithm that uses the learning procedure as a subroutine to drive agents to a desired equilibrium without prior knowledge of utilities.

Significance. If the polynomial-round guarantee holds for arbitrary no-regret algorithms, the result would be significant for mechanism design and learning in games, as it separates utility learning from knowledge of specific regret rates and enables steering without utility knowledge. The zero-sum reduction technique is a potentially useful primitive. The manuscript does not appear to ship machine-checked proofs or reproducible code.

major comments (2)
  1. [Abstract] Abstract and §1 (central claim): the stated guarantee of a number of rounds polynomial in the game size, independent of the agents' specific no-regret algorithm, is load-bearing for the main result. No-regret only guarantees regret = o(T); the rate can be slower than any fixed function of T. The zero-sum construction relies on observed average play being ε-close to a coarse correlated equilibrium of the payment-modified game, and this closeness is controlled by (regret bound)/T. Without a uniform or known regret bound the T(ε) required cannot be bounded by any polynomial that is independent of the unknown algorithm.
  2. [Abstract] Abstract and introduction (zero-sum reduction): the manuscript provides no proof sketch, error analysis, or explicit dependence on the regret bound in the abstract or early sections. It is therefore impossible to verify whether the reduction actually produces a polynomial independent of the algorithm or whether an implicit assumption on the regret rate has been introduced post-hoc.
minor comments (1)
  1. [Abstract] The abstract states the steering result but does not indicate whether the polynomial bound carries over to the steering subroutine or whether additional rounds are required.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful comments on our manuscript. We address each of the major comments below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract and §1 (central claim): the stated guarantee of a number of rounds polynomial in the game size, independent of the agents' specific no-regret algorithm, is load-bearing for the main result. No-regret only guarantees regret = o(T); the rate can be slower than any fixed function of T. The zero-sum construction relies on observed average play being ε-close to a coarse correlated equilibrium of the payment-modified game, and this closeness is controlled by (regret bound)/T. Without a uniform or known regret bound the T(ε) required cannot be bounded by any polynomial that is independent of the unknown algorithm.

    Authors: We agree that our stated claim requires clarification. While the result holds for any no-regret algorithm, the number of rounds needed to achieve the desired precision depends on the specific regret bound of the algorithm in question. The polynomial is in the game size and 1/ε, but the leading constants and the choice of T incorporate the regret rate. We will revise the abstract and Section 1 to explicitly state the dependence on the regret bound and avoid any implication of a uniform polynomial independent of the algorithm. revision: yes

  2. Referee: [Abstract] Abstract and introduction (zero-sum reduction): the manuscript provides no proof sketch, error analysis, or explicit dependence on the regret bound in the abstract or early sections. It is therefore impossible to verify whether the reduction actually produces a polynomial independent of the algorithm or whether an implicit assumption on the regret rate has been introduced post-hoc.

    Authors: We acknowledge the lack of a detailed proof sketch in the abstract and introduction. The full proof in the body of the paper includes the error analysis relating the distance to the CCE to the per-round regret. We will add a high-level proof sketch and explicit discussion of the regret dependence to the introduction to facilitate verification of the result. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent zero-sum construction

full rationale

The paper's central result is a polynomial-round algorithm for learning utilities to precision ε via a zero-sum game between principal (choosing payment functions) and agents. No equations, fitted parameters, or self-citations are shown reducing the bound to a definition or tautology. The technique is presented as an external construction that works for any no-regret algorithm whose regret is o(T), without the bound itself being forced by redefinition of inputs. This is a standard non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or additional axioms beyond standard repeated-game assumptions are visible.

axioms (1)
  • domain assumption Agents follow no-regret learning dynamics whose regret is bounded over repeated play.
    Invoked to guarantee that the principal's payments can elicit distinguishable behavior.

pith-pipeline@v0.9.0 · 5691 in / 1027 out tokens · 29630 ms · 2026-05-23T01:13:52.982189+00:00 · methodology

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