Title: The Complexity of Correlated Equilibria in Generalized Games
Venue: NeurIPS 2025
Accepted: yes
Keywords: computational complexity

PPAD

correlated equilibrium

generalized games

equilibrium

game theory

nash
Abstract: Correlated equilibria —and their generalization $\Phi$-equilibria— are a fundamental object of study in game theory, offering a more tractable alternative to Nash equilibria in multi-player settings. While computational aspects of equilibrium computation are well-understood in some settings, fundamental questions are still open in _generalized games_, that is, games in which the set of strategies allowed to each player depends on the other players' strategies. These classes of games model fundamental settings in economics and have been a cornerstone of economics research since the seminal paper of Arrow and Debreu [1954].
Recently, there has been growing interest, both in economics and in computer science, in studying correlated equilibria in generalized games. It is known that finding a social welfare maximizing correlated equilibrium in generalized games is NP-hard. However, the existence of efficient algorithms to find _any_ equilibrium remains an important open question. In this paper, we answer this question negatively, showing that this problem is PPAD-complete.


Peer reviews:
comments: SUMMARY:
The paper studies the computational complexity of finding certain constrained approximate $\Phi$-equilibria in generalized games. It is shown that this search problem is PPAD-complete and thus presumably not solvable in polynomial time. This answers a recent open question from the literature. PPAD-hardness is based on a reduction from the PolyMatrix problem (finding approximate equilibria in polymatrix games) and membership in PPAD is shown via reduction to some problem related to quasi-variational inequalities.

STRENGTHS AND WEAKNESSES:
Strengths: The paper settles the complexity status of a seemingly important search problem in (generalized) game theory.

Weakness:
- The result is purely theoretic and I do not see a direct connection to the NeurIPS community. There is no discussion on possible practical implications of the result.
- The presentation is not understandable for non-experts in generalized games. Many important notions are not properly explained and defined (e.g. What is $\Delta(\mathcal A)$ and $\Delta(A)$? What is the idea behind the mediator recommendation and deviation? What is correlated distribution?).
- From a theoretical perspective, since I am not an expert in the field, I am not sure how relevant or surprising the result is.
- The paper misses a concluding discussion (other directions or open cases).

QUESTIONS:
1. Since your problem is formulated as a promise problem, do you not need to show for your construction that it satisfies the promise?

2. For the correctness, you only show one direction. Is this enough because we consider total search problems (where existence of a solution is guaranteed)?

3. Are there any interesting special cases which are efficiently solvable?

LIMITATIONS:
No limitations are explicitly mentioned.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the notion of \\Phi-correlated equilibrium in n player games under cost constraints for agents which are emposed on the their joint actions.\nMore precisely, the set \\Phi describes the set of (possibly randomized) deviations each player can make from action a for each action a. The equilibrium notion then restricts the players to use these deviations to deviate from their recommended strategy which is sampled by a mediator who correlates the agents joint actions.\n\nAgents should not want to use any of the allowed deviations.\nMoreover, each agent has an m dimensional cost vector associated with each joint action profile. For each cost dimension each agent has a threshold cost which she does not want to cross in expectation.\nIf this holds for all cost dimensions the correlated action profile is considered safe and one can define the induced set of safe deviations naturally.\n\nBecause safe deviations may not always exist, the authors constrain to a setting where each agent has at least one safe deviation - that is we are promised that an equilibrium exists.\n\nThe paper then shows that finding a constrained \\Phi-correlated equilibrium is PPAD-complete.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is well written, although at times a bit verbose (e.g. 67-69).\nOverall, I think the paper is fine, but do not see the usefulness of the result and further it doesn't seem technically\nchallenging enough to accept it despite this. \n\nAlthough not my main concern, it is not clear why \\Phi equilibrium in its generality is useful, especially since the deviation agents are only not incentivized to deviate in an average sense.\n\n__\n\nMinor errors:\n\nIn Line 201, in the second line and in Line 212 i and j seem to be flipped\n\nQUESTIONS:\nYou mention some applications in the introduction such as repeated auctions/ online advertising. Can you explain how your specific hardness result is relevant for the specific applications you mention?\n\nWhich sets of functions \\Phi are relevant other than CCE and CE? \nCan you name some applications in which other restricted type of deviations would only be feasible?\n\nLIMITATIONS:\nLack of directly relevant applications", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the complexity of computing correlated equilibria (CE) in generalized games, where players’ strategies are interdependent. While CE is typically easier to compute than Nash equilibria, the authors show that in these settings with coupled constraints, even to approximate CE computation is PPAD-complete. The authors establish it via a reduction from polymatrix games using two-team zero-sum game constructions. The results hold even with small approximation or constraint violations. Under standard assumptions, they further show that the problem likely requires quasi-polynomial time, resolving a key open question. This is a great paper to read overall.\n\nSTRENGTHS AND WEAKNESSES:\nBelow are my comments on this paper:\n\n(a) This paper presents deep theoretical insights but lacks experiments or simulations to illustrate practical implications... such as how these hardness results manifest in real-world economic or multi-agent systems? Can you highlight one or two real-life usecases wherein these results are of significant use?\n\n(b) The hardness proofs rely on structured reductions (eg: from polymatrix games) and certain types of deviation sets.. thus, I believe that these results may not generalize easily to broader classes of games as well as different equilibrium refinements... if not, can you please elaborate more on this?\n\n(c) The results in this paper only highlight worst-case intractability.. they do not explore whether practical heuristics or approximation schemes may still be effective in realistic settings (and thereby offering limited guidance for algorithmic design in applied domains)?\n\nQUESTIONS:\nPlease answer the following questions:\n\n(a) This paper presents deep theoretical insights but lacks experiments or simulations to illustrate practical implications... such as how these hardness results manifest in real-world economic or multi-agent systems? Can you highlight one or two real-life usecases wherein these results are of significant use?\n\n(b) The hardness proofs rely on structured reductions (eg: from polymatrix games) and certain types of deviation sets.. thus, I believe that these results may not generalize easily to broader classes of games as well as different equilibrium refinements... if not, can you please elaborate more on this?\n\n(c) The results in this paper only highlight worst-case intractability.. they do not explore whether practical heuristics or approximation schemes may still be effective in realistic settings (and thereby offering limited guidance for algorithmic design in applied domains)?\n\nLIMITATIONS:\n(a) This paper presents deep theoretical insights but lacks experiments or simulations to illustrate practical implications... such as how these hardness results manifest in real-world economic or multi-agent systems? Can you highlight one or two real-life usecases wherein these results are of significant use?\n\n(b) The hardness proofs rely on structured reductions (eg: from polymatrix games) and certain types of deviation sets.. thus, I believe that these results may not generalize easily to broader classes of games as well as different equilibrium refinements... if not, can you please elaborate more on this?\n\n(c) The results in this paper only highlight worst-case intractability.. they do not explore whether practical heuristics or approximation schemes may still be effective in realistic settings (and thereby offering limited guidance for algorithmic design in applied domains)?", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper studies the computational complexity of computing Correlated constrained equilibria in generalized games. Previously, it was shown that constrained-phi-equilibria are PPAD-complete to compute in generalized games, but this kept open the question of whether the introduction of correlation could be a sufficient relaxation to make the problem poly-time computable. This paper resolves this question, showing that correlation is insufficient. The proof proceeds by a reduction to Nash Equilibrium computation in polymatrix games. The result also implies that the quasi-polynomial time algorithm for computing approximate phi-equilibria of Bernasconi et al. [2023] is tight.

STRENGTHS AND WEAKNESSES:
Strengths:
- Resolves a fundamental problem in the complexity of computing equilibria for game theoretic problems, an active and important area of research
- Proof is technically involved
- Has broad consequences

Weaknesses:
- Unclear what the practical consequences are -- are there practical settings where the fact that these equilibria are hard to compute may cause issues? Since this paper has been submitted to NeurIPS, it would be nice to talk about *some* potential practical consequences, even if they are stylized. Are there ways to get around this hardness in practice?

QUESTIONS:
- What are the practical consequences of this work?
- Are there other relaxations of constrained equilibria that may be easy to compute?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-06