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arxiv: 2606.25707 · v1 · pith:US55XHA4new · submitted 2026-06-24 · 💻 cs.GT · math.OC

Equilibrium and Infeasibility: A new solution concept for games

Anne Reulke (LIA) , Mika\"el Touati , Rachid El-Azouzi (FR 3621 , LIA) This is my paper

Pith reviewed 2026-06-25 19:59 UTC · model grok-4.3

classification 💻 cs.GT math.OC
keywords generalized Nash equilibriuminfeasible constraintspenalty termssolution conceptnon-cooperative gamesNash bargaining
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The pith

Games with possibly infeasible constraints admit a new solution concept based on limits of penalized generalized Nash equilibria.

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

The paper introduces a solution concept for generalized games where individual constraints may be infeasible. It defines the solution as the limit of a sequence of generalized Nash equilibria from versions of the game that use penalty terms to relax those constraints. Existence of such solutions is shown for a wide class of games. Conditions are given under which the solution can be seen as each player choosing a strategy that maximizes their utility among those that minimize the penalties they incur. The concept matches the usual generalized Nash equilibrium when constraints are feasible and also aligns with the Nash bargaining solution.

Core claim

A ψ-penalized solution is defined as the limit of a sequence of generalized Nash equilibria induced by games with penalty terms relaxing the individual constraints. Existence is established for a broad range of games and conditions are provided to characterize a ψ-penalized solution as a strategy profile maximizing every player's utility over all her penalty minimizing strategies. The concept is compatible with the GNE and the solution to the Nash bargaining problem, as illustrated by a variation of Divide-the-Dollar.

What carries the argument

The ψ-penalized solution, constructed as the limit of penalized generalized Nash equilibrium sequences that relax infeasible constraints via penalties.

If this is right

  • Solutions exist for many games that would otherwise lack equilibria due to infeasibility.
  • Under specified conditions, the solution maximizes utility subject to minimizing penalties for each player.
  • The new concept reduces to the standard generalized Nash equilibrium when constraints are feasible.
  • It is compatible with the Nash bargaining solution in appropriate settings.

Where Pith is reading between the lines

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

  • The penalty limit method supplies an operational way to locate equilibria in games where direct feasibility fails.
  • The characterization as utility maximization over penalty minimizers offers a practical test for candidate solutions.
  • Compatibility with Nash bargaining indicates the concept can extend to cooperative settings with shared constraints.

Load-bearing premise

The limit of the penalized GNE sequence exists, and the given conditions are sufficient to ensure the solution maximizes each player's utility among penalty-minimizing strategies.

What would settle it

A concrete game with infeasible constraints where no sequence of penalized GNE has a limit point, or where an existing limit fails to maximize utility over penalty-minimizing strategies under the stated conditions.

Figures

Figures reproduced from arXiv: 2606.25707 by Anne Reulke (LIA), LIA), Mika\"el Touati, Rachid El-Azouzi (FR 3621.

Figure 1
Figure 1. Figure 1: ψ-penalized solutions (red) and best response correspondences for player 1 (purple) and player 2 (green) under the coupled constraint L (blue) and their respective individual constraints for game Gce . 3.3 Sufficient conditions and exact penalization In this section, we give conditions under which every strategy profile solving Problem (3) for all players is a ψ-penalized solution. Theorem 3. Let a ∗ ∈ L. … view at source ↗
Figure 2
Figure 2. Figure 2: ψ-penalized solutions (red) and best response correspondences for player 1 (purple) and player 2 (green) under the coupled constraint L (blue) and their respective individual constraints for game GDD. 5.1 Generalized Nash equilibrium First, we remark that if the individual constraints are induced by the coupled constraint L, i.e. for all players i ∈ N Ci := Li , the penalty function is null by definition. … view at source ↗
Figure 3
Figure 3. Figure 3: Example 2a with the modified individual constraint correspondence for player 2. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Addressing infeasibility in non-cooperative games has become an important topic, as many problems across different applications face this issue. In this paper, we propose a new solution concept for generalized games with possibly infeasible individual constraints. A solution is defined as the limit of a sequence of generalized Nash equilibria induced by games with penalty terms relaxing the individual constraints. Existence is established for a broad range of games and we provide conditions allowing to characterize a $\psi$-penalized solution as a strategy profile maximizing every player's utility over all her penalty minimizing strategies. A variation of Divide-the-Dollar serves as an illustrative example. We further establish the compatibility with the GNE and the solution to the Nash bargaining.

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 / 0 minor

Summary. The paper proposes a new solution concept for generalized games with possibly infeasible individual constraints. A solution is defined as the limit of a sequence of generalized Nash equilibria induced by games with penalty terms relaxing the individual constraints. Existence is established for a broad range of games and conditions are provided to characterize a ψ-penalized solution as a strategy profile maximizing every player's utility over all her penalty minimizing strategies. Compatibility with the GNE and the Nash bargaining solution is shown, with a variation of Divide-the-Dollar as an example.

Significance. If the limit construction yields a well-defined (sequence-independent) solution concept with the stated characterization and compatibility results, the work would provide a useful extension for addressing infeasibility in non-cooperative games, with potential applications in constrained game settings.

major comments (1)
  1. [Abstract] Abstract: the definition takes a solution to be 'the limit of a sequence of generalized Nash equilibria induced by games with penalty terms'. No argument is supplied that this limit is independent of the specific penalty sequence (e.g., different rates at which the penalty parameter tends to infinity). This is load-bearing for the central claim, as sequence-dependence would render both the existence result and the utility-maximization characterization path-dependent rather than intrinsic to the original game.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify aspects of our proposed solution concept. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the definition takes a solution to be 'the limit of a sequence of generalized Nash equilibria induced by games with penalty terms'. No argument is supplied that this limit is independent of the specific penalty sequence (e.g., different rates at which the penalty parameter tends to infinity). This is load-bearing for the central claim, as sequence-dependence would render both the existence result and the utility-maximization characterization path-dependent rather than intrinsic to the original game.

    Authors: The referee correctly notes that the abstract (and, upon review, the main text) does not supply an explicit argument establishing that the limit is independent of the particular penalty sequence chosen. The manuscript defines the solution as such a limit, proves existence for a broad class of games, and provides conditions under which a ψ-penalized solution maximizes each player's utility over her penalty-minimizing strategies. This characterization is sequence-independent by construction, but we do not demonstrate that every convergent sequence yields the same limit profile. We therefore agree that an additional argument or set of sufficient conditions for sequence-independence is needed to make the concept intrinsic rather than path-dependent. We will revise the manuscript by adding a remark or short subsection after the definition that either (i) proves independence under the existing assumptions or (ii) explicitly restricts the solution concept to cases where the limit is unique across sequences. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definition relies on external limit construction

full rationale

The paper defines its solution concept explicitly as the limit of a sequence of penalized generalized Nash equilibria, an external construction that does not reduce to a self-referential equation, fitted parameter renamed as prediction, or load-bearing self-citation. Existence and characterization claims are presented as separate results under stated conditions, without evidence that any central claim collapses by construction to its inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard game-theoretic existence results for penalized games and on the technical conditions for the limit characterization; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Existence of generalized Nash equilibria in each penalized game
    Required for the sequence whose limit defines the solution.
  • domain assumption The limit of the penalized GNE sequence exists under the stated broad conditions
    Directly invoked to define the solution concept.

pith-pipeline@v0.9.1-grok · 5659 in / 1095 out tokens · 17212 ms · 2026-06-25T19:59:43.430327+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

25 extracted references

  1. [1]

    Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms , 2025

    Roland Andrews, Justin Carpentier, and Adrien Taylor. Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms , 2025

    2025

  2. [2]

    Arrow and Gerard Debreu

    Kenneth J. Arrow and Gerard Debreu. Existence of an Equilibrium for a Competitive Economy . Econometrica , 22(3):265--290, 1954

    1954

  3. [3]

    Byrd, Frank E

    Richard H. Byrd, Frank E. Curtis, and Jorge Nocedal. Infeasibility Detection and SQP Methods for Nonlinear Optimization . SIAM Journal on Optimization , 20(5):2281--2299, 2010

    2010

  4. [4]

    Topological Spaces

    Claude Berge. Topological Spaces . Oliver and Boyd , Edinburgh and London, UK, 1963

    1963

  5. [5]

    A generalized Nash equilibrium problem arising in banking regulation: An existence result with Tarski's theorem

    Yann Braouezec and Keyvan Kiani. A generalized Nash equilibrium problem arising in banking regulation: An existence result with Tarski's theorem . Operations Research Letters , 51(1):105--110, 2023

    2023

  6. [6]

    Yann Braouezec and Keyvan Kiani. Economic foundations of generalized games with shared constraint: Do binding agreements lead to less Nash equilibria? European Journal of Operational Research , 308(1):467--479, 2023

    2023

  7. [7]

    Bartle and Donald R

    Robert G. Bartle and Donald R. Sherbert. Introduction to Real Analysis . J. Wiley, New York, NY, 2000

    2000

  8. [8]

    How the augmented Lagrangian algorithm can deal with an infeasible convex quadratic optimization problem

    Alice Chiche and Jean Charles Gilbert. How the augmented Lagrangian algorithm can deal with an infeasible convex quadratic optimization problem . Journal of Convex Analysis , 23:30, 2010

    2010

  9. [9]

    Chinneck

    John W. Chinneck. Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods . International Series in Operations Research & Management Science. Springer, New York, NY, 2008

    2008

  10. [10]

    Variational methods for the solution of problems of equilibrium and vibrations

    Richard Courant. Variational methods for the solution of problems of equilibrium and vibrations . Bulletin of the American Mathematical Society , 49(1):1 -- 23, 1943

    1943

  11. [11]

    A Social Equilibrium Existence Theorem

    Gerard Debreu. A Social Equilibrium Existence Theorem . Proceedings of the National Academy of Sciences of the USA , 38(10):886--893, 1952

    1952

  12. [12]

    Topology

    James Dugundji. Topology . Allyn and Bacon, Boston, MA, 1978

    1978

  13. [13]

    Existence theorems for generalized Nash equilibrium problems: an analysis of assumptions

    Christophe Dutang. Existence theorems for generalized Nash equilibrium problems: an analysis of assumptions . Journal of Nonlinear Analysis and Optimization , 4(2):115--126, 2013

    2013

  14. [14]

    The augmented Lagrangian method can approximately solve convex optimization with least constraint violation , 2023

    Yu-Hong Dai and Liwei Zhang. The augmented Lagrangian method can approximately solve convex optimization with least constraint violation , 2023

    2023

  15. [15]

    Constrained Traffic Equilibrium in Routing

    Rachid El-Azouzi and Eitan Altman. Constrained Traffic Equilibrium in Routing . IEEE Transactions on Automatic Control , 48(9):1656--1660, 2003

    2003

  16. [16]

    Generalized Nash Equilibrium Problems

    Francisco Facchinei and Christian Kanzow. Generalized Nash Equilibrium Problems . Annals of Operations Research , 175(1):177--211, 2010

    2010

  17. [17]

    Penalty Methods for the Solution of Generalized Nash Equilibrium Problems

    Francisco Facchinei and Christian Kanzow. Penalty Methods for the Solution of Generalized Nash Equilibrium Problems . SIAM Journal on Optimization , 20(5):2228--2253, 2010

    2010

  18. [18]

    Partial penalization for the solution of generalized Nash equilibrium problems

    Francisco Facchinei and Lorenzo Lampariello. Partial penalization for the solution of generalized Nash equilibrium problems . Journal of Global Optimization , 50:39--57, 2011

    2011

  19. [19]

    William W. Hogan. Point-to-Set Maps in Mathematical Programming . SIAM Review , 15(3):591--603, 1973

    1973

  20. [20]

    Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium

    Hélène Le Cadre , Paulin Jacquot, Cheng Wan, and Clémence Alasseur. Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium. European Journal of Operational Research , 282(2):753--771, 2020

    2020

  21. [21]

    The Bargaining Problem

    John Nash. The Bargaining Problem . Econometrica , 18:155--162, 1950

    1950

  22. [22]

    Two-Person Cooperative Games

    John Nash. Two-Person Cooperative Games . Econometrica , 21(1):128--140, 1953

    1953

  23. [23]

    J. B. Rosen. Existence and Uniqueness of Equilibrium Points for Concave N-Person Games . Econometrica , 33(3):520--534, 1965

    1965

  24. [24]

    Steen and J

    Lynn A. Steen and J. Arthur Seebach Jr. Counterexamples in Topology . Holt, Rinehart and Winston, Inc., New York, NY, 1970

    1970

  25. [25]

    Stability and Perfection of Nash Equilibria

    Eric van Damme. Stability and Perfection of Nash Equilibria . Springer Berlin, Heidelberg, Germany, 1991

    1991