Solving Infinite-Player Games with Player-to-Strategy Networks

Carlos Martin; Tuomas Sandholm

REVIEW 3 major objections 2 minor 88 references

Solving Infinite-Player Games with Player-to-Strategy Networks

T0 review · 3 major / 2 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Neural networks that map players to strategies can find approximate Nash equilibria in games with infinitely many players.

desk verdict Useful representation and training rule for continuum-player games, but the empirical convergence claim is undermined by an unstated grid resolution and an untested blind spot for opposing-interest games. read the letter →

arxiv 2501.09330 v1 pith:HUPPV3MZ submitted 2025-01-16 cs.GT

Carlos Martin , Tuomas Sandholm This is my paper · ORCID

classification cs.GT MSC 91A1091A1668T07

keywords infinite-playergamesNashequilibriumPlayer-to-StrategyNetworkShared-ParameterSimultaneousGradientmeanfieldmultiagentsystemscontinuous-actionexploitability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that games with infinitely many players can be solved by representing the entire strategy profile as a neural network that maps each player to a strategy, then training that network with a new equilibrium-seeking update. The update, Shared-Parameter Simultaneous Gradient (SPSG), re-expresses the classical simultaneous gradient as a functional derivative of the integral of players' utilities and estimates it by sampling players and differentiating through a hybrid profile in which only the sampled player's output is varied. On five families of continuum-player games (two Ising variants, two Cournot variants, and a crowding game), the paper reports that mean regret decreases toward zero over training, meaning the learned profiles approach approximate Nash equilibria. If the method works as reported, it offers a general, symmetry-free route to equilibrium computation in infinite-player games.

What carries the argument

The two load-bearing objects are the Player-to-Strategy Network (P2SN), a neural network that maps a player's feature vector (plus optional observation and noise inputs) to that player's strategy, and the Shared-Parameter Simultaneous Gradient (SPSG), the functional derivative in Equation 5. The P2SN makes the infinite strategy profile a finite set of shared parameters; a learnable Fourier feature map on player coordinates lets it represent high-frequency structure on low-dimensional player spaces that plain networks underfit. SPSG estimates the functional derivative by sampling one player per gradient call, forming the hybrid profile $s[i\mapsto r(i)]$, and backpropagating through the sampled player's output; because parameters are shared, the update is a stochastic estimate of the whole-profile derivative rather than a per-player gradient. For non-differentiable utilities the paper also adopts pseudo-gradient estimators, which perturb strategies with Gaussian noise and differentiate a smoothed utility, and noise-injected network inputs to represent mixed strategies over continuous action sets.

What would settle it

Run SPSG on a two-player zero-sum game (or a continuum analogue, such as two symmetric populations with opposing payoffs) with a single shared network: footnote 10 predicts the parameters do not move and regret never decreases. More directly, take one of the test games with a known closed-form equilibrium, compute the learned profile's regret on a much finer grid or exactly, and check whether the reported near-zero mean regret survives; if a narrow deviation outside the evaluation grid yields high utility, the observed convergence is an artifact of discretization.

Watch

Extended reading notes

Core claim

The paper's central claim is that a strategy profile for countably or uncountably many players can be carried by a single Player-to-Strategy Network, and that the update rule SPSG drives this network toward approximate Nash equilibrium. SPSG replaces the per-player gradient with the functional derivative $v(s) = \frac{d}{dr}\int_{i\sim\mu} u(s[i\mapsto r(i)], i)\,|_{r=s}$, effectively differentiating the mean utility through only one player at a time while holding the rest of the profile fixed; a Monte Carlo sample of players gives an unbiased estimator of the integral, and differentiating through the hybrid profile $s[i\mapsto r(i)]$ gives the gradient. The paper reports convergence of mean regret toward zero in one- and two-dimensional Ising games, distance-based variants of each, Cournot competition with a global demand function, Cournot competition with a local demand function, and a crowding game, with eight trials per experiment and training times of 9 to 14 minutes per run on one GPU.

Load-bearing premise

The load-bearing premise is that repeatedly sampling one player, differentiating that player's utility through the shared network, and stepping along the averaged result moves the whole infinite-player profile toward low regret—an assumption that is not proven, and that the paper's own footnote 10 shows fails when shared parameters face exactly opposing pulls.

Editorial extensions

If this is right

If the reported convergence holds, continuum-player games can be solved without symmetry or mean-field assumptions, so boundary effects and heterogeneous player features are handled automatically.
The method applies in principle to games with infinitely many players, states, and actions, including mixed strategies on continuous action spaces and utility functions with discontinuities.
Because SPSG generalizes simultaneous gradient ascent, existing variants such as optimistic updates and pseudo-gradients can be swapped in as the optimizer inside the same P2SN representation.
For games whose utilities decompose into pairwise or higher-order interactions as in Equation 6, unbiased utility estimates can be obtained by sampling the interacting players, making the method feasible for interaction-driven models.
The paper's regret curves constitute evidence that approximate Nash equilibria are reachable in the tested families, including Ising games, Cournot competition, local Cournot competition, and crowding games.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the convergence is empirical and the paper itself notes that shared parameters freeze in a two-player zero-sum game, an implicit scope restriction is that the games must not have exactly opposing players pulling shared parameters in opposite directions; a natural extension would partition parameters per role or use symplectic or optimistic updates for adversarial components.
The regret certification used in the experiments evaluates best responses on a finite grid, which lower-bounds true regret; if a narrow profitable deviation falls between grid points, reported convergence could overstate equilibrium quality, so a direct test is to compare learned profiles against known analytic equilibria in the same games.
If SPSG is an unbiased functional gradient, then the method should extend beyond pairwise aggregative utilities to games with three-way interactions and to continuous-time dynamics; testing on a game with a known closed-form asymmetric equilibrium would separate representation error from optimization error.
A practical consequence left implicit by the authors is that the same player-to-strategy network could be reused for equilibrium selection and mechanism design: once a social planner can cheaply evaluate equilibria of a continuum game, they could search over game parameters to steer the equilibrium.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: the P2SN representation and the SPSG training rule are a genuinely useful way to think about infinite-player games, and the paper is worth engaging. But the experiments don't yet support the abstract's convergence claim, because the regret measure is computed on a finite grid whose resolution is never reported, and the method has a real blind spot for opposing-interest games that the paper doesn't test.

What's new: representing a continuum of players as a single network over player features, and deriving the SPSG update via a functional derivative of the total-utility integral using a hybrid-profile Monte Carlo estimator. That's a clean formalization, and the paper explains it well. The related-work survey is solid. The five test games are sensible benchmarks, and the regret curves do fall on all of them. To its credit, the paper says in Section 7 that broad convergence proofs are future work; it does not pretend to have them.

The soft spots, in proportion:

1. Regret evaluation is the load-bearing issue. The paper discretizes player space and strategy space into N points but never states N. Expected utility uses N=200 samples, but the grids are unspecified. A finite grid can only produce a lower bound on true regret, and with bias fields oscillating at 6π, that bound can be arbitrarily loose. Without a finer-grid verification or comparison to known equilibria—both are feasible for Ising and Cournot—the observed 'convergence' is not established. I agree with the stress-test note here.

2. The method's scope is narrower than claimed. SPSG is the derivative of the integral of all players' utilities. In a zero-sum or constant-sum game that integral is a constant, so the gradient is zero and the method cannot move. The paper's footnote 10 gestures at shared-parameter cancellation, but the bigger issue is the integral objective itself. All five test games are alignment/potential-type games, so the paper gives no evidence that SPSG handles competition. The abstract's general claim about 'games with infinitely many players' needs caveats.

3. No baselines, no code. The empirical results could be checked if code were released and if the regrets were compared against an independent solver or an analytic equilibrium.

Who this is for: researchers working on neural methods for large or continuum games will find the representation and the estimator a useful starting point, and the evaluation pitfalls are instructive. I would not cite it as evidence of convergence yet.

Recommendation: send to peer review—the ideas deserve referee time—but the experiments need a major revision: report the grid resolution, verify with finer grids or closed-form equilibria, and test at least one game with opposing incentives. If that is done, it could be a solid methods paper.

Referee Report

3 major / 2 minor

Summary. The paper introduces a Player-to-Strategy Network (P2SN), a neural network mapping player features to strategies, and Shared-Parameter Simultaneous Gradient (SPSG), an equilibrium-seeking algorithm that trains the network by estimating the gradient of the integrated simultaneous-gradient vector field. The method is tested on five infinite-player games (Ising, distance-based Ising, Cournot, local Cournot, crowding), and the authors report that regrets decrease toward zero, which they interpret as convergence to approximate Nash equilibria. The paper claims the approach handles infinitely many players, states, and actions, with no symmetry assumptions.

Significance. If the claims hold, the paper offers a genuinely new way to compute approximate Nash equilibria in continuum-player games without symmetry or mean-field structure, using a finite-dimensional neural network representation. The idea of treating the strategy profile as a function learned by a network is natural and useful, and the experiments span several nontrivial games. The paper is clearly written and provides a useful formalization of regret and exploitability in this setting. However, because the empirical evaluation relies on a lower-bound regret estimator with unspecified resolution and no external validation, and because the central unbiasedness claim is not justified, the significance of the contribution is not yet established.

major comments (3)

[§6] The reported regrets are computed by discretizing the player space and each strategy space into N points, but the value of N is never stated. Since a finite-grid best response is a lower bound on the true supremum over the full strategy set, and a finite player grid can miss high-regret players, the plotted regrets in Figures 3, 6, 8, 10, 12, 14, and 15 are lower bounds on the true mean regret. Without specifying N or demonstrating that the regrets are insensitive to an increase in N, the central empirical claim of convergence to approximate Nash equilibria is not established. This is particularly concerning for the high-frequency bias fields in §6.2 and §6.5, where a coarse grid could miss narrow basins of large unilateral gains.
[§5, Eq. (5)] The paper claims that the player-sampling procedure yields an unbiased estimator of the SPSG because 'the integral and derivative commute with each other' and offers no justification. For a functional derivative of an integral with respect to a function, this interchange requires regularity conditions that are not stated and may fail for the discontinuous utility functions the paper claims to handle. Additionally, footnote 10 states that in a two-player zero-sum game with shared parameters the parameters do not change at all, so the method excludes adversarial settings; this directly contradicts the unqualified abstract claim that the approach can handle infinite-player games. The authors should state sufficient conditions for Eq. (5) and restrict the applicability claims accordingly.
[§6] The experimental validation contains no comparison against known equilibria or existing mean-field-game solvers. The only evidence is the internal regret metric, which uses the same finite-grid approximation that the method itself is being judged against. For example, in the 1D Ising game of §6.1, the best response is analytically computable (s(i) = sign(b(i) + ∫ s(j) dν(j))), so the learned profile could be checked directly against an exact best response. Without such an independent check, the plotted decreasing regrets do not demonstrate that the learned strategies are close to Nash equilibria in the true infinite-player game.

minor comments (2)

[§6] The symbol N is used for two different things: the number of discretization points for the player and strategy grids, and the number of Monte Carlo utility samples (N = 200). The grid sizes for the player and strategy discretizations should be given explicitly and consistently.
[§5, footnote 11] Footnote 11 assumes 0 < μ(I) < ∞, which excludes the counting measure on a countably infinite player set. The abstract's claim that the method handles countably infinite players should be qualified, since the estimator as written applies to normalizable measures.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: SPSG and the regret certificate are independently defined, and the minor self-citations are not load-bearing.

full rationale

I walked the derivation chain from Equation (5) through the SPSG update and the evaluation protocol. Equation (5) is a definition of the functional derivative of aggregate utility with respect to the strategy profile; it does not presuppose the Nash equilibrium condition, and regret E(s) is defined independently via best responses to the current profile. No fitted parameter is later repackaged as a prediction: the paper states that game constants 'were chosen (before training)', and the networks are not trained to match any target equilibrium profile. The self-citations to Martin and Sandholm (2023, 2024) concern randomized policy networks and joint-perturbation pseudo-gradients; neither component is required by the differentiable experiments in Section 6, so these citations are not load-bearing for the central convergence observation. The finite-grid regret estimate in Section 6 is a lower bound on true regret and may overstate convergence, but that is a validation weakness rather than circularity, because the observed decrease is not forced by construction. The paper itself concedes that convergence theory is left to future work and notes a shared-parameter freeze in opposing-player games; those are limitations, not circular steps. Overall, no circular step is present; the score of 2 reflects only the existence of minor self-citations that do not carry the paper's central claim.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard game-theoretic existence results, neural network approximation, unbiased gradient estimation, and the fidelity of discretized regret. No new particles, forces, dimensions, or conserved quantities are introduced; P2SN is a representation and SPSG is an algorithm. The weakest links are the unproven unbiasedness of the SPSG estimator and the lower-bound nature of the regret metric.

free parameters (3)

learning rate and optimizer schedule
SPSG updates require step sizes; exact schedule and values are not reported, so convergence behavior depends on unstated choices.
Fourier feature dimension N=64 = 64
Player features are mapped to 64 Fourier features; this capacity is hand-chosen and affects expressivity of the P2SN.
Fourier initialization scale sigma=64 = 64
Entries of B are initialized as normal with std 64, a hand-set hyperparameter that controls the frequency content of the learned mapping.

assumptions (5)

domain assumption Pure-strategy Nash equilibria exist in all test games (Glicksberg and Dasgupta-Maskin conditions).
Section 6 relies on existence to interpret the learned profiles; the games have compact convex strategy sets and continuous utilities.
domain assumption The P2SN is expressive enough to approximate the equilibrium strategy profile (universal approximation).
Section 5 cites Cybenko, Hornik, Leshno et al.; the trained network's capacity is assumed sufficient to represent the equilibrium.
domain assumption Integration and differentiation commute, and the one-player-per-iteration Monte Carlo estimator of Eq. (5) is unbiased.
Section 5 steps 2-6 sample a single player and claim an unbiased estimator of the whole integral; this requires regularity of the utility and a finite integral.
domain assumption The measure mu is normalizable, with 0 < mu(I) < infinity.
Stated in Section 5 note 11; needed to define the sampling distribution over players.
ad hoc to paper Finite-grid best responses and N=200 Monte Carlo utility samples give a faithful approximation of true regret.
Section 6 estimates regret by discretizing players and strategies; this yields a lower bound on regret, so near-zero reported values are assumed to certify approximate equilibrium.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Solving Infinite-Player Games with Player-to-Strategy Networks." pith.science (2026). https://pith.science/paper/HUPPV3MZ

@misc{pith2026250109330,
  author       = {Pith},
  title        = {Pith review of: Solving Infinite-Player Games with Player-to-Strategy Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUPPV3MZ}},
  note         = {Machine review of arXiv:2501.09330}
}

read the original abstract

We present a new approach to solving games with a countably or uncountably infinite number of players. Such games are often used to model multiagent systems with a large number of agents. The latter are frequently encountered in economics, financial markets, crowd dynamics, congestion analysis, epidemiology, and population ecology, among other fields. Our two primary contributions are as follows. First, we present a way to represent strategy profiles for an infinite number of players, which we name a Player-to-Strategy Network (P2SN). Such a network maps players to strategies, and exploits the generalization capabilities of neural networks to learn across an infinite number of inputs (players) simultaneously. Second, we present an algorithm, which we name Shared-Parameter Simultaneous Gradient (SPSG), for training such a network, with the goal of finding an approximate Nash equilibrium. This algorithm generalizes simultaneous gradient ascent and its variants, which are classical equilibrium-seeking dynamics used for multiagent reinforcement learning. We test our approach on infinite-player games and observe its convergence to approximate Nash equilibria. Our method can handle games with infinitely many states, infinitely many players, infinitely many actions (and mixed strategies on them), and discontinuous utility functions.

Figures

Figures reproduced from arXiv: 2501.09330 by the authors.

**Figure 1.** High-level structure of the P2SN. 5 Proposed Method In this section, we present our method. Player-to-Strategy Network. Suppose we are interested in tackling a game with infinitely many players. This raises the problem of how to represent a strategy profile that has an infinite number of players. We propose a way to do this which we coin a Player-to-Strategy Network (P2SN). This is a neural network that takes as inp… view at source ↗

**Figure 2.** Strategy profile for 1D Ising game [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Regrets for 1D Ising game. Each player i picks a point s(i) and seeks to minimize the Euclidean distance to a target point b(i), but also to minimize the Euclidean distance to the points chosen by its neighbors. Each player’s utility is concave in its own strategy. The strategy profile learned by our algorithm is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 6.** Figure 6: Regrets for 2D Ising game [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Strategy profile for 1D distance-based Ising [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 11.** Figure 11: The regrets of our algorithm over the course [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]

**Figure 8.** Figure 8: Regrets for 1D distance-based Ising game. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 11.** Figure 11: Strategy profile for Cournot competition. [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 12.** Figure 12: Regrets for Cournot competition [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

**Figure 15.** Figure 15: Regrets for crowding game. vergence of our method can be theoretically proven. As described in §2, many games have a unique NE. However, some games could have multiple NE. NE refinements and equilibrium selection are outside the scope of this paper, but potentially a…

**Figure 16.** Figure 16: Learned strategy profile for crowding game. [PITH_FULL_IMAGE:figures/full_fig_p011_16.png]

**Figure 17.** Figure 17: Histogram of strategies for crowding game. [PITH_FULL_IMAGE:figures/full_fig_p011_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

88 extracted references · 72 canonical work pages

[1]

Robert J. Aumann. Markets with a continuum of traders. Econometrica, 32 0 (1): 0 39--50, 1964

1964
[2]

The mechanics of n-player differentiable games

David Balduzzi, Sebastien Racaniere, James Martens, Jakob Foerster, Karl Tuyls, and Thore Graepel. The mechanics of n-player differentiable games. In International Conference on Machine Learning (ICML), 2018

2018
[3]

Berahas, Liyuan Cao, Krzysztof Choromanski, and Katya Scheinberg

Albert S. Berahas, Liyuan Cao, Krzysztof Choromanski, and Katya Scheinberg. A theoretical and empirical comparison of gradient approximations in derivative-free optimization. Foundations of Computational Mathematics, 22 0 (2): 0 507--560, 2022

2022
[4]

Learning equilibria in symmetric auction games using artificial neural networks

Martin Bichler, Maximilian Fichtl, Stefan Heidekr \"u ger, Nils Kohring, and Paul Sutterer. Learning equilibria in symmetric auction games using artificial neural networks. Nature Machine Intelligence, 3 0 (8): 0 687--695, 2021

2021
[5]

JAX : Composable transformations of Python + NumPy programs, 2018

James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, et al. JAX : Composable transformations of Python + NumPy programs, 2018

2018
[6]

Superhuman AI for multiplayer poker

Noam Brown and Tuomas Sandholm. Superhuman AI for multiplayer poker. Science, 365 0 (6456): 0 885--890, 2019

2019
[7]

Peter E. Caines. Mean field games. In John Baillieul and Tariq Samad, editors, Encyclopedia of Systems and Control. Springer London, Gewerbestrasse 11, 6330 Cham, Switzerland, 2013

2013
[8]

Caines, Minyi Huang, and Roland P

Peter E. Caines, Minyi Huang, and Roland P. Malham \'e . Mean field games. In Tamer Ba s ar and Georges Zaccour, editors, Handbook of Dynamic Game Theory. Springer International Publishing, Gewerbestrasse 11, 6330 Cham, Switzerland, 2018

2018

Show all 88 references

[9]

The dyson and coulomb games

Ren \'e Carmona, Mark Cerenzia, and Aaron Zeff Palmer. The dyson and coulomb games. In Annales Henri Poincar \'e , volume 21, pages 2897--2949. Springer, 2020

2020
[10]

Bertrand and Cournot mean field games

Patrick Chan and Ronnie Sircar. Bertrand and Cournot mean field games. Applied Mathematics & Optimization, 71 0 (3): 0 533--569, 2015

2015
[11]

Principes de la th \'e orie des richesses

Antoine Augustin Cournot. Principes de la th \'e orie des richesses . Hachette, 1863
[12]

Approximation by superpositions of a sigmoidal function

George Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems, 2 0 (4): 0 303--314, 1989

1989
[13]

The existence of equilibrium in discontinuous economic games 1: Theory

Partha Dasgupta and Eric Maskin. The existence of equilibrium in discontinuous economic games 1: Theory. Review of Economic Studies, 53: 0 1--26, 1986

1986
[14]

Training GAN s with optimism

Constantinos Daskalakis, Andrew Ilyas, Vasilis Syrgkanis, and Haoyang Zeng. Training GAN s with optimism. In International Conference on Learning Representations (ICLR), 2018

2018
[15]

A social equilibrium existence theorem

Gerard Debreu. A social equilibrium existence theorem. Proceedings of the National Academy of Sciences, 38 0 (10), 1952

1952
[16]

The D eep M ind JAX E cosystem, 2020

DeepMind, Igor Babuschkin, Kate Baumli, Alison Bell, Surya Bhupatiraju, Jake Bruce, Peter Buchlovsky, David Budden, Trevor Cai, Aidan Clark, Ivo Danihelka, Antoine Dedieu, Claudio Fantacci, Jonathan Godwin, Chris Jones, Ross Hemsley, Tom Hennigan, Matteo Hessel, Shaobo Hou, St...

2020
[17]

Duchi, Michael I

John C. Duchi, Michael I. Jordan, Martin J. Wainwright, and Andre Wibisono. Optimal rates for zero-order convex optimization: the power of two function evaluations. IEEE Transactions on Information Theory, 61 0 (5), 2015

2015
[18]

Beitrag zur theorie des ferromagnetismus

Ising Ernst. Beitrag zur theorie des ferromagnetismus. Zeitschrift f \"u r Physik A Hadrons and Nuclei , 31 0 (1): 0 253--258, 1925

1925
[19]

Fixed point and minimax theorems in locally convex topological linear spaces

Ky Fan. Fixed point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences, 38 0 (2): 0 121--126, 1952

1952
[20]

Feldman, Inwon C

William M. Feldman, Inwon C. Kim, and Aaron Zeff Palmer. The sharp interface limit of an Ising game. ESAIM: Control, Optimisation and Calculus of Variations, 30: 0 35, 2024

2024
[21]

Ising model versus normal form game

Serge Galam and Bernard Walliser. Ising model versus normal form game. Physica A: Statistical Mechanics and its Applications, 389 0 (3): 0 481--489, 2010

2010
[22]

Computing equilibria by incorporating qualitative models

Sam Ganzfried and Tuomas Sandholm. Computing equilibria by incorporating qualitative models. In International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), 2010

2010
[23]

A further generalization of the K akutani fixed point theorem, with application to N ash equilibrium points

Irving Leonard Glicksberg. A further generalization of the K akutani fixed point theorem, with application to N ash equilibrium points. Proceedings of the American Mathematical Society, 3 0 (1): 0 170--174, 1952

1952
[24]

Monte Carlo sampling methods using Markov chains and their applications

Wilfred Keith Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57 0 (1): 0 97--109, 1970

1970
[25]

Delving deep into rectifiers: Surpassing human-level performance on I mage N et classification

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on I mage N et classification. In International Conference on Computer Vision (ICCV), 2015

2015
[26]

Flax : A neural network library and ecosystem for JAX , 2023

Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Rondepierre, Andreas Steiner, and Marc van Z ee. Flax : A neural network library and ecosystem for JAX , 2023. URL http://github.com/google/flax

2023
[27]

Approximation capabilities of multilayer feedforward networks

Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4 0 (2): 0 251--257, 1991

1991
[28]

Multilayer feedforward networks are universal approximators

Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approximators. Neural networks, 2 0 (5): 0 359--366, 1989

1989
[29]

The limits of min-max optimization algorithms: Convergence to spurious non-critical sets

Ya-Ping Hsieh, Panayotis Mertikopoulos, and Volkan Cevher. The limits of min-max optimization algorithms: Convergence to spurious non-critical sets. In International Conference on Machine Learning (ICML), 2021

2021
[30]

On the convergence of single-call stochastic extra-gradient methods

Yu-Guan Hsieh, Franck Iutzeler, J \'e r \^o me Malick, and Panayotis Mertikopoulos. On the convergence of single-call stochastic extra-gradient methods. In Conference on Neural Information Processing Systems (NeurIPS), 2019

2019
[31]

John D. Hunter. Matplotlib : A 2D graphics environment. Computing in Science & Engineering, 9 0 (3): 0 90--95, 2007

2007
[32]

Self-gated rectified linear unit for performance improvement of deep neural networks

Israt Jahan, Md Faisal Ahmed, Md Osman Ali, and Yeong Min Jang. Self-gated rectified linear unit for performance improvement of deep neural networks. ICT Express, 9 0 (3): 0 320--325, 2023

2023
[33]

Ali Khan

M. Ali Khan. Equilibrium points of nonatomic games over a nonreflexive Banach space. Journal of Approximation Theory, 43 0 (4): 0 370--376, 1985

1985
[34]

Ali Khan

M. Ali Khan. Equilibrium points of nonatomic games over a Banach space. Transactions of the American Mathematical Society, 293 0 (2): 0 737--749, 1986

1986
[35]

Ali Khan and Nikolaos S

M. Ali Khan and Nikolaos S. Papageorgiou. On Cournot -- Nash equilibria in generalized qualitative games with an atomless measure space of agents. Proceedings of the American Mathematical Society, 100 0 (3): 0 505--510, 1987 a

1987
[36]

Ali Khan and Nikolaos S

M. Ali Khan and Nikolaos S. Papageorgiou. On Cournot -- Nash equilibria in generalized qualitative games with a continuum of players. Nonlinear Analysis: Theory, Methods & Applications, 11 0 (6): 0 741--756, 1987 b

1987
[37]

Ali Khan and Yeneng Sun

M. Ali Khan and Yeneng Sun. Non-cooperative games with many players. Handbook of Game Theory with Economic Applications, 3: 0 1761--1808, 2002

2002
[38]

Yannelis

Taesung Kim, Karel Prikry, and Nicholas C. Yannelis. Equilibria in abstract economies with a measure space of agents and with an infinite dimensional strategy space. Journal of Approximation Theory, 56 0 (3): 0 256--266, 1989

1989
[39]

Auction Theory

Vijay Krishna. Auction Theory. Academic Press, 2002

2002
[40]

A unified game-theoretic approach to multiagent reinforcement learning

Marc Lanctot, Vinicius Zambaldi, Audrunas Gruslys, Angeliki Lazaridou, Karl Tuyls, Julien P \'e rolat, David Silver, and Thore Graepel. A unified game-theoretic approach to multiagent reinforcement learning. In Conference on Neural Information Processing Systems (NeurIPS), pag...

2017
[41]

Non-differentiable supervised learning with evolution strategies and hybrid methods

Karel Lenc, Erich Elsen, Tom Schaul, and Karen Simonyan. Non-differentiable supervised learning with evolution strategies and hybrid methods. arXiv:1906.03139, 0, 2019

1906 arXiv
[42]

Beitr s ge zum verst s ndnis der magnetischen eigenschaften in festen k s rpern

Wilhelm Lenz. Beitr s ge zum verst s ndnis der magnetischen eigenschaften in festen k s rpern. Physikalische Zeitschrift, 21 0 (1): 0 613--615, 1920

1920
[43]

Andrey Leonidov, Alexey Savvateev, and Andrew G. Semenov. QRE in the Ising game. In CEUR Workshop, 2020

2020
[44]

Andrey Leonidov, Alexey Savvateev, and Andrew G. Semenov. Ising game on graphs. Chaos, Solitons & Fractals, 180: 0 114540, 2024

2024
[45]

Multilayer feedforward networks with a nonpolynomial activation function can approximate any function

Moshe Leshno, Vladimir Ya Lin, Allan Pinkus, and Shimon Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural networks, 6 0 (6), 1993

1993
[46]

Differentiable game mechanics

Alistair Letcher, David Balduzzi, S \'e bastien Racaniere, James Martens, Jakob Foerster, Karl Tuyls, and Thore Graepel. Differentiable game mechanics. Journal of Machine Learning Research, 20 0 (84): 0 1--40, 2019 a

2019
[47]

Stable opponent shaping in differentiable games

Alistair Letcher, Jakob Foerster, David Balduzzi, Tim Rockt \"a schel, and Shimon Whiteson. Stable opponent shaping in differentiable games. In International Conference on Learning Representations (ICLR), 2019 b

2019
[48]

Computing approximate equilibria in sequential adversarial games by exploitability descent

Edward Lockhart, Marc Lanctot, Julien P \'e rolat, Jean-Baptiste Lespiau, Dustin Morrill, Finbarr Timbers, and Karl Tuyls. Computing approximate equilibria in sequential adversarial games by exploitability descent. In Proceedings of the International Joint Conference on Artifi...

2019
[49]

Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks

Carlos Martin and Tuomas Sandholm. Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), 2023

2023
[50]

Joint-perturbation simultaneous pseudo-gradient

Carlos Martin and Tuomas Sandholm. Joint-perturbation simultaneous pseudo-gradient. arXiv:2408.09306, 0, 2024

2024
[51]

Ratliff, and S

Eric Mazumdar, Lillian J. Ratliff, and S. Shankar Sastry. On gradient-based learning in continuous games. SIAM Journal on Mathematics of Data Science, 2 0 (1): 0 103--131, 2020

2020
[52]

Mazumdar, Michael I

Eric V. Mazumdar, Michael I. Jordan, and S. Shankar Sastry. On finding local N ash equilibria (and only local N ash equilibria) in zero-sum games. arXiv:1901.00838, 0, 2019

1901 arXiv
[53]

Learning in games with continuous action sets and unknown payoff functions

Panayotis Mertikopoulos and Zhengyuan Zhou. Learning in games with continuous action sets and unknown payoff functions. Mathematical Programming, 173: 0 465--507, 2019

2019
[54]

Equation of state calculations by fast computing machines

Nicholas Metropolis, Arianna Rosenbluth, Marshall Rosenbluth, Augusta Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21 0 (6): 0 1087--1092, 1953

1953
[55]

Generic Uniqueness of Equilibria in Nonatomic Congestion Games

Igal Milchtaich. Generic Uniqueness of Equilibria in Nonatomic Congestion Games. Hebrew University of Jerusalem, 1996

1996
[56]

Generic uniqueness of equilibrium in large crowding games

Igal Milchtaich. Generic uniqueness of equilibrium in large crowding games. Mathematics of Operations Research, 25 0 (3): 0 349--364, 2000

2000
[57]

Topological conditions for uniqueness of equilibrium in networks

Igal Milchtaich. Topological conditions for uniqueness of equilibrium in networks. Mathematics of Operations Research, 30 0 (1): 0 225--244, 2005

2005
[58]

Learning equilibria in mean-field games: Introducing mean-field PSRO

Paul Muller, Mark Rowland, Romuald Elie, Georgios Piliouras, Julien Perolat, Mathieu Lauriere, Raphael Marinier, Olivier Pietquin, and Karl Tuyls. Learning equilibria in mean-field games: Introducing mean-field PSRO . In Autonomous Agents and Multi-Agent Systems, page 926–934,...

2022
[59]

Equilibrium points in n-person games

John Nash. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36: 0 48--49, 1950

1950
[60]

Non-cooperative games

John Nash. Non-cooperative games. Annals of Mathematics, 54: 0 289--295, 1951

1951
[61]

Random gradient-free minimization of convex functions

Yurii Nesterov and Vladimir Spokoiny. Random gradient-free minimization of convex functions. Foundations of Computational Mathematics, 17 0 (2): 0 527--566, 2017

2017
[62]

Perfectly competitive markets as the limits of Cournot markets

William Novshek. Perfectly competitive markets as the limits of Cournot markets. Journal of Economic Theory, 35 0 (1): 0 72--82, 1985

1985
[63]

Graphon games

Francesca Parise and Asuman Ozdaglar. Graphon games. In Proceedings of the ACM Conference on Economics and Computation (EC), page 457–458, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450367929. doi:10.1145/3328526.3329638. URL https://doi.org/10.1145...

2019
[64]

Graphon games: A statistical framework for network games and interventions

Francesca Parise and Asuman Ozdaglar. Graphon games: A statistical framework for network games and interventions. Econometrica, 91 0 (1): 0 191--225, 2023

2023
[65]

Scaling mean field games by online mirror descent

Julien P\' e rolat, Sarah Perrin, Romuald Elie, Mathieu Lauri\` e re, Georgios Piliouras, Matthieu Geist, Karl Tuyls, and Olivier Pietquin. Scaling mean field games by online mirror descent. In Autonomous Agents and Multi-Agent Systems, page 1028–1037, Richland, SC, 2022. Inte...

2022
[66]

Fictitious play for mean field games: Continuous time analysis and applications

Sarah Perrin, Julien P \'e rolat, Mathieu Lauri \`e re, Matthieu Geist, Romuald Elie, and Olivier Pietquin. Fictitious play for mean field games: Continuous time analysis and applications. Conference on Neural Information Processing Systems (NeurIPS), 33: 0 13199--13213, 2020

2020
[67]

Approximation theory of the MLP model in neural networks

Allan Pinkus. Approximation theory of the MLP model in neural networks. Acta numerica, 8: 0 143--195, 1999

1999
[68]

A modification of the A rrow- H urwicz method for search of saddle points

Leonid Denisovich Popov. A modification of the A rrow- H urwicz method for search of saddle points. Mathematical notes of the Academy of Sciences of the USSR, 28: 0 845--848, 1980

1980
[69]

On the spectral bias of neural networks

Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. In International Conference on Machine Learning (ICML), pages 5301--5310. PMLR, 2019
[70]

Kali P. Rath. A direct proof of the existence of pure strategy equilibria in games with a continuum of players. Economic Theory, 2: 0 427--433, 1992

1992
[71]

The unreasonable effectiveness of quasirandom sequences, 2018

Martin Roberts. The unreasonable effectiveness of quasirandom sequences, 2018

2018
[72]

Ben Rosen

J. Ben Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33 0 (3): 0 520--534, 1965

1965
[73]

An automatic method for finding the greatest or least value of a function

HoHo Rosenbrock. An automatic method for finding the greatest or least value of a function. The computer journal, 3 0 (3): 0 175--184, 1960

1960
[74]

Evolution strategies as a scalable alternative to reinforcement learning, 2017

Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever. Evolution strategies as a scalable alternative to reinforcement learning, 2017

2017
[75]

Equilibrium points of nonatomic games

David Schmeidler. Equilibrium points of nonatomic games. Journal of Statistical Physics, 7: 0 295--300, 1973

1973
[76]

An optimal algorithm for bandit and zero-order convex optimization with two-point feedback

Ohad Shamir. An optimal algorithm for bandit and zero-order convex optimization with two-point feedback. Journal of Machine Learning Research, 18 0 (52): 0 1--11, 2017

2017
[77]

Fourier features let networks learn high frequency functions in low dimensional domains

Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Conference on Neural Information Pr...

2020
[78]

Approximate exploitability: Learning a best response

Finbarr Timbers, Nolan Bard, Edward Lockhart, Marc Lanctot, Martin Schmid, Neil Burch, Julian Schrittwieser, Thomas Hubert, and Michael Bowling. Approximate exploitability: Learning a best response. In Proceedings of the International Joint Conference on Artificial Intelligenc...

2022
[79]

Munchausen reinforcement learning

Nino Vieillard, Olivier Pietquin, and Matthieu Geist. Munchausen reinforcement learning. Conference on Neural Information Processing Systems (NeurIPS), 33: 0 4235--4246, 2020

2020
[80]

Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St \'e fan J

Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St \'e fan J. van der Walt , Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nels...

2020
[81]

Multi-agent reinforcement learning in O pen S piel: A reproduction report

Michael Walton and Viliam Lisy. Multi-agent reinforcement learning in O pen S piel: A reproduction report. arXiv:2103.00187, 0, 2021

2021 arXiv
[82]

Yongzhao Wang and Michael P. Wellman. Empirical game-theoretic analysis for mean field games. In Autonomous Agents and Multi-Agent Systems, page 1025–1033, Richland, SC, 2023. International Foundation for Autonomous Agents and Multiagent Systems. ISBN 9781450394321

2023
[83]

Natural evolution strategies

Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, Jan Peters, and J\" u rgen Schmidhuber. Natural evolution strategies. Journal of Machine Learning Research, 15 0 (1): 0 949--980, 2014

2014
[84]

COLA : Consistent learning with opponent-learning awareness

Timon Willi, Alistair Letcher, Johannes Treutlein, and Jakob Foerster. COLA : Consistent learning with opponent-learning awareness. In International Conference on Machine Learning (ICML), 2022

2022
[85]

Open and closed loop Nash equilibria in games with a continuum of players

Agnieszka Wiszniewska-Matyszkiel. Open and closed loop Nash equilibria in games with a continuum of players. Journal of Optimization Theory and Applications, 160: 0 280--301, 2014

2014
[86]

Population-aware online mirror descent for mean-field games by deep reinforcement learning

Zida Wu, Mathieu Lauri\` e re, Samuel Jia Cong Chua, Matthieu Geist, Olivier Pietquin, and Ankur Mehta. Population-aware online mirror descent for mean-field games by deep reinforcement learning. In Autonomous Agents and Multi-Agent Systems, page 2561–2563, Richland, SC, 2024....

2024
[87]

C. Xin, G. Yang, and J. P. Huang. Ising game: Nonequilibrium steady states of resource-allocation systems. Physica A: Statistical Mechanics and its Applications, 471: 0 666--673, 2017

2017
[88]

Stochastic search using the natural gradient

Sun Yi, Daan Wierstra, Tom Schaul, and J\" u rgen Schmidhuber. Stochastic search using the natural gradient. In International Conference on Machine Learning (ICML), 2009

2009

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Robert J. Aumann. Markets with a continuum of traders. Econometrica, 32 0 (1): 0 39--50, 1964

1964

[2] [2]

The mechanics of n-player differentiable games

David Balduzzi, Sebastien Racaniere, James Martens, Jakob Foerster, Karl Tuyls, and Thore Graepel. The mechanics of n-player differentiable games. In International Conference on Machine Learning (ICML), 2018

2018

[3] [3]

Berahas, Liyuan Cao, Krzysztof Choromanski, and Katya Scheinberg

Albert S. Berahas, Liyuan Cao, Krzysztof Choromanski, and Katya Scheinberg. A theoretical and empirical comparison of gradient approximations in derivative-free optimization. Foundations of Computational Mathematics, 22 0 (2): 0 507--560, 2022

2022

[4] [4]

Learning equilibria in symmetric auction games using artificial neural networks

Martin Bichler, Maximilian Fichtl, Stefan Heidekr \"u ger, Nils Kohring, and Paul Sutterer. Learning equilibria in symmetric auction games using artificial neural networks. Nature Machine Intelligence, 3 0 (8): 0 687--695, 2021

2021

[5] [5]

JAX : Composable transformations of Python + NumPy programs, 2018

James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, et al. JAX : Composable transformations of Python + NumPy programs, 2018

2018

[6] [6]

Superhuman AI for multiplayer poker

Noam Brown and Tuomas Sandholm. Superhuman AI for multiplayer poker. Science, 365 0 (6456): 0 885--890, 2019

2019

[7] [7]

Peter E. Caines. Mean field games. In John Baillieul and Tariq Samad, editors, Encyclopedia of Systems and Control. Springer London, Gewerbestrasse 11, 6330 Cham, Switzerland, 2013

2013

[8] [8]

Caines, Minyi Huang, and Roland P

Peter E. Caines, Minyi Huang, and Roland P. Malham \'e . Mean field games. In Tamer Ba s ar and Georges Zaccour, editors, Handbook of Dynamic Game Theory. Springer International Publishing, Gewerbestrasse 11, 6330 Cham, Switzerland, 2018

2018