Generative Adversarial Equilibrium Solvers
Authors: Denizalp Goktas
David C. Parkes
Ian Gemp
Luke Marris
Georgios Piliouras
Romuald Elie
Guy Lever
Andrea Tacchetti
Venue: ICLR 2024 poster (2024)
Keywords: Game Theory
Amortized Optimization
Generalized Nash equilibrium
Economics
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We reformulate the problem of computing generalized Nash equilibrium in pseudo-games (and competitive equilibrium in Arrow-Debreu competitive economies) as a learning problem for a generative adversarial network.

Abstract: We introduce the use of generative adversarial learning to compute equilibria in general game-theoretic settings, specifically the generalized Nash equilibrium (GNE) in pseudo-games, and its specific instantiation as the competitive equilibrium (CE) in Arrow-Debreu competitive economies. Pseudo-games are a generalization of games in which players' actions affect not only the payoffs of other players but also their feasible action spaces. Although the computation of GNE and CE is intractable in the worst-case, i.e., PPAD-hard, in practice, many applications only require solutions with high accuracy in expectation over a distribution of problem instances. We introduce Generative Adversarial Equilibrium Solvers (GAES): a family of generative adversarial neural networks that can learn GNE and CE from only a sample of problem instances. We provide computational and sample complexity bounds for Lipschitz-smooth function approximators in a large class of concave pseudo-games, and apply the framework to finding Nash equilibria in normal-form games, CE in Arrow-Debreu competitive economies, and GNE in an environmental economic model of the Kyoto mechanism.

Published as a conference paper at ICLR 2024
GENERATIVE ADVERSARIAL EQUILIBRIUM SOLVERS
Denizalp Goktas
David C. Parkes Ian Gemp, Luke Marris,
Brown University den www.ll...otssbbroon... edu
Georgios Piliouras, Romuald Elie, Guy Lever, Andrea Tacchetti Google DeepMind
(parkesd, imgemp, marris, parkesd, imgemp, marris, gpil, relie, guylever, atacchet atacchet/espoglee /egoogle. com
ABSTRACT
We introduce the use of generative adversarial learning to compute equilibria in general game-theoretic settings, specifically the generalized Nush equilibrium (GNE) in aseudo-games, and its specific instantiation as the competitive equilib- rium (CE) in Arrow-Debreu competitive economies. Pseudo-games are a gener- alization of games in which players' actions affect not only the payoff's of other players but also their feasible action spaces. Although the computation of GNE and CE is intractable in the worst-case, i.e., PPAD-hard, in practice, many applica- tions only require solutions with high accuracy in expectation over a distribution of problem instances. We introduce Generative Adversarial Equilibrium Solvers (GAES): a family of generative adversarial neural networks that can learn GNE and CE from only a sample of problem instances. We provide computational and sample complexity bounds for Lipschitz-smost class concave pseudo-games, and apply the framework finding Nash equilib- ria in normal-form games, CE in Arrow-Debres competitive economies, and GNE in an environmental economic model of the Kyoto mechanism.
function approximators in a large
INTRODUCTION
Economic models and equilibrium concepts are critical tools to solve practical problems, including capacity allocation in wireless and network communication (Han al., 2011 Pang et al. 2008).
energy resource allocation (Hobbs & Pangl 2007: Ding- Yuan & Smeers] 1999,, and cloud comput-
ing (Gutman Nisan 2012 Lai et al, 2005| Zahedi et al. 2018 Ardagna et al. 2017). Many aveudo-games in which the actions taken by each plaver affect not only the other players' payoffs, as in games, but also the other play- ers' strategy sets The formalism of pseudo-ge games was introduced by Arrow & Debreu (1954), who used it in studying their foundational microeconomic equilibrium model, the competitive economy.
of these economic models are instances of what are known
The standard solution concept for pseudo- games is the generalized Nash equilibrium (GNE) (Arrow & Debreu 1954, Facchinei & Kanzow 2010a), which is an action profile from which no player can improve their payoff by unilaterally deviating another action in the space of admissible actions determined by the actions of other players. Important economic models can often be formulated as a pseudo-game, with their set of solutions equal the set of GNE of the pseudo-game: for instance, the set of competitive equilibria (CE) (Walras) 1896. Arrow & Debreu 1954) of an Arrow-Debreu competitive economy corresponds the set of GNE of an associated pseudo-game.
A large literature has been devoted to the computation of GNE in certain classes of pseudo-games but unfortunately many algorithms that are guaranteed converge theory have in practice been observed to converge slowly in ill-conditioned or large problems or fail numerically (Facchinei &
Research conducted while author was an intern at Google DeepMind.
Also, School of Engineering and Applied Sciences, Harvard U University.
many games, such as chess, the action taken by one player affects the actions available to the others. but these games are sequential, while in pseudo-gand games actions are chosen simultaneously. Additionally, even if one constructs game with payoffss that penalize the players for actions that are not allowed, the NE of the ensuing game will in general not correspond to the GNE of the original pscudo-game and can often be trivial. We refer the reader Appendix for mathematical example.

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Additionally, all known algo- rithms have hyperparameters that have to be optimized individualyy y for every geeudo-game instance (Facchinei& & Kanzow 2010a), deteriorating the performance of these algorithms when used to solve multiple pseudo-games. These issues point to a need to develop methods to compute GNE, for a distribution of pseudo-gamen games, reliably and quickly.
Kanzow 2010b; Jordan et al. 2022 Goktas & Greenwald 2022
We reformulate the problem of computing GNE in peeudo-games (and CE in Arrow-Debrr com- petitive economies) as learning problem for generative adversarial network (GAN) called the Generative Adversarial Equilibrium Solver (GAES), consisting of a generator and discriminator network. The generator takes as input a parametric representation of a pseudo -game, and predicts a solution that consists of tuple of actions. one per player. The discriminator takes as input both the pseudo-game and the output of the generator, and outputs a best-response for each player, seeking to find a useful unilateral deviation for all players; this also gives the sum of regrets, with which to evaluate the generator (see Figure D. GAES predicts GNE and CE in batches and in order to minimize the expected exploitability, across distribution of pseudo-games GAES amortizes com- putational cost up-front in training, and allows for near constant evaluation time for inference. Our approach is inspired by previous methods that cast the computation of an equilibrium in normal- form games as an unsupervised learning problem (Duan et al. 2021a Marris et al. (022). These methods train network to predict a strategy profile that minimizes the exploitability (i.e., the sum of the players' payoff-maximize bution of games. These methods become inefficient in pseudo-games, since in contrast to regular games, the exploitability in pseudo-gama (2) is not hipschitz-continuurs in turn making it hard to learn from samples, and (3) has unbounded gradients, which might lead to exploding gradients in neighborhoods of GNE. Our GAN formulation circumvents all three of these issues.
unilateral deviations w.r.t. a given strategy profile) over a distri-
(1) requires solving non-linear optimization problem,
Although the computation of GNE is intractable in the worst-cast (Chen & Deng, 2006 Daskalakis
et al., 2009 Chen & Teng 2009 Vazirani & Yannakakis) 2011 Garg et al. 2017). 7 in practice, ap- plications may only require solver that gives solutions with high accuracy in expectation over a realistic distribution of problem instances. In particular, a decision maker may need to compute a GNE for sequence of pseudo-games from some family or en masse over a set of pseudo-gama sampled from some distribution of interest. An example of such an application is the problem of resource allocation cloud computing platforms (Hindman al, 2011 Isard et al. 2009 Burns jtt al. 2016; Vavilapalli et al.] 2013)) where a significant number of methods make use of repeated
computation of competitive equilibrium (Gutman & Nisan 2012 Lai al. 2005 Budish 2011 Zahedi et al. 2018 Varian 1973) and generalized Nash equilibrium (Ardagna et al. 2017| 201 a
Anselmi al. (014)) In such settings, as consumers request resources from the platform. the plat-
forms have to find a new equilibrium while handling all numerical failures within a given time frame.
Another example policy makers who often want to understand the equilibria induced by a policy for different distributions of agent preferences in a pseudo gamee allowing them to study the impact of a policy for a distribution on different possible kinds of participants. For example, in studying the impact of protocol such as the Kyoto joint implementation mechanism (see Section 55. one might be interested understanding how the emission levels of countries would change based on their productivity levels (Jones et al. 2000). Other applications include computing competitive equilibria in stochastic market environments. For example, recently proposed algorithms work through a series of equilibrium problems, each of which has be solved quickly (Liu al. 2022
Contributions. Earlier approaches Duan et al. (2021a) Marris et al. (022) do not extend even to continuous (non pseudo-)gand gaames since eval- uating the expected exploitability and its gra- dient over a distribution of pseudo-games -ames re- quires solving as many convex programs as ex- amples in the data set. Additionally, in pscudo- not Lipschitz-
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games, the exploitability continuous, and thus its gradient is unbounded (Appendix D). hindering the use of standard tools to prove sample complexity and conver- Figure 1: Summary of the Architecture of GAES. gence bounds, and making training hard due to exploding gradients. By delegating the task of computing best-response to discriminator, our method circumvents the issue of solving convex
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program, yielding training problem given by a min-max optimization problem whose objective is Lipschitz-contimuus, for which gradients can be guaranteed to be bounded under standard assump- tions on the discriminator and the payoffs of players.
Our approach also extends the class of (non (roudo-)games that can be solved through deep learn- ing methods from normal-form games simultaneous- move continuous action games, since the non-linear program involved in the computation of exploitability in previous methods makes them inefficient in application continuous-action games. We prove polynomial-time convergence of our training algorithm for arbitrary Lipschitz-smooth function approximators with the joint action space dimensions larger than parameter space dimensions (Theorem 4.1 Section 4 and provide general- ization bounds for arbitrary function approximators (Theorem empirical evidence that GAES outperforms state of the art baselines in Arrow-Debreu competitive economies, and show that GAES can replicate existing qualitative analyses for pseudo-gand games, sug- gesting that GAES makes predictions that not only have low expected exploitability, but also are qualitatively correct, i.c., close to the true GNE in action space (Section
Section Finally, we provide
5
Additional related work. We refer the reader to Appendix B for a survey of methods to compute GNEs, and to Appendix C for survey of applications of GNE, Our contributions generally relate to line of work on differentiable economics, which seeks to use methods of neural computation for problems of economic design and equilibrium computation. In regard to finding optimal economic
designs, deep learning has been used for problems of auction design (Dütting et al.) 2019 Curry
et al. 2022c Tacchetti al.) 2019; Curry al., 2022a Gemp et al. 2022 Rahme et al. 2020 Duan
ct al. 2022 and matching (Ravindranath et al. 2021). In regard to solving for equilibria, some
recent works have tried to solve for Nash equilibria in auctions (Heidekrüger al. 2019] Bichler
et al.
202Th. and dynamic stochastic general equilibrium models (Curry et al. 2022b Chen et al.
2021
Hill al. 202T)
2 PRELIMINARIES
Notation. All notation for variable types, e.g., vectors, are clear from context, if any confusions arise see Appendix E We denote the set of integers {0...., n - 1} by [n), the set of natural numbers by N. the set of real numbers by IR, and the positive and strictly positive elements of a set by a subscript + and ++, e.g., R+ and 14+. We denote by An = {x E R E Ij = 1}, and by (AA) the set of probability measures on the set A.
= (n, A, he u²), denoted (n, X. h, u) when clear from context, comprises n N+ players, where player i E [n] chooses an action a, from a non-empty, compact, and convex action space A, C R We denote the players' joint action space by A = Xigin A; C R"m Each player i E [n] aims to maximize their continuous payoff. u, : A - R, which is concave in a, by choosing feasible action from set of actions, X((a C A., this depending on the actions a_, A_; C (===1jm of the other players. Here, X, A_i x= A, is a non-empty, continuous, compact- and convex-valued (feasible) action correspondence. It is this dependence on each others' actions that makes this a pseudo-game, and not just a game. In applications. we will represent X; as X,(a_,) = {a, E A, I hi (a, a 2 0, for all p E [d]), where for all i E [m]. and p E [d]. h.p is continuous and quasi-coneave function in a, which defines the constraints. We denote the product (feasible) action correspondence by (a) = X.e(n) X,(a_,), which we note is guaranteed to be continuous, and
Pseudo-Games. A (concave) pseudo-game (Arrow & Debreu 1954
non-empty-, compact-valued We denote X' the set of jointly feasible strategies, i.e., X = {a E Al
(a) 2 0, vi E [n], E [d]] We denote the class of all pseudo-games by 1
Given pseudo-game G, generalized Nash equilibrium (GNE) is an action profile a" E X, s.t. for all E [n] and a; E X,(a",), u,(a*) 2 u, (a,a, a a²,). An equilibrium mapping, h : + X' is a mapping that takes input a pseudo-game G I and outputs GNE, hG),, for that game. Given pseudo-gamd gamoog-ame@g G, we define the regret for player i E [m for action a, as compared to another action b,, given the action profile a_; of other players, as Regret, (a,, b,; a_;) = uy (b, a.;) ))- --u u (a. a_:). Additionally, the cumulative regret, between two action profiles a E A and b A is given by
A game Nash 1950 is pseudo-game where, for all players E [n]. X, is a constant correspondence
with value A,. discrete action game game where A, = Am and u, multilinear for all i [m].
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(a, Eve(n) Regrety (a,, b;; a.). Further, the exploitabilitty Nikaido & Isoda 1955p. of an action profile is defined as - (a) = Liein] Ex Regret, (a b,: a). Note that an action profile R is GNE iff 8p (a
Mathematical preliminaries. For any function f X' + V we denote its Lipschitz-continitty constant by Ey. For two arbitrary sets H. H1' C F, the set H' r-covers H. (w.rt. some norm I|-ID for any E H there exists h' E 'H' such that || h'll S r. The r-covering number, p(h, r), of a is the cardinality of the smallest set H" C F that r-covers H. A set H. is said to have a bounded covering number, if for all R+, we have that the logarithm of its covering number polynomially bounded in 1/r, that is log(p(h,r)) S poly(1/r). Additional background can be found in Appendix E
3 GENERATIVE ADVERSARIAL LEARNING OF EQUILIBRIUM MAPPINGS
In this section, we revisit previous formulations of the problem of learning equilibrium mapping, discuss the computational difficulties associated with these formulations when used to learn GNE, and introduce our generative adversarial learning formulation.
As creating a sufficiently diverse sample of (pseudo-game, GNE) pairs, while performing adequate equilibrium selection, is intractable both theoretically and computationally, forgo of supervised learning methods, and formulate the equilibrium mapping learning problem as an unsupervised learning problem, following the approach adopted by Marris al. (2022); Duan et al. 202Ib) for finding Nash equilibria. Given a hypothesis class H S and a distribution over pseudo- -amdess D E A (T), the unsupervised learning problem for an equilibrium mapping consists of finding hy- pothesis h- arg minleи HE [E(G, h h())) where é I A + R is loss function that outputs the distance of a A from GNE, such that for any pseudo-game gamee G E I,, (( a*) 0 iff a is GNE of g. In particular, Marris et al. (022); Duan et al. (021b) suggest to use exploitability as the loss function. However, number of issues arise when trying to minimize the expected exploitability over distribution of pseudo-games:
(1) Computing the gradient of the exploitability, when it exists, for only one pseudo-game requires solving concave maximization problem (this reasoning also applies to continuous games).
(2) The exploitability in sneudo-games, is in general not Lipschitz-continuuus (unlike in regular games), even when payoff's are Lipschitz-contimuuss, since the inputs of the exploitability parame- terize the constraints in the optimization problem defining each player's maximal regret computa- tion. This makes it unclear how to efficiently approximate Eg-D (a)) from samples, without knowledge of the distribution.
(3) The exploitability in pscudo- garioo is absolutely continuous and hence differentiable almost everywhere Afriai (971), but in contrast to games, the gradients cannot be bounded. This in turn precludes the convergence of first-order methods
To address the aforementioned issues, we propose a generative adversarial learning formulation of the associated unsupervised learning problem for equilibrium mappings. The formulation relies on the following observation, whose proof is deferred to Appendix F the exploitability can be computed post after computing the expected cumulative regret by optimizing over the space of best-response functions from pseudo-games to actions, rather than the space of actions individually for every pseudo-game.
Observation For any D A(T), we have minnex" Eg (h())) min, EXT max JeA (GG." (h())) EgND (
By Arrow-Debreu's lemma on abstract economies Arrow & Debreu h" is guaranteed to exist and is an equilibrium mapping iff max
1954
arg min hEXT
EgwD ( (().
We refer the reader to Appendix has unbounded gradients.
P for example in which exploitability is not Lipschitz-comtinuuus and
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This problem formulation allows us overcome issues (1) and (2). For (I), rather than solve cave program to compute the exploitability for each pseudo-gante and action profile, we can learn a function that maps action profiles to their associated best-response profiles (see for example Lanctot Lanctot et al. (201)) for training best-response oracles). For (2), the objective function in the Equation on the right hand side of Observation || i.e., the cumulative regret, is Lipschitz-continuuus in the action profiles when payoff functions are, which opens the doors to use standard proof techniques to learn the objective from polynomial sample of pseudo-games.
Still, the gradient of maxy Eg-D (w) (h(G), (G; with respect to h is in gen- eral unbounded even when it exists, due to the constraint bA E F.f(G) E X (h())). However, since any solution f ((())(())) to the inner optimization problem E [ ( equilibrium mapping h, we can represent this dependence explicitly in the optimization problem, and restrict selection of f continuously differentiable hypothesis class F A and overcome issue (3).
is implicitly parameterized by the choice of
With these observations in mind, given hypothesis classes H. X and F C A the generative adversarial learning problem to find tuple (h f*) E H x F that consists of generator and discriminator to solve the following optimization problem:
min NERSEFNSET (G:h (A()))
max
Eg-D [v ( (MG)))(;
(1)
This problem can be interpreted as a zero-sum game between the generator and the discriminator. The generator takes input parametric representation of sreudo-game, and predicts solution that consists of an action profile, i.e., a tuple of actions, one per agent. The discriminator takes the game and the output of the generator input, and outputs best-response for each agent (Fig- ure Db. The optimal mappings (h" x ff for Equation [1 are then called the Generative Adversarial Equilibrium Solver (GAES).
4 CONVERGENCE AND SAMPLE COMPLEXITY
For training, we propose stochastic variant of the multistep gradient descent ascent algorithm (Nouiched et al. 2019: Sanjabi et al., 2018), which we call stochastic exploitability descent. Our al- gorithm computes the optimal generator and discriminator by estimating the gradient of the expected cumulative regret and exploitability on a training set of pseudo-games games.
Algorithm Stochastic Exploitability Descent
Inputs: B, Th, Mf Thy Ty. w 1100) wf.(0)
Outputs: (tt) w (()) Th
)==0
1: for t == 0..... ... Th 1 do
2: Receive batch ((t) S.
3: h.++11 wh.(1 1/
(t) 1/66)) Egerfe) (wu (00e:0t0,,,f000 w 20f
4: for s :=.....,
do
5: Receive batch 13()) C S.
6: w = w + +11 (x)
1/18901 EgeBy) V Vun (wr wo))
7: end for
8: wt (c+1) wt
9: end for
10: Return (w), wt .11) Tn (0))
Training Algorithm. For purposes of applicability, going forward, will assume that have access to the distribution of pseudo-gamm -eeudo-games games D only indirectly through training set S D of k E N, sampled aseudo-games Additionally, will assume that the generator h. H. and discriminator
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f F are parameterized by parameter vectors, wh, w E R- such that for all geeudo-games E r. and weight vectors w E IR." ((g: E X' and fG; hG; w²), w²) E A.
For notational simplicity, define the expected cumulative regrer and the empirical cumulative gret, respectively, as (w"", w = E (h(G; w"), fG, h(G; w))))) and ((w", w w wr]]]] where the expectation is over the distribution of pseudo- games G ~ D and the uniform distribution over the training set, ~ uS) respectively. Simi- larly, we define the expected exploitability and the empirical exploitability respectively as:
[w w²)) fG; (G;
F(w")
(ww", w
p ((w²1)=
(10",
max w EE VgEr, 1C: (EX(h(
f f (
Putting this together, our training problem becomes:
max 1 ER= w (IAGEEEE
min
(ee wt).
We propose Algorithm | to solve this optimization problem. This is a nested stochastic gradient descent-ascent algorithm, which for each generator descent step runs multiple stochastic gradient ascent steps on the weights of the discriminator to approximate the empirical cumulative regret by processing the peruuo--gmess in the training set in batches, i.c., mutually exclusive subsets of the training set. After the stochastic gradient ascent steps are done, the algorithm then takes a step of stochastic gradient descent the empirical exploitability w.r.t. the weights of the generator using the discriminator's 's weights computed by the stochastic gradient ascent steps to compute the gradient. Note that when the hypothesis class of the discriminator is assumed contain only differentiable functions, the gradient of the generator with respect to its weights exist, and they are given by the implicit function theorem.
Convergence and generalization bounds. We give assumptions under which our algorithm con- verges to a stationary point of the empirical exploitability in polynomial-timat
their payoff uf is &v. -Lipschitz smooth, 2. (Strong concavity) their payoff uy is uu, -strongly-concave in a and 3. (Lipschitz-smooth hypothesis classes) For all E H C xUxR" .EEFF FC X hG; ((;:)) are injective, and Lipschitz-mmooth functions.
Assumption 1. For any player i [n] and E supp(D): 1. Siipshitz--smootheeet
We note that strong concavity can further be weakened to concavity, and our results can directly extend to any concave pseudo-game by replacing the cumulative regret with its regularized coun- terpart &i (a. b) vha (a. b) 0/2 a bll bll²- which strongly-concave in b without modifying the solutions of the optimization (Von Heusinger & Kanzow 2009). However, as in our experiments nonrregularized cumulative regret performed better, for consistency, we present our theoretical re- sults under the strong concavity assumption. 'hat said, strong concavity in each player's 's action is a much weaker assumption than strong monotonicity of the pseudo-game which is commonly used in the GNE literature (Jordan et al. 2022). For omitted definitions, and proofs/results refer the reader to Appendix and Appendix respectively.
Theorem 4. tells us that our algorithm converges to stationary point of the empirical exploitability at 00% /Th) rate, up to error term that depends linearly the distance, e, of the discriminator computed by the algorithm w.r.t. to the optimal discriminator. A smaller e results in higher accuracy, at the expense of longer run time.
Theorem 4.1 (Convergence to Stationary Point). Suppose that Assumption 7 holds. Ler > O. (1) If Algorithm is run with inputs satisfying Th (1/V), (8) E (V/s), Th E N++. and T E (1)e), for all t E [Ta], E [Ty]. Then, the outputs (t),.(), wt (1)(1)) Th (t))Tn === satisfy
E
min Th |/ G (wo EO
(og(Th) (Th +
*Our min-max problem is non-convex-PL. for which single loop stochastic gradient descent ascent algo- rithms are not guaranteed to converge stationary points of the exploitability (Lin et al. 2020 Daskalakis
ct al. 2009). Additionally, our algorithm's 's computational complexity ((1-2) is orders magnitude faster
than the best known complexity of O(1/,)) for such problems (Lin 2020

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We also give sample complexity result showing how cumulative regret can be approximated with a sample of aseudo-games that is polynomial in the parameters of the game distribution, 1/e and 1/6. The novelty of the result comes from the context: we mentioned earlier that expected exploitability need not be Lipschitz-contimuuus, making it hard to use any standard and simple machinery to prove sample complexity bound. However, by reframing this problem as one of learning the expected cumulative regret, which is Lipschitz-continuuur we can obtain the result.
Theorem 4.2 (Sample Complexity of Expected Cumulative Regret), Suppose that part 1. of As- sumption I holds. Let s, 8 E (o, 1), r E O (ffta) and consider hypothesis classes H.F given by the minimum r-covering sets of , AA'x1I closest hypotheses h" E arg EH h - -k'll and f E arg minn ff'll to respectively. Then, for any pseudo-game distribution D E (T) with compact support, with probability at least 1 - 8 over draws of the training set S ~ D* with k E 9 (1/2 log p(H,,/tt))le ))e)))):
respectively. For any h E X and f E A take the
[Bg-unitis) [v (9), h (G))) Bg (10 ((9,) SE.
EXPERIMENTAL RESULTS
Arrow-Debreu exchange economies Our first set of experiments aim to solve CE Arrow- Debreu exchange economies (Arrow & Debreu 1954). The difficulty in solving the pseudo-game game associated with Arrow-Debreu exchange economies hereafter exchange economies-aris from the fact that it does not fit into any well-defined categories of pseudo-games, e.g., monotone or jointly convex, for which there are algorithms that converge to GNEs.
An exchange economy (u, E) consists of finite set of m E N+ goods and E N+ consumers (or traders). Every consumer E [n]] has a set of possible consumptions X; C IR,, an endowment of goods e, (ea),...,, Cim)
R.m and a utility function u, I + R. We denote E
(ei...., e..) Any exchange economy can be formulated as a pseudo-game whose set of GNE is equal to the set of competitive equilibria (CE of the original economy (Arrow & Debreul 1954.. This pseudo-game game consists of n + agents, who correspond to the n buyers and a seller. The pseudo-game given by the following optimization problem, for each buyer E [n]]. and the seller,
respectively:
(E D; -EEE)) (i))) max Dm
max
U4 (x(3):))
E.X,
Let Us E RT, E 1j", be vector of parameters for the utility function of buyer i E [n]. our experiments, we consider the following utility function classes: Linear: u,(x))) = Eje(m UtjIaj. Cobb-Douglas, u, u,(x:) = =I 2.j Leontief: u, ux(x,) = minje[m] {zw//eg) and constant elasticity of substitution (CES): (x.))) = (E(m) Pogly)'e Ve. When we take p 1, 0, and p -o for CES utilities, we obtain linear, Cobb-Dougla and Leontief utilities respectively. We denote V = (UI...., v₂,) As is standard in the literature (Chung et al. 2013 Brânzei et al., 2021 we assume that for all buyers i E [n]. X RT. Once utility function class is fixed, an exchange economy is referred to with the name of the utility function, and can be sampled as tuple (V, E) E Raxm x Raxm for linear,
and Leontief exchange economies, and tu- ple (V, P. E) BERR Raxm x RR" x R=xm for CES exchange economies. For CES exchange economies, we have gross substitute (GS) and gross complements (GC) CES economy, either when p. 2 0 for all buyers or P: < 0 for all buyers, respectively. Otherwise, this is mixed CES economy.
Baselines. For special cases of exchange economies, the computation of CE is well-studied (e.g., Bei et al. (215)). allowing us to compare the performance of GAES to known specialized methods. We benchmark GAES to tâtonnement (Walras 1896). which an auction-like algorithm that is guaranteed to converge for CES utilities with p 0, 1) (Bei et al. 2015). including Cobb-Douglas and excluding Linear utilities. We also benchmark to exploitability descen! (Goktas & Greenwald, 2022). For each of these baselines. we run an extensive grid search over decreasing learning rates during validation (see Appendix G.2. Each bascline was run to convergence.
We report the results of two experiments. First, we run our algorithms in linear, Cobb-Douglas. Leontief, GS CES, GC CES, and mixed CES exchange economies (we defer the results from the GS
We include experiments on normal-form games, as well as all missing additional implementation details and network architecture diagrams in Appendix G
We refer the reader to Appendix G.2 for definition.

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CES, and GC CES experiments to the Appendix). We report the distribution of the exploitability the test set for GAES and the baselines in Figure Io (additional plots can be found in Appendix[G2 We measure performance w.r.t the normalized exploitability, which is the exploitability of the method divided by the average exploitability over the action space. We observe that in all economies, GAES outputs action profile that is on average better than at least 99% of the action profiles in terms of exploitability. In all four markets, GAES on average achieves lower exploitability than the baselines (see Figure 2 We also see in Figure TO (Appendix G.2) that GAES outperforms the baselines, in distribution, in every economy except Cobb-Doughe Douglas. This is not surprising since
tâtonnement guaranteed to converge in Cobb-Douglas economies (Bei et al. 2015). That said, tâtonement does not outperform GAES on average, since tâtonnement's 's convergence guarantees hold for different learning rates in each market.
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Figure 2: Training and Testing Exploitability of GAES in linear. Cobb- boob-Do Leontief, and mixed CES economies. GAES outperforms all baselines on average in all economies.
Second, we compare GAES with the performance of tâtonnment in pathological and well-know Leontief exchange economy, the Scarf economy (Scarl) 1960) (Figure 3. Here, we soft start tâtonnement with the output of GAES with some added uniform noise. This additional noise en- sures that the starting point of tâtonnement is distinct from the output of GAES). We see on Figure Bl that the prices generated by tâtonnement spiral out and settle into orbit. This makes sense as, unlike pure Nash equilibria, GNE and CE are not locally stable (Flokas et al. 2020). meaning that soft-starting the algorithm with the output of GAES might be worse than using GAES alone. The success of GAES in Leontief exchange economies as well as the Scarf economy is notable, and suggests that GAES might be smoothing out the loss landscape since for these economies the exploitability is non non-differentiable.
Eoplatabliy for scad Sconcin
Kyoto joint implementation mechanism. We solve a pseudo- game model of the joint implementation mechanism proposed the Kyoto protocols (Protocol 1997p. The Kyoto Joint Im- plementation Mechanism is cap-an and-trade mechanism that bounds each country that signed onto the protocol to emit an- thropogenic gases below a particular emission cap. Countries bound by the mechanism can also invest in green projects in other countries, which in return increases their emission caps. Breton et al. (006) introduce a model of the Kyoto Joint Im- plementation Mechanism, using this predict the impact of the mechanism. The model that the authors propose is par- tially solvable analytically, that is, one can characterize equi- libria qualitatively using comparative statics (Nachbar, (002). but cannot obtain closed-form solutions for GNE, Moreover, there is no algorithm that is known to converge to the GNE of this aseudo-game, since it is not monotone.
Figure 3: equilibrium prices in the Scarf Economy. While the output of GAES is close to the equilibrium prices, the final prices outputted by tâtonnement prices are further than the starting prices.
A phase portrait of
Formally. the (Kyoto) Joint Implementation Mechanism (JI) consists of n E N countries. Each country E [n], can make decisions that result in environmentally damaging an- thropogenic emissions, e, E R.,, and can make investments 2; E Rd to offset their emissions. The investments x, E 1² that each country E [n] makes in another country j E [m] offsets that country's 's emissions by Fij where this is in proportion to investment return rate Ti > O for country j, with y E RT. Each country i has revenue r, R7 + R. which is function of its emissions e,, cost function C2 Ra + R that is function of all investments, and a negative externality function, d, R - R. that is a function of the net emissions of all countries.
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Each country aims to maximize their surplus, i..,, their revenue minus costs and and negative exter- nalities, constrained by keeping their net emissions under an emission cap, 1), E R,. Additionally, require the emission transfer balance to hold, i.c., vi, c; - % Σjeimi Tij 2 0, that is no country can transfer more emission reduction than they have, giving for each country, i:
The ri(e;) - 4(x) - (
(fe:
max xR"", E Sty vie(m],ei>> Eye(n) 2,.20
literature has traditionally assumed that ri(e;;
es c; e1/2), (x(X)
[ Ejzz [(my + + -
" and di(e Eie(n) Di Bi) = Bi Diemijle Ejem Tty).
Fixing these functional forms, we can sample (0, y; 7:)) E ET Rn Rmxm R and obtain a representation of the JI mechanism.
We run two different experiments replicate and extend the analysis of [Beeton et al. 2006 We first replicate their qualitative analysis of equilibria (Figure Breton et al.] (006) introduce a comparative static analysis, in which they fix all parameters of JI except for O, and characterize the kinds of GNE as 0 varies. In Figure 4 the six regions correspond to different kinds of GNE. For instance in Region 1, both countries emit strictly less than their emission cap (see Section 4 of Breton et al. (2066). ) The parts of the plot that are not shaded correspond to pseudo-games for which GNE are not unique and for which equilibria cannot be characterized analytically. We superpose on top of this plot set of pseudo- games from an unseen test set. and color each aseudo-game by the color of the region whose condition they fulfill.
We observe that with the exception of Re- gions and 2a, the structure of the GNE gen- erated by GAES lines up well with the type of GNE. We believe that failure to predict Re- gions 1 and 2a is due the closeness of the ac- 2b
Regions of Kyoto| Pseudo-games
tion profiles that fit either of these equilibrium types to other ones. For instance, GAES pre- dicts a GNE of type 4a for pseudo-games in Re- gion 2a but these GNE, although qualitatively different, are very close in action space: the only difference between the two regions that for the type 2a GNE, one player emits strictly less than its cap and the other emits exactly its cap, while for the type 4a GNE, both players emit exactly their emission cap. A similar con- clusion holds for GNE in Region 3a, pre- dicted in Region 1.
30
9: 50
40
Figure 4: A taxonomy of different equilibrium types for various pseudo-games obtained by fix- ing all parameters, and varying 0. The x and axes represent the revenue parameters f1, 022 of the countries. The colored regions (obtained analyti- cally) correspond to different qualitative types of GNE while the dots correspond to pseudo-games in the test set colored by the GNE types that were predicted for them by GAES.
We also solve the JI aseudo-game for dis- tribution of JI pseudo-games (Figure [4] Ap- pendix G and these results confirm that the testing normalized exploitability is very low. We note that normalized exploitability is given as the exploitability divided by the average exploitability over the action space, which means that GAES has on average lower exploitability than 99.5% of the feasible action profiles. This con- firms our hypothesis that failure to predict Regions and 2a in Figure g arises from the proximity between GNE of these types and GNE of types 3a or 4a respectively, since according to exploitabil- ity there is very little improvement left for GAES.
6 CONCLUSION
We introduced GAES, GAN that learns mappings from pseudo-gamed to GNE. GAES outperforms existing methods to compute GNE or CE exchange economies, and solves even pathological examples, i.e., Scarf economies. Our approach extends the use of exploitability-baeed learning methods from normal-form games to continuous games, exchange markets, and beyond. GAES adds to the growing list of differentiable economics methods that aim to provide practitioners with computational tools for the study of economic properties. GAES extends the range of models for which we have approximate and reliable solvers.

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Published as a conference paper at ICLR 2024
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Reviews:
[{"review_id": "0lcNa2lVfL", "reviewer": "Reviewer_ni1H", "summary": "This work proposes a novel algorithm, General Adversarial Equilibrium Solvers, for training general GNE solvers.\n\nThe goal of equilibrium solvers is, given a strategic game between multiple players, to find a (generalized) Nash equilibrium of the game. \n\nWhile there has been a few previous work that proposes algorithms to train equilibrium solvers, they all suffers from three technical challenges:\n- the gradient of the exploitability requires solving a concave maximization problem\n- the exploitability of pseudo-games is in general not Lipschitz-continuous\n- the gradients cannot be bounded in general\n\nThe authors formulates equilibrium solver training as training a generative adversarial networks, where the generator takes a pseudo-game representation, and outputs a tuple of actions (one per player), and the discriminator takes both the pseudo-game, and the output of the generator, and outputs a best-response for each player.\n\nThe goal of discriminator is to output a best-response actions that produces the exploitability, and the goal of generator is to output actions that minimizes the exploitability, i.e., GNE.", "questions": "- Would it be possible to scale GAES to modern games that consists of multiple neural networks (e.g., GANs, multi-agent RL problems, etc.)? If not, what would be the main technical challenges to do so?", "limitations": "", "rating": 10, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "# Presentation\n- The paper is well-organized and easy-to-follow: Sec.1 motivates the readers by illustrating the possible applications of GNE solvers, including network communication, cloud computing, and economic models (e.g., Arrow-Debreu exchange economy, Kyoto joint implementation mechanism)\n\n# Novelty, Technical Contribution\n- The formulation of GAES establishes a novel, efficient, simple, and scalable algorithm to train generic GNE solvers.\n    - To the best of my knowledge, most of the previous work relied on supervised learning, and suffered from a few technical challenges in terms of computational tractability and stability. \n    - GAES beautifully solves these problems, and provides a simple yet powerful framework for training GNE solvers.\n- The formulation is strongly backed up by theoretical guarantees; convergence of the networks towards a stationary point of exploitability, and sample complexity. \n- The experiments are conducted on non-trivial games, namely Arrow-Debreu exchange economies and Kyoto joint implementation mechanism — which are non-monotone or non-jointly convex. Strong empirical results on these games verifies the efficiency of GAES.", "weaknesses": "# Weaknesses\n- I don’t see any special weakness in this paper. The authors establish a simple yet powerful framework for training GNE solvers, and backs up their algorithm both with strong theoretical guarantees and empirical results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vXMaWKwBpf", "reviewer": "Reviewer_Db1F", "summary": "This paper study the generalized Nash equilibrium of pseudo games where a player’s action not only affects his utility, but also other players’ action sets. The authors use GAN and employ exploitability as the loss function. The solver is applied to compute the GNE of Arrow-Debreu competitive economies and the Kyoto mechanism.", "questions": "1. In which paper was the name \"pseudo game\" and \"GNE\" made? It seems that the cited paper by Arrow Debreu mentioned the game first, but did not name it.\n2. Do you measure the difference between the results and the optimal action in the feasible set? Notice that the results “is on average better than at least 99% of the action profiles” in the experiments.\n3. Is there any guarantee when the utility function only satisfy convexity?\n4. What is the title of this paper?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. Introduce a novel method to compute GNE by GAN.\n2. Provide theoretical guarantee on convergence and generalization bounds.\n3. The performance is better according to the experiments.\n4. The literature review in the appendix summarizes the current methods to solve GNE and the application of pseudo games.", "weaknesses": "1. Assume strong concavity in assumption 1, however, the utility function is not strong convex in Arrow-Debreu competitive economy.\n2. Do not provide guarantee for the performance on the training set.\n3. Use different network architecture in two experiments which means GAES is not a general solver for GNE.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rCNKi8T07D", "reviewer": "Reviewer_Wcqa", "summary": "The paper introduces generative adversarial equilibrium solvers (GAES), a GAN that learns to map pseudo-games to their generalized Nash equilibria from a sample of problem instances. In particular, they provide a formulation that makes the problem amenable to standard stochastic first-order methods. They use GAES to compute in a scalable way competitive equilibria in exchange economies and an environmental economic model of the Kyoto mechanism, outperforming earlier methods.", "questions": "Some minor comments for the authors:\n\n1. The title of the submission document is the default one.\n2. The references have to be polished. There are many papers that are published many years ago and only the arXiv version is cited. There is also an issue with consistency: sometimes URLs are included, sometimes they are not. Please fix those issues.\n3. There are underfull equations in Observation 1 and immediately below.\n4. I don't understand how the paper of Daskalakis et al. (2009) is relevant in the context of Footnote 4 about min-max optimization.\n5. The appendix has many overfull equations that need to be formatted properly.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "Pseudo-games are very general game-theoretic models with a number of applications, most notably Arrow-Debreu competitive economies. Yet, there is a lack of scalable techniques for computing generalized Nash equilibria in such settings. This paper makes a concrete contribution in that direction by providing a method with promising performance across a number of benchmarks. The proposed method is natural and the experimental results are overall convincing and quite thorough. Indeed, the paper appears to attain state of the art performance in a number of important applications, and could have significant impact in this area.", "weaknesses": "There are some soundness issues that the authors have to address. First, there appears to be a significant gap between the theoretical analysis and the experimental settings. Specifically, it is not clear how a stationary point in the sense of Theorem 4.1 translates to a GNE. If stationary points are not necessarily GNE, the narrative of the paper has to be restructured. In particular, it is often claimed that the method maps pseudo-games to GNE, and it is not clear whether that claim is theoretically sound. Of course, computing GNE is intractable, but it is alluded (for example in the abstract) that under a distribution over problem instances the problem could be easier. Theorem 4.1 also makes a strong concavity assumption which appears to be violated in all settings of interest. It should be the case that a \"small\" regularizer can always be incorporated without affecting the equilibria by much, but I think this should be discussed in more detail. \n\nI am also confused about Footnote 4. It is claimed that the method obtains the state of the art $O(1/\\epsilon^3)$ complexity, a major improvement over $O(1/\\epsilon^6)$, which the authors claim was the previous state of the art. The authors have to explain more precisely the class of problems this applies to; there are many variants of the PL condition studied in the literature. In particular, the following papers seem to obtain a much better dependency: \"Faster single-loop algorithms for minimax optimization without strong concavity,\" \"faster stochastic algoritms for minimax optimization under Polyak-Lojasiewicz Conditions\" and \"Doubly smoothed GDA for constrained nonconvex-nonconcave minimax optimization.\"\n\nBesides the issues above, the algorithmic approach is very close to the paper \"Exploitability minimization in games and beyond,\" which limits to some extent the algorithmic contribution of the present paper. The authors have to highlight the comparisons in more detail.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MA8W896ldk", "reviewer": "Reviewer_pH5J", "summary": "In this paper, the authors present a novel neural network-based method for approximating the generalized Nash equilibrium (GNE) within pseudo-games derived from a specific distribution. These pseudo-games consist of players operating within compact and convex action spaces, where the choices made by each player can influence the feasible action space of others.\n\nTo facilitate the training of the GNE neural network solver, the authors introduce exploitability as the loss function.  Exploitability quantifies the total utility gains that all players would achieve by deviating to their own best responses. However, calculating exploitability poses a challenge due to the potentially infinite action space. To tackle this issue, an adversarial network is employed to approximate each player's best response.\n\nThe authors also establish a theoretical framework by providing a generalization bound for this neural solver. Furthermore, in practical experiments, they apply this approach to identify Nash equilibria in normal-form games, compute competitive equilibria in Arrow-Debreu economic models, and determine GNE in an environmental economic model involving the Kyoto mechanism.", "questions": "- How does the performance of GAES degrade with approximate discriminators in practice? Is there a way to quantify the required discriminator accuracy?\n- Could you apply GAES to find other solution concepts like correlated equilibria? How would the formulation need to change?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- The versatility of this method is evident because it can be applied to a wide range of games, thanks to the inherent generality of pseudo-games.\n- The methodology looks strong. The use of two neural networks (a generator and a discriminator) and adversarial training is intriguing to me.\n- The concept of employing a neural network as a function approximator to compute GNE is innovative. I believe it has the potential to expedite equilibrium computation in practice.", "weaknesses": "- The title in the PDF is still the template title.\n- The paper concentrates on finding a single equilibrium, yet many games have multiple equilibria. Incorporating a discussion on equilibrium selection would enhance the work.\n- The overall presentation of this paper would benefit from further refinement. The figures within the paper are small and appear blurry due to the absence of vector graphics formats such as .pdf or .svg. Upgrading the figures to vector graphics would improve their clarity and overall visual impact.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)