Stochastic generalized Nash equilibrium seeking under partial-decision information

We consider for the first time a stochastic generalized Nash equilibrium problem, i.e., with expected-value cost functions and joint feasibility constraints, under partial-decision information, meaning that the agents communicate only with some trusted neighbours. We propose several distributed algorithms for network games and aggregative games that we show being special instances of a preconditioned forward-backward splitting method. We prove that the algorithms converge to a generalized Nash equilibrium when the forward operator is restricted cocoercive by using the stochastic approximation scheme with variance reduction to estimate the expected value of the pseudogradient.


Introduction
In a stochastic Nash equilibrium problem (SNEP), some agents interact with the aim of minimizing their expected value cost function which is affected by the decision variables of the other agents. The characteristic feature is the presence of uncertainty, represented by a random variable with unknown distribution. Due to this complication, the equilibrium problem is typically hard to solve [20,28]. However, many practical problems must be modelled with uncertainty, for instance, electricity markets with unknown demand [18] and transportation systems with erratic travel time [31]. Another recurrent engineering aspect is that agents may be subject to shared feasibility constraints. For instance, consider the gas market where the companies participate in a bounded capacity market [1] or more generally any network Cournot game with market capacity constraints and uncertainty in the demand [35,32]. In this case, we have a stochastic generalized NEP (SGNEP), i.e., the problem of finding a Nash equilibrium where not only the cost function but also the feasible set depend on the decisions of the other agents [7,12,32]. This class of problems is of high interest for the decision and control community, in both deterministic [7,27,32,14] and stochastic cases [20,34]. Notably, the ⋆ This paper was not presented at any IFAC meeting. Corresponding author B. Franci. Tel. +31-152785019.
presence of shared constraints makes the computation of an equilibrium very challenging, especially when searching for distributed algorithms, where each agent only knows its local cost function and its local constraints. Perhaps the most elegant way to design a solution algorithm for a SGNEP is to recast the problem as an inclusion, leveraging on monotone operator theory. In particular, operator splitting methods, paired with a primal-dual analysis on the pseudogradient mapping, can be used to obtain fixed-point iterations that converge to an equilibrium, i.e., a collective strategy that simultaneously solve the interdependent optimization problems of the agents while reaching consensus on the dual variables associated to the coupling constraints [10,22]. Unfortunately, these methods do not necessarily lead to distributed iterations, thus, for this purpose, a suitable "preconditioning" is required in some problem classes [4,32]. Although computationally convenient, distributed algorithms have one main flaw: the information that each agent must know about the others. Nevertheless, in most cases, it is assumed that the agents have direct access to the decision variables of the other agents that affect their cost function [12,20,28]. This is the so-called fulldecision information setup, where the agents must share information with all the competing agents. From a more realistic point of view however, it is more natural to assume that the agents agree to share information with some trusted neighbors only. In this case, we have the so-called partial-decision information setup [15,27]. In the literature of deterministic GNEPs, there are several algorithms for both the full information setup [10,17,32] and the partial-decision information one [4,5,27]. On the contrary, to the best of our knowledge, the only few algorithms for SGNEPs are in full information [12,28,35]. Among others, one of the fastest and less computationally demanding algorithms that may be exploited is the forward-backward (FB) splitting method [3,Section 26.5], for which a suitable preconditioning is needed to obtain distributed iterations [12,32]. In the stochastic case, the FB algorithm converges [12,30,34] when the operator used for the forward step is strongly monotone [12,30] or cocoercive as we preliminarily show in the full decision information setup [13]. There is another important issue in SGNEPs due to the presence of shared constraints: the monotonicity properties of the involved mappings are not necessarily preserved in the extended primal-dual operators obtained to decouple the shared constraints. Hence, ensuring convergence can be difficult because of the lack of a strongly monotone forward operator, not even when the original pseudogradient mapping is strongly monotone. Instead, cocoercivity can be obtained from a strongly monotone or cocoercive pseudogradient [12,13,32]. Nonetheless, in partial-decision information, the extended operator can only be at most restricted cocoercive with respect to the solution set and only when the pseudogradient mapping is strongly monotone [14,27]. Besides the monotonicity properties, another challenging aspect is the uncertainty. The presence of a random variable with unknown distribution implies that the agents cannot compute the exact cost function but only an approximation [13,12,19,20]. This results in a stochastic error which complicates the analysis and prevents from applying the proofing techniques used in the deterministic case [32,27,14,33]. In this paper we propose the first distributed algorithms specifically tailored for SGNEPs in partial-decision information and show their convergence to an equilibrium under restricted cocoercivity of the stochastic forward operator. Our contributions are summarized next: • We model and study for the first time SGNEPs under partial-decision information. • We propose two distributed algorithms for network games and two for aggregative games. The algorithms are characterized by the specific way we impose consensus on the dual variables, i.e., nodebased or edge-based. While both the approaches are present in the literature of deterministic GNEPs [5,27,33] they have been only partially used in the stochastic case and only in full information [12,13]. • We show that our algorithms are instances of a preconditioned FB splitting and we prove their convergence when the forward operator is restricted cocoercive with respect to the solution set. The restricted cocoercivity assumption is much weaker than the monotonicity assumptions usually adopted in the stochastic literature [30,13].
As a special case, we also consider aggregative games, where the cost function of each agent does not depend explicitly on the individual decision of the other agents but it is related instead to some aggregate value of all the decisions [14,17,21]. Illustrative examples are traffic networks where the time delay depends on the overall congestion [26] and energy markets where the price of electricity depends on the aggregate demand [8]. In this case as well, the literature is not extensive [23,24,25]. While the authors in [23,24] propose a stochastic proximal gradient response for aggregative games, relatively similar to ours, they do not consider shared constraints. Moreover, they assume a strongly monotone mapping to prove convergence. However, when dealing with SGNEPs in partial information, also in the aggregative case, the extended operator can be at most restricted cocoercive; therefore, the results in [23,24] are not applicable to our generalized setting. We remark that, despite our proposed algorithms are all instances of a FB scheme, hence inspired by the literature on the topic [13,12,27,14], this is the first time SGNEPs in partial decision information are addressed from an algorithmic point of view. Moreover, the edgebased approach is loosely studied even in the deterministic case, while here we show that it can be a valid alternative to the more classic node-based algorithm. Paper organization. The next section recalls some preliminary notions on operator and graph theory. SGNEPs in partial-decision information are described in Section 3. The first two algorithms for network games are presented in Section 4 while the aggregative case is discussed in Section 5. Sections 6, 7 and 8 are devoted to the theoretical convergence results. Specifically, in Section 6 a fundamental lemma is proven and then, it is used in Sections 7 and 8 to show that the algorithms for network games and aggregative games, respectively, converge to an equilibrium. Numerical simulations (Section 9) and conclusion (Section 10) end the paper.

Preliminaries and Notation
We use the same notation as in [13] and, with a slight abuse of notation, given two sets A and B, we may indicate the Cartesian product as [ A B ] = A × B, to ease the reading. The definitions are taken from [3,11,16].
for all x, y ∈ dom(F ); maximally monotone if there exists no monotone operator G : C → R n such that the graph of G properly contains the graph of F [3,Def. 20.20]. We use the adjective restricted if a property holds for all (x, y) ∈ dom(F ) × fix(F ). Graph theory. Basic definitions can be found in [16].
It is connected if there is a path between every pair of vertices. Let W be the weighted adjacency matrix, L be the associated Laplacian matrix and V be the incidence matrix. Then, if the graph is undirected W and L are symmetric, i.e., W = W ⊤ and L = L ⊤ . If the graph is also connected, it holds that L = V ⊤ V and that namely, the null space of V and L is the consensus subspace. The laplacian matrix L has an eigenvalue equal zero and let the other eigenvalues ordered as 0 < λ 2 (L) ≤ · · · ≤ λ N (L) where d max ≤ λ N (L) ≤ 2d max and d max = max i∈I {d i }. Given d max , it follows from the Baillon-Haddad Theorem that the Laplacian is 1 2dmax -cocoercive.
3 Stochastic generalized Nash equilibrium problems under partial-decision information

Problem setup
We consider a stochastic generalized Nash equilibrium problem (SGNEP), i.e., the problem of finding a Nash equilibrium when the cost functions are expected value functions and the agents, indexed by I = {1, . . . , N }, are subject to coupling constraints. Each agent i ∈ I has a decision variable x i ∈ Ω i ⊆ R ni and a local cost function defined as for some measurable function f i : R n × R d → R and n = i∈I n i . Each agent i aims at minimizing its local cost function within its feasible strategy set Ω i . The cost function is split in smooth (f i ) and non smooth parts (g i : R ni →R). The latter may also model local constraints via an indicator function (g i (x i ) = ι Ωi (x i )).
Assumption 1 (Local cost functions) For each i ∈ I, the function g i in (2) is lower semicontinuous and convex and dom(g i ) = Ω i is nonempty, compact and convex.
For each agent i, the cost function in (2) depends on the local variable x i , on the decision of the other agents Besides the local constraints x i ∈ Ω i ⊆ R ni , the agents are also subject to coupling constraints Ax ≤ b, therefore, the feasible decision set of each agent i ∈ I is 1 From now on, we use ξ instead of ξ(ω) and E instead of E ξ .
where A i ∈ R m×n and b ∈ R m , and the collective feasible set reads as where A = [A 1 , . . . , A N ] ∈ R m×n , A i ∈ R m×ni and b ∈ R m . We suppose that the constraints are deterministic and satisfy the following assumption [10,28].
Assumption 2 (Constraint qualification) For each i ∈ I, the set Ω i is nonempty, compact and convex. The set X satisfies Slater's constraint qualification.
Given the decision variables of the other agents x −i , the goal of each agent i is to choose a strategy x i that solves its local optimization problem, i.e., By simultaneously solving all the coupled optimization problems, we have a stochastic generalized Nash equilibrium (SGNE).
Definition 3 A stochastic generalized Nash equilibrium is a collective strategy x * ∈ X such that for all i ∈ I To guarantee the existence of a SGNE [28, Section 3.1], we make further assumptions on the cost function, typical of the deterministic setup as well [28,9,10].
Assumption 4 (Cost functions convexity) For every i ∈ I and x −i ∈ X −i the function f i (·, x −i ) is convex and continuously differentiable. For every i ∈ I and for every ξ i ∈ Ξ i , the function f i (·, x −i , ξ i ) is convex, continuously differentiable, and Lipschitz continuous and for each Among all the possible equilibria, we focus on the class named variational equilibria (v-SGNE), i.e., those equilibria that are also solution of a suitable stochastic variational inequality (SVI). To describe this class, let us introduce the pseudogradient mapping where the exchange between the expected value and the gradient is possible because of Assumption 4 [28]. A standard assumption on the pseudogradient in partialdecision information is postulated next [14,27].
Assumption 5 (Strongly monotone pseudogradient) F in (6) is η-strongly monotone and ℓ F -Lipschitz continuous, for some constants η, ℓ F > 0 respectively. As in [10,2,28], the SGNEP can be recasted as the monotone inclusion i.e., as the problem of finding a zero of the setvalued mapping T : The operator in (7) can be obtained via a primal-dual characterization of the equilibria: the i-th component of the first row of T in (7) To overcome the lack of knowledge of the decision that affects its cost, each agent keeps an estimate of the action of the other players [14,27]. Let us denote withx i,j the estimate of the decision of agent j stored by agent i and let us collect all the estimates stored by agent i in . Thus, to compute the equilibria of the game in (5), the agents should reach consensus not only on the dual variables, but also on the estimates, i.e., λ i = λ j andx i =x j for all i, j ∈ I. With this aim, let us introduce the consensus subspace of dimension q ∈ N C q = {y ∈ R N q : y = 1 N ⊗ y, y ∈ R q } and its orthogonal complement C ⊥ q . Then, C n is the consensus subspace of the estimated decisions while C m is the consensus subspace of the dual variables. To compute the v-SGNE of the game in (5), the agents use the estimates, i.e., the pseudogradient mapping is modified according to

Approximation scheme
Since the random variables have an unknown distribution, i.e., the expected values in (6) are virtually impossible to compute, we take an approximation of the pseudogradient mapping. We suppose that the agents have access to an increasing number M k of samples of the random variables ξ i and that they are able to compute an approximation of ) for all i ∈ I and ξ = col(ξ 1 , . . . ,ξ n ) is an i.i.d. sequence of random variables drawn from P. Approximations of the form (9), using a finite (increasing) number of samples, are known as stochastic approximations (SA) and they are very common in Monte-Carlo simulation, machine learning and computational statistics [19]. From now on, we indicate with the superscript SA the operators where the mapping F is sampled with F SA as in (9).
From the last assumption it follows that 1/M k is summable, which is standard when using a SA scheme, especially in combination with the forthcoming variance reduction assumption (Assumption 8) [19]. Since we use an approximation, for k ≥ 0, let us introduce the stochastic error Let us define the filtration Then, standard assumptions for the stochastic error are to have zero mean and bounded variance [19,20].

Assumption 8 (Zero mean and bounded variance) For
Remark 3 Under Assumptions 7 and 8, it holds that for where M k is the batch size sequence used in the approximation (9) (see [19,Lem. 3.12], [13,Lem. 6] for a proof ).
Since (12) implies that the second moment of the error diminishes with the number of samples M k , algorithms using the approximation in (9) are also known as variancereduced methods [19].

Stochastic preconditioned forward-backward algorithms for network games
In this section, we present two distributed algorithms for network games. We suppose that each agent i ∈ I only knows its own cost function J i , its feasible set Ω i , and its own portion of the coupling constraints A i and b i . Moreover, through the graph G, the agents have access to some of the variables of the other agents. In Section 7 we show that the algorithms are instances of a preconditioned forward-backward (pFB) algorithm [32,3] and we show how to choose suitable operators to derive them.

Node-based algorithm for network games
We start with the distributed iterations presented in Algorithm 1. Its steps involve: a proximal step to update each decision variable x i ; an updating rule for the estimates that pushesx i toward consensus; the auxiliary variable z i which helps reaching the dual variable consensus [32]; a projection step into the positive orthant for the dual variable λ i .
Algorithm 1 (Node-based fully-distributed preconditioned forward-backward) The algorithm is inspired by the preconditioned FB iterations proposed in [27]. The main difference is that Algorithm 1 is not deterministic, thus, for the update of the primal variable, the approximation of the extended pseudogradient mapping in (8) is used. The algorithm is fully distributed since each agent i knows its own variables and shares its information only with the neighbors in N i . The algorithm is characterized by the choice of the consensus constraint for the dual variables. In this case, exploiting (1), it is imposed as Lλ = 0 where L is the laplacian matrix associated to G. Since we use the laplacian matrix, we call the algorithm node-based. Similarly, it is also imposed the consensus constraint on the estimates, Lx = 0, in the update ofx. We now state our first convergence result.
PROOF. See Section 7.1 where we also provide explicit bounds for the step sizes. ✷

Edge-based algorithm for network games
Let us now describe another instance of the pFB algorithm that differs from Algorithm 1 in the way we impose the consensus constraint on the dual variables. Specifically, following (1), we impose the constraint V λ = 0.
The details on how we exploit this edge-based constraint, i.e., using the incidence matrix, are presented in Section 7.2 while the iterations are presented in Algorithm 2.
Algorithm 2 (Edge-based fully-distributed preconditioned forward-backward) A consequence of the edge-based constraint is that only one communication round is required because each λ i depends only on local variables. The updating rule of x i andx i , instead, are the same as in Algorithm 1 because they are not affected by the dual variable constraint. The use of the incidence matrix is not common, even in the deterministic case. Similar iterations has been considered in [33] which however proposes a deterministic asynchronous Krasnoselskii-Mann iteration (see also Section 7.2). We can state the convergence result for Algorithm 2.
PROOF. See Section 7.2 where we also provide explicit bounds for the step sizes. ✷

Stochastic aggregative games
With aggregative games we mean a class of games where the cost function explicitly depends on the aggregate decision of all the agents. Formally, given the actions of all the players x i ∈ R ni where n i =n for all i ∈ I, let be the average strategy. Then, the cost function of each agent i ∈ I in the aggregative case can be written as Notice that in this case, since n i =n for all i ∈ I, avg(x) ∈ Rn and n =nN . Since this is a particular case of the classical SGNEP in (5), existence and uniqueness of an equilibrium hold under the same assumptions and the v-SGNE can be characterized using the KKT conditions in (7). Accordingly, Algorithm 1 and Algorithm 2 can be used to reach an equilibrium. However, the previous algorithms require the agents to exchange the estimates of all the other actions, i.e., a vector of dimension (N − 1)n, while the aggregate value has dimensionn (independent of the number of agents). To reduce the computational complexity, we propose two algorithms, depending on the consensus constraint, tailored for aggregative games. Let us introduce the pseudogradient mapping for the aggregative case as F a (x, The variable y indicates the dependency on the aggregate value. In fact, Remark 4 It follows from Assumptions 5 and Remark 2 that F a in (13) is Lipschitz continuous in both the arguments with constants ℓ x a , ℓ u a > 0, respectively.
Due to the partial-decision information setup, the agents cannot compute the exact average strategy. To overcome this problem, each agent updates an auxiliary variable s i = avg(x) − x i ∈ Rn [5]. The variable s = col(s i ) i∈I is used to reconstruct the true aggregate value, controlling only the information received from the neighboring agents. Specifically, it should hold that s k → 1 N ⊗ avg(x k ) − x k asymptotically. Moreover, let and u := col((u i ) i∈I ). The variable u k represents the approximated average through the iterations. We remark that the explicit tracking of u k is not necessary in our algorithms since, to estimate di aggregative value, we update iteratively the variable s k . Moreover, from the updating rule of s k , it follows that avg(s k ) = 0n. Thus, provided that the algorithm is initialized appropriately, i.e., s 0 i = 0n, for all i ∈ I, an invariance property holds for the approximated average, i.e., for all k ∈ N,

Node-based algorithm for aggregative games
We first consider the node-based consensus constraint introduced in Section 4.1. Since in this case we have to take into consideration also the aggregative value, the state variable is ω = col(x, s, z, λ), where x is the exact decision variable, s is the tracking variable, z is the auxiliary variable for consensus of the dual variables and λ is the dual variable.
Algorithm 3 (Node-based fully-distributed preconditioned forward-backward for aggregative games) Initialization: x 0 i ∈ Ω i , λ 0 i ∈ R m ≥0 , and z 0 i ∈ R m . Iteration k: Agent i (1) Receives x k j , s k j and λ k j for j ∈ N i , then updates (2) Receives z k+1 j for j ∈ N i , then updates Compared to Algorithm 1, besides the presence of the variable s, Algorithm 3 has a different updating rule for x k , that now includes the estimated aggregative value x k + s k . The remaining variables (z k and λ k ) are not influenced by the average strategy, therefore the updating rules are the same as in Algorithm 1. The operators used to obtain the iterations in Algorithm 3 are given in Section 8.1 but here we state the convergence result.
PROOF. See Section 8.1 where we also provide explicit bounds for the step sizes. ✷

Remark 5
In [14], the authors propose an algorithm in which the agents keep track of the whole aggregative value u (instead of s). Specifically, the aggregative value is updated as where Ln := L ⊗ In. The rule in (16) can be derived from the updating rule of s k in Algorithm 3 or 4 and its definition in (14) and it can be regarded as a dynamic tracking of the time-varying quantity avg(x) [14,21]. The algorithm in [14] is still an instance of a pFB but the operators and preconditioning matrix are different from ours. See Section 8 for further technical details.

Edge-based algorithm for aggregative games
In this section, we consider an edge-based pFB algorithm, similarly to Section 4.2. The iterations in the aggregative case read as in Algorithm 4.
Algorithm 4 (Edge-based fully-distributed preconditioned forward-backward for aggregative games) Initialization: , and z 0 i = 0. Iteration k: Agent i (1) Receives x k j , s k j and λ k j for j ∈ N i , then updates We note that the updating rule for x i and for the auxiliary variable s i are the same as in Algorithm 3 while the difference is in the auxiliary variable z i and the dual variable λ i that now depends only on local variables. This follows using the edge-based consensus constraint as in Section 4.2. More details on how to obtain the iterations and the proof of the following result can be found in Section 8.2.
PROOF. See Section 8.2 where we also provide explicit bounds for the step sizes. ✷

Convergence analysis: A fundamental lemma
In this section we show that the classic pFB splitting converges a.s. in the stochastic case to the zeros of the operator T in (7) when the forward operator is restricted cocoercive. First of all, let us rewrite the operator T into the summation of the two operators Then, finding a solution of the SGNEP translates in finding a pair (x * , λ * ) ∈ X × R m ≥0 such that (x * , λ * ) ∈ zer(A + B). The zeros of the mapping T = A + B can be obtained through a FB splitting [3, Section 26.5], [4,32], that for any matrix Φ ≻ 0, leads to the FB iteration: where ω k = col(x k , λ k ), (Id + Φ −1 B) −1 is the backward step and (Id − Φ −1 A) is the forward step. We note that the convergence of (18) is independent on the specific choice of the operators A and B as long as some monotonicity conditions are satisfied. For this reason, we postulate the following assumption.

Assumption 9
The forward operator is restricted βcocoercive for some β > 0 and the backward operator is maximally monotone.
Remark 6 An affine map F (x) = Ax+b with A ∈ R n×n symmetric and positive semidefinite is cocoercive [11,Page 79]. More generally, every η-strongly monotone, ℓ-Lipschitz continuous function is η ℓ 2 -cocoercive, but the vice-versa is not true in general.
Checking if Assumption 9 holds for suitable operators will be the key to prove convergence of the algorithms presented in the previous sections. Before stating the convergence result of the pFB iteration, however we need to postulate some further assumptions and to consider the approximation scheme. In fact, since the random variable has an unknown distribution, we replace A with A SA , the operator obtained using the approximation in (9). Thus, the pFB iteration reads as We note that there is no uncertainty in the constraints, therefore, the corresponding error of the approximated extended operator is To guarantee convergence, the preconditioning matrix Φ should be positive definite. Since this property may de-pend on the specific choice of the matrix Φ, here we postulate it as an assumption and in the following sections we ensure that it holds for the proposed algorithms.
Moreover, to guarantee convergence and independently on the choice of Φ, the step sizes should be bounded.
Assumption 11 Φ −1 < 2β where β is the cocoercivity constant of the forward operator as in Assumption 9.
We can now state and prove the convergence result for the iteration in (19).
PROOF. For brevity, we letÂ = A SA . We start by using that the resolvent is firmly nonexpansive [3, Cor. 23.9] and that if ω * is a solution then it is a fixed point of the FB iteration in (18): Choosing ζ > 1 and such that Assumption 10 is satisfied, we can use Young's inequality to obtain (20) Then, by using restricted cocoercivity of A and including (20), we obtain: Φ and let us recall that res Φ (ω) = 0 if and only if ω is a solution [11,Proposition 1.5.8]. It holds that where the first equality follows by adding and subtracting ω k+1 and using its definition and the last inequality follows from nonexpansivity. Then, Finally, equation (21) becomes By Assumption 10 and by monotonicity, the second last term is smaller or equal than zero, hence, by taking the expected value and by using Assumption 8 and Remark 3 we have that Using Robbins-Siegmund Lemma [29] we conclude that (ω k ) k∈N is bounded and has a cluster pointω. Moreover, it follows that k 1 2 1 ζ − 1 res Φ (ω k ) 2 < ∞, hence, res(ω k ) → 0 as k → ∞ and res(ω) = 0. ✷ Remark 7 We note that the operators A and B is (17) satisfy Assumption 9 [4, Lemma 1] and that the matrix Lemma 3], therefore, the pFB algorithm in (19) obtained with these operators converges to a v-SGNE of the game in (5). However, expanding (19), the iterations that we obtain require full information on the decision of the other agents [4].

Convergence analysis for network games
We now show how to obtain suitable forward and backward operators that lead to Algorithms 1 and 2, presented in Section 4. Later, we show that such operators satisfy the assumptions of Lemma 1, i.e., that the algorithms converge a.s. to a v-SGNE of the game in (5).
Let us first introduce some notation. Similarly to [27], let us define, for all i ∈ I, the matrices 0n <i ×n i 0n <i ×n >i 0n >i ×n <i 0n >i ×n i In >i (22) where n <i := j<i,j∈I n j , n >i := j>i,j∈I n j . The two matrices in (22) can be interpret as follows: R i selects the i-th n i dimensional component from an n-dimensional vector, while S i removes it. Namely, We can now analyze the two algorithms separately.

Convergence of Algorithm 1
Let L ∈ R N ×N be the Laplacian of the communication graph G and let L m = L ⊗ Id m ∈ R N m×N m and L n = L ⊗ Id n ∈ R N n×N n . Let also A = diag{A 1 , . . . , A N } ∈ R N m×n and λ = col(λ 1 , . . . , λ N ) ∈ R N m ; similarly let b ∈ R N m . As already mentioned in Section 4.1, exploiting (1), to impose consensus on the primal and dual variables, one can consider the Laplacian constraints Lx = 0 and Lλ = 0 [32,27]. To include these constraints, similarly to [27], let us define the operators . We note that we consider the estimatesx as a state variable and we use the matrix R to select the variables corresponding to each agent. Compared to the operators in [27], we have the expected valued extended pseudogradient in (8) that, in the iterative process, is replaced by a stochastic approximation of the form in (9). The term L nx is a measure of the disagreement between the decision variables of the agents and the estimates. Each agent can use this term to move towards consensus of the estimates while it uses the gradient to minimize the cost. We note that in a full information setup, the disagreement term can be removed (setting c = 0) [27]. Given the operators in (23), the pFB algorithm in compact form reads as (24) where the operator A SA p is the operator A p with the approximated pseudogradient mapping F SA as in (9) and ω k = col(x k , z k , λ k ). To obtain the distributed iterations in Algorithm 1, a suitable preconditioning matrix should be taken [27]. Specifically, let where α = diag{α 1 Id n1 , . . . , α N Id nN } ∈ R n×n , and similarly ν and δ are block diagonal matrices collecting the step sizes. Then, expanding the iterations in (24) with A p and B p as in (23) and Φ p as in (25), we obtain , that is, the update for each agent decision variable x i as in Algorithm 1. Instead, if we premultiply by S we obtain Sx k+1 = Sx k − αcSLnx, i.e., the update of the estimates. For the sake of the convergence analysis, we have to guarantee that the preconditioning matrix Φ p is positive definite (Assumption 10) and bounded in norm (Assumption 11), therefore, we take some bounds on the step sizes [27].
Assumption 12 For a given τ > 0, the step sizesᾱ,ν andδ are such that, for all i ∈ I, i ] jk indicates the entry (j, k) of the matrix A ⊤ i , and such that Φ p satisfies Assumption 11.
Then, it follows from the Gershgorin Theorem and [27, Lemma 5] that Φ p ≻ 0. We can now prove the convergence result.
PROOF. [Proof of Theorem 1] First, we show that the zeros of A p + B p correspond to a v-SGNE of the game in (5). Expanding the inclusion ω = col(x * , z * , λ * ) ∈ zer(A p + B p ) we obtain Then, from the second line of (27) it follows that λ * ∈ null(L m ), i.e., λ * = 1 N ⊗ λ * by (1). Similarly to [27,Theorem 1], from the first line of (27) it follows that x * ∈ null(L n ), i.e.,x * = 1 N ⊗x * and that the first KKT condition in (7) is satisfied. From the third line, we obtain the second KKT condition in (7) [27, Theorem 1]. Moreover, it also holds that zer(A p +B p ) = ∅. In fact, by Remark 1, there exists a unique solution x * and, therefore, there exists λ * such that the KKT conditions in (7) are satisfied [11, Proposition 1.2.1] and (x * , λ * ) ∈ zer(T ). The first two lines of (27) are satisfied and using (7) we can prove that there exists z such that the third line is satisfied as well [27,Theorem 1]. We now show that the operators in (23) have the properties in Assumption 9. To this aim, we define some preliminary quantities [27,Lemma 3]. Let and let µ p = λ min (Υ p ) be its smaller eigenvalue. Let c > c min , where c min λ 2 (L) = (ℓp+ℓ F ) 2 4η + ℓ p . Let us indicate with Z = C n × R N m × R N m the set where there is consensus on the first component, i.e., on the primal variable. First, let us recall that from Assumption 12 it follows that Φ ≻ 0 [27, Lemma 5]. Then, the fact that A p is β p -restricted cocoercive with respect to Z, with constant β p ∈ (0, µp θ 2 p ], where µ p = λ min (Υ p ) and θ p = ℓ p +2cd max follows similarly to [27,Lemma 3 and 4]. [27,Lemma 6]. Concerning B p , it is monotone similarly to [27,Lemma 4]. Consequently, Φ −1 p B p is maximally monotone in the Φ p -induced norm [27,Lemma 6]. Since by Assumption 12, Assumptions 9 -11 hold, the pFB iterations presented in Algorithm 2 converge to a v-SGNE of the game in (5) by Lemma 1. ✷

Convergence of Algorithm 2
Let us now focus on how to obtain Algorithm 2. Let us consider the incidence matrix V of the communication graph G. Then, another possibility to force consensus on the dual variables, according to (1), is to consider the constraint V λ = 0 (instead of Lλ = 0). Exploiting this constraint, we define the two operators where v = col((v l ) l∈{1,...,E} ) ∈ R Em , L n := L ⊗ I n and V m := V ⊗ Id m . We note that the variable v = col((v l ) l∈{1,...,E} ) ∈ R Em is used to help reaching consensus on the dual variables. Moreover, it can be interpreted as the network flow. In fact, if we consider A i x i as in-flow and b i as out-flow for each node i ∈ I, then Ax = b can be read as a conservative flow balancing constraint. Therefore, v l can be seen as flow on each edge l to ensure such constraint. In other words, the variable v estimates the contribution of the other agents to the coupling constraints, and ensures that the dual variables reach consensus. Since we consider the incidence matrix V instead of the Laplacian L, the preconditioning matrix is given by where α, ν and δ are defined analogously to (25). Then, by replacing the operator C p with C SA p , i.e. the operator approximated via the SA scheme according to (9), the pFB iteration reads as where ω k = col(x k , v k , λ k ). Then, by expanding (30), First, we note that also in this case we separate the update of the local decision variables x k i and of the estimatesx k i,−i . Moreover, in (31), two communication rounds are required: one at the beginning of each iteration k to update x k+1 and v k+1 and one before updating λ k+1 . To avoid this second round, let us introduce, with a little abuse of notation, the variable z = col((z i ) i∈I ) such that z k = V ⊤ m v k , for all k ≥ 0. Given an appropriate initialization, e.g., v 0 = 0, the following equivalences hold: After this change of variables, the iterations in (31) can be rewritten as in Algorithm 2. Moreover, since the iterations in (30) are equivalent to Algorithm 2, for the analysis we use the operator in (28). Let us also note that the operators are similar to [33] but, due to the change of variable and the approximation scheme, Algorithm 2 is different from the asynchronous one proposed in [33] and requires less communications. We now proceed in showing that Algorithm 2 converges to an equilibrium. First, to obtain a positive definite preconditioning matrix (using Gershgorin Theorem), let us bound the step sizes.
Assumption 13 Given τ > 0, the step sizes sequence is such thatᾱ is as in Assumption 12 andν andδ are such that, for all i ∈ I i ] jk indicates the entry (j, k) of the matrix A ⊤ i and such that Ψ p satisfies Assumption 11.
PROOF. [Proof of Theorem 2] First, we relate the unique v-SGNE of the game in (5) to the zeros of C p + D p in (28). Namely, given anyω * := col(x * , v * , λ * ) ∈ zer(C p + D p ), it holds thatx * = 1 N ⊗ x * and λ * = 1 N ⊗ λ * , where the pair (x * , λ * ) satisfies the KKT conditions (7), i.e., x * is a v-SGNE of the game in (5). This follows analogously to Theorem 1 noting that, expandingω ∈ zer(C p + D p ), from the second line we have λ ∈ null(V ) and we can premultiply by (1 ⊤ E ⊗ Id m ) the third line to obtain the KKT conditions in (7). Similarly, zer(C p + D p ) = ∅. Then, we show the monotonicity properties of the operators. Note that C p is the same as A p therefore it is β p -restricted cocoercive with respect to Z, where β p ∈ (0, µp θ 2 p ], µ = λ min (Υ p ) and θ p = ℓ p + 2cd max by Theorem The operator D p is maximally monotone analogously to the proof of Theorem 1. It follows that Ψ −1 p D p is maximally monotone in the Ψ p -induced norm. Convergence follows by Lemma 1. ✷

Convergence analysis for aggregative games
Analogously to Section 7, we show in this section that the algorithms proposed for the aggregative case in Section 5 converge a.s. to a v-SGNE of the game in (5). We recall that we keep track of the aggregate value through the variable s = 1 N ⊗ avg(x) − x and that the approximated average is given by u = x + s as in (14).

Convergence of Algorithm 3
Let us start by defining the operators that leads to the iterations in Algorithm 3. Specifically, the forward and backward operators should be defined according to where Ln = L ⊗ Idn, L m = L ⊗ Id m . We note that, compared to the operators A p and B p in (23), in (32) instead of the estimates we can take the true decision variables x as a state variable because we track the average value through the variable s. The preconditioning matrix reads similarly to Φ p in (25) with the addition of a line (corresponding to the variable s): where α −1 = diag{α −1 1 Idn, . . . , α −1 N Idn} ∈ R n×n and similarly γ −1 , ν −1 and δ −1 are block diagonal matrices of suitable dimensions. Given the operators in (32) and the preconditioning matrix in (33), Algorithm 3 in compact form reads as the pFB iteration is the forward step where the pseudogradient mapping is approximated according to (9).

Remark 8
In line with Remark 5, let us note that in [14] a different splitting and preconditioning are used. Since the paper tracks the aggregate value u (instead of s), to prove convergence, it uses an auxiliary iterative scheme on the orthogonal complement of the consensus subspace [14, Lemma 2]. Hence, the extended operators and preconditioning matrix depend on the projection on the consensus subspace (P ) and on its orthogonal complement (P ⊥ ), while ours are a generalization of the operators in (23) and (25) to the aggregative case. We can avoid the auxiliary iteration because we track the aggregative value with s instead of measuring the disagreement with u ⊥ . We also note that, in general, s = u ⊥ . Now we prove that using the operators A a and B a in (32) we can reach a v-SGNE of the game in (5). To this aim, we restrict our analysis to the invariant subspace Σ := {(x, s, z, λ) ∈ R 2n+2N m | avg(s) = 0n}, (34) i.e., the space where the agents are able to compute the exact aggregative value. Moreover, to ensure that A a and B a have the monotonicity properties of Lemma 1, the preconditioning matrix should be positive definite. For this reason, we take some bounds on the step sizes.
We are ready to prove convergence.

≥0
(1 N ⊗ λ * ), Since range(L m ) = null(1 ⊤ N ⊗ I m ) = C ⊥ m , there exists z * such that also the last line of (35) is satisfied, i.e., ω * ∈ zer(A a + B a ) ∩ Σ. We now prove the monotonicity properties of the operators A a and B a . Similarly to the proof of Theorem 1 and [14, Lemma 4], let us introduce where η is the strong monotonicity constant as in Assumption 5 and ℓ u a the Lipschitz constant as in Remark 4. Then, the two operators A a and B a in (32) have the properties of Assumption 9. To prove this let us note that each vector can be decomposed as u = u + u ⊥ where u = P u ∈ Cn and u ⊥ = P ⊥ u ∈ C ⊥ n and P and P ⊥ are the projection operators defined as P = 1 N 1 N ⊗ 1 ⊤ N ⊗ Idn and P ⊥ = Id N n −P [14]. We note that P u k = 1 N ⊗ avg(u k ) = P x k by the property in (15). First, we prove that A a is β a -restricted cocoercive with constant 0 < β a ≤ µa θa , where µ a = λ min (Υ a ) and θ a = max{(2ℓ x a ) 2 , (2ℓ u a ) 2 + 3λ 2 (L)}. To this aim, we first prove that the operator Lnu is restricted strongly monotone, then restricted cocoercivity of A a follows. Let w = col(x, s), then [14,Lemma 4]. Notice that u − avg(x) = u + u ⊥ − avg(x) = u ⊥ by the invariance property (15). Therefore, similarly to [14,Lemma 4], To prove cocoercivity, we use an argument similar to [14,Lemma 4] to obtain The constant θ a is defined as θ a = max{(2ℓ x a ) 2 , (2ℓ u a ) 2 + 3λ 2 (L)}. Pairing (36) and (37), we have thatĀ a is µa θacocoercive with µ a = λ min (Υ a ). The fact that Φ −1 A a is restricted cocoercive follows from A a being restricted cocoercive and the maximal monotonicity of B a and Φ −1 B a follows analogously to Theorem 1. Since Assumptions 9-11 are verified, convergence follows from Lemma 1. ✷

Convergence of Algorithm 4
We now consider Algorithm 4 and show how to obtain its iterates. Later we prove Theorem 4. Including the variable s to track the aggregative value and considering the consensus constraint V λ = 0, the operators of the edge-based algorithm read as D a : while the preconditioning matrix is given by where V m = V ⊗ Id m and α, γ, ν and δ are the block diagonal step sizes. Similarly to the previous sections, the pFB iteration is given by where C SA a is approximated according to (9). The iterations in Algorithm 4 can be obtained expanding (40) and using a change of variables z k = V ⊤ m v k , as in Section 4.2.
To prove convergence of Algorithm 4, we consider the invariant subspace Σ := {(x, s, v, λ) ∈ R 2n+Em+N m | avg(s) = 0n}, similarly to Section 5.1. We note that the dimension of this set is different from (34). Before stating the result, analogously to the previous sections, we must guarantee that the preconditioning matrix is positive definite.

Assumption 15
Given τ > 0, the step sizes are such that γ > 0,ᾱ is as in Assumption 12,ν andδ are such that, for all i ∈ I, i ] jk indicates the entry (j, k) of the matrix A ⊤ i and such that Ψ a satisfies Assumption 11.
Then, the convergence result holds.
To this aim, let us consider any ω * ∈ zer(C a + D a ) ∩ Σ and let u * = x * + s * ; then we have The fact that the KKT conditions in (7) are satisfied, i.e., that x * is a v-SGNE of the game in (5), follows analogously to Theorem 3. Moreover, it holds that zer(C a + D a ) ∩ Σ = ∅, similarly to Theorem 3 and using the fact that range(V ⊤ q ) ⊇ range(L q ) = null(1 ⊤ N ⊗ I q ) = E ⊥ q . The fact that C a is β a -restricted cocoercive where 0 < β a ≤ µa θa , µ a = λ min (Υ a ) and θ a = max{(2ℓ x a ) 2 , (2ℓ u a ) 2 + 3λ 2 (L)} and that the operator D a is maximally monotone and that Ψ −1 a C a is β a δ a -cocoercive with δ a = 1 |Φ −1 a | Fig. 1. Distance from a solution (network game). and that Ψ −1 a D a is maximally monotone in the Φ ainduced norm follows analogously to Theorem 3. Convergence follows by Lemma 1. ✷

Numerical simulations
Let us present some numerical results to validate the convergence analysis of our proposed algorithms. For network games, we consider a Nash-Cournot game [21,27,35] while for the aggregative case we use a charging scheduling problem [17]. In both cases we consider two different topologies for the communication graph, i.e., a complete graph and a cycle graph, to show how connectivity affects the results. For the tests, we take the step sizes to be half of those that generate instability.

Nash-Cournot games
In a Nash-Cournot game [21,27,35] a set I = {1, . . . , N } of N = 20 companies (agents) produce a commodity to  the node-base algorithm for the cycle graph is the slowest in terms of both number of iterations ( Figure 3) and computational time (Figure 4).

Conclusion
The preconditioned forward-backward (pFB) algorithm can be used to find stochastic generalized Nash equilibria in a partial-decision information setup. Leveraging on the estimation of the unknown variables, the pFB algorithm can be tailored for network games and for aggregative games. Thanks to the preconditioning almost sure convergence holds under restricted cocoercivity of the forward operator with respect to the solution set.