Generalized Nash Equilibrium Problems with Mixed-Integer Variables

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido-Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.


Introduction
The generalized Nash equilibrium problem constitutes a fundamental class of noncooperative games with applications in economics [7], transport systems [2] and electricity markets [1]. The differentiating feature of GNEPs compared to classical games in strategic form is the flexibility to model dependencies among the strategy spaces of players, that is, the individual strategy space of every player depends on the strategies chosen by the rival players. Examples in which this aspect is crucial appear for instance in market games where discrete goods are traded and the buyers have hard spending budgets: effectively, the strategy space of a buyer depends on the market price (set by the seller) as only those bundles of goods remain affordable that fit into the budget. Other examples appear in transportation systems, where joint capacities (e.g. road-, production-or storage capacity) constrain the strategy space of a player. For further applications of the GNEP and an overview of the general theory, we refer to the excellent survey articles of Fachinei and Kanzow [15] and Fischer et al. [17].
While the GNEP is a research topic with constantly increasing interest, the majority of work is concerned with the continuous and convex GNEP, i.e., instances of the GNEP where the strategy sets of players are convex or at least connected and the cost functions are continuous. Our focus in this paper is to derive insights into non-convex or discrete GNEPs including GNEPs with mixed-integer variables. Our main approach is to reformulate the GNEP via a convexification approach and then to identify expressive subclasses of GNEPs which can be reformulated as standard optimization problems.
Let us introduce the model formally and first recap the standard pure Nash equilibrium problem (NEP). For an integer k ∈ N, let [k] := {1, . . . , k}. Let N = [n] be a finite set of players. Each player i ∈ N controls the variables x i ∈ X i ⊂ R k i . We call x = (x 1 , . . . , x n ) with x i ∈ X i for all i ∈ N a strategy profile and X = X 1 × · · · × X n ⊆ R k the strategy space, where k := n i=1 k i . We use standard game theory notation; for a strategy profile x ∈ X, we write x = (x i , x −i ) meaning that x i is the strategy that player i plays in x and x −i is the partial strategy profile of all players except i. The private cost of player i ∈ N in strategy profile x ∈ X is defined by a function π i : X → R with x → π i (x) for any x ∈ X. A (pure) Nash equilibrium is a strategy profile x * ∈ X with π i (x * ) ≤ π i (y i , x * −i ) for all y i ∈ X i , i ∈ N. The GNEP generalizes the model by allowing that the strategy sets of every player may depend on the rival players' strategies. More precisely, for any x −i ∈ R k −i (using the notation k −i := n j =i k j ), there is a feasible strategy set X i (x −i ) ⊆ R k i . In this regard, one can think of the strategy space of player i ∈ N represented by a set valued mapping X i : R k −i ⇒ R k i . This leads to the notation of the combined strategy space represented by a mapping X : R k ⇒ R k with y ∈ X(x) ⇔ y i ∈ X i (x −i ) for all i ∈ N . The private cost function is given by π i : R k → R for every player i ∈ N . The problem of player i ∈ N -given the rival's strategies x −i -is to solve the following minimization problem: A generalized Nash equilibrium (GNE) is a strategy profile x * ∈ X(x * ) with We can compactly represent a GNEP by the tuple I = (N, (X i (·)) i∈N , (π i ) i∈N ). In the sequel of this paper, we will heavily use the Nikaido-Isoda function (short: NI-function), see [26].
Definition 1 (NI-Function). Let an instance I = (N, (X i (·)) i∈N , (π i ) i∈N ) of a GNEP be given. For any two vectors x, y ∈ R k , the NI-function is defined as: By definingV (x) := sup y∈X(x) Ψ(x, y) we can recap the following well-known characterization of a generalized Nash equilibrium, see for instance Facchinei and Kanzow [14]. Theorem 1. For an instance I of the GNEP the following statements are equivalent.
1. x is a generalized Nash equilibrium for I.
This characterization does not rely on any convexity assumptions on the strategy spaces nor on the private cost functions of the players. Yet, the characterization seems computationally of limited interest as neither the Nikaido-Isoda function itself nor the fixed-point condition x ∈ X(x) seems computationally tractable.

Our Results
Our approach relies on a convexification technique applied to the original non-convex game leading to a new characterization of the existence of Nash equilibria for GNEPs. In particular, we convexify the strategy space of every player using the convex hull conv(X i (x −i )) and we replace the private cost function π i (x i , x −i ) by its convex envelope φ . This way, we obtain from an instance I a new convexified instance I conv . Under mild assumptions on the private cost functions, our main result (Theorem 2) states that a strategy profile x ∈ X(x) is a GNE for I if and only if it is a GNE for I conv and the convexified cost functions coincide with the original ones. The proof is based on using the Nikaido-Isoda functions for both games I conv and I. While the convexified GNEP may admit an equilibrium under certain circumstances, this equilibrium might still not be feasible for the original non-convex game. The advantage of our convex reformulation, however, lies in the possibility that for some problems, it is computationally tractable to solve the convexified version while preserving feasibility with respect to the original game. In this regard, we study three subclasses of GNEPs for which this methodology applies.
In Section 3, we consider quasi-linear GNEPs in which the cost functions of players are quasilinear and the convex hull of the players' strategy spaces are quasi-polyhedral sets, that is, for fixed strategies of the other players, the private cost functions are linear and convex hulls of strategy spaces are polyeder. Under these assumptions, we show in Theorem 3 that minimizing the convexified NI-function over the convexified strategy space can be modeled as a standard (non-linear) optimization problem.
We then consider in Section 4 jointly constrained GNEPs which are sometimes called GNEPs with shared constraints. In these type of GNEPs, the players' strategy sets are restricted via a shared feasible set X ⊂ R k . Note that we do not impose a priori convexity assumptions for X, i.e., the set is for example allowed to be discrete. This setting is a generalization of the jointly convex GNEP, where X needs to be convex. We discuss different modifications of our convexification technique such that I conv belongs to the class of jointly convex GNEPs and still falls under our main Theorem 2. We introduce a subclass of the jointly constrained GNEP which we call projectiveclosed. A projective-closed set X requires that for fixed x −i , the projection of conv(X) onto R k i is equal to the convex hull of the projection of X onto R k i for fixed x −i . This property allows for a computationally tractable adaptation of our convexification technique yielding a jointly convex GNEP. The required property of the shared feasible set is for example fulfilled for all {0, 1} k games and thus admits interesting applications.
In Section 5, we discuss a special class of GNEPs which we call quasi-separable. Here, the n players' individual minimization problems are almost separable, that is, only a neutral n + 1-th player may have an impact on the players' cost functions with the goal to enforce a certain property of the strategy profile of the n players. We show that in this case, the set of GNE correspond to certain optima of a convex optimization problem.
In Section 6 we present numerical results on the computation of equilibria for a class of discrete flow games with shared constraint which are shown to belong to projective-closed and quasi-linear GNEPs. To find equilibria of an instance I, the challenge is to compute an integral GNE of its jointly convex instance I conv . Based on known methods for solving jointly convex GNEPs, we implemented different procedures from the literature for solving a jointly convex GNEP which are then enhanced by a simple rounding procedure in order to obtain an integral equilibrium. We then compared these procedures to a solution approach, where our quasi-linear reformulation is plugged into a standard non-convex solver (BARON). Perhaps surprisingly, it turned out that our approach was not only faster (on average) in finding specifically integral GNE for the original non-convex GNEP but also for computing (not necessarily integral) GNE for the convexified instances.

Related Work
Continuous and Convex GNEPs. GNEPs have been studied intensively in terms of equilibrium existence and numerical algorithms. It is fair to say, that the majority of works focus on the continuous and convex case, that is, the utility functions of players are convex (or at least continuous) and the strategy spaces are convex. The main reason for these restrictive assumptions lies in the lack of tools to prove existence of equilibria. Indeed, most existence results rely on an application of Kakutani's fixed point theorem which in turn requires those convexity assumptions (e.g. Rosen [27]). We refer to the survey articles of Fachinei and Kanzow [15] and Fischer et al. [17] for an overview of the general theory.
Based on reducing the GNEP to the standard NEP, Facchinei and Sagratella [16] described an algorithm to compute all solutions of a jointly convex GNEP, where the joint restrictions are given by linear equality constraints. However, this algorithm does not terminate in finite time whenever there are infinitely many equilibria. Dreves [8,9] tackled this problem via an algorithm which computes in finite time the whole solution set of linear (not necessarily jointly convex) GNEPs, i.e. GNEPs where the cost functions are linear and the strategy sets are described by linear functions. Returning to the jointly convex GNEP, Heusinger and Kanzow [33] presented an optimization reformulation using the Nikaido-Isoda function, assuming that the cost functions π i (x i , x −i ) of the players are (at least) continuous in x and convex in x i . Under the same assumptions concerning the cost functions, Dreves, Kanzow and Stein [11] generalized this approach to player-convex GNEPs, where additionally to the assumptions on the cost functions, the strategy sets are assumed to be described by for a restriction function g i which is (at least) continuous in x and convex in x i . In comparison to this optimization reformulation, Dreves et al. [10] took a different approach to finding equilibria via the KKT conditions of the GNEP. Under sufficient regularity, e.g. C 2 cost-and restriction functions, they discuss how the KKT system of the GNEP may be solved in order to find generalized Nash equilibria.
While the assumptions concerning the cost-and restriction functions in the above papers [33,11,10] are mild in the context of continuous GNEPs, they are rather restrictive when it comes to solving our convexification I conv . This is due to the fact, that for discrete sets X i (x −i ), the convex envelopes are typically not C 1 functions and more crucially, the players' strategy sets conv(X i (x −i )) for I conv will in general not admit a smooth behaviour and typically jump under variation of x −i as soon as a point leaves or joins the set X i (x −i ). Therefore, in general, the techniques described in these papers are not applicable to I conv right away.
Non-Convex and Discrete GNEPs. In contrast to the continuous/convex case, the existence and computability of equilibria for non-convex and discrete GNEPs are not well understood, yet, they are extremely important for modeling real-world systems. The analysis of markets with discrete goods and budget constrained buyers are among the core topics in economics. These models involve discrete decision variables (bundles of goods) and, thus, violate the convexity assumption. In electricity markets, standard dispatch models involve binary decision variables and thus also lead to non-convex GNEPs. In transportation systems, the design of tolling schemes so as to enforce routing patterns obeying predefined emission bounds also leads to a GNEP. If a traffic routing model is based on a discrete formulation (as is the case for most simulation-based software) we arrive at a non-convex GNEP involving integrality constraints. While these research areas do have a substantial literature regarding the existence and computability of equilibria, these result do not hold for a general GNEP formulation and we refrain from discussing further specific references.
One of the few approaches towards the non-convex and discrete GNEP (resp. NEP) was recently introduced by Sagratella [30] (resp. [29]). For the discrete standard NEP, Sagratella [29] presented a branching method to compute all solutions of the Nash equilibrium problem. Regarding the GNEP, he described in [30] mainly two different techniques to find GNE for the subclass of so called generalized potential games with mixed-integer variables. Similar to the jointly convex GNEPs, in these potential games the players are restricted through a common convex set X with the further restriction that some strategy components need to be integral. Additionally, the cost functions admit a potential function over the set X. On the one hand Sagratella introduced certain optimization problems with mixed-integer variables based on the fact that minimizer of the potential function correspond to a subset of generalized Nash equilibria. On the other hand, he showed that a Gauss-Seidel best-response algorithm may approximate equilibria arbitrary well within a finite amount of steps in this setting. Although the class of generalized potential games is rather restrictive, several interesting models have emerged based upon the results presented by Sagratella as for example in the domain of Automated Driving [13], Traffic Control [5] or Transportation Problems [31].

Convexification
For any instance I = (N, (X i (·)) i∈N , (π i ) i∈N ) of the GNEP, we will now introduce a convexified game by defining I conv = (N, conv(X i (·)) i∈N , (φ i ) i∈N ), where the strategy space of player i for given x −i is defined as the convex hull of X i (x −i ). The cost function φ i (x) of player i is given by where φ , see Horst and Tuy [22, §4.3.]. Outside of conv(X i (x i )), the costs φ i (·, x −i ) are represented by an arbitrary function ξ i (·, x −i ) : R k i → R. Notice that I conv is again an instance of the GNEP. In the following, we impose a mild assumption on the cost functions π i , i ∈ N and the players' strategy spaces which we assume to hold throughout the paper.
It follows by a result of Grotzinger [19] that the (unique) convex envelope exists and reads as: where Λ j := α ∈ R j + | j k=1 α k = 1 for any j ∈ N. The following theorem derives a connection between I and the convexified instance I conv .
For any x ∈ X(x) = n i=1 X i (x −i ), the following assertions are equivalent. 1) x is a generalized Nash equilibrium for I.
2) x is a generalized Nash equilibrium for I conv and φ( Proof. We first show that for every x ∈ X(x) the inequalityV conv (x) ≤V (x) holds, whereV conv (x) is theV function for I conv . Analogously, we define for X(x) w.r.t. the instance I conv the set is obviously separable in y. Therefore the following is true: As conv( holds. This is a well known property of convex envelopes, yet, we prove it in the following to keep the exposition self-contained. The fact that the left side can't be smaller follows immediately by Grotzinger's description in (2) of the convex envelope as it yields for an arbitrary Therefore we get where the last equality follows from the fact that α il ∈ Λ k i +1 . Since y ′ i ∈ conv(X i (x −i )) was arbitrary, we arrive at To see that ≤ holds, we use the fact that for every Therefore the equality in (5) holds. We thus arrive at: (4) and (5) by the same argumentation as for (4) =V (x) Therefore we have the inequalitŷ which allows us to prove the equivalence of 1) and 2). We start with ⇒: Let x ∈ X(x) be a generalized Nash equilibrium of I. Theorem 1 and inequality (8) imply that V (x) = 0 ≥V conv (x). Obviously x ∈ X conv (x) as x ∈ X(x) and by observing thatV conv (x) ≥ 0 for all x ∈ X conv (x) we conclude thatV conv (x) = 0. Summarizing we have a x ∈ X conv (x) withV conv (x) = 0 which is equivalent to x being a generalized Nash equilibrium for I conv by Theorem 1. FurthermoreV conv (x) = 0 =V (x) implies that the inequality in (7) must be tight, i.e.
be a generalized Nash equilibrium of I conv and φ(x) = π(x). Theorem 1 implies thatV conv (x) = 0 while the equality φ(x) = π(x) implies that the inequality in (7) is tight and thereforeV (x) =V conv (x) = 0 holds. Again, Theorem 1 implies that x is a generalized Nash equilibrium for I which finishes our proof.
Remark 2. We remark here that Assumption 1 is only needed for the existence of a unique convex envelope as well as for the equality stated in (6). If an instance I of the GNEP satisfies the latter two properties, then Theorem 2 remains true without Assumption 1.
Theorem 2 allows us to formulate the following characterization of a generalized Nash equilibrium: Corollary 1. For an instance I of the GNEP the following statements are equivalent.
1. x is a generalized Nash equilibrium for I.

Quasi-Linear GNEPs
is of particular interest for an instance I of the GNEP, sinceV conv is bounded from below by zero for any x ∈ X conv (x) and thus every optimal solution x with objective value zero satisfying x ∈ X(x) and φ(x) = π(x) is a generalized Nash equilibrium of I.
In what follows we identify a subclass of the GNEP such that the optimization problem (9) becomes more accessible. The main idea is that the evaluation of the functionV conv at a strategy profile x is substantially easier if it is a linear optimization problem. A sufficient condition for this which is relatively simple to verify is described in the following Definition 3. Roughly speaking, for fixed x −i the player i has a linear cost function as well as a strategy set whose convex hull conv(X i (x −i )) is a polyhedron. The latter property is for example fulfilled whenever the strategy sets of the players consist of only finitely many points. We call an instance I fulfilling these assumptions quasi-linear.
Definition 3. An Instance I = (N, (X i (·)) i∈N , (π i ) i∈N ) of the GNEP is called quasi-linear, if it fulfills for every i ∈ N the following two statements:

There exists a vector-valued function
The following Theorem shows that for quasi-linear GNEPs the optimization problem (9) can be described by a standard (non-linear) optimization problem. This description relies heavily on the aforementioned fact, thatV conv (x) is a linear maximization problem depending on x ∈ X conv (x). Using the corresponding dual minimization problem allows us to then rewrite the optimization problem (9) as one combined minimization problem.
Theorem 3. Let I = (N, (X i (·)) i∈N , (π i ) i∈N ) be a quasi-linear GNEP. Every optimal solution x ∈ X conv (x) of optimization problem (9) corresponds to an optimal solution (x, ν) of (R) with the same objective value and vice versa: Proof. Let I = (N, (X i (·)) i∈N , (π i ) i∈N ) be a quasi-linear GNEP. In what follows let i ∈ N and x −i ∈ R k −i be arbitrary. We start by observing that the cost functions π i (·, x −i ) of player i is linear on his feasible strategy set X i (x −i ). Since the cost function φ i (·, x −i ) of the convexified game I conv is the convex envelope of the linear function π i (·, Now consider for an arbitrary but fixed x ∈ X conv (x) the functionV conv (x). From the proof of Theorem 2 we already know: Since Assumption 1 holds, X i (x −i ) and therefore also the convex hull conv(X i (x −i )) are compact sets. Furthermore the sets are not empty, since ). Using Assumption 3.1 we arrive at the following linear optimization Since we already know that there exists an optimal solution for (LP i (x −i )), strong duality holds, i.e. the dual (DP i (x −i )) to (LP i (x −i )) admits an optimal solution with objective value equal to the optimal objective value of (LP i (x −i )). In the following let us denote by DP i (x −i ) and LP i (x −i ) also the corresponding optimal objective values. It will be clear by context whether we're talking about the actual optimization problem or just the optimal value of it. We can now reformulatê Since the n maximization problems (DP i (x −i )) are completely separable we can combine them to one maximization problem and end up with the following representation ofV conv (x): Using this description forV conv (x), the representation of the convex envelopes derived in (13) as well as Definition 3.1 for the condition x ∈ X(x), we arrive at the desired result.
Remark 4. Similar to Remark 2, the Assumption 1 can be dropped, if the existence of an optimal solution for LP i (x −i ) is guaranteed for all i ∈ N and x ∈ X conv (x). Furthermore by rewriting the part of the objective function of (R) corresponding to player i by for all i ∈ N , it turns out that Theorem 3 describes a way of solving the KKT conditions of the convexified GNEP I conv which are exactly described by (10), (11), (12) and (14) being equal to zero for all i ∈ N . Thus, Theorem 3 is an alternative to other techniques solving the KKT conditions for the GNEP, see e.g. [10].
Example 1 (Capacitated Discrete Flow Games (CDFG)). We consider a directed capacitated graph G = (V, E, c), where V are the nodes, E the edge set with |E| = m and c ∈ N m 0 denotes the edge capacities. There is a set of players N = {1, . . . , n} and each i ∈ N is associated with an end-to-end pair (s i , t i ) ∈ V × V . We call a strategy x i an integral feasible flow for player i ∈ N if x i represents an integral flow which sends d i ∈ N flow units from his source s i to his sink t i . A player is further restricted in his strategy choice by the capacity constraints, i.e. for given integral feasible flows x −i , his flow x i has to satisfy the restriction x i ≤ c − s =i x s . Thus the strategy set of a player i ∈ N is -for given integral feasible flows x −i -described by For vectors x −i that do not represent integral feasible flows, the strategy set is empty. Thus the first condition in Definition 3 is fulfilled by the CDFG as the strategy spaces only consist of finitely many points. We define the cost functions by Here, the first term can be interpreted as costs that arise through congestion whereas the second term represent congestion independent costs for player i. These cost functions fulfill the second condition in Definition 3 and thus, the CDFG is a quasi-linear GNEP.
In general the optimization problem in Theorem 3 might still be quite complex since we did not assume any conditions for the matrix-and vector-valued functions M i , C i and e i . But the given representation already gives key insights in the conditions a quasi-linear instance I has to additionally fulfill such that the optimization problem (R) can be solved efficiently on the one hand, and on the other hand yield meaningful results for the instance I. Namely the following criteria play a decisive role: properties that guarantee optimal solutions (x, ν) of (R) with x ∈ X(x).

The dimension
3. The behavior of the matrix-/vector-valued functions M i , C i and e i under variation of x −i .
Under further assumptions, Theorem 3 implies that the existence of general Nash equilibria of the instance I can be determined by solving a convex optimization problem.
Then I has a generalized Nash equilibrium iff the following convex optimization problem has the optimal value 0.
Proof. This follows immediately by Theorem 2 and 3 as well as the fact that the optimization problem in (15) attains it minimum at an extreme point of F as the set F is convex and the objective function is linear.

Jointly Constrained GNEPs
In several interesting applications, the players' strategy sets are restricted by coupled constraints, that is, the strategy sets of every player i ∈ N are given by for one joint set X ⊆ R k which doesn't have to be convex and may even be discrete. We call these type of GNEPs jointly constrained w.r.t. X. This type of GNEP occurs for example in the domain of Automated Driving [13], Traffic Control [5] or Transportation Problems [31]. Before we investigate the structure of the convexification I conv of a jointly constrained GNEP, let us motivate this special type of GNEP further by the following example.
Example 2 (Jointly Constrained Atomic Congestion Games). We describe the atomic (resourceweighted) congestion game, which is a generalization of the model of Rosenthal [28], without joint restrictions first. The set of strategies available to player i ∈ N = {1, . . . , n} is given by X i ⊆ × j∈E {0, d ij } for weights d ij > 0 and resources j ∈ E = {1, . . . , m}. Note that by assuming x i ∈ {0, 1} m for all i ∈ N , that is, d ij = 1, we obtain the standard congestion game model of Rosenthal. The cost functions on resources are given by player-specific functions c ij (ℓ j (x)), j ∈ E, i ∈ N , where ℓ(x) = i∈N x i . The private cost of a player i ∈ N for strategy profile x ∈ i∈N X i is defined by This model can be generalized by allowing joint restrictions in the players' strategy sets, that is, extending the above model to a jointly constrained GNEP with respect to a set X ⊆ i∈N X i , e.g., if the the usage of resources is bounded by hard capacities.
A jointly constrained GNEP w.r.t. a convex set X is often referred to as jointly convex in the literature. The jointly convex GNEP is one of the best understood subclasses of the GNEP and thus it seems quite natural to hope for I conv to be a jointly convex GNEP, given a jointly constrained instance I. We remark in the following that this is in general not the case. Remark 5 (I conv is not jointly convex in general). If I conv is jointly convex w.r.t. a joint restriction set X conv , then conv(X) ⊆ X conv has to hold. Thus the example in Figure 1 already shows, that I conv will in general not be jointly convex as conv(X 1 ( In fact, I conv is not even a jointly constrained GNEP w.r.t. some X conv as the sets displayed in picture 2 and 3 of Figure 1 would both have to describe the same set X conv as the following Lemma implicates.
Lemma 1. An instance I of the GNEP is jointly constrained w.r.t. X if and only if the identity holds for all i ∈ N , where we understand We start with the only if direction, i.e. let I be jointly constrained w.r.t. X and i ∈ N arbitrary.
We first show that the inclusion ⊆ in (17) holds. For any For the other inclusion, we take an arbitrary (x i , x −i ) ∈ X. Again by definition of the jointly constrained GNEPs it follows that x i ∈ X i (x −i ) and therefore ( For the if direction, let I be a GNEP which satisfies (17) for all i ∈ N . We have to show that for an arbitrary i ∈ N the identity in (16) holds. The inclusion ⊆ follows as X i (x −i ) × x −i ⊆ X holds by (17). Conversely for any (x i , x −i ) ∈ X it follows directly by (17)

Extending I conv
The fact that for a jointly constrained instance I, the convexification I conv may neither be jointly convex nor jointly constrained, shows that our convexification method needs to be adapted in order to obtain a jointly convex convexification.
The main idea is to extend I conv to some instance I ext in such a way such that I ext is jointly convex and Theorem 2 still applies for the extension. One naive approach would be to simply enlarge the strategy spaces for I ext such that I ext is jointly convex w.r.t. some convex set X ext ⊇ conv(X) and adjust the cost functions to +∞ on the new strategies, that is It is not hard to see that the equilibria of I can be characterized by I ext in the same fashion as in Theorem 2 with I conv . Yet this approach of extending I conv seems computationally of limited interest as the extended cost functions do not have any regularity. One may try to extend the cost functions in a smooth manner instead of just setting them to +∞ outside of the original strategy space. Yet, it is not clear how to extend these functions reasonably in a computational regard as one wants as much regularity of the cost functions as possible while putting as little effort as possible in the computation of the cost-functions themselves. We remark here that the cost functions φ i (x i , x −i ) of I conv are by definition arbitrary outside of conv(X i (x −i )) and thus a similar problem as described above occurs w.r.t. the convexified cost functions φ i . But it is substantially easier to find any arbitrary smooth extension compared to finding a smooth extension which preserves original GNE. On top of that, the functions conv( may have a natural and smooth extension to the whole domain, as it is the case for most quasi-linear GNEPs for example. This gives rise to the question whether or not one can modify I conv without changing the costfunctions and thus only extending the strategy spaces. Clearly, this will in general lead to a loss of original GNE. To see this, let's take a look back at the Example in Figure 1. Assume that the cost function of player 2 is represented by φ 2 (x 1 , x 2 ) = −x 2 on the whole R 2 . Let I ext be a jointly convex extension of I conv as described above, but without changing the cost functions. Then [1, 8/3] ⊆ X ext 2 (2) as conv(X) ⊆ X ext and therefore the generalized Nash equilibrium (x * 1 , x * 2 ) = (2, 1) for I would not remain a GNE for the extension I ext .
As this example shows, the possibility to lose original equilibria occurs when we enlarge non empty strategy sets X conv i (x −i ) = ∅. In contrast, the following lemma shows that we can not lose original GNE if we enlarge empty strategy sets.
Lemma 2. Let I be any instance of the GNEP and assume that for an i ∈ N and x * −i ∈ R k −i the set X i (x * −i ) = ∅ is empty. Define by I ext the extended GNEP which is identical to I with the exception that X ext i (x * −i ) := F is set to an arbitrary F ⊆ R k i . Then the following two statements are equivalent for any x ∈ X(x): • x is a generalized Nash equilibrium for I • x is a generalized Nash equilibrium for I ext and together with the definition of I ext , the equality X j (x −j ) = X ext j (x −j ) holds for all j ∈ N and in particular x ∈ X ext (x) follows. As furthermore the cost functions coincide, the equivalence follows by the definition of a generalized Nash equilibrium.
As the example in Figure 1 shows, for general jointly constrained GNEPs, there are non-empty sets which we necessarily have to enlarge in order to get a jointly convex GNEP. This is due to the fact, that conv(X) ⊆ X ext has to hold for a jointly convex extension and therefore any set with has to be extended, e.g. the set X conv 2 (2) in Figure 1

Projective-closed GNEPs
The insights of the previous subsection motivate the definition of projective-closed GNEPs. These special jointly constrained GNEPs have the property, that their common feasible set X does not admit non-empty sets which we necessarily have to enlarge in order to get a jointly convex extension. Definition 6. Let I be jointly constrained w.r.t. X. Then X is called projective-closed (with respect to the convex hull operator), if for all i ∈ N and x −i ∈ R k −i one of the following statements holds: We call a jointly constrained instance I w.r.t. X projective-closed if the set X is projective-closed.
The above concept of projective-closed sets requires that for fixed x −i , the projection of conv(X) onto R k i is equal to the convex hull of the projection of X onto R k i for fixed x −i . We can give a geometric interpretation of projective-closed sets via the following lemma.
Lemma 3. Let I be jointly constrained w.r.t. X. Then X is projective-closed if and only if for all i ∈ N and x −i ∈ R k −i either the intersection conv(X) ∩ (R k i × x −i ) contains no points of X (1.) or any extreme point of the intersection is contained in X (2.).
where E(·) denotes again the extreme points of a set.
Before we start the proof, we remark that with this geometric interpretation, one can easily verify that the Example in Figure 1 is not projective-closed, as the intersection of conv(X) ∩ (R k 2 × x 1 ) = conv(X) ∩ ({2} × R) (marked in red in the first picture) for x 1 = 2 is not empty, yet the extreme point (2, 8/3) (marked as red cross) is not contained in X.
Proof. The equivalence between the first statement of Definition 6 and the lemma is clear. Thus it suffices to show that the respective second conditions are equivalent. We start by showing that Definition 6 (2.) ⇒ Lemma 3 (2.): The condition stated in Definition 6 (2.) implies that the inclusion holds. To see this, we argue by contradiction and assume there exists By the equality stated in Definition 6 (2.), x * i ∈ conv(A) and therefore there exists a convex combination x * i = l s=1 λ i x s i with λ ∈ Λ l and x s i ∈ A. But since A ⊆ A conv and x * i ∈ E(A conv ) it follows that x s i = x * i for all s. This is a contradiction as x * i / ∈ A but x s i ∈ A. Now the inclusion in (18) yields: which concludes this direction of the proof. For Definition 6 (2.) ⇐ Lemma 3 (2.) we argue: Since the inclusion ⊇ in the last line always holds, the proof is finished.
The following Theorem shows that for projective-closed GNEPs I, the convexification is indeed extendable by only enlarging the strategy spaces leading to a jointly convex GNEP.
Theorem 4. Let I be a jointly constrained GNEP w.r.t. a projective-closed X and let I conv be its convexification. We define the jointly convex extension I ext w.r.t. X ext by π ext i := φ i for all i ∈ N and X ext representing any convex set, e.g. conv(X), that satisfies for any Then the following two statements are equivalent for x ∈ X(x).
1) x is a generalized Nash equilibrium for I.
2) x is a generalized Nash equilibrium for I ext and π ext (x) = π(x), where π ext (x) = n i=1 π ext i (x). Proof. The strategy sets of the convexified game I conv are either empty or fulfill the equality stated in Definition 6 (2.) since I is projective-closed. If X conv i (x −i ) = ∅ isn't empty, then clearly X i (x −i ) = ∅ holds as well which implies that Thus by I ext being jointly convex and in particular X ext satisfying (19), the strategy sets X ext i (x −i ) and X conv i (x −i ) coincide in the case of X conv i (x −i ) = ∅ not being empty. Therefore, the only strategy sets that change when we look at I ext instead of I conv are the empty sets X conv i (x −i ) = ∅. Lemma 2 then implies that for all x ∈ X conv (x) holds, that x is a GNE for I conv if and only if it is a GNE for I ext . As any x ∈ X(x) satisfies x ∈ X conv (x), our main Theorem 2 finishes the proof.
it is not hard to see that the CDFG is a jointly constrained GNEP w.r.t. X. It turns out, that it even belongs to the class of projective-closed GNEPs. To verify this, we show that X fulfills the condition stated in Definition 6. Let i ∈ N and We have to show that the equality in Definition 6 (2.) holds. Since ⊆ always holds we just have to show the inclusion ⊇. To prove this, define the relaxation of X bŷ  We argue that the following two steps are valid: By the definition of X it follows immediately that X ⊆X. SinceX is convex, the inclusion conv(X) ⊆X and thus also the inclusion (21) holds. By rewriting the sets in (22) via the definition of X andX, the inclusion is equivalent to: must be integral. Thus the polytope on the left has integral vertices since the flow polyhedron is box-tdi, see Edmonds and Giles [12] and Schrijver [32] for a definition of box-tdi and the aforementioned property of the flow polyhedron. These integral vertices are clearly contained in the right set and therefore the inclusion follows and hence the model is a projective-closed GNEP.
Furthermore, the above proof shows that the relaxed setX fulfills the equality stated in (19) and thus can be used instead of conv(X) in Theorem 4 which is extremely convenient in a computational regard. Note that conv(X) =X in general as the following instance of the capacitated discrete flow game shows.
Let N = {1, 2} and G be the graph displayed in Figure 3. We set the capacity on every edge and the amount of flow each player wants to send to one. Then X consists of only two elements, namely X = {(x * 1 , x u 2 ), (x * 1 , x l 2 )} where we denote by x * 1 the flow sending one flow unit over the edge (s 1 , t 1 ) and by x u 2 resp. x l 2 the unique path from s 2 to t 2 starting with the upper edge e 2 u resp. lower edge e 2 l . The setX contains for example the point 1 2 · (x u 1 + x l 1 , x u 2 + x l 2 ) / ∈ conv(X) where we define x u 1 and x l 1 analogously to x u 2 and x l 2 , thus showing that conv(X) X . Furthermore note, that the above argumentation works not only for the linear joint restriction j∈N x j ≤ c but for all joint restrictions g(x) ≤ 0 which describe a convex set and have the property, that for all i ∈ N and integral feasible flows In what follows we show that also several other interesting jointly constrained GNEPs lie in the class of projective-closed GNEPs, e.g. all 0, 1 games. Lemma 4. Let I be a jointly constrained GNEP w.r.t. X. If the projection of X to the strategy space of any player i does only consist of extreme points, where the projection is defined by We have to show that the equality in Definition 6 (2.) holds. As mentioned before, the ⊆ inclusion always holds. For the other inclusion ⊇ we argue as follows. For a given x i ∈ R k i with (x i , x −i ) ∈ conv(X) there exists a convex combination l j=1 α j x j = (x i , x −i ) with x j ∈ X for all j = 1, . . . , l and α ∈ Λ l . Since Thus x s ∈ P s (X) for all s = i which implies by our assumption, that x s is an extreme point of the projection. Furthermore, as x j ∈ X for all j = 1, . . . , l and thus x j s ∈ P s (X) for all s, the fact that l j=1 α j x j s = x s for s = i implies that x j s = x s for all j = 1, . . . , l. This concludes the proof as l j=1 Since any integral point of the {0, 1} k i hypercube is an extreme point, we get as a direct consequence of Lemma 4 the following statement: Corollary 3. For any jointly constrained GNEP I w.r.t. X ⊆ {0, 1} k the set X is projective-closed.
As another example for projective-closed GNEPs and our extension technique we revisit Example 2.
Example 2 (continued). Assume that the weights d ij = 1 equal to one for all i ∈ N , j ∈ E. Then X ⊆ {0, 1} n·m and thus by Corollary 3, our Theorem 4 is applicable. Furthermore the cost functions π i are quasi-linear, i.e. they admit the same structure as described in Definition 3.2: ) and therefore we may extend the latter description of φ i to the whole R k . With this definition of I conv and assuming that c i : R m → R m is a smooth function, the extended version I ext is a jointly convex GNEP w.r.t. X ext with smooth cost-functions for the players and thus various methods to solve I ext are known. For example, for an appropriate choice of X ext , one way to solve I ext is given by our Theorem 3, cf. Section 6.

Quasi-separable GNEPs
Several interesting problems can be modelled by quasi-separable GNEPs, that is, roughly speaking, GNEPs where the players' individual minimization problems are almost separable. To motivate these type of GNEPs, we take a look back at the CDFG and modify the example slightly. Instead of the joint capacity restriction i∈N x i ≤ c on the players strategy sets, we now assume that a central authority wants to influence the players' choice of a flow such that the players voluntarily meet the capacity constraints. The way the central authority may influence the players is by defining tolls λ j ∈ R + for each edge j ∈ E, where we additionally assume that edges may only be priced if the demand would otherwise exceed the supply. This central authority is modelled by a n + 1-th player whose strategy set is thus given by where we denote by x the strategy profile of the first n player and by ℓ(x) := i∈[n] x i . Furthermore, the cost functions of the first n player may only depend on their strategy as well as exhibit an affine dependency on the prices λ, that is The cost function of the the central authority is set to a constant as his only goal is to enforce the capacity constraints. Summarizing we end up with an instance of the GNEP I qs = (N, (X i (·)) i∈N , (π i ) i∈N ) with N := [n + 1] of the following form, which we call quasi-separable: . . , n and π n+1 (x, λ) ≡ 0 where we even allow for a function g i : R m → R m mapping a strategy to its actual resource consumption. The strategy sets are given by with ℓ(x) := i∈[n] g i (x i ). Note that in the above modification of the CDFG, the strategy spaces X i for the first n player would be given by their respective flow polyhedron. Pricing in resource allocation games is a prime example which can be modelled by such a quasiseparable GNEP. In various different domains pricing problems are considered as for example in the realm of network and congestion games [18], (electricity) market models [3,6,23] as well as trading and communication networks [21,24]. Translated to the GNEP setting, the main question in this topic is to determine whether or not a GNE exists and if so, how to compute it. In [20] a unified framework for pricing in nonconvex resource allocation games was presented together with a broad spectrum of applications for this framework as for example for tolls in network routing, Walrasian market models, trading networks or congestion control. The main theorem in [20] establishes a connection between the existence of a GNE and the primal dual gap of a certain optimization problem. Furthermore, similar techniques as presented here lead to convexified instances as well as a relationship between the convexified and the original instances. It turns out that these relationships follow almost immediately by Theorem 2 and a special structure the NI-function admits in quasiseparable GNEPs.
In what follows we show that for a quasi-separable GNEP I qs , the NI-function measures the Lagrangian primal dual gap of a minimization problem. To see this we setπ(x) := n i=1π i (x i ), X := n i=1 X i and calculate: Since (x, λ) ∈ X(x, λ) holds, we have x i ∈ X i for all i = 1, . . . , n and λ ⊤ (ℓ(x) − c) = 0 and thus: where µ(λ) is the Lagrangian dual to the optimization problem Summarizing, I has a generalized Nash equilibrium if and only if the above optimization problem has zero duality gap. Now in the case of g i being affine for all i, it is an immediate consequence of Grotzinger's description that the cost functions of the convexified game I conv admit the same structure as π i (x, λ), that is: has zero duality gap. We denote analogously by µ conv the Lagrangian dual to this problem. As (24) is a convex optimization problem, the duality gap is often known to be zero, for example whenever some constraint qualifications are satisfied. In this case, Theorem 2 then implies that I has a generalized Nash equilibrium if and only if there exists an optimal solution x ∈X for (24) withφ(x) =π(x). Altogether we get: Theorem 5. For I qs = (N, (X i (·)) i∈N , (π i ) i∈N ) with affine functions g i , the following assertions are equivalent for (x, λ) ∈ X(x, λ): • (x, λ) is a generalized Nash equilibrium for I.
In particular, if (24) always has zero duality gap, the existence of a generalized Nash equilibrium of I is equivalent to (24) having an optimal solution inX withφ(x) =π(x).

Computational Study
In this section we present numerical results on the computation of generalized Nash equilibria for Example 1, the capacitated discrete flow game. Four different methods to find equilibria are presented with their respective computation time and their solution quality.
Concerning the source sink pair of each player, we generated two types of the CDFG. Namely on the one hand a single source single sink type in which every player gets the same randomly selected (connected) source sink pair. On the other hand a multi source multi sink type in which each player has an individual randomly selected (connected) source sink pair. Similar, the weight of each player, i.e. the integral amount of flow each player wants to send, is either chosen uniformly at random from the range of 1 to 10 or set to 1 for each player. To generate arc capacities that actually have an impact on the strategy sets, we first chose the capacities uniformly at random from a relative small range of 1 to max(n, d 1 , . . . , d n ). If the resulting strategy space X is empty, the capacities are reassigned. This random reassignment is executed until either the strategy space is not empty anymore or a limit for the amount of reassignments is exceeded. In the latter case, the range of values in which the capacities are chosen is incremented by one and the procedure is repeated.
Concerning the cost functions, we use depending on the weight type of the instance one of the following quasi-linear descriptions: In the first case, C cong i ∈ N m×m 0 is a randomly generated diagonal matrix with values in the range of 0 to 20. Thus the CDFG is also a jointly constrained atomic congestion game in this case, cf.
Example 2. For the other case, C 1 i ∈ N m×m 0 and C 2 i ∈ N m 0 are randomly generated with values in the range of 0 to 20.
In conclusion, we generated 10 graphs for each instance-type, i.e. for each combination of |V |, |N |, the two types concerning the source/sink assignment as well as the weight assignment, leading to a total of 360 test instances of the CDFG.

Computing generalized equilibria
As a consequence of the quasi-linear cost functions, we may define the convexified cost functions as φ i (x) = π i (x) on the whole R k . We have shown in the previous section that the CDFG belongs to the class of projective closed GNEPs and thus Theorem 4 is applicable. Furthermore we have seen that we can use the relaxationX as the joint constraint set X ext for the extended GNEP I ext .
For this version of I ext , it follows by Theorem 4 that the set of equilibria of the original instance of the CDFG is exactly described by the set of integral GNE of I ext . To see this, we observe that the condition x ∈ X(x) in Theorem 4 is in fact equivalent to x ∈X ∩ N m 0 . This is due to the fact, that the CDFG is a jointly constrained GNEP w.r.t. X and thus the equivalence x ∈ X(x) ⇔ x ∈ X =X ∩ N m 0 holds. Furthermore we have set φ i (x) = π i (x) on the whole R k , thus, the requirement of π ext (x) = π(x) is always satisfied.
To find integral equilibria of the jointly convex GNEP I ext , we implemented four different methods in MATLAB ® . The first three methods are based on finding (local) minima ofV -like functions, i.e. functions that are bounded from below by zero and characterize the complete set of equilibria via the set of feasible points that have an objective value of zero. This property does not change when one multiplies the respective objective functions by a penalty term that penalizes non-integrality. In this regard, we also implemented such a penalized run for the first three methods. The solvers which we used to find (local) minima request a starting point. Thus we computed an ordered and common set of random starting points by projecting random vectors in [0, max(n, d 1 , . . . , d n )] k to the setX. Beginning with the first starting point, a (local) minimum is then computed of the respective objective function. Each component of this local minimum is then rounded to the nearest integer. The resulting integral vector is then checked for feasibility and whether or not it is a GNE of I ext by evaluating theV function for I ext at that point. If the rounded solution is not a GNE, the next (local) minimum is computed with the usage of the next starting point. This procedure is executed until either a GNE has been found or a time limit of three hours is exceeded, in which case the current computation is exited and no further (local) minima are computed.

Standard Approaches
Minimizing theV function. For the first method we implemented theV function for I ext . The evaluation ofV at a x ∈X requires to solve the n linear programs: This is done via the linprog solver of the MATLAB ® Optimization Toolbox. The local minima ofV over the setX are computed via the fmincon solver of the Optimization Toolbox with an increased maximum function evaluation limit of 15000 (default: 3000) as the solver would otherwise typically prematurely exit the computation.
Minimizing the regularizedV α function. The second method is completely analogous to the first method and only differs in the function we are minimizing. Instead of the standardV function, we use the following regularization: where || · || is the Euclidean norm and α > 0 denotes a regularization parameter which we set to 0.02 for our computations. Evaluating this regularization requires to solve n quadratic programs, which is done by the quadprog solver of the Optimization Toolbox. The properties of this regularizedV α function were extensively studied in [11] by Dreves, Kanzow and Stein. They proved that under suitable assumptions, which are fulfilled in our case, the function V α is bounded from below by zero, characterizes the GNE in the same manner as the standardV function and most importantly is piecewise continuously differentiable. The latter allows us to provide an analytic gradient for the fmincon solver, which speeds up the minimum-computation significantly.
Minimizing the unconstrainedV αβ function. Dreves, Kanzow and Stein introduced in [11] the functionV where PX [x] denotes the projection of x to the feasible setX and 0 < α < β and c > 0 are regularization parameters. Similar to the regularizedV α function, they showed that under suitable assumptions, this function is well-defined for all x ∈ R k , bounded from below by zero and piecewise continuously differentiable (even for c = 0 in the case of jointly convex GNEPs). Furthermorē V αβ (x) = 0 holds if and only if x is a generalized Nash equilibrium for any x ∈ R k . Again, it can be shown that I ext fulfills the required assumptions. To find (local) minima ofV αβ (x), we used the same procedure as presented in [11] with the only difference that we set c := 0 as we're dealing with a jointly convex GNEP. Therefore, we set the parameters α := 0.02, β := 0, 05 and used the robust gradient sampling algorithm from [4] to minimizeV αβ (x) over R k . The MATLAB ® implementation of the gradient sampling algorithm is provided online by the authors of [4] under the following address: http://www.cs.nyu.edu/overton/papers/gradsamp. We modified the latter slightly by checking at every function evaluation with a value below 1e-03 if the rounded point is an equilibrium (by evaluatingV ) and if so, exiting the computation early. The function evaluation is done in the same manner as for the regularized functionV α with the additional computation of the projection PX [x] via quadprog.

Penalizer
For the first three methods we also implemented a penalized run, that is, we multiplied the respective objective function with the term which penalizes non-integrality of strategy profiles with an objective value bigger than zero. This does not change the characteristics of the functions concerning their ability to characterize equilibria. Yet, this penalty term must be viewed with caution. On the one hand, the local minima found by the solver are more likely to be integral. Thus the rounding of the solution is less problematic concerning the possibility to round to an unfeasible strategy profile as well as to round too far away, making the computation of the local minimum redundant. On the other hand, the computation of a single local minimum is likely to be more time consuming. Furthermore, the weight of the penalty term has to be chosen carefully as new local minima with an objective value bigger than zero may be generated through this penalty term.

Quasi-Linear reformulation
As I ext is a convex quasi-linear GNEP, i.e. the convexification of I ext is I ext again, we can use the optimization problem (R) to find equilibria. The advantage that comes with this reformulation is the possibility to implement the problem with integrality constraints regarding the x-variable and use a global MINLP-solver such as the BARON-solver, see [25]. These solvers typically require that the objective and restriction functions have a computationally tractable (algebraic) description, i.e. only consist of solver-supported operations like +, −, · etc.. This contrasts the previous methods where in each function call (e.g.V (x)) a separate optimization problem has to be solved. The BARON-solver provides a MATLAB ® -interface which we used to compute global minima.

Results
All four methods have been implemented in MATLAB ® R2019b on Windows 10 Enterprise. The computations have been performed on a machine with Intel Core i5-8250U and 8 GB of memory. For the instances with 4 or less players we tried to find an equilibrium with each method. Minimizing the standardV function (resp. its penalized version) was the least performing method and was only able to find in 82.5% (resp. 87.92%) of instances an equilibrium within an average time of 810.43 seconds (resp. 1027.50 seconds). We remark that, if no equilibrium was found, the time was not included in the average time computation. In contrast, the method using the regularized V α function (resp. its penalized version) was able to find an equilibrium in 100% (resp. 99.58%) of the time and needed only 8.90 seconds (resp. 6.24 seconds). The unconstrained reformulation V αβ (resp. its penalized version) found an equilibrium with a chance of 96.25% (resp. 97.92%) within 294.65 seconds (resp. 360.28 seconds). Finally, the BARON-solver applied to our quasilinear reformulation found an equilibrium with a probability of 99.58% in an average time of 1.92 seconds. Note that we only included instances where at least one equilibrium was found by any of the methods since otherwise instances where no equilibrium exists may deflect the methods' performances. However, this occurred only once.
In regard of the instances with 10 players, we only present results for the regularizedV α function and the quasi-linear reformulation, as the other two methods were far off being competitive. Minimizing theV α function (resp. its penalized version) found an equilibrium in 95.83% (resp. 99.17%) of the time in an average of 437.03 seconds (resp. 364.65 seconds). In Comparison, using the quasi-linear reformulation had a chance of 99.17% to find an equilibrium within an average time of 57.72 seconds. We refer to Section 6.3.1 for a more extensive table showing the performance of the methods per instance type.
Concluding, the BARON-solver applied to our quasi-linear reformulation was performing best in both aspects: the success rate of finding equilibria as well as in terms of the computation time. Furthermore the computation of a single local minimum (not necessary equilibrium) by any of the other methods often took longer than the global minimum computation of the BARON-solver. Thus, the BARON-solver applied to our quasi-linear reformulation does not only gain its advantage by being able to implement a priori integrality constraints, but would also outperform the other methods if one was only concerned in a (not necessary integral) equilibrium of the GNEP I ext . On top of that, our reformulation has the striking advantage to prove the non-existence of equilibria for an instance, because with BARON we obtain lower bounds on the Nikaido-Isoda function serving as a certificate for non-existence.  Table 2) shows for each instance type with 4 or less players (resp. 10 players) the performance of each method (resp. only of theV α and quasi-linear methods), that is, the percentage of how often an equilibrium was found as well as how long it took to compute the GNE on average. If no equilibrium was found, the time was not included in the average time computation. The different instance types are described by a tuple of the form (|N |, |V |, a, b) where a ∈ {s, m} determines whether it's the single (s) or multi (m) source/sink type and b ∈ {1, 10} indicates whether the weights of the player are all set to 1 or randomly set in the range of 1 to 10.

Detailed Results
To demonstrate the behaviour of the various methods, we present in Figure 4 boxplots of the performance of all methods. These boxplots are based on 100 randomly generated instances of the type (2,20,m,10). The diagrams show the distribution of the computation time (in seconds) of an integral GNE. The mark inside each box denotes the median, boxes represent lower and upper quartiles, and the whisker ends show the minimum and maximum, respectively, apart from possible outliers marked by a cycle.  Table 1: The GNE column displays the percentage of how often an equilibrium was found over the ten graphs per instance. The Time column shows how long it took (in seconds) to compute the equilibrium on average over the ten graphs per instance.  Table 2: The GNE column displays the percentage of how often an equilibrium was found over the ten graphs per instance. The Time column shows how long it took (in seconds) to compute the equilibrium on average over the ten graphs per instance. (g) BARON Figure 4: Boxplots of the performance of all methods with respect to the instance type (2,20,m,10). The diagrams show the distribution of the computation time (in seconds) of an integral GNE. We did not include the time when no equilibrium was found. In this regard, the methods (a)-(g) found (100,100,100,99,97,97,100) equilibria respectively.

Conclusions
We derived a new characterization of generalized Nash equilibria by convexifying the original instance I, leading to a more structured instance I conv of the GNEP. We then derived for the three problem classes of quasi-linear, projective-closed and quasi-separable GNEPs, respectively, new characterizations of the existence and computability of generalized Nash equilibria. We demonstrated the applicability of the latter by presenting various methods and corresponding numerical results for the computation of equilibria in the CDFG. We see our approach as an initial step to systematically approach non-convex and discrete GNEPs which are still poorly understood. We believe that there is still untapped potential in our convexification method in order to obtain structural insights into the problem as well as pave the way for a more tractable computational approach.