Generalized Nash Equilibrium Problems with Mixed-Integer Variables

TL;DR

The work addresses generalized Nash equilibrium problems with non-convex and mixed-integer strategy spaces by introducing a convexification framework based on the Nikaido-Isoda function. It constructs convexified instances \\mathcal{I}^{\\mathrm{conv}} using convex hulls and convex envelopes, proving that a strategy profile is a GNE for the original game iff it is a GNE for a convexified instance with costs aligned, enabling tractable analysis in otherwise intractable settings. The paper develops three strands—quasi-linear GNEPs, jointly constrained GNEPs via k-restrictive-closed and restrictive-closed notions, and a computational study on integral network flows and discrete markets—demonstrating both theoretical characterizations and practical algorithms (MINLP reformulations, LP dual representations, BR heuristics). Overall, the approach provides a principled pathway to leverage convexification for non-convex and discrete GNEPs, with concrete applicability to integral networks and market equilibria and potential for broader adoption in non-convex game analysis.

Abstract

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.

Figures (10)

• Figure 1: Representation of the strategy sets and the resulting set of feasible strategy profiles $x \in X(x)$ marked via red circles.
• Figure 2: Example for a 2-player jointly constrained GNEP $I$ w.r.t. $X\subseteq \mathbb{R}^{(1,1)}$ represented by the four black dots in the first picture. The prescribed strategy sets $X_i^\mathrm{conv}(x_{-i}) = \mathrm{conv}(X_i(x_{-i})), x_{-i} \in \mathrm{rdom} X_i$ which rule out the possibility for $I^\mathrm{conv}$ to be jointly constrained are represented in picture 2 and 3.
• Figure 3: Example for a 2-player GNEP $I$ which is not jointly constrained but admits a jointly constrained convexified instance $I^\mathrm{conv} \in \mathcal{I}^\mathrm{conv}$. The prescribed strategy sets are represented in picture 1 and 2 where the dots correspond to the original strategies. In the third picture is an example for a possible joint restriction set $X^\mathrm{conv}$.
• Figure 4: A 2-player jointly constrained GNEP $I$ w.r.t. $X\subseteq \mathbb{R}^{k}, k := (1,1)$ represented by the black set in the first picture. In picture 2 the union of the prescribed strategy sets is represented which equals the $k$-convex hull (and even the regular convex hull) of $X$. Yet, for $i = 2$ and $x_{-2} = x_1 := 3$ we have $\mathrm{conv}^k(\mathrm{res}(X,x_{-i})) = \{(3,1)\} \subsetneq \{3\}\times [1,4] =\mathrm{res}(\mathrm{conv}^k(X),x_{-i})$. Remember that $X = \mathcal{S}_{i}=\mathcal{S}$ holds for any $i \in N$ in a jointly constrained instance.
• Figure 5: Example for a 2-player jointly constrained GNEP $I$ w.r.t. a $(1,1)$-restrictive-closed $X\subseteq \mathbb{R}^{(1,1)}$ represented by the four black dots in the first picture. In picture 2 and 3 are two suitable choices of $X^\mathrm{conv}$ which fulfill \ref{['eq:strongJCthm']}.

Theorems & Definitions (50)

• Definition 1: NI-function
• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 2
• proof
• Corollary 1
• Example 1