Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

In this paper we consider the Nash equilibrium problem for infinite player games with vector payoffs in a topological vector space setting. By employing new concepts of relative (pseudo)monotonicity, we establish several existence results of solutions for usual and normalized vector equilibria. The results strengthen existence results for vector equilibrium problems, which were based on classical pseudomonotonicity concepts. They also extend previous results for vector variational inequalities and finite player games under relative (pseudo)monotonicity.

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

Let (X, =) be a topological space. A subset A of X is called pre-open if A ⊆ Int(Cl(A)). Let P O(X) denote the family of all pre-open sets in X. In general, P O(X) does not form a topology on X. Furthermore, in topological vector spaces, it is not always true that P O(L) forms a topology on L where L is a topological vector space. In this note, we prove that the class of strongly preirresolute topological vector spaces is that subclass of topological vector spaces in which P O(L) forms a topology and thereby we see that all proved results in [5] concerning strongly preirresolute topological vector spaces are obvious.

This article considers the (two-player) composite Nash equilibrium (CNE) problem with a separable nonsmooth part, which is known to include the composite saddle-point (CSP) problem as a special case. Due to its two-block structure, this problem can be solved by any algorithm belonging to the block-decomposition hybrid proximal-extragradient (BD-HPE) framework proposed in [R. D. C. Monteiro and B. F. Svaiter, SIAM J. Optim., 23 (2013), pp. 475--507]. The framework consists of a family of inexact proximal point methods for solving a more general two-block structured monotone inclusion problem which, at every iteration, solves two prox subinclusions according to a certain relative error criterion. By exploiting the fact that the two prox subinclusions in the context of the CNE problem are equivalent to two composite convex programs, this article proposes a new instance of the BD-HPE framework that approximately solves them using an accelerated gradient method. It is shown that the new instance is able to take significantly larger prox stepsizes than other instances from this framework that perform single composite gradient steps for solving the subinclusions. As a result, it is shown that the first instance has better iteration-complexity than the latter ones. Finally, it is also shown that the new accelerated BD-HPE instance computationally outperforms several state-of-the-art algorithms on many relevant classes of CSP and CNE instances.

We consider the problem of finding Nash equilibrium for two-player turn-based zero-sum games. Inspired by the AlphaGo Zero (AGZ) algorithm, we develop a Reinforcement Learning based approach. Specifically, we propose Explore-Improve-Supervise (EIS) method that combines exploration, improvement and supervised learning to find the value function and policy associated with Nash equilibrium. We identify sufficient conditions for convergence and correctness for such an approach. For a concrete instance of EIS where random policy is used for exploration, Monte-Carlo Tree Search is used for improvement and Nearest Neighbors is used for supervised learning, we establish that this method finds an $\varepsilon$-approximate value function of Nash equilibrium in $\widetildeO(\varepsilon^-(d+4))$ steps when the underlying state-space of the game is continuous and d-dimensional. This is nearly optimal as we establish a lower bound of $\widetildeOmega (\varepsilon^-(d+2) )$ for any policy.

In this paper, we formulate the random bimatrix game as a chance-constrained game using chance constraint. We show that a Nash equilibrium problem, corresponding to independent normally distributed payoffs, is equivalent to a nonlinear complementarity problem. Further if the payoffs are also identically distributed, a strategy pair where each player’s strategy is the uniform distribution over his action set, is a Nash equilibrium. We show that a Nash equilibrium problem corresponding to independent Cauchy distributed payoffs, is equivalent to a linear complementarity problem.

The problem of seeking Nash equilibrium (NE) based on aggregative games under quantization constraints is full of challenges. Although the NE seeking algorithm in continuous-time systems has been studied, this problem in discrete-time systems still needs to be solved urgently. To address this problem, three distributed algorithms are first proposed under three quantization cases, adaptive, random, and time-varying quantizations, based on doubly stochastic communication topology networks. Then, the actions of players would eventually converge to NE under the conditions of vanishing step size and strong monotonicity are proved. Moreover, the convergence rate of the three quantization cases are analyzed, respectively. Finally, numerical experiments are implemented on plug-in hybrid electric vehicles (PHEVs) to validate the effectiveness of the proposed distributed algorithms. Comparing the convergence rates of the three proposed algorithms, the convergence effect of the adaptive quantization is better than that of the other two quantization cases.

A note on multiplier convergent series

In this paper, we introduce the notion of almost \(s-\)topological vector spaces and present some examples and counterexamples of almost \(s-\)topological vector spaces. Some general properties of almost \(s-\)topological vector spaces are investigated.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We address the Nash equilibrium problem in a partial-decision information scenario, where each agent can only observe the actions of some neighbors, while its cost possibly depends on the strategies of other agents. Our main contribution is the design of a fully-distributed, single-layer, fixed-step algorithm, based on a proximal best-response augmented with consensus terms. To derive our algorithm, we follow an operator-theoretic approach. First, we recast the Nash equilibrium problem as that of finding a zero of a monotone operator. Then, we demonstrate that the resulting inclusion can be solved in a fully-distributed way via a proximal-point method, thanks to the use of a novel preconditioning matrix. Under strong monotonicity and Lipschitz continuity of the game mapping, we prove linear convergence of our algorithm to a Nash equilibrium. Furthermore, we show that our method outperforms the fastest known gradient-based schemes, both in terms of guaranteed convergence rate, via theoretical analysis, and in practice, via numerical simulations.

Our aim is to give other proofs of some slight generalizations of results from Baron (Aequat Math, 2013). We describe larger classes of discontinuous additive involutions $${a:X\to X}$$ on a topological vector space X such that $${a(H)\setminus H\neq\emptyset}$$ holds for a sufficiently numerous set $${H\subset X}$$ of vectors linearly independent over $${{\mathbb{Q}}}$$ . We also consider the topological vector space $${{\mathcal{A}}_X}$$ of all additive functions $${a:X\to X}$$ with the topology induced by the Tychonoff topology of the space X X . We prove in a simple way that some classes of discontinuous additive involutions are dense in the topological vector space $${{\mathcal{A}}_X}$$ .

The spaces of test functions we are going to use are vector spaces of smooth (i.e., sufficiently often continuously differentiable) functions on open nonempty subsets Ω ⊆ ℝ n equipped with a ‘natural’ topology. Accordingly we start with a general method to equip a vector space V with a topology such that the vector space operations of addition and scalar multiplication become continuous, i.e., such that $$ \begin{array}{*{20}c} {_{M\,:\,\,\mathbb{K}\, \times \,V\, \to \,V,}^{A\,:\,\,V\, \times \,V\, \to \,V,} } & {_{M(\lambda ,\,x)\, = \,\lambda x,\,}^{A(x,\,y)\, = \,x\, + \,y,} } & {_{\lambda \, \in \,\mathbb{K},\,\,\,x\, \in \,V}^{x,\,y\, \in \,V} } \end{array} $$ become continuous functions for this topology. This can be done in several different but equivalent ways. The way we describe has the advantage of being the most natural one for the spaces of test functions we want to construct. A vector space V which is equipped with a topology T such that the functions A} and M are continuous is called a topological vector space, usually abbreviated as TVS. The test function spaces used in distribution theory are concrete examples of topological vector spaces where, however, the topology has the additional property that every point has a neighborhood basis consisting of (absolutely) convex sets. These are called locally convex topological vector spaces, abbreviated as LCVTVS.

Tropical convexity in Riesz spaces

With the advancement of power-to-gas (P2G) technology and local energy markets, new opportunities emerge for small-scale gas distributors (GDs) to unlock additional value. However, research on multiple GDs within the same power network participating in local energy markets remains unexplored, potentially leading to suboptimal collaborative efficiency and market imbalance. To bridge this gap, this paper proposes a heterogeneous game framework that integrates mixed-integer Nash equilibrium problem (MI-NEP) and generalized Nash equilibrium problem (GNEP). The framework assumes rational GDs simultaneously engage in non-convex GNEP for optimal power flow (OPF) coordination in distribution systems while addressing MI-NEP governing pipeline flow control in local gas market transactions. We develop a collaborative computing solution combining the best response (BR) method with a Korpelevich-inspired algorithm to obtain Nash equilibrium (NE) solutions ensuring fairness and operational stability. Comprehensive case studies on integrated 10-node power distribution networks with 7-node and 20-node gas networks demonstrate that the algorithm converges within 10 iterations with a residual error below [Formula: see text]. The proposed framework effectively ensures market fairness by reducing the cost disparity between isomorphic agents to zero.