Stabilization of a class of congestion games via intermittent control

Abstract

Congestion games, firstly proposed in [1], have been widely applied in many practical problems such as electric vehicles charging, road pricing, and wireless spectrum sharing. It was proved in [1] that a congestion game admits at least one pure Nash equilibrium (NE). In reality, players may own different loads, which gives rise to weighted congestion games. In addition, players may also generate extra utilities by themselves, which brings more challenge to analyze the game. We call this model as the weighted congestion game with player-specific utilities (WCGPSU). In contrast to existing studies on the existence and seeking of NEs of WCGPSU [2], this study focuses on the dynamic behaviors of evolutionary WCGPSU via semi-tensor product (STP) of matrices [3]. STP provides a convenient platform for the analysis and control of finite-valued dynamic systems, such as Boolean networks [4] and finite games [5]. In this study, the intermittent control of evolutionary WCGPSU is investigated, in which some players act as controllers only when some special profiles occur. This kind of control can reduce both the control time and the cost. It should be pointed out that the intermittent control is similar to the event-triggered control in [6,7], where Ref. [6] considered the networked evolutionary games using external controllers and Ref. [7] focused on Boolean networks. The main contribution of this study is as follows: (1) Using STP, the algebraic expression of the dynamic equation for the evolutionary WCGPSU is established and the property on NEs is presented. (2) By designing open-loop intermittent control and state feedback intermittent control, respectively, two necessary and sufficient conditions are obtained to stabilize the evolutionary WCGPSU to the NE. In fact, using our method, the game can be globally stabilized to any profile we want. Notations. For a matrix A, Col(A) denotes the set of columns of A and Coli(A) is the ith column of A. δ k := Coli(Ik) and ∆k := {δ i k | i = 1, 2, . . . , k}. L = [δ1 m , δ i2 m , . . . , δ in m ] is a logical matrix and can be briefly represented as L = δm[i1, i2, . . . , in]. Lm×n denotes the set of m × n logical matrices. |S| denotes the cardinality of the set S. Throughout this article, STP is the default matrix product, and thus the STP symbol ‘⋉’ is mostly omitted if no confusion arises. A brief introduction on STP is listed in Appendix A. Problem formulation. A WCGPSU is defined as a tuple G = (N,M,Si,Ξj , wi, ri, C), where N = {1, 2, . . . , n} is the set of players and M = {1, 2, . . . ,m} is the set of facilities. The player i chooses ki facilities to achieve his/her purpose and the number of strategies that player i can choose is Ki = C ki m . Let si denote player i’s strategy and Si ⊂ 2 M