Distributed Generalized Nash Equilibrium Seeking: An Operator-Theoretic Perspective

Abstract

Generalized games model interactions between a set of selfish decision makers, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">players</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">agents</i> , where both the objective function <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">and</i> the feasible decision set of each player may depend on strategies of the competitors. Such games arise, for example, when agents compete for a share of some common but limited resources. For instance, consider a set of vehicles sharing the road, set of radio channels competing for bandwidth, or set of companies servicing an economic market. They can each be modeled as players/agents in a game, competing for a portion of available resources, that is, road capacity, total bandwidth, or market share. As resources are limited, player choices are bound together by a coupling-capacity constraint. Essentially, in a generalized game, or one with coupling constraints, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">set of available</i> choices an agent has depends on the choices of other agents. Thus, a player cannot simply optimize their own objective function without considering the decisions of the others, even though this objective might not depend on him or her. The relevant equilibrium (namely, the solution concept) for noncooperative decision making with coupling constraints is called a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">generalized</i> Nash equilibrium (GNE).