The Equilibrium Existence Problem in Finite Network Congestion Games

Abstract

Abstract. An open problem is presented regarding the existence of pure strategy Nash equilibrium (PNE) in network congestion games with a finite number of non-identical players, in which the strategy set of each player is the collection of all paths in a given network that link the player's origin and destination vertices, and congestion increases the costs of edges. A network congestion game in which the players differ only in their origin-destination pairs is a potential game, which implies that, regardless of the exact functional form of the cost functions, it has a PNE. A PNE does not necessarily exist if (i) the dependence of the cost of each edge on the number of users is player-as well as edgespecific or (ii) the (possibly, edge-specific) cost is the same for all players but it is a function (not of the number but) of the total weight of the players using the edge, with each player i having a different weight w i . In a parallel two-terminal network, in which the origin and the destination are the only vertices different edges have in common, a PNE always exists even if the players differ in either their cost functions or weights, but not in both. However, for general twoterminal networks this is not so. The problem is to characterize the class of all two-terminal network topologies for which the existence of a PNE is guaranteed even with player-specific costs, and the corresponding class for player-specific weights. Some progress in solving this problem is reported.

On the Existence of Pure Nash Equilibria in Weighted Congestion Games

“…Weighted congestion games are among the core topics in the game theory, operations research, computer science, and economics literature. This class of games has several applications such as scheduling games, routing games, facility location games, network design games, etc; see Ackermann et al [1], Anshelevich et al [5], Chen and Roughgarden [11], Gairing et al [18], Ieong et al [23], and Milchtaich [27]. In all of the above applications there are two fundamental goals from a system design perspective: (i) the system must be stabilizable, that is, there must exist a stable point (PNE) from which no player wants to unilaterally deviate; (ii) myopic play of the players should guide the system to a stable state.…”

“…Gairing et al [18] showed that best response dynamics do not cycle if the player-specific cost functions are linear without a constant term. Milchtaich [27] further showed that general network games with player-specific costs need not admit a PNE in general. In fact, the corresponding decision problem turns out to be NP-complete, as shown by Ackermann and Skopalik [2].…”

Strong equilibrium in cost sharing connection games

“…Topological characterizations for single-commodity network games have been recently provided for various equilibrium properties, including equilibrium existence [12,7,8], equilibrium uniqueness [10] and equilibrium efficiency [17,11]. The existence of pure Nash equilibrium in single-commodity network congestion games with player-specific costs or weights was studied in [12].…”

Weighted congestion games with separable preferences

“…This observation points to a link between the questions discussed in this paper and the equilibrium existence problem in network congestion games (Milchtaich, 2006b(Milchtaich, , 2009). However, the latter is a quite different problem in that the network topology figures prominently in it.…”