Topological Conditions for Uniqueness of Equilibrium in Networks

Abstract

Equilibrium flow in a physical network with a large number of users (e.g., transportation, communication, and computer networks) need not be unique if the costs of the network elements are not the same for all users. Such differences among users may arise if they are not equally affected by congestion or have different intrinsic preferences. Whether or not, for all assignments of strictly increasing cost functions, each user's equilibrium cost is the same in all Nash equilibria can be determined from the network topology. Specifically, this paper shows that in a two-terminal network, the equilibrium costs are always unique if and only if the network is one of several simple networks or consists of several such networks connected in series. The complementary class of all two-terminal networks with multiple equilibrium costs for some assignment of (user-specific) strictly increasing cost functions is similarly characterized by an embedded network of a particular simple type. 1. Introduction. Different kinds of networks, such as transportation, communication, and computer networks exhibit congestion effects, whereby increased demand for certain network elements (e.g., roads, telecommunication lines, and servers) tends to downgrade their performance or increase the cost of using them. In such networks, the users' decisions (e.g., choice of routes) are interdependent in that their optimal choices (e.g., the fastest routes) depend on what the others do. If they all choose optimally, given the others' choices, then the users' choices constitute a Nash equilibrium. Even if the users are identical in all respects, due to the congestion externalities, their choices at equilibrium may differ. However, if the number of users is very large and each of them has a negligibly small effect on the others, then they have equal equilibrium payoffs or costs. Moreover, the payoffs or costs in different equilibria are the same (Aashtiani and Magnanti [1]). With a heterogeneous population of users (i.e., a multiclass network; Dafermos [5]), this need not be so. As the following example shows, if the users are not identical, and are differently affected by congestion, equilibrium costs may vary not only across users, but also from one Nash equilibrium to another. Example 1.1. A continuum of three classes of users travels on the two-terminal network shown in Figure 2(a) below. Each user has to choose one of the four routes connecting the users' common point of origin o and the common destination d. The cost of each route is the sum of the costs of its edges. For each user class, the cost of each edge e is given by an increasing affine function of the fraction x of the total population with a route that includes e. The fraction of the population in each class and the corresponding cost functions are given in Figure 1, where blank cells indicate prohibitively high costs. Clearly, users in each class effectively have a choice of only two routes from o to d e 1 and e 2 e 5 for Class I users, e 2 e 5 and e 3 for Class II, and e 3 and e 4 e 5 ...

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].…”

Network topology and the efficiency of equilibrium

“…In Section 6, some of the definitions and assumptions underlying these results are discussed. In particular, the advantages of dealing with undirected rather than directed networks are explained, and the similarities and differences between the topological conditions for efficiency and uniqueness of the equilibrium (Milchtaich, 2005) are described. The proofs of all the propositions and theorems in this paper are given in Section 7.…”

The Equilibrium Existence Problem in Finite Network Congestion Games

“…1(e) but with the reverse directions for some of the edges joining u and v. These networks and those in Fig. 1 are the directed versions of the nearly parallel networks [17], which are essentially the only two-terminal networks for which uniqueness of the equilibrium in nonatomic network congestion games with playerspecific costs is guaranteed. This adds weight to Konishi's [11] observation that the conditions for topological existence (for a finite number of players with different cost functions) are similar to the conditions for topological uniqueness (for a continuum of such players).…”

“…"Embedding" is used here in a somewhat generic sense. There are at least two different meanings for this term that may be relevant in the present context [17,18]. Very roughly, they correspond to the notions of a minor and topological minor of a graph [5].…”

“…With a finite number of non-identical players and unsplittable flow, it is virtually impossible to guarantee uniqueness.) The class of all two-terminal networks for which uniqueness is guaranteed is defined by five simple kinds of networks, called the nearly par-allel networks [17]. The complementary class of all two-terminal networks for which player-specific costs can result in multiple equilibria consists of all the networks in which one of four known "forbidden" networks is embedded.…”