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MATHEMATICS OF OPERATIONS RESEARCH Vol. 30, No. 1, February 2005, pp. 225–244 issn 0364-765X ± eissn 1526-5471 ± 05 ± 3001 ± 0225 inf orms ® doi 10.1287/moor.1040.0122 © 2005 INFORMS Topological Conditions for Uniqueness of Equilibrium in Networks Igal Milchtaich Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel, [email protected] , http://faculty.biu.ac.il/ milchti 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. Speci±cally, 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-speci±c) strictly increasing cost functions is similarly characterized by an embedded network of a particular simple type. Key words : congestion externalities; nonatomic games; transportation networks; network topology; uniqueness of equilibrium MSC2000 subject classifcation : Primary: 90B10; secondary: 91A13 OR/MS subject classifcation : Primary: Networks/graphs; secondary: games/group decisions, noncooperative History : Received August 28, 2003; revised March 31, 2004. 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.
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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