Topological Conditions for Uniqueness of Equilibrium in Networks

Milchtaich, Igal

doi:10.1287/moor.1040.0122

Cited by 68 publications

(69 citation statements)

References 16 publications

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“…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].…”

Section: Related Workmentioning

confidence: 99%

Strong equilibrium in cost sharing connection games

Epstein

Feldman

Mansour

2007

Proceedings of the 8th ACM Conference on Electronic Commerce

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In this work we study cost sharing connection games, where each player has a source and sink he would like to connect, and the cost of the edges is either shared equally (fair connection games) or in an arbitrary way (general connection games). We study the graph topologies that guarantee the existence of a strong equilibrium (where no coalition can improve the cost of each of its members) regardless of the specific costs on the edges.Our main existence results are the following: (1) For a single source and sink we show that there is always a strong equilibrium (both for fair and general connection games).(2) For a single source multiple sinks we show that for a series parallel graph a strong equilibrium always exists (both for fair and general connection games). (3) For multi source and sink we show that an extension parallel graph always admits a strong equilibrium in fair connection games.As for the quality of the strong equilibrium we show that in any fair connection games the cost of a strong equilibrium is Θ(log n) from the optimal solution, where n is the number of players. (This should be contrasted with the Ω(n) price of anarchy for the same setting.) For single source general connection games and single source single sink fair connection games, we show that a strong equilibrium is always an optimal solution.

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Section: Related Workmentioning

confidence: 99%

Strong equilibrium in cost sharing connection games

Epstein

Feldman

Mansour

2007

Proceedings of the 8th ACM Conference on Electronic Commerce

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Network topology and the efficiency of equilibrium

Milchtaich

2006

Games and Economic Behavior

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Abstract. Different kinds of networks, such as transportation, communication, computer, and supply networks, are susceptible to similar kinds of inefficiencies. These arise when congestion externalities make the cost for each user depend on the other users' choice of routes. If each user chooses the least expensive (e.g., the fastest) route from the users' common point of origin to the common destination, the result may be Pareto inefficient in that an alternative choice of routes would reduce the costs for all users. Braess's paradox represents an extreme kind of inefficiency, in which the equilibrium costs may be reduced by raising the cost curves. As this paper shows, this paradox occurs in an (undirected) twoterminal network if and only if it is not series-parallel. More generally, Pareto inefficient equilibria occur in a network if and only if one of three simple networks is embedded in it. JEL Classification: C72, R41.

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“…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).…”

Section: Or Can Be Obtained By Connecting Several Of These Network Imentioning

confidence: 96%

“…"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].…”

Section: Examplementioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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The Equilibrium Existence Problem in Finite Network Congestion Games

Milchtaich

2006

Lecture Notes in Computer Science

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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.

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Topological Conditions for Uniqueness of Equilibrium in Networks

Cited by 68 publications

References 16 publications

Strong equilibrium in cost sharing connection games

Strong equilibrium in cost sharing connection games

Network topology and the efficiency of equilibrium

The Equilibrium Existence Problem in Finite Network Congestion Games

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