Exact and approximate equilibria for optimal group network formation

Abstract We consider a process called the Group Network Formation Game, which represents the scenario when strategic agents are building a network together. In our game, agents can have extremely varied connectivity requirements, and attempt to satisfy those requirements by purchasing links in the network. We show a variety of results about equilibrium properties in such games, including the fact that the price of stability is 1 when all nodes in the network are owned by players, and that doubling the number of players creates an equilibrium as good as the optimum centralized solution. For the general case, we show the existence of a 2-approximate Nash equilibrium that is as good as the centralized optimum solution, as well as how to compute good approximate equilibria in polynomial time. Our results essentially imply that for a variety of connectivity requirements, giving agents more freedom can paradoxically result in more efficient outcomes.

[1]  Elias Koutsoupias,et al.  On the Price of Anarchy and Stability of Correlated Equilibria of Linear Congestion Games , 2005, ESA.

[2]  Haim Kaplan,et al.  On the Price of Stability for Designing Undirected Networks with Fair Cost Allocations , 2006, ICALP.

[3]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[4]  Elliot Anshelevich,et al.  Exact and Approximate Equilibria for Optimal Group Network Formation , 2009, ESA.

[5]  Fabrizio Grandoni,et al.  An improved LP-based approximation for steiner tree , 2010, STOC '10.

[6]  Martin Hoefer,et al.  Geometric Network Design with Selfish Agents , 2005, COCOON.

[7]  Tim Roughgarden,et al.  Network Design with Weighted Players , 2006, SPAA '06.

[8]  Éva Tardos,et al.  Near-optimal network design with selfish agents , 2003, STOC '03.

[9]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[10]  Robert W. Rosenthal,et al.  The network equilibrium problem in integers , 1973, Networks.

[11]  Alex Zelikovsky,et al.  Tighter Bounds for Graph Steiner Tree Approximation , 2005, SIAM J. Discret. Math..

[12]  Christos H. Papadimitriou,et al.  The complexity of pure Nash equilibria , 2004, STOC '04.

[13]  Tim Roughgarden,et al.  Designing networks with good equilibria , 2008, SODA '08.

[14]  Elliot Anshelevich,et al.  Price of Stability in Survivable Network Design , 2011, Theory of Computing Systems.

[15]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[16]  M. Jackson.  A Survey of Models of Network Formation: Stability and Efficiency , 2003 .

[17]  Gabrielle Demange,et al.  A Survey of Network Formation Models: Stability and Efficiency , 2005 .

[18]  Joseph Naor,et al.  Non-Cooperative Multicast and Facility Location Games , 2007, IEEE J. Sel. Areas Commun..

[19]  Yishay Mansour,et al.  Strong equilibrium in cost sharing connection games , 2007, EC '07.

[20]  Elliot Anshelevich,et al.  On Survivable Access Network Design: Complexity and Algorithms , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.

[21]  David P. Williamson,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[22]  R. Holzman,et al.  Strong Equilibrium in Congestion Games , 1997 .

[23]  Tim Roughgarden,et al.  The price of stability for network design with fair cost allocation , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[24]  Martin Hoefer,et al.  Non-cooperative facility location and covering games , 2006, Theor. Comput. Sci..

[25]  Martin Hoefer,et al.  Non-Cooperative Tree Creation , 2006, Algorithmica.

[26]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[27]  Tim Roughgarden,et al.  Selfish routing and the price of anarchy , 2005 .

[28]  Elliot Anshelevich,et al.  Terminal Backup, 3D Matching, and Covering Cubic Graphs , 2011, SIAM J. Comput..