research-article

On Nash Equilibria for a Network Creation Game

Authors:

Susanne Albers,

Yishay Mansour,

Liam RodittyAuthors Info & Claims

ACM Transactions on Economics and Computation (TEAC), Volume 2, Issue 1

Article No.: 2, Pages 1 - 27

https://doi.org/10.1145/2560767

Published: 01 March 2014 Publication History

Abstract

We study a basic network creation game proposed by Fabrikant et al. [2003]. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.

Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all nontransient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this article, we disprove the tree conjecture. More precisely, we show that for any positive integer n₀, there exists a graph built by n ≥ n₀ players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1 < α ≤ √n/2. Our construction makes use of some interesting results on finite affine planes. On the other hand, we show that, for α ≥ 12n ⌈log n⌉, every Nash equilibrium forms a tree.

Without relying on the tree conjecture, Fabrikant et al. proved an upper bound on the price of anarchy of O(√α), where α ∈ [2, n². We improve this bound. Specifically, we derive a constant upper bound for α ∈ O(√n) and for α ≥ 12n ⌈log n⌉. For the intermediate values, we derive an improved bound of O(1 + (min{α²/n, n²/α})^1/3). Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.

References

[1]

N. Alon, E. D. Demaine, M. T. Hajiaghayi, and T. Leighton. 2010. Basic network creation games. In Proceedings of the 22nd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). 106--113.

Digital Library

[2]

N. Andelman, M. Feldman, and Y. Mansour. 2009. Strong price of anarchy. Games Econ. Behav. 65, 2, 289--317.

Digital Library

[3]

E. Anshelevich, A. Dasgupta, J. M. Kleinberg, E. Tardos, T. Wexler, and T. Roughgarden. 2008a. The price of stability for network design with fair cost allocation. SIAM J. Comput. 38, 4, 1602--1623.

Digital Library

[4]

E. Anshelevich, A. Dasgupta, E. Tardos, and T. Wexler. 2008b. Near-optimal network design with selfish agents. Theory Comput. 4, 1, 77--109.

Digital Library

[5]

V. Bala and S. Goyal. 2000. A non-cooperative theory of network formation. Econometrica 68, 5, 1181--1229.

[6]

A. Blokhuis and A. E. Brouwer. 1988. Geodetic graphs of diameter two. Geometricae Dedicata 25, 527--533.

[7]

H.-L. Chen and T. Roughgarden. 2009. Network design with weighted players. Theory Comput. Syst. 45, 2, 302--324.

Digital Library

[8]

J. Corbo and D. C. Parkes. 2005. The price of selfish behavior in bilateral network formation. In Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing (PODC). 99--107.

Digital Library

[9]

J. R. Correa, A. S. Schulz, and N. E. Stier Moses. 2004. Selfish routing in capacitated networks. Math. Oper. Res. 29, 4, 961--976.

Digital Library

[10]

A. Czumaj and B. Vöcking. 2007. Tight bounds for worst-case equilibria. ACM Trans. Algor. 3, 1.

Digital Library

[11]

A. Czumaj, P. Krysta, and B. Vöcking. 2002. Selfish traffic allocation for server farms. In Proceedings of the 34th Symposium on Theory of Computing. 287--296.

Digital Library

[12]

E. D. Demaine, M. T. Hajiaghayi, H. Mahini, and M. Zadimoghaddam. 2009. The price of anarchy in cooperative network creation games. SIGecom Exchanges 8, 2, 2.

Digital Library

[13]

E. D. Demaine, M. T. Hajiaghayi, H. Mahini, and M. Zadimoghaddam. 2012. The price of anarchy in network creation games. ACM Trans. Algor. 8, 2, 13.

Digital Library

[14]

R. Distel. 2000. Graph Theory. Graduate Texts in Mathematics. Springer.

[15]

S. Ehsani, M. Fazli, A. Mehrabian, S. S. Sadeghabad, M. A. Safari, M. Saghafian, and S. ShokatFadaee. 2011. On a bounded budget network creation game. In Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). 207--214.

Digital Library

[16]

A. Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, and S. Shenker. 2003. On a network creation game. In Proceedings of the 22nd Annual Symposium on Principles of Distributed Computing (PODC). 347--351.

Digital Library

[17]

D. Fotakis, S. C. Kontogiannis, E. Koutsoupias, M. Mavronicolas, and P. G. Spirakis. 2009. The structure and complexity of Nash equilibria for a selfish routing game. Theor. Comput. Sci. 410, 36, 3305--3326.

Digital Library

[18]

A. Gupta, A. Srinivasan, and E. Tardos. 2008. Cost-sharing mechanisms for network design. Algorithmica 50, 1, 98--119.

Digital Library

[19]

Y. Halevi and Y. Mansour. 2007. A network creation game with nonuniform interests. In Proceedings of the 3rd International Workshop on Internet and Network Economics (WINE). Lecture Notes in Computer Science, vol. 4858, Springer-Verlag, Berlin Heidelberg, 287--292.

Digital Library

[20]

H. Haller and S. Sarangi. 2003. Nash networks with heterogeneous agents. Working Paper, Department of Economics, Louisiana State University, no. 2003-06.

[21]

M. Jackson. 2003. A survey of models of network formation: Stability and efficiency. In Group Formation in Economics: Networks, Clubs and Coalitions, G. Demange and M. Wooders Eds.

[22]

K. Jain and V. Vazirani. 2001. Applications of approximation algorithms to cooperative games. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing. 364--372.

Digital Library

[23]

R. Johari, S. Mannor, and J. Tsitsiklis. 2006. A contract-based model for directed network formation. Games Econ. Behav. 56, 2, 201--224.

[24]

E. Koutsoupias and C. H. Papadimitriou. 2009. Worst-case equilibria. Comput. Sci. Rev. 3, 2, 65--69.

Digital Library

[25]

N. Laoutaris, L. J. Poplawski, R. Rajaraman, R. Sundaram, and S.-H. Teng. 2008. Bounded budget connection (BBC) games or how to make friends and influence people, on a budget. In Proceedings of the 27th Annual ACM Symposium on Principles of Distributed Computing (PODC). 165--174.

Digital Library

[26]

H. Lin. 2003. On the price of anarchy of a network creation game. Class final project, University of California, Berkeley.

[27]

F. J. MacWilliams and N. J. A. Sloane. 1978. The Theory of Error-Correcting Codes. North Holland Publishing.

[28]

M. Mihalak and J. C. Schlegel. 2010. The price of anarchy in network creation games is (mostly) constant. In Proceedings of the 3rd International Symposium on Algorithmic Game Theory (SAGT). Lecture Notes in Computer Science, vol. 6386. Springer-Verlag, Berlin Heidelberg, 276--287.

Digital Library

[29]

J. F. Nash. 1951. Non-cooperative games. Ann. Math. 54, 286--295.

[30]

M. Pál and E. Tardos. 2003. Group strategyproof mechanisms via primal-dual algorithms. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science. 584--593.

Digital Library

[31]

T. Roughgarden and E. Tardos. 2002. How bad is selfish routing? J. ACM 49, 2, 236--259.

Digital Library

[32]

O. Rozenfeld and M. Tennenholtz. 2010. Near-strong equilibria in network creation games. In Proceedings of the 6th International Workshop on Internet and Network Economics (WINE). Lecture Notes in Computer Science, vol. 6484. Springer-Verlag, Berlin Heidelberg, 339--353.

Digital Library

Cited By

Bilò DFriedrich TLenzner PMelnichenko A(2024)Geometric Network Creation GamesSIAM Journal on Discrete Mathematics10.1137/20M137666238:1(277-315)Online publication date: 11-Jan-2024
https://doi.org/10.1137/20M1376662
Abiad AÀlvarez CMessegué A(2024)The diameter of sum basic equilibria gamesTheoretical Computer Science10.1016/j.tcs.2024.1148071018(114807)Online publication date: Nov-2024
https://doi.org/10.1016/j.tcs.2024.114807
Friedrich TGawendowicz HLenzner PZahn AOshman RNolin AHalldorsson MBalliu A(2023)The Impact of Cooperation in Bilateral Network CreationProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594588(321-331)Online publication date: 19-Jun-2023
https://dl.acm.org/doi/10.1145/3583668.3594588
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Index Terms

On Nash Equilibria for a Network Creation Game
1. Theory of computation

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Published In

cover image ACM Transactions on Economics and Computation

ACM Transactions on Economics and Computation Volume 2, Issue 1

March 2014

93 pages

ISSN:2167-8375

EISSN:2167-8383

DOI:10.1145/2597758

Editors:
Vincent Conitzer
Duke University
,
Preston McAfee
Google

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Copyright © 2014 ACM.

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Publication History

Published: 01 March 2014

Accepted: 01 June 2013

Received: 01 April 2013

Published in TEAC Volume 2, Issue 1

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Cited By

Bilò DFriedrich TLenzner PMelnichenko A(2024)Geometric Network Creation GamesSIAM Journal on Discrete Mathematics10.1137/20M137666238:1(277-315)Online publication date: 11-Jan-2024
https://doi.org/10.1137/20M1376662
Abiad AÀlvarez CMessegué A(2024)The diameter of sum basic equilibria gamesTheoretical Computer Science10.1016/j.tcs.2024.1148071018(114807)Online publication date: Nov-2024
https://doi.org/10.1016/j.tcs.2024.114807
Friedrich TGawendowicz HLenzner PZahn AOshman RNolin AHalldorsson MBalliu A(2023)The Impact of Cooperation in Bilateral Network CreationProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594588(321-331)Online publication date: 19-Jun-2023
https://dl.acm.org/doi/10.1145/3583668.3594588
Àlvarez CBuisan A(2023)On the PoA Conjecture: Trees versus Biconnected ComponentsSIAM Journal on Discrete Mathematics10.1137/21M146642637:2(1030-1052)Online publication date: 15-Jun-2023
https://doi.org/10.1137/21M1466426
Avarikioti ZLizurej TMichalak TYeo M(2023)Lightning Creation Games2023 IEEE 43rd International Conference on Distributed Computing Systems (ICDCS)10.1109/ICDCS57875.2023.00037(1-11)Online publication date: Jul-2023
https://doi.org/10.1109/ICDCS57875.2023.00037
Dresler MMansour STalhaoui SYamauchi YTixeuil S(2023)On Dynamics of Basic Network Creation Games with Non-Uniform Communication Interest2023 Eleventh International Symposium on Computing and Networking Workshops (CANDARW)10.1109/CANDARW60564.2023.00023(86-92)Online publication date: 27-Nov-2023
https://doi.org/10.1109/CANDARW60564.2023.00023
Friedrich TGawendowicz HLenzner PMelnichenko A(2023)Social Distancing Network CreationAlgorithmica10.1007/s00453-022-01089-685:7(2087-2130)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1007/s00453-022-01089-6
Wang Q(2022)On Tree Equilibria in Max-Distance Network Creation GamesAlgorithmic Game Theory10.1007/978-3-031-15714-1_17(293-310)Online publication date: 12-Sep-2022
https://dl.acm.org/doi/10.1007/978-3-031-15714-1_17
Dippel JVetta A(2022)An Improved Bound for the Tree Conjecture in Network Creation GamesAlgorithmic Game Theory10.1007/978-3-031-15714-1_14(241-257)Online publication date: 12-Sep-2022
https://dl.acm.org/doi/10.1007/978-3-031-15714-1_14
Bilò DGualà LLeucci SProietti G(2021)Network Creation Games with Traceroute-Based StrategiesAlgorithms10.3390/a1402003514:2(35)Online publication date: 26-Jan-2021
https://doi.org/10.3390/a14020035
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