zbMATH — the first resource for mathematics

Anarchy is free in network creation. (English) Zbl 1342.68036
Bonato, Anthony (ed.) et al., Algorithms and models for the web graph. 10th international workshop, WAW 2013, Cambridge, MA, USA, December 14–15, 2013. Proceedings. Berlin: Springer (ISBN 978-3-319-03535-2/pbk). Lecture Notes in Computer Science 8305, 220-231 (2013).
Summary: The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game theoretic study of network formation, and our paper considers one of the most-well studied models, originally proposed by A. Fabrikant et al. [in: Proceedings of the 22nd annual ACM symposium on principles of distributed computing, PODC ’03, Boston, MA, USA, July 13–16, 2003. New York, NY: Association for Computing Machinery (ACM). 347–351 (2003; Zbl 1322.91013)]. In it, each of \(n\) agents corresponds to a vertex, which can create edges to other vertices at a cost of \(\alpha \) each, for some parameter \(\alpha \). Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent’s cost function is further augmented by the sum total of all (graph theoretic) distances to all other vertices.
Previous research proved that for many regimes of the \((\alpha ,n)\) parameter space, the total social cost (sum of all agents’ costs) of every Nash equilibrium is bounded by at most a constant multiple of the optimal social cost. In algorithmic game theoretic nomenclature, this approximation ratio is called the price of anarchy. In our paper, we significantly sharpen some of those results, proving that for all constant non-integral \(\alpha > 2\), the price of anarchy is in fact \(1 + o(1)\), i.e., not only is it bounded by a constant, but it tends to 1 as \(n \rightarrow \infty \). For constant integral \(\alpha \geq 2\), we show that the price of anarchy is bounded away from 1. We provide quantitative estimates on the rates of convergence for both results.
For the entire collection see [Zbl 1298.68022].

68M10 Network design and communication in computer systems
05C57 Games on graphs (graph-theoretic aspects)
91A43 Games involving graphs
91A10 Noncooperative games
68M11 Internet topics
68R10 Graph theory (including graph drawing) in computer science
Full Text: DOI arXiv