8 research outputs found
The Price of Anarchy in Cooperative Network Creation Games
In general, the games are played on a host graph, where each node is a
selfish independent agent (player) and each edge has a fixed link creation cost
\alpha. Together the agents create a network (a subgraph of the host graph)
while selfishly minimizing the link creation costs plus the sum of the
distances to all other players (usage cost). In this paper, we pursue two
important facets of the network creation game. First, we study extensively a
natural version of the game, called the cooperative model, where nodes can
collaborate and share the cost of creating any edge in the host graph. We prove
the first nontrivial bounds in this model, establishing that the price of
anarchy is polylogarithmic in n for all values of α in complete host
graphs. This bound is the first result of this type for any version of the
network creation game; most previous general upper bounds are polynomial in n.
Interestingly, we also show that equilibrium graphs have polylogarithmic
diameter for the most natural range of \alpha (at most n polylg n). Second, we
study the impact of the natural assumption that the host graph is a general
graph, not necessarily complete. This model is a simple example of nonuniform
creation costs among the edges (effectively allowing weights of \alpha and
\infty). We prove the first assemblage of upper and lower bounds for this
context, stablishing nontrivial tight bounds for many ranges of \alpha, for
both the unilateral and cooperative versions of network creation. In
particular, we establish polynomial lower bounds for both versions and many
ranges of \alpha, even for this simple nonuniform cost model, which sharply
contrasts the conjectured constant bounds for these games in complete (uniform)
graphs
Quality of Service in Network Creation Games
Network creation games model the creation and usage costs of networks formed
by n selfish nodes. Each node v can buy a set of edges, each for a fixed price
\alpha > 0. Its goal is to minimize its private costs, i.e., the sum (SUM-game,
Fabrikant et al., PODC 2003) or maximum (MAX-game, Demaine et al., PODC 2007)
of distances from to all other nodes plus the prices of the bought edges.
The above papers show the existence of Nash equilibria as well as upper and
lower bounds for the prices of anarchy and stability. In several subsequent
papers, these bounds were improved for a wide range of prices \alpha. In this
paper, we extend these models by incorporating quality-of-service aspects: Each
edge cannot only be bought at a fixed quality (edge length one) for a fixed
price \alpha. Instead, we assume that quality levels (i.e., edge lengths) are
varying in a fixed interval [\beta,B], 0 < \beta <= B. A node now cannot only
choose which edge to buy, but can also choose its quality x, for the price
p(x), for a given price function p. For both games and all price functions, we
show that Nash equilibria exist and that the price of stability is either
constant or depends only on the interval size of available edge lengths. Our
main results are bounds for the price of anarchy. In case of the SUM-game, we
show that they are tight if price functions decrease sufficiently fast.Comment: An extended abstract of this paper has been accepted for publication
in the proceedings of the 10th International Conference on Web and Internet
Economics (WINE
The Price of Anarchy for Network Formation in an Adversary Model
We study network formation with n players and link cost \alpha > 0. After the
network is built, an adversary randomly deletes one link according to a certain
probability distribution. Cost for player v incorporates the expected number of
players to which v will become disconnected. We show existence of equilibria
and a price of stability of 1+o(1) under moderate assumptions on the adversary
and n \geq 9.
As the main result, we prove bounds on the price of anarchy for two special
adversaries: one removes a link chosen uniformly at random, while the other
removes a link that causes a maximum number of player pairs to be separated.
For unilateral link formation we show a bound of O(1) on the price of anarchy
for both adversaries, the constant being bounded by 10+o(1) and 8+o(1),
respectively. For bilateral link formation we show O(1+\sqrt{n/\alpha}) for one
adversary (if \alpha > 1/2), and \Theta(n) for the other (if \alpha > 2
considered constant and n \geq 9). The latter is the worst that can happen for
any adversary in this model (if \alpha = \Omega(1)). This points out
substantial differences between unilateral and bilateral link formation
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..