29 research outputs found
The Price of Anarchy in Network Creation Games Is (Mostly) Constant
We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2, ,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1, ,n, is to build some edges adjacent to i. The cost of building an edge is α>0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α>273⋅n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α>129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field—is price of anarchy of the network creation game constant for all values of α?—in an affirmative way, up to a tiny range of
On the Structure of Equilibria in Basic Network Formation
We study network connection games where the nodes of a network perform edge
swaps in order to improve their communication costs. For the model proposed by
Alon et al. (2010), in which the selfish cost of a node is the sum of all
shortest path distances to the other nodes, we use the probabilistic method to
provide a new, structural characterization of equilibrium graphs. We show how
to use this characterization in order to prove upper bounds on the diameter of
equilibrium graphs in terms of the size of the largest -vicinity (defined as
the the set of vertices within distance from a vertex), for any
and in terms of the number of edges, thus settling positively a conjecture of
Alon et al. in the cases of graphs of large -vicinity size (including graphs
of large maximum degree) and of graphs which are dense enough.
Next, we present a new swap-based network creation game, in which selfish
costs depend on the immediate neighborhood of each node; in particular, the
profit of a node is defined as the sum of the degrees of its neighbors. We
prove that, in contrast to the previous model, this network creation game
admits an exact potential, and also that any equilibrium graph contains an
induced star. The existence of the potential function is exploited in order to
show that an equilibrium can be reached in expected polynomial time even in the
case where nodes can only acquire limited knowledge concerning non-neighboring
nodes.Comment: 11 pages, 4 figure
Tree Nash Equilibria in the Network Creation Game
In the network creation game with n vertices, every vertex (a player) buys a
set of adjacent edges, each at a fixed amount {\alpha} > 0. It has been
conjectured that for {\alpha} >= n, every Nash equilibrium is a tree, and has
been confirmed for every {\alpha} >= 273n. We improve upon this bound and show
that this is true for every {\alpha} >= 65n. To show this, we provide new and
improved results on the local structure of Nash equilibria. Technically, we
show that if there is a cycle in a Nash equilibrium, then {\alpha} < 65n.
Proving this, we only consider relatively simple strategy changes of the
players involved in the cycle. We further show that this simple approach cannot
be used to show the desired upper bound {\alpha} < n (for which a cycle may
exist), but conjecture that a slightly worse bound {\alpha} < 1.3n can be
achieved with this approach. Towards this conjecture, we show that if a Nash
equilibrium has a cycle of length at most 10, then indeed {\alpha} < 1.3n. We
further provide experimental evidence suggesting that when the girth of a Nash
equilibrium is increasing, the upper bound on {\alpha} obtained by the simple
strategy changes is not increasing. To the end, we investigate the approach for
a coalitional variant of Nash equilibrium, where coalitions of two players
cannot collectively improve, and show that if {\alpha} >= 41n, then every such
Nash equilibrium is a tree
Multilevel Network Games
We consider a multilevel network game, where nodes can improve their
communication costs by connecting to a high-speed network. The nodes are
connected by a static network and each node can decide individually to become a
gateway to the high-speed network. The goal of a node is to minimize its
private costs, i.e., the sum (SUM-game) or maximum (MAX-game) of communication
distances from to all other nodes plus a fixed price if it
decides to be a gateway. Between gateways the communication distance is ,
and gateways also improve other nodes' distances by behaving as shortcuts. For
the SUM-game, we show that for , the price of anarchy is
and in this range equilibria always exist. In range
the price of anarchy is , and
for it is constant. For the MAX-game, we show that the
price of anarchy is either , for ,
or else . Given a graph with girth of at least , equilibria always
exist. Concerning the dynamics, both the SUM-game and the MAX-game are not
potential games. For the SUM-game, we even show that it is not weakly acyclic.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
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 Max-Distance Network Creation Game on General Host Graphs
In this paper we study a generalization of the classic \emph{network creation
game} in the scenario in which the players sit on a given arbitrary
\emph{host graph}, which constrains the set of edges a player can activate at a
cost of each. This finds its motivations in the physical
limitations one can have in constructing links in practice, and it has been
studied in the past only when the routing cost component of a player is given
by the sum of distances to all the other nodes. Here, we focus on another
popular routing cost, namely that which takes into account for each player its
\emph{maximum} distance to any other player. For this version of the game, we
first analyze some of its computational and dynamic aspects, and then we
address the problem of understanding the structure of associated pure Nash
equilibria. In this respect, we show that the corresponding price of anarchy
(PoA) is fairly bad, even for several basic classes of host graphs. More
precisely, we first exhibit a lower bound of
for any . Notice that this implies a counter-intuitive lower
bound of for very small values of (i.e., edges can
be activated almost for free). Then, we show that when the host graph is
restricted to be either -regular (for any constant ), or a
2-dimensional grid, the PoA is still , which is proven to be tight for
. On the positive side, if , we show
the PoA is . Finally, in the case in which the host graph is very sparse
(i.e., , with ), we prove that the PoA is , for any
.Comment: 17 pages, 4 figure
Selfish Network Creation with Non-Uniform Edge Cost
Network creation games investigate complex networks from a game-theoretic
point of view. Based on the original model by Fabrikant et al. [PODC'03] many
variants have been introduced. However, almost all versions have the drawback
that edges are treated uniformly, i.e. every edge has the same cost and that
this common parameter heavily influences the outcomes and the analysis of these
games.
We propose and analyze simple and natural parameter-free network creation
games with non-uniform edge cost. Our models are inspired by social networks
where the cost of forming a link is proportional to the popularity of the
targeted node. Besides results on the complexity of computing a best response
and on various properties of the sequential versions, we show that the most
general version of our model has constant Price of Anarchy. To the best of our
knowledge, this is the first proof of a constant Price of Anarchy for any
network creation game.Comment: To appear at SAGT'1
On a Bounded Budget Network Creation Game
We consider a network creation game in which each player (vertex) has a fixed
budget to establish links to other players. In our model, each link has unit
price and each agent tries to minimize its cost, which is either its local
diameter or its total distance to other players in the (undirected) underlying
graph of the created network. Two versions of the game are studied: in the MAX
version, the cost incurred to a vertex is the maximum distance between the
vertex and other vertices, and in the SUM version, the cost incurred to a
vertex is the sum of distances between the vertex and other vertices. We prove
that in both versions pure Nash equilibria exist, but the problem of finding
the best response of a vertex is NP-hard. We take the social cost of the
created network to be its diameter, and next we study the maximum possible
diameter of an equilibrium graph with n vertices in various cases. When the sum
of players' budgets is n-1, the equilibrium graphs are always trees, and we
prove that their maximum diameter is Theta(n) and Theta(log n) in MAX and SUM
versions, respectively. When each vertex has unit budget (i.e. can establish
link to just one vertex), the diameter of any equilibrium graph in either
version is Theta(1). We give examples of equilibrium graphs in the MAX version,
such that all vertices have positive budgets and yet the diameter is
Omega(sqrt(log n)). This interesting (and perhaps counter-intuitive) result
shows that increasing the budgets may increase the diameter of equilibrium
graphs and hence deteriorate the network structure. Then we prove that every
equilibrium graph in the SUM version has diameter 2^O(sqrt(log n)). Finally, we
show that if the budget of each player is at least k, then every equilibrium
graph in the SUM version is k-connected or has diameter smaller than 4.Comment: 28 pages, 3 figures, preliminary version appeared in SPAA'1