4,791 research outputs found
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
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
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
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
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
Inequality and Network Formation Games
This paper addresses the matter of inequality in network formation games. We
employ a quantity that we are calling the Nash Inequality Ratio (NIR), defined
as the maximal ratio between the highest and lowest costs incurred to
individual agents in a Nash equilibrium strategy, to characterize the extent to
which inequality is possible in equilibrium. We give tight upper bounds on the
NIR for the network formation games of Fabrikant et al. (PODC '03) and Ehsani
et al. (SPAA '11). With respect to the relationship between equality and social
efficiency, we show that, contrary to common expectations, efficiency does not
necessarily come at the expense of increased inequality.Comment: 27 pages. 4 figures. Accepted to Internet Mathematics (2014
The Evolutionary Price of Anarchy: Locally Bounded Agents in a Dynamic Virus Game
The Price of Anarchy (PoA) is a well-established game-theoretic concept to shed light on coordination issues arising in open distributed systems. Leaving agents to selfishly optimize comes with the risk of ending up in sub-optimal states (in terms of performance and/or costs), compared to a centralized system design. However, the PoA relies on strong assumptions about agents\u27 rationality (e.g., resources and information) and interactions, whereas in many distributed systems agents interact locally with bounded resources. They do so repeatedly over time (in contrast to "one-shot games"), and their strategies may evolve.
Using a more realistic evolutionary game model, this paper introduces a realized evolutionary Price of Anarchy (ePoA). The ePoA allows an exploration of equilibrium selection in dynamic distributed systems with multiple equilibria, based on local interactions of simple memoryless agents.
Considering a fundamental game related to virus propagation on networks, we present analytical bounds on the ePoA in basic network topologies and for different strategy update dynamics. In particular, deriving stationary distributions of the stochastic evolutionary process, we find that the Nash equilibria are not always the most abundant states, and that different processes can feature significant off-equilibrium behavior, leading to a significantly higher ePoA compared to the PoA studied traditionally in the literature
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