17 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
A Game-Theoretic Model Motivated by the DARPA Network Challenge
In this paper we propose a game-theoretic model to analyze events similar to
the 2009 \emph{DARPA Network Challenge}, which was organized by the Defense
Advanced Research Projects Agency (DARPA) for exploring the roles that the
Internet and social networks play in incentivizing wide-area collaborations.
The challenge was to form a group that would be the first to find the locations
of ten moored weather balloons across the United States. We consider a model in
which people (who can form groups) are located in some topology with a
fixed coverage volume around each person's geographical location. We consider
various topologies where the players can be located such as the Euclidean
-dimension space and the vertices of a graph. A balloon is placed in the
space and a group wins if it is the first one to report the location of the
balloon. A larger team has a higher probability of finding the balloon, but we
assume that the prize money is divided equally among the team members. Hence
there is a competing tension to keep teams as small as possible.
\emph{Risk aversion} is the reluctance of a person to accept a bargain with
an uncertain payoff rather than another bargain with a more certain, but
possibly lower, expected payoff. In our model we consider the \emph{isoelastic}
utility function derived from the Arrow-Pratt measure of relative risk
aversion. The main aim is to analyze the structures of the groups in Nash
equilibria for our model. For the -dimensional Euclidean space ()
and the class of bounded degree regular graphs we show that in any Nash
Equilibrium the \emph{richest} group (having maximum expected utility per
person) covers a constant fraction of the total volume
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
Stars and celebrities: A network creation game
CoRRCelebrity games, a new model of network creation games is introduced. The specific features of this model are that players have different celebrity weights and that a critical distance is taken into consideration. The aim of any player is to be close (at distance less than critical) to the others, mainly to those with high celebrity weights. The cost of each player depends on the cost of establishing direct links to other players and on the sum of the weights of those players at a distance greater than the critical distance. We show that celebrity games always have pure Nash equilibria and we characterize the family of subgames having connected Nash equilibria, the so called star celebrity games. Exact bounds for the PoA of non star celebrity games and a bound of O(n/Ă+Ă) for star celebrity games are provided.
The upper bound on the PoA can be tightened when restricted to particular classes of Nash equilibria graphs. We show that the upper bound is O(n/Ă) in the case of 2-edge-connected graphs and 2 in the case of trees.Preprin
Celebrity games
We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the player's weights) and there is a critical distance Ă as well as a link cost a. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than Ă ) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if Ă>1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance Ă introduced in BilĂČ et al. [6]. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA=PoS=max{1,W/a}; for star celebrity games PoS=1 and PoA=O(min{n/Ă,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when Ă=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for Ă=2.Peer ReviewedPostprint (author's final draft