767 research outputs found
Multicast Network Design Game on a Ring
In this paper we study quality measures of different solution concepts for
the multicast network design game on a ring topology. We recall from the
literature a lower bound of 4/3 and prove a matching upper bound for the price
of stability, which is the ratio of the social costs of a best Nash equilibrium
and of a general optimum. Therefore, we answer an open question posed by
Fanelli et al. in [12]. We prove an upper bound of 2 for the ratio of the costs
of a potential optimizer and of an optimum, provide a construction of a lower
bound, and give a computer-assisted argument that it reaches for any
precision. We then turn our attention to players arriving one by one and
playing myopically their best response. We provide matching lower and upper
bounds of 2 for the myopic sequential price of anarchy (achieved for a
worst-case order of the arrival of the players). We then initiate the study of
myopic sequential price of stability and for the multicast game on the ring we
construct a lower bound of 4/3, and provide an upper bound of 26/19. To the
end, we conjecture and argue that the right answer is 4/3.Comment: 12 pages, 4 figure
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
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