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

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

Statics and dynamics of selfish interactions in distributed service systems

We study a class of games which model the competition among agents to access some service provided by distributed service units and which exhibit congestion and frustration phenomena when service units have limited capacity. We propose a technique, based on the cavity method of statistical physics, to characterize the full spectrum of Nash equilibria of the game. The analysis reveals a large variety of equilibria, with very different statistical properties. Natural selfish dynamics, such as best-response, usually tend to large-utility equilibria, even though those of smaller utility are exponentially more numerous. Interestingly, the latter actually can be reached by selecting the initial conditions of the best-response dynamics close to the saturation limit of the service unit capacities. We also study a more realistic stochastic variant of the game by means of a simple and effective approximation of the average over the random parameters, showing that the properties of the average-case Nash equilibria are qualitatively similar to the deterministic ones.Comment: 30 pages, 10 figure

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 ß 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 ß ) 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 ß>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 ß introduced in Bilò 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/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft