Stars and Celebrities: A Network Creation Game

Celebrity 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 subgraphs 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/β + β) 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/β) in the case of 2-edge-connected graphs and 2 in the case of trees.