Mobile Network Creation Games

We introduce a new class of network creation games, called mobile network creation games, modelling the spontaneous creation of communication networks by the distributed and uncoordinated interaction of k selfish mobile devices. Each device is owned by a player able to select a node in an underlying positions graph so as to minimize a cost function taking into account two components: the distance from her home position, and the number of players she is not connected to, with the connectivity costs being prevailing, i.e., the Nash Equilibria are stable solution states in which communication is possible among all the players. We show that the game always admits a Pure Nash equilibrium, even if the convergence after a finite number of improving movements is guaranteed only when players perform their best possible moves. More precisely, if initial positions are arbitrary, that is not necessarily coinciding with the home ones, an order of kD best moves is necessary (and sufficient) to reach an equilibrium, where D is the diameter of the positions graph. As for the Nash equilibria performances, we first prove that the price of stability is 1 (i.e. an optimal solution is also a Nash equilibrium). Moreover, we show that the lack of centralized control of mobile devices is a major issue in terms of final performance guaranteed. Namely, the price of anarchy is Θ(kD). Nevertheless, we are able to prove that if players start at their home positions, in Θ(k min {k2,D}) best moves they reach an equilibrium approximating the optimal solution by a factor of Θ(k min {k,D}).