Geometric Network Design with Selfish Agents

We study a geometric version of a simple non-cooperative network creation game introduced in [2], assuming Euclidean edge costs on the plane. The price of anarchy in such geometric games with k players is Θ(k). Hence, we consider the task of minimizing players incentives to deviate from a payment scheme, purchasing the minimum cost network. In contrast to general games, in small geometric games (2 players and 2 terminals per player), a Nash equilibrium purchasing the optimum network exists. This can be translated into a (1+e)-approximate Nash equilibrium purchasing the optimum network under more practical assumptions, for any e > 0. For more players there are games with 2 terminals per player, such that any Nash equilibrium purchasing the optimum solution is at least $\left(\frac{4}{3}-\epsilon\right)$-approximate. On the algorithmic side, we show that playing small games with best-response strategies yields low-cost Nash equilibria. The distinguishing feature of our paper are new techniques to deal with the geometric setting, fundamentally different from the techniques used in [2].