Network Creation Games with Local Information and Edge Swaps

In the swap game (SG), selfish players, each of which is associated with a vertex, form a graph by edge swaps, i.e., a player changes its strategy by simultaneously removing an adjacent edge and forming a new edge (Alon et al. 2013). The cost of a player considers the average distance to all other players or the maximum distance to other players. Any SG by n players starting from a tree converges to an equilibrium with a constant Price of Anarchy (PoA) within O(n(3)) edge swaps (Lenzner 2011). We focus on SGs where each player knows the subgraph induced by players within distance k. Therefore, each player cannot compute its cost nor a best response. We first consider pessimistic players who consider the worst-case global graph. We show that any SG starting from a tree (i) always converges to an equilibrium within O(n3) edge swaps irrespective of the value of k, (ii) the PoA is Theta(n) for k = 1, 2, 3, and (iii) the PoA is constant for k >= 4. We then introduce weakly pessimistic players and optimistic players and show that these less pessimistic players achieve constant PoA for k <= 3 at the cost of best response cycles.