Online multicast with egalitarian cost sharing

We consider a multicast game played by a set of selfish noncoop-erative players (i.e., nodes) on a rooted undirected graph. Players arrive one by one and each connects to the root by greedily choosing a path minimizing its cost; the cost of using an edge is split equally among all users using the edge. How large can the sum of the players' costs be, compared to the cost of a "^{socially optimal} solution, defined to be a minimum Steiner tree connecting the players to the root? We show that the ratio is O(log² n) and Ω(log n), when there are n players. One can view this multicast game as a variant of ONLINE STEINER TREE with a different cost sharing mechanism. Furthermore, we consider what happens if the players, in a second phase, are allowed to change their paths in order to decrease their costs. Thus, in the second phase players play best response dynamics until eventually a Nash equilibrium is reached. We show that the price of anarchy is O(log³ n) and Ω(log n). We also make progress towards understanding the challenging case where arrivals and path changes by existing terminals are interleaved. In particular, we analyze the interesting special case where the terminals fire in random order and prove that the cost of the solution produced (with arbitrary interleaving of actions) is at most O(polylog(n) √n) times the optimum.