Selfishness, Collusion and Power of Local Search for the ADMs Minimization Problem

We consider non-cooperative games in all-optical networks where users share the cost of the used ADM switches for realizing given communication patterns. We show that the two fundamental cost sharing methods, Shapley and Egalitarian, induce polynomial converging games with price of anarchy at most 5/3, regardless of the network topology. Such a bound is tight even for rings. Then, we show that if collusion of at most k players is allowed, the Egalitarian method yields polynomially converging games with price of collusion between 3/2 and 3/2 + 1/k. This result is very interesting and quite surprising, as the best known approximation ratio, that is 3/2 + epsilon, can be achieved in polynomial time by uncoordinated evolutions of collusion games with coalitions of increasing size. Finally, the Shapley method does not induce well defined collusion games, but can be exploited in the definition of local search algorithms with local optima arbitrarily close to optimal solutions. This would potentially generate PTAS, but unfortunately the arising algorithm might not converge. The determination of new cost sharing methods or local search algorithms reaching a compromise between Shapley and Egalitarian is thus outlined as being a promising and worth pursuing investigating direction.