On Nash Equilibria for Multicast Transmissions in Ad-Hoc Wireless Networks

We study a multicast game in ad-hoc wireless networks in which a source sends the same message or service to a set of receiving stations via multi-hop communications and the overall transmission cost is divided among the receivers according to given cost sharing methods. Receivers enjoy a benefit equal to the difference between the utility they get from the transmission and the shared cost they are asked to pay. Assuming a selfish and rational behavior, each user is willing to receive the transmission if and only if his shared cost does not exceed his utility. Moreover, given the strategies of the other users, he wants to select a strategy of minimum shared cost. A Nash equilibrium is a solution in which no user can increase his benefit by seceding in favor of a different strategy. We consider the following reasonable cost sharing methods: the overall transmission cost is equally shared among all the receivers (egalitarian), the cost of each intermediate station is divided among its downstream receivers equally (semi-egalitarian) or proportionally to the transmission powers they require to reach their next-hop stations (proportional), and finally each downstream user at an intermediate station equally shares only his required next-hop power among all the receivers asking at least such a level of power (Shapley value). We prove that, while the first three cost sharing methods in general do not admit a Nash equilibrium, the Shapley value yields games always converging to a Nash equilibrium. Moreover, we provide matching upper and lower bounds for the price of anarchy of the Shapley game with respect to two different global cost functions, that is the overall cost of the power assignment, that coincides with the sum of all the shared costs, and the maximum shared cost paid by the receivers. Finally, in both cases we show that finding the best Nash equilibrium is computationally intractable, that is NP-hard.