This is a study of a multicast game in ad-hoc wireless networks. In studying such games, an objective is to achieve a solution, called a Nash equilibrium, in which no player can increase his benefit by choosing to adopt a different strategy. Another key concept is the measuring of the price of anarchy, the ratio between the maximum global cost of a Nash equilibrium and minimal global cost of the optimal solution that could be achieved if all players cooperate.

A contribution here is the study of games induced by six reasonable cost-sharing methods: an egalitarian way, a semi-egalitarian way, a next-hop proportional way, a path proportional way, an egalitarian path proportional way, and applying the definition of Shapley value from a referenced work. It is proven that the first five methods do not admit a Nash equilibrium, and the Shapley value method always converges to a Nash equilibrium. By adopting a special case, two additional cost-sharing methods from the above list yield convergent games.

Other contributions are the open questions and suggestions for directions for further research. Some open questions are: achieving convergence in a polynomial number of steps (the determination of any Nash equilibrium, even nonoptimal, would solve this too), the fact that there are convergent cost-sharing methods yielding a better price of anarchy, and extending the work of the paper to particular cases (the paper suggests two). These give the reader insight into the status of this research.