Coalition formation resource sharing games in networks

Cooperative game theory deals with systems where players want to cooperate to improve their payoffs. But players may choose coalitions in a non-cooperative manner, leading to a coalition-formation game. We consider such a game with several players (willing to cooperate) and a possible adamant player (unwilling to cooperate) involved in resource-sharing. Here, the strategy of a player is the set of players with whom it wants to form a coalition. Given a strategy profile, an appropriate partition of coalitions is formed; players in each coalition maximize their collective utilities leading to a non-cooperative resource-sharing game among the coalitions, the (unique) utilities at the resulting equilibrium are shared via Shapley-value; these shares define the utilities of players for the given strategy profile in the coalition-formation game. We also consider the utilitarian solution to derive the price of anarchy. With a sufficient number of almost similar players, no collaborative (players working together) partition emerges at equilibrium. But the scenario reverses when the players are significantly different: the system is in a ‘lethargic state’ where almost all partitions are stable against unilateral deviations. In the intermediate cases, the higher and lower influential players have more affinity to form successful collaborations. Grand coalition is the utilitarian solution and is also stable against coalitional deviations under certain conditions. The price of anarchy increases as the number of significant players increases or as the relative strength of adamant player increases/decreases.