Efficient effort equilibrium in cooperation with pairwise cost reduction

There are multiple situations in which bilateral interaction between agents results in considerable cost reductions. The cost reduction that an agent obtains depends on the effort made by other agents. We model this situation as a bi-form game with two states. In the first stage, agents decide how much effort to exert. We model this first stage as a non-cooperative game, in which these efforts will reduce the cost of their partners in the second stage. This second stage is modeled as a cooperative game in which agents reduce each other’s costs as a result of cooperation, so that the total reduction in the cost of each agent in a coalition is the sum of the reductions generated by the rest of the members of that coalition. The proposed cost allocation for the cooperative game in the second stage determines the payoff function of the non-cooperative game in the first stage. Based on this model, we explore the costs, benefits, and challenges associated with setting up a pairwise effort situation. We identify a family of cost allocations with weighted pairwise reductions which are always feasible in the cooperative game and contain the Shapley value. We also identify the cost allocation with weighted pairwise reductions that generate an efficient equilibrium effort level.