Efficient Effort Equilibrium in Cooperation with Pairwise Cost Reduction

There are multiple situations in which bilateral interaction between agents results in considerable cost reductions. Such interaction can occur in settings where agents are interested in sharing resources, knowledge or infrastructures. Their common purpose is to obtain individual advantages, e.g. by reducing their respective individual costs. Achieving this pairwise cooperation often requires the agents involved to make some level of effort. It is natural to think that the amount by which one agent could reduce the costs of the other may depend on how much effort the latter exerts. In the first stage, agents decide how much effort they are to exert, which has a direct impact on their pairwise cost reductions. We model this first stage as a non-cooperative game, in which agents determine the level of pairwise effort to reduce the cost of their partners. In the second stage, agents engage in a bilateral interaction between independent partners. We study this bilateral cooperation 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. In the non-cooperative game that precedes cooperation with pairwise cost reduction, the agents anticipate the cost allocation that results from the cooperative game in the second stage by incorporating the effect of the effort exerted into their cost functions. Based on this model, we explore the costs, benefits, and challenges associated with setting up a pairwise effort network. We identify a family of cost allocations with weighted pairwise reduction which are always feasible in the cooperative game and contain the Shapley value. We show that there are always cost allocations with weighted pairwise reductions that generate an optimal level of efficient effort and provide a procedure for finding the efficient effort equilibrium.