A three-stage game model of the supply chain in disaster relief operations

We propose a three-stage non-cooperative game theoretic model to describe the distribution of relief supplies in response to slow-onset disasters. The first-stage game is modeled as a finite standard game, where each humanitarian organization (HO) has to decide whether to form a coalition with other HOs to negotiate framework agreements with the carriers before the disaster occurs. The second-stage game models the actual negotiation between HOs and carriers: it is a generalized Nash game between carriers, the HOs outside of the coalition, and the coalition formed in the first stage. After the disaster occurs, the third-stage game models the competition between HOs which have to purchase and distribute the relief items: it is formulated as a generalized Nash game where the HOs which had taken part in the coalition in the first stage share some constraints. First, we prove that the second-stage and third-stage games have a unique variational equilibrium. Next, under suitable assumption on the parameters, we prove that the variational equilibrium of the second-stage game can be written in closed form and the grand coalition formed by all the HOs in the first stage is the most efficient in terms of the social welfare computed at the third-stage equilibrium. Moreover, we provide sufficient conditions for the grand coalition to be a Nash equilibrium of the first-stage game. Finally, some preliminary numerical experiments are shown.