An integrated financial and logistical game theory model for humanitarian organizations with purchasing costs, multiple freight service providers, and budget, capacity, and demand constraints

In this paper, a game theory model for disaster relief is constructed that incorporates both financial and logistical aspects of humanitarian organizations involved in the purchasing and delivery of relief items, post-disaster, using freight services. The model allows for the purchasing of the relief items, both locally and nonlocally, includes a budget constraint for each humanitarian organization, along with imposed lower and upper bound demand constraints at each point of demand by a higher level organization. The governing concept is that of Generalized Nash Equilibrium, since not only does the utility function of a given humanitarian organization depend on its own strategies and the strategies of the other humanitarian organizations, but the constraints do as well. The concept of a variational equilibrium is utilized to derive the variational inequality formulation of the governing equilibrium conditions and the model is analyzed qualitatively. Lagrange analysis of the marginal utilities is conducted to gain insights on the impact of the constraints and an alternative variational inequality constructed, with nice features for computations. An algorithm is proposed and explicit formulae provided for the logistical flows and Lagrange multipliers at each iteration. Numerical examples, inspired by Hurricane Harvey hitting Houston, Texas, as occurred in August 2017, illustrate the framework. This work adds to the still nascent literature on game theory and disaster relief and also to the literature on variational inequality models with nonlinear constraints, which is limited.