In this paper, we present a new Generalized Nash Equilibrium (GNE) model for post-disaster humanitarian relief by introducing novel financial funding functions and altruism functions, and by also capturing competition on the logistics side among humanitarian organizations. The common, that is, the shared, constraints associated with the relief item deliveries at points of need are imposed by an upper level humanitarian organization or regulatory body and consist of lower and upper bounds to ensure the effective delivery of the estimated volumes of supplies to the victims of the disaster. We identify the network structure of the problem, with logistical and financial flows, and propose a variational equilibrium framework, which allows us to then formulate, analyze, and solve the model using the theory of variational inequalities (rather than quasivariational inequality theory). We then utilize Lagrange analysis and investigate qualitatively the humanitarian organizations’ marginal utilities if and when the equilibrium relief item flows are (or are not) at the imposed demand point bounds. We illustrate the game theory model through a case study focused on tornadoes hitting western Massachusetts, a highly unusual event that occurred in 2011. This work significantly extends the original model of Nagurney, Alvarez Flores, and Soylu (2016), which, under the imposed assumptions therein, allowed for an optimization formulation, and adds to the literature of game theory and disaster relief, which is nascent.