Diffusion control of wildland fire via a cooperative game-theoretic model

Various modeling approaches have been proposed in the literature to forecast the evolution of wildland fires over time and space and to control them. Here, a methodology based on cooperative game theory is proposed to optimize the use of limited resources to control a fire, such as firefighters, Canadair, drones, etc. In particular, the Shapley value, i.e., a solution concept of cooperative games, is exploited. It provides a measure of the value of each player in a so-called transferable-utility game, and recently it has been used to evaluate the importance of edges or nodes in a network. In the proposed approach, the wildland area is modeled as a network, in which nodes represent areas of particular interest. Some of them may become fire outbreaks, from which the fire can reach other nodes. The edges represent possible directions of fire evolution and can be weighted by taking into account several features, including elevation of the terrain, kind of vegetation, and estimated rate of spread. A temperature diffusion process is considered, where the global rate of diffusion is given by the second smallest eigenvalue of the weighted Laplacian matrix of the network. Then, a transferable-utility game is defined, where players form a subset of the edge set, and the utility function is related to the decrease of the global rate of diffusion of the fire when one or more edges are removed. This corresponds to control diffusion paths from a fire outbreak. The Shapley values are exploited to decide the order in which the edges should be removed, that is, how one should act with fire-extinguishing resources, in such a way as to effectively reduce the global fire diffusion rate.