Given a graph , we define a space of subgraphs M with the binary operation of union and the unique decomposition property into blocks. This space allows us to discuss a notion of minimal subgraphs (minimal coalitions) that are of interest for the game. Additionally, a partition of the game is defined in terms of the gain of each block, and subsequently, a solution to the game is defined based on distributing to each player (node and edge) present in each block a payment proportional to their contribution to the coalition.
References
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Myerson, R.B. (1977) Graphs and Cooperation in Games. Mathematics of Operations Research, 2, 225-229. http://www.jstor.org/stable/3689511
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Owen, G. (1986) Values of Graph-Restricted Games. Society for Industrial and Applied Mathematics, 7, 210-221. https://doi.org/10.1137/0607025
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Caulier, J.-F., Skoda, A. and Tanimura, E. (2017) Allocation Rules for Networks Inspired by Cooperative Game-Theory. Revue d’économie Politique, 127, 517-558. https://doi.org/10.3917/redp.274.0517
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Alarcón, A.C., Gallardo, J.M. and Jiménez-Losada, A. (2022) A Value for Graph-Restricted Games with Middlemen on Edges. Mathematics, 10, 1856. https://www.mdpi.com/2227-7390/10/11/1856 https://doi.org/10.3390/math10111856
