图博弈的过程比例解

doi:10.15960/j.cnki.issn.1007-6093.2021.04.009

Abstract

Abstract:

This paper studies cooperative games with graph structure, shortly graph games. By applying the proportional rule and procedural method in feasible coalitions, a so-called proportional procedure is defined and studied regarding concepts and properties. And then, according to the over-efficiency of the proposed procedure, an efficient solution concept, the procedural proportional solution, is provided by compromising with equalratio. Moreover, the stability of the solution is also investigated. Finally, the bankruptcy problems is discussed by using the proportional procedure and the procedural proportional solution, and we obtain that the proposed solution of the related graph game coincides with the proportional rule of the bankruptcy problem.

Key words: cooperative game, graph structure, proportional procedure, stability

CLC Number:

O225
F224.32

Guang ZHANG, Wenjun XIAO, Donghua WU. The procedural proportional solution for graph games[J]. Operations Research Transactions, 2021, 25(4): 101-110.

Figures/Tables 3

References 15

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$ S(\in C^L(N)) $	$ \{1\} $	$ \{2\} $	$ \{3\} $	$ \{1, 2\} $	$ \{1, 3\} $	$ \{1, 2, 3\} $
$ v(S) $	$ 0 $	$ 4 $	$ 0 $	$ 6 $	$ 0 $	$ 7 $
$ \chi^{1} $	$ (0, \cdot, \cdot) $^*	$ (\cdot, \underline{4}, \cdot) $	$ (\cdot, \cdot, 0) $	$ (3, 3, \cdot) $	$ (0, \cdot, 0) $	$ (\frac{7}{4}, \frac{7}{4}, \frac{14}{4}) $
$ \chi^{2} $	$ (0, \cdot, \cdot) $	$ (\cdot, \boldsymbol{4}, \cdot) $	$ (\cdot, \cdot, 0) $	$ (\underline{2}, {\boldsymbol{4}}, \cdot) $	$ (0, \cdot, 0) $	$ (1, \boldsymbol{4}, 2) $
$ \chi^{3} $	$ (\boldsymbol{2}, \cdot, \cdot) $	$ (\cdot, \boldsymbol{4}, \cdot) $	$ (\cdot, \cdot, 0) $	$ (\boldsymbol{2}, \boldsymbol{4}, \cdot) $	$ (\boldsymbol{2}, \cdot, -2) $	$ (\boldsymbol{2}, \boldsymbol{4}, \underline{1}) $
$ \gamma^{3} $						$ (\boldsymbol{2}, \boldsymbol{4}, \boldsymbol{1}) $

$ S(\in C^L(N)) $	$ \{1\} $	$ \{2\} $	$ \{3\} $	$ \{1, 2\} $	$ \{1, 3\} $	$ \{1, 2, 3\} $
$ v(S) $	$ 0 $	$ 4 $	$ 2 $	$ 6 $	$ 0 $	$ 7 $
$ \chi^{1} $	$ (0, \cdot, \cdot) $	$ (\cdot, \underline{4}, \cdot) $	$ (\cdot, \cdot, 2) $	$ (3, 3, \cdot) $	$ (0, \cdot, 0) $	$ (\frac{7}{4}, \frac{7}{4}, \frac{14}{4}) $
$ \chi^{2} $	$ (0, \cdot, \cdot) $	$ (\cdot, \bf{4}, \cdot) $	$ (\cdot, \cdot, 2) $	$ (\underline{2}, \bf{4}, \cdot) $	$ (0, \cdot, 0) $	$ (1, {\bf}, 2) $
$ \chi^{3} $	$ (\bf{2}, \cdot, \cdot) $	$ (\cdot, {\bf4}, \cdot) $	$ (\cdot, \cdot, \underline{2}) $	$ (\bf{2}, \bf{4}, \cdot) $	$ ({\bf 2}, \cdot, -2) $	$ ({\bf2}, {\bf4}, 1) $
$ \chi^{4} $	$ ({\bf2}, \cdot, \cdot) $	$ (\cdot, {\bf4}, \cdot) $	$ (\cdot, \cdot, \bf{2}) $	$ ({\bf2}, {\bf4}, \cdot) $	$ ({\bf2}, \cdot, {\bf2}) $	$ ({\bf2}, {\bf4}, \bf{2}) $

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The procedural proportional solution for graph games

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