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Recent Advances in Game Theory and Applications pp 67–89Cite as

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The Shapley Value as a Sustainable Cooperative Solution in Differential Games of Three Players

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Part of the book series: Static & Dynamic Game Theory: Foundations & Applications ((SDGTFA))

Abstract

The contribution of the paper is twofold: first, it has been shown that the Yeung’s conditions can be used to construct a strongly time-consistent core. In this core there is a supporting imputation which has the property that a single deviation from this imputation in favor of any other imputation from the core still leads to the payment from the core. The obtained results were formulated for the Shapley value taken as the supporting imputation. Second, a particular class of differential games was considered. For this class of games the δ-characteristic function turns out to be superadditive and the Yeung’s conditions are satisfied without any additional restrictions on the parameters of the model. All results are presented in the analytic form.

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Acknowledgements

The author acknowledges the grants 9.38.245.2014, 9.41.723.2015, and 9.42.1043.2016 from St. Petersburg State University. A part of this research was conducted while visiting prof. G. Zaccour at GERAD, Montreal, Canada, February 2015.

The author is grateful to the anonymous reviewers for the valuable comments.

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    Ekaterina Gromova

Authors
  1. Ekaterina Gromova
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Correspondence to Ekaterina Gromova .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics and Control Processes, Saint Petersburg State University, Saint Petersburg, Russia

    Leon A. Petrosyan

  2. Institute of Applied Mathematical Research, Karelia Research Center of Russian Academy of Sciences, Petrozavodsk, Russia

    Vladimir V. Mazalov

Appendices

Appendix 1

The expression for the Shapley value calculated for a game of pollution control.

$$\displaystyle{\begin{array}{l} Sh_{1}(x(t),T - t) = \\ = \frac{\left (T-t\right )} {36} (18b_{1}^{2} - 36d_{ 1}x(t) + 12T^{2}d_{ 1}^{2} + 3T^{2}d_{ 2}^{2} + 3T^{2}d_{ 3}^{2} + 12d_{ 1}^{2}t^{2}+ \\ + 3d_{2}^{2}t^{2} + 3d_{3}^{2}t^{2} - 18Tb_{1}d_{1} - 18Tb_{2}d_{1} - 18Tb_{3}d_{1} + 18b_{1}d_{1}t+ \\ + 18b_{2}d_{1}t + 18b_{3}d_{1}t + 16T^{2}d_{1}d_{2} + 16T^{2}d_{1}d_{3} + 4T^{2}d_{2}d_{3} - 24Td_{1}^{2}t- \\ - 6Td_{2}^{2}t - 6Td_{3}^{2}t + 16d_{1}d_{2}t^{2} + 16d_{1}d_{3}t^{2} + 4d_{2}d_{3}t^{2} - 32Td_{1}d_{2}t- \\ - 32Td_{1}d_{3}t - 8Td_{2}d_{3}t); \\ Sh_{2}(x(t),T - t) = \\ = \frac{\left (T-t\right )} {36} (18b_{2}^{2} - 36d_{ 2}x(t) + 3T^{2}d_{ 1}^{2} + 12T^{2}d_{ 2}^{2} + 3T^{2}d_{ 3}^{2} + 3d_{ 1}^{2}t^{2}+ \\ + 12d_{2}^{2}t^{2} + 3d_{3}^{2}t^{2} - 18Tb_{1}d_{2} - 18Tb_{2}d_{2} - 18Tb_{3}d_{2} + 18b_{1}d_{2}t+ \\ + 18b_{2}d_{2}t + 18b_{3}d_{2}t + 16T^{2}d_{1}d_{2} + 4T^{2}d_{1}d_{3} + 16T^{2}d_{2}d_{3} - 6Td_{1}^{2}t- \\ - 24Td_{2}^{2}t - 6Td_{3}^{2}t + 16d_{1}d_{2}t^{2} + 4d_{1}d_{3}t^{2} + 16d_{2}d_{3}t^{2} - 32Td_{1}d_{2}t- \\ - 8Td_{1}d_{3}t - 32Td_{2}d_{3}t); \\ Sh_{3}(x(t),T - t) = \\ = \frac{\left (T-t\right )} {36} (18b_{3}^{2} - 36d_{ 3}x(t) + 3T^{2}d_{ 1}^{2} + 3T^{2}d_{ 2}^{2} + 12T^{2}d_{ 3}^{2} + 3d_{ 1}^{2}t^{2}+ \\ + 3d_{2}^{2}t^{2} + 12d_{3}^{2}t^{2} - 18Tb_{1}d_{3} - 18Tb_{2}d_{3} - 18Tb_{3}d_{3} + 18b_{1}d_{3}t+ \\ + 18b_{2}d_{3}t + 18b_{3}d_{3}t + 4T^{2}d_{1}d_{2} + 16T^{2}d_{1}d_{3} + 16T^{2}d_{2}d_{3}- \\ - 6Td_{1}^{2}t - 6Td_{2}^{2}t - 24Td_{3}^{2}t + 4d_{1}d_{2}t^{2} + 16d_{1}d_{3}t^{2} + 16d_{2}d_{3}t^{2} \\ - 8Td_{1}d_{2}t - 32Td_{1}d_{3}t - 32Td_{2}d_{3}t).\end{array} }$$

Appendix 2

The expression for the IDP calculated for a game of pollution control.

$$\displaystyle{\begin{array}{l} \beta _{1}(t) = \frac{b_{1}^{2}} {2} - d_{1}\left (x_{0} + \left (b_{s} - 3Td_{s}\right )(t - t_{0}) + \frac{3t^{2}d_{ s}} {2} -\frac{3t_{0}^{2}d_{ s}} {2} \right ) -\frac{\left (T-t\right )} {36} (18b_{s}d_{1}- \\ - 18Td_{1}^{2} - 6\tilde{d}_{s}(T - t) + 18d_{1}^{2}t - 36d_{1}\left (b_{s} - 3Td_{s} + 3td_{s}\right ) - 32Td_{1}d_{2} - 32Td_{1}d_{3}- \\ - 8Td_{2}d_{3} + 32d_{1}d_{2}t + 32d_{1}d_{3}t + 8d_{2}d_{3}t) + \frac{T^{2}d_{ 1}^{2}} {3} + \frac{T^{2}d_{ 2}^{2}} {12} + \frac{T^{2}d_{ 3}^{2}} {12} + \frac{d_{1}^{2}t^{2}} {3} + \frac{d_{2}^{2}t^{2}} {12} + \\ + \frac{d_{3}^{2}t^{2}} {12} -\frac{b_{s}d_{1}(T-t)} {2} + \frac{4T^{2}d_{ 1}d_{2}} {9} + \frac{4T^{2}d_{ 1}d_{3}} {9} + \frac{T^{2}d_{ 2}d_{3}} {9} -\frac{2Td_{1}^{2}t} {3} -\frac{Td_{2}^{2}t} {6} -\frac{Td_{3}^{2}t} {6} + \frac{4d_{1}d_{2}t^{2}} {9} + \\ + \frac{4d_{1}d_{3}t^{2}} {9} + \frac{d_{2}d_{3}t^{2}} {9} -\frac{8Td_{1}d_{2}t} {9} -\frac{8Td_{1}d_{3}t} {9} -\frac{2Td_{2}d_{3}t} {9};\end{array} }$$
$$\displaystyle{\begin{array}{l} \beta _{2}(t) = \frac{b_{2}^{2}} {2} - d_{2}\left (x_{0} + \left (b_{s} - 3Td_{s}\right )(t - t_{0}) + \frac{3t^{2}d_{ s}} {2} -\frac{3t_{0}^{2}d_{ s}} {2} \right ) -\frac{\left (T-t\right )} {36} (18b_{s}d_{2}- \\ - 18Td_{2}^{2} - 6\tilde{d}_{s}(T - t) + 18d_{2}^{2}t - 36d_{2}(b_{s} - 3Td_{s} + 3td_{s}) - 32Td_{1}d_{2} - 32Td_{2}d_{3}- \\ - 8Td_{1}d_{3} + 32d_{1}d_{2}t + 8d_{1}d_{3}t + 32d_{2}d_{3}t) + \frac{T^{2}d_{ 1}^{2}} {12} + \frac{T^{2}d_{ 2}^{2}} {3} + \frac{T^{2}d_{ 3}^{2}} {12} + \frac{d_{1}^{2}t^{2}} {12} + \frac{d_{2}^{2}t^{2}} {3} + \\ + \frac{d_{3}^{2}t^{2}} {12} -\frac{Tb_{1}d_{2}} {2} -\frac{Tb_{2}d_{2}} {2} -\frac{Tb_{3}d_{2}} {2} + \frac{b_{1}d_{2}t} {2} + \frac{b_{2}d_{2}t} {2} + \frac{b_{3}d_{2}t} {2} + \frac{4T^{2}d_{ 1}d_{2}} {9} + \frac{T^{2}d_{ 1}d_{3}} {9} + \frac{4T^{2}d_{ 2}d_{3}} {9} - \\ -\frac{Td_{1}^{2}t} {6} -\frac{2Td_{2}^{2}t} {3} -\frac{Td_{3}^{2}t} {6} + \frac{4d_{1}d_{2}t^{2}} {9} + \frac{d_{1}d_{3}t^{2}} {9} + \frac{4d_{2}d_{3}t^{2}} {9} -\frac{8Td_{1}d_{2}t} {9} -\frac{2Td_{1}d_{3}t} {9} -\frac{8Td_{2}d_{3}t} {9}; \\ \beta _{3}(t) = \frac{b_{3}^{2}} {2} - d_{3}\left (x_{0} + \left (b_{s} - 3Td_{s}\right )(t - t_{0}) + \frac{3t^{2}d_{ s}} {2} -\frac{3t_{0}^{2}d_{ s}} {2} \right ) -\frac{(T-t)} {36} (18b_{s}d_{3}- \\ - 18Td_{3}^{2} - 6\tilde{d}_{s}(T - t) + 18d_{3}^{2}t - 36d_{3}(b_{s} - 3Td_{s} + 3td_{s}) - 8Td_{1}d_{2} - 32Td_{1}d_{3}- \\ - 32Td_{2}d_{3} + 8d_{1}d_{2}t + 32d_{1}d_{3}t + 32d_{2}d_{3}t) + \frac{T^{2}d_{ 1}^{2}} {12} + \frac{T^{2}d_{ 2}^{2}} {12} + \frac{T^{2}d_{ 3}^{2}} {3} + \frac{d_{1}^{2}t^{2}} {12} + \frac{d_{2}^{2}t^{2}} {12} + \\ + \frac{d_{3}^{2}t^{2}} {3} -\frac{Tb_{1}d_{3}} {2} -\frac{Tb_{2}d_{3}} {2} -\frac{Tb_{3}d_{3}} {2} + \frac{b_{1}d_{3}t} {2} + \frac{b_{2}d_{3}t} {2} + \frac{b_{3}d_{3}t} {2} + \frac{T^{2}d_{ 1}d_{2}} {9} + \frac{4T^{2}d_{ 1}d_{3}} {9} + \frac{4T^{2}d_{ 2}d_{3}} {9} - \\ -\frac{Td_{1}^{2}t} {6} -\frac{Td_{2}^{2}t} {6} -\frac{2Td_{3}^{2}t} {3} + \frac{d_{1}d_{2}t^{2}} {9} + \frac{4d_{1}d_{3}t^{2}} {9} + \frac{4d_{2}d_{3}t^{2}} {9} -\frac{2Td_{1}d_{2}t} {9} -\frac{8Td_{1}d_{3}t} {9} -\frac{8Td_{2}d_{3}t} {9}. \end{array} }$$

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Gromova, E. (2016). The Shapley Value as a Sustainable Cooperative Solution in Differential Games of Three Players. In: Petrosyan, L., Mazalov, V. (eds) Recent Advances in Game Theory and Applications. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43838-2_4

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