# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2020021

## Shapley value for differential network games: Theory and application

 1 St. Petersburg State University, Univereitetskaya Nab. 7/9 Saint-Petersburg, Russia 2 SRS Consortium for Advanced Study in Cooperative Dynamic Games, Shue Yan University, 10 Wai Tsui Cres, North Point, Hong Kong

* Corresponding author: Leon Petrosyan

Received  December 2019 Revised  December 2019 Published  July 2020

Fund Project: The first author is supported by Russian Science Foundation the grant Optimal Behavior in Conflict-Controlled Systems 17-11-01079

This paper presents a time-consistent dynamic Shapley value imputation for a class of differential network games. A novel form for measuring the worth of coalitions – named as cooperative-trajectory characteristic function – is developed for the Shapley value imputation. This new class of characteristic functions is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under min-max confrontation or the Nash non-cooperative stance. The resultant dynamic Shapley value imputation yields a new cooperative solution in differential network games.

Citation: Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics & Games, doi: 10.3934/jdg.2020021
##### References:
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##### References:
 [1] H. Cao, E. Ertin and A. Arora, MiniMax equilibrium of networked differential games, ACM Transactions on Autonomous and Adaptive Systems, 3 (1963). doi: 10.1145/1452001.1452004.  Google Scholar [2] Y. V. Chirkova, Optimal calls to a 2-server with loss and random access, Autom. Remote Control, 78 (2017), 557-580.  doi: 10.1134/s0005117917030146.  Google Scholar [3] H. Gao and Y. Pankratova, Cooperation in dynamic network games, Contributions to Game Theory and Management, 10 (2017), 42-67.   Google Scholar [4] E. Gromova, The shapley value as a sustainable cooperative solution in differential games of three players, in Recent Advances in Game Theory and Applications, Static Dyn. Game Theory Found. Appl., Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-43838-2\_4.  Google Scholar [5] R. Isaacs, Differential Games, Wiley, New York, 1965. Google Scholar [6] N. N. Krasovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Control of a Dynamic System, Nauka, Moskow, 1985.  Google Scholar [7] V. Mazalov and J.V. Chirkova, Networking Games: Network Forming Games and Games on Networks, Academic Press, 2019.   Google Scholar [8] M. A. G. Meza and J. D. Lopez-Barrientos, A Differential game of a duopoly with network externalities, in Recent Advances in Game Theory and Applications, Static Dyn. Game Theory Found. Appl., Birkhäuser/Springer, Cham, 2016, 49–66. doi: 10.1007/978-3-319-43838-2.  Google Scholar [9] H.-M. Pai, A differential game formulation of a controlled network, Queueing Syst., 64 (2010), 325-358.  doi: 10.1007/s11134-009-9161-6.  Google Scholar [10] L. A. Petrosian, E. V. Gromova and S. V. Pogozhev, Strong time-consistent subset of core in cooperative differential games with finite time horizon, Autom. Remote Control, 79 (2018), 1912-1928.  doi: 10.5555/3288409.3288431.  Google Scholar [11] L. A. Petrosjan, The shapley value for differential games, in New trends in dynamic games and applications, Ann. Internat. Soc. Dynam. Games, 3, Birkhäuser Boston, Boston, MA, 1995, 409–417.  Google Scholar [12] L. A. Petrosyan, Cooperative differential games on networks, Trudy Inst. Mat. i Mekh. UrO RAN, 16 (2010), 143-150.   Google Scholar [13] L. A. Petrosyan and A. A. Sedakov, Multistage networking games with full information, Matematicheskaya Teoriya Igr I ee Prilozheniya, 2 (2009), 66-81.   Google Scholar [14] L. Petrosyan and G. Zaccour, Time-consistent shapley value allocation of pollution cost reduction, J. Econom. Dynam. Control, 27 (2003), 381-398.  doi: 10.1016/S0165-1889(01)00053-7.  Google Scholar [15] L. S. Shapley, A value for n-person games,in Contributions to the Theory of Games, vol. 2, Annals of Mathematics Studies, 28, Princeton University Press, Princeton, NJ, 1953.   Google Scholar [16] B.-W. Wie, A differential game model of Nash equilibrium on a congested traffic network, Networks, 23 (1993), 557-565.  doi: 10.1002/net.3230230606.  Google Scholar [17] B. W. Wie, A differential game approach to the dynamic mixed behavior traffic network equilibrium problem, European J. Oper. Res., 83 (1995), 117-136.   Google Scholar [18] D. W. K. Yeung, Subgame consistent shapley value imputation for cost-saving joint ventures, Mathematical Game Theory and Applications, 2 (2010), 137-149.   Google Scholar [19] D. W. K. Yeung and L. A. Petrosyan, Subgame consistent cooperative solutions in stochastic differential games, J. Optim. Theory Appl., 120 (2004), 651-666.  doi: 10.1023/B:JOTA.0000025714.04164.e4.  Google Scholar [20] D. W. K. Yeung and L. A. Petrosyan, Subgame consistent cooperation: A comprehensive treatise, in Theory and Decision Library C., 47, Springer, Singapore, 2016.  Google Scholar [21] D. W. K. Yeung and L. A. Petrosyan, Dynamic Shapley Value and Dynamic Nash Bargaining, Nova Science, New York, 2018. Google Scholar [22] H. Zhang, L. V. Jiang, S. Huang, J. Wang and Y. Zhang, Attack-defense differential game model for network defense strategy selection, IEEE Access, (2018). doi: 10.1109/ACCESS.2018.2880214.  Google Scholar
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