doi: 10.3934/jdg.2022017
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On the symmetry relation between different characteristic functions for additively separable cooperative games

1. 

Department of Mathematics, Higher School of Economics St. Petersburg Campus, St. Petersburg Russia

2. 

Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, St. Petersburg Russia

*Corresponding author: Ekaterina Gromova, e-mail: ekaterina.shevkoplyas@gmail.com

Received  January 2022 Revised  March 2022 Early access July 2022

Fund Project: This study was partially done while E. Gromova was with St. Petersburg State University

We analyze 4 characteristic functions $ V^\alpha $, $ V^\delta $, $ V^\zeta $, and $ V^\eta $, and give a necessary condition for these functions to satisfy the relation $ V^\alpha - V^\delta = V^\zeta - V^\eta $ for all coalitions $ S $. To do so, we define and formally analyze the class of additively separable games. It is shown that many important types of games, both static and dynamic, belong to this class.

Citation: Ekaterina Gromova, Kirill Savin. On the symmetry relation between different characteristic functions for additively separable cooperative games. Journal of Dynamics and Games, doi: 10.3934/jdg.2022017
References:
[1]

E. BacchiegaL. Lambertini and A. Palestini, On the time consistency of equilibria in a class of additively separable differential games, J. Optim. Theory Appl., 145 (2010), 415-427.  doi: 10.1007/s10957-010-9673-6.

[2]

P. Chander and H. Tulkens, The core of an economy game with multilateral externalities, Internat. J. Game Theory, 26 (1997), 379-401.  doi: 10.1007/s001820050041.

[3]

E. J. Dockner, S. J orgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press: Cambridge, 2000. doi: 10.1017/CBO9780511805127.

[4]

W. M. Gorman, Conditions for additive separability, Econometrica: Journal of the Econometric Society, (1968), 605–609.

[5]

E. GromovaE. Marova and D. Gromov, A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games, J. Dyn. Games, 7 (2020), 105-122.  doi: 10.3934/jdg.2020007.

[6]

E. Gromova and L. Petrosyan, On a approach to the construction of characteristic function for cooperative differential games, Autom. Remote Control, 78 (2017), 1680-1692.  doi: 10.1134/s0005117917090120.

[7]

T. Kamihigashi and T. Furusawa, Global dynamics in repeated games with additively separable payoffs, Review of Economic Dynamics, 13 (2010), 899-918. 

[8]

E. MarovaE. GromovaP. Barsuk and A. Shagushina, On the effect of the absorption coefficient in a differential game of pollution control, Mathematics, 8 (2020), 961-984. 

[9]

L. Petrosjan and E. Gromova, Two-level cooperation in coalitional differential games, Tr. Inst. Mat. Mekh., 20 (2014), 193-203. 

[10]

L. Petrosjan 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.

[11]

L. Petrosyan and N. Zenkevich, Game Theory, 2$^nd$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. doi: 10.1142/9824.

[12]

P. V. Reddy and G. Zaccour, A friendly computable characteristic function, Math. Social Sci., 82 (2016), 18-25.  doi: 10.1016/j.mathsocsci.2016.03.008.

[13]

L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, 28 (1953), 307-317. 

[14]

J. Von Neumann and O. Morgenstern, Game Theory and Economic Behavior, Princeton University Press: Princeton, NJ, USA, 1944.

show all references

References:
[1]

E. BacchiegaL. Lambertini and A. Palestini, On the time consistency of equilibria in a class of additively separable differential games, J. Optim. Theory Appl., 145 (2010), 415-427.  doi: 10.1007/s10957-010-9673-6.

[2]

P. Chander and H. Tulkens, The core of an economy game with multilateral externalities, Internat. J. Game Theory, 26 (1997), 379-401.  doi: 10.1007/s001820050041.

[3]

E. J. Dockner, S. J orgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press: Cambridge, 2000. doi: 10.1017/CBO9780511805127.

[4]

W. M. Gorman, Conditions for additive separability, Econometrica: Journal of the Econometric Society, (1968), 605–609.

[5]

E. GromovaE. Marova and D. Gromov, A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games, J. Dyn. Games, 7 (2020), 105-122.  doi: 10.3934/jdg.2020007.

[6]

E. Gromova and L. Petrosyan, On a approach to the construction of characteristic function for cooperative differential games, Autom. Remote Control, 78 (2017), 1680-1692.  doi: 10.1134/s0005117917090120.

[7]

T. Kamihigashi and T. Furusawa, Global dynamics in repeated games with additively separable payoffs, Review of Economic Dynamics, 13 (2010), 899-918. 

[8]

E. MarovaE. GromovaP. Barsuk and A. Shagushina, On the effect of the absorption coefficient in a differential game of pollution control, Mathematics, 8 (2020), 961-984. 

[9]

L. Petrosjan and E. Gromova, Two-level cooperation in coalitional differential games, Tr. Inst. Mat. Mekh., 20 (2014), 193-203. 

[10]

L. Petrosjan 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.

[11]

L. Petrosyan and N. Zenkevich, Game Theory, 2$^nd$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. doi: 10.1142/9824.

[12]

P. V. Reddy and G. Zaccour, A friendly computable characteristic function, Math. Social Sci., 82 (2016), 18-25.  doi: 10.1016/j.mathsocsci.2016.03.008.

[13]

L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, 28 (1953), 307-317. 

[14]

J. Von Neumann and O. Morgenstern, Game Theory and Economic Behavior, Princeton University Press: Princeton, NJ, USA, 1944.

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