# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2022017
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Bacchiega, L. 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. Gromova, E. 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. Marova, E. Gromova, P. 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. Bacchiega, L. 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. Gromova, E. 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. Marova, E. Gromova, P. 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.
 [1] Ekaterina Gromova, Ekaterina Marova, Dmitry Gromov. A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games. Journal of Dynamics and Games, 2020, 7 (2) : 105-122. doi: 10.3934/jdg.2020007 [2] İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics and Games, 2021, 8 (3) : 267-275. doi: 10.3934/jdg.2021010 [3] Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics and Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555 [4] Beatris Adriana Escobedo-Trujillo, Alejandro Alaffita-Hernández, Raquiel López-Martínez. Constrained stochastic differential games with additive structure: Average and discount payoffs. Journal of Dynamics and Games, 2018, 5 (2) : 109-141. doi: 10.3934/jdg.2018008 [5] Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497 [6] Matthew Bourque, T. E. S. Raghavan. Policy improvement for perfect information additive reward and additive transition stochastic games with discounted and average payoffs. Journal of Dynamics and Games, 2014, 1 (3) : 347-361. doi: 10.3934/jdg.2014.1.347 [7] Zeyang Wang, Ovanes Petrosian. On class of non-transferable utility cooperative differential games with continuous updating. Journal of Dynamics and Games, 2020, 7 (4) : 291-302. doi: 10.3934/jdg.2020020 [8] Deng-Feng Li, Yin-Fang Ye, Wei Fei. Extension of generalized solidarity values to interval-valued cooperative games. Journal of Industrial and Management Optimization, 2020, 16 (2) : 919-931. doi: 10.3934/jimo.2018185 [9] Vittorio Martino. On the characteristic curvature operator. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911 [10] Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, 2021  doi: 10.3934/fods.2021027 [11] Vladimir Dragović, Katarina Kukić. Discriminantly separable polynomials and quad-equations. Journal of Geometric Mechanics, 2014, 6 (3) : 319-333. doi: 10.3934/jgm.2014.6.319 [12] Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial and Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031 [13] Pablo Álvarez-Caudevilla, Julián López-Gómez. The dynamics of a class of cooperative systems. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 397-415. doi: 10.3934/dcds.2010.26.397 [14] Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109 [15] Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112 [16] Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053 [17] Anwa Zhou, Jinyan Fan. A semidefinite relaxation algorithm for checking completely positive separable matrices. Journal of Industrial and Management Optimization, 2019, 15 (2) : 893-908. doi: 10.3934/jimo.2018076 [18] Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283 [19] Xin Yang, Nan Wang, Lingling Xu. A parallel Gauss-Seidel method for convex problems with separable structure. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 557-570. doi: 10.3934/naco.2020051 [20] Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1

Impact Factor: