# American Institute of Mathematical Sciences

April  2020, 7(2): 105-122. doi: 10.3934/jdg.2020007

## A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games

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

* Corresponding author: Ekaterina Gromova

Received  September 2019 Published  April 2020

Fund Project: The reported study was funded by RFBR under the grant 18-00-00727 (18-00-00725)

In this paper, we present a systematic overview of different endogenous optimization-based characteristic functions and discuss their properties. Furthermore, we define and analyze in detail a new, $\eta$-characteristic function. This characteristic function has a substantial advantage over other characteristic functions in that it can be obtained with a minimal computational effort and has a reasonable economic interpretation. In particular, the new characteristic function can be seen as a reduced version of the classical Neumann-Morgenstern characteristic function, where the players both from the coalition and from the complementary coalition use their previously computed strategies instead of solving respective optimization problems. Our finding are illustrated by a pollution control game with $n$ non-identical players. For the considered game, we compute all characteristic functions and compare their properties. Quite surprisingly, it turns out that both the characteristic functions and the resulting cooperative solutions satisfy some symmetry relations.

Citation: 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
##### References:
 [1] T. Başar, On the uniqueness of the Nash solution in linear-quadratic differential games, Internat. J. Game Theory, 5 (1976), 65-90.  doi: 10.1007/BF01753310. [2] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. [3] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. [4] M. Breton, G. Zaccour and M. Zahaf, A differential game of joint implementation of environmental projects, Automatica J. IFAC, 41 (2005), 1737-1749.  doi: 10.1016/j.automatica.2005.05.004. [5] P. Chander and H. Tulkens, The core of an economy with multilateral environmental externalities, Internat. J. Game Theory, 26 (1997), 379-401.  doi: 10.1007/BF01263279. [6] J. A. Filar and P. S. Gaertner, A regional allocation of world CO2 emission reductions, Mathematics and Computers in Simulation, 43 (1997), 269-275.  doi: 10.1016/S0378-4754(97)00009-8. [7] G. Freiling, G. Jank and H. Abou-Kandil, On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games, IEEE Trans. Automat. Control, 41 (1996), 264-269.  doi: 10.1109/9.481532. [8] A. Friedman, Differential games, in Handbook of Game Theory with Economic Applications, Vol. 2, North-Holland, Amsterdam, 1994,781–799. [9] J. Greenberg, Coalition structures, in Handbook of Game Theory with Economic Applications, Vol. 2, North-Holland, Amsterdam, 1994, 1306–1337. [10] D. Gromov and E. Gromova, On a class of hybrid differential games, Dyn. Games Appl., 7 (2017), 266-288.  doi: 10.1007/s13235-016-0185-3. [11] E. Gromova, A. Malakhova and E. Marova, On the superadditivity of a characteristic function in cooperative differential games with negative externalities, in 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), St. Petersburg, 2017, 1–4. doi: 10.1109/CNSA.2017.7973963. [12] E. Gromova, The Shapley value as a sustainable cooperative solution in differential games of three players, in Recent Advances in Game Theory and Applications, Birkhäuser/Springer, Cham, 2016, 67–89. [13] E. V. Gromova and E. V. Marova, Coalition and anti-coalition interaction in cooperative differential games, IFAC-PapersOnLine, 51 (2018), 479-483.  doi: 10.1016/j.ifacol.2018.11.466. [14] E. V. Gromova and L. A. Petrosyan, On an approach to constructing a characteristic function in cooperative differential games, Autom. Remote Control, 78 (2017), 1680-1692.  doi: 10.1134/s0005117917090120. [15] J. Hajduková, Coalition formation games: A survey, Int. Game Theory Rev., 8 (2006), 613-641.  doi: 10.1142/S0219198906001144. [16] S. Hart, Shapley value, in Game Theory, Palgrave Macmillan, 1989,210–216. [17] A. Haurie and G. Zaccour, Differential game models of global environmental management, in Control and Game-Theoretic Models of the Environment, Vol. 2, Birkhäuser Boston, Boston, MA, 1995, 3–23. doi: 10.1007/978-1-4612-0841-9_1. [18] C.-Y. Huang and T. Sjöström, The recursive core for non-superadditive games, Games, 1 (2010), 66-88.  doi: 10.3390/g1020066. [19] D. G. Hull, Optimal Control Theory for Applications, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4757-4180-3. [20] S. Jørgensen and E. Gromova, Sustaining cooperation in a differential game of advertising goodwill accumulation, European J. Oper. Res., 254 (2016), 294-303.  doi: 10.1016/j.ejor.2016.03.029. [21] H. Moulin, Equal or proportional division of a surplus, and other methods, Internat. J. Game Theory, 16 (1987), 161-186.  doi: 10.1007/BF01756289. [22] M. J. Osborne and A. Rubinstein, A Course in Game Theory, MIT press, Cambridge, MA, 1994. [23] 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. [24] L. A. Petrosyan and N. N. Danilov, Stability of solutions in non-zero sum differential games with transferable payoffs, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1 (1979), 52-59. [25] L. A. Petrosyan and E. V. Gromova, Two-level cooperation in coalitional differential games, Tr. Inst. Mat. Mekh., 20 (2014), 193-203. [26] 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. [27] Alvin E. Roth (ed.), Introduction to the Shapley value, in The Shapley value: Essays in honor of Lloyd S. Shapley, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511528446.002. [28] A. Sedakov, Characteristic functions in a linear oligopoly TU game, in Frontiers of Dynamic Games, Birkhäuser/Springer, Cham, 2018,219–235. [29] L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, New Jersey, 1953,307-317. [30] J. Von Neumann and O. Morgenstern, Game Theory and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944. [31] E. Winter, R. J. Aumann and S. Hart (eds.), The Shapley value, in Handbook of Game Theory with Economic Applications, Vol. 3, Elsevier/North-Holland, Amsterdam, 2002, 1521–2351.

show all references

##### References:
 [1] T. Başar, On the uniqueness of the Nash solution in linear-quadratic differential games, Internat. J. Game Theory, 5 (1976), 65-90.  doi: 10.1007/BF01753310. [2] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. [3] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. [4] M. Breton, G. Zaccour and M. Zahaf, A differential game of joint implementation of environmental projects, Automatica J. IFAC, 41 (2005), 1737-1749.  doi: 10.1016/j.automatica.2005.05.004. [5] P. Chander and H. Tulkens, The core of an economy with multilateral environmental externalities, Internat. J. Game Theory, 26 (1997), 379-401.  doi: 10.1007/BF01263279. [6] J. A. Filar and P. S. Gaertner, A regional allocation of world CO2 emission reductions, Mathematics and Computers in Simulation, 43 (1997), 269-275.  doi: 10.1016/S0378-4754(97)00009-8. [7] G. Freiling, G. Jank and H. Abou-Kandil, On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games, IEEE Trans. Automat. Control, 41 (1996), 264-269.  doi: 10.1109/9.481532. [8] A. Friedman, Differential games, in Handbook of Game Theory with Economic Applications, Vol. 2, North-Holland, Amsterdam, 1994,781–799. [9] J. Greenberg, Coalition structures, in Handbook of Game Theory with Economic Applications, Vol. 2, North-Holland, Amsterdam, 1994, 1306–1337. [10] D. Gromov and E. Gromova, On a class of hybrid differential games, Dyn. Games Appl., 7 (2017), 266-288.  doi: 10.1007/s13235-016-0185-3. [11] E. Gromova, A. Malakhova and E. Marova, On the superadditivity of a characteristic function in cooperative differential games with negative externalities, in 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), St. Petersburg, 2017, 1–4. doi: 10.1109/CNSA.2017.7973963. [12] E. Gromova, The Shapley value as a sustainable cooperative solution in differential games of three players, in Recent Advances in Game Theory and Applications, Birkhäuser/Springer, Cham, 2016, 67–89. [13] E. V. Gromova and E. V. Marova, Coalition and anti-coalition interaction in cooperative differential games, IFAC-PapersOnLine, 51 (2018), 479-483.  doi: 10.1016/j.ifacol.2018.11.466. [14] E. V. Gromova and L. A. Petrosyan, On an approach to constructing a characteristic function in cooperative differential games, Autom. Remote Control, 78 (2017), 1680-1692.  doi: 10.1134/s0005117917090120. [15] J. Hajduková, Coalition formation games: A survey, Int. Game Theory Rev., 8 (2006), 613-641.  doi: 10.1142/S0219198906001144. [16] S. Hart, Shapley value, in Game Theory, Palgrave Macmillan, 1989,210–216. [17] A. Haurie and G. Zaccour, Differential game models of global environmental management, in Control and Game-Theoretic Models of the Environment, Vol. 2, Birkhäuser Boston, Boston, MA, 1995, 3–23. doi: 10.1007/978-1-4612-0841-9_1. [18] C.-Y. Huang and T. Sjöström, The recursive core for non-superadditive games, Games, 1 (2010), 66-88.  doi: 10.3390/g1020066. [19] D. G. Hull, Optimal Control Theory for Applications, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4757-4180-3. [20] S. Jørgensen and E. Gromova, Sustaining cooperation in a differential game of advertising goodwill accumulation, European J. Oper. Res., 254 (2016), 294-303.  doi: 10.1016/j.ejor.2016.03.029. [21] H. Moulin, Equal or proportional division of a surplus, and other methods, Internat. J. Game Theory, 16 (1987), 161-186.  doi: 10.1007/BF01756289. [22] M. J. Osborne and A. Rubinstein, A Course in Game Theory, MIT press, Cambridge, MA, 1994. [23] 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. [24] L. A. Petrosyan and N. N. Danilov, Stability of solutions in non-zero sum differential games with transferable payoffs, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1 (1979), 52-59. [25] L. A. Petrosyan and E. V. Gromova, Two-level cooperation in coalitional differential games, Tr. Inst. Mat. Mekh., 20 (2014), 193-203. [26] 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. [27] Alvin E. Roth (ed.), Introduction to the Shapley value, in The Shapley value: Essays in honor of Lloyd S. Shapley, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511528446.002. [28] A. Sedakov, Characteristic functions in a linear oligopoly TU game, in Frontiers of Dynamic Games, Birkhäuser/Springer, Cham, 2018,219–235. [29] L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, New Jersey, 1953,307-317. [30] J. Von Neumann and O. Morgenstern, Game Theory and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944. [31] E. Winter, R. J. Aumann and S. Hart (eds.), The Shapley value, in Handbook of Game Theory with Economic Applications, Vol. 3, Elsevier/North-Holland, Amsterdam, 2002, 1521–2351.
Partial order diagram
Computational effort required for computing different characteristic functions
 C.F. # of optimization problems (# of variables) # of Nash equilibrium problems $\alpha$ ($\beta$) $2^n-1 (n)$ 0 $\delta$ $2^n-n-1 (2\div n)$ 1 $\zeta$ $2^n-1 (1\div n)$ 0 $\eta$ 1 (n) 1
 C.F. # of optimization problems (# of variables) # of Nash equilibrium problems $\alpha$ ($\beta$) $2^n-1 (n)$ 0 $\delta$ $2^n-n-1 (2\div n)$ 1 $\zeta$ $2^n-1 (1\div n)$ 0 $\eta$ 1 (n) 1
Possible strategic interactions between the coalition $S$ and its complement $N\setminus S$ and the respective characteristic functions
 $S$ $\max\limits_{u_i\in \mathcal{U}_i\atop i\in S} \sum\limits_{i\in S} J_i$ $u_i=u_i^{NE}$, $i\in S$ $u_i=u_i^*$, $i\in S$ $\min\limits_{u_j\in \mathcal{U}_j\atop j\in N\setminus S} \sum\limits_{i\in S} J_i$ $\alpha/\beta$ $F_1$ $\zeta$ $N\setminus S$ $u_j=u_j^{NE}$, $j\in N\setminus S$ $\delta$ Nash equilibrium $\eta$ $u_j=u_j^*$, $j\in N\setminus S$ $F_2$ $F_3$ Cooperative agreement
 $S$ $\max\limits_{u_i\in \mathcal{U}_i\atop i\in S} \sum\limits_{i\in S} J_i$ $u_i=u_i^{NE}$, $i\in S$ $u_i=u_i^*$, $i\in S$ $\min\limits_{u_j\in \mathcal{U}_j\atop j\in N\setminus S} \sum\limits_{i\in S} J_i$ $\alpha/\beta$ $F_1$ $\zeta$ $N\setminus S$ $u_j=u_j^{NE}$, $j\in N\setminus S$ $\delta$ Nash equilibrium $\eta$ $u_j=u_j^*$, $j\in N\setminus S$ $F_2$ $F_3$ Cooperative agreement
 [1] Jewaidu Rilwan, Poom Kumam, Onésimo Hernández-Lerma. Stability of international pollution control games: A potential game approach. Journal of Dynamics and Games, 2022, 9 (2) : 191-202. doi: 10.3934/jdg.2022003 [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] 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 [4] Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131 [5] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [6] John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics and Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018 [7] 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 [8] Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889 [9] Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719 [10] Ellina Grigorieva, Evgenii Khailov. Hierarchical differential games between manufacturer and retailer. Conference Publications, 2009, 2009 (Special) : 300-314. doi: 10.3934/proc.2009.2009.300 [11] Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics and Games, 2021, 8 (2) : 151-166. doi: 10.3934/jdg.2020021 [12] Alexei Korolev, Gennady Ougolnitsky. Optimal resource allocation in the difference and differential Stackelberg games on marketing networks. Journal of Dynamics and Games, 2020, 7 (2) : 141-162. doi: 10.3934/jdg.2020009 [13] Weihua Ruan. Markovian strategies for piecewise deterministic differential games with continuous and impulse controls. Journal of Dynamics and Games, 2019, 6 (4) : 337-366. doi: 10.3934/jdg.2019022 [14] 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 [15] 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 [16] Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95 [17] Lesia V. Baranovska. Pursuit differential-difference games with pure time-lag. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1021-1031. doi: 10.3934/dcdsb.2019004 [18] Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375 [19] Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045 [20] Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics and Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002

Impact Factor: