# 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 & Games, 2020, 7 (2) : 105-122. doi: 10.3934/jdg.2020007
##### References:

show all references

##### References:
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] İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021010 [2] Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026 [3] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [4] Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006 [5] Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014 [6] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020 [7] Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $G$-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072 [8] M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202 [9] Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026 [10] Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 [11] Yongkun Wang, Fengshou He, Xiaobo Deng. Multi-aircraft cooperative path planning for maneuvering target detection. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021050 [12] Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 [13] Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 [14] Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004 [15] Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 [16] Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 [17] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [18] Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019 [19] Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021070 [20] Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

Impact Factor: