# 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
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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
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