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

## TL;DR

This paper introduces a new, computationally efficient characteristic function for cooperative differential games, offering an alternative to the classical Neumann-Morgenstern function with practical and interpretative advantages.

## Contribution

It defines and analyzes an $ ext{eta}$-characteristic function that simplifies computation and retains economic interpretability, expanding the toolkit for cooperative differential game analysis.

## Key findings

- The $ ext{eta}$-characteristic function is easier to compute than classical functions.
- The new function maintains symmetry properties in the pollution control game.
- All characteristic functions yield solutions with symmetry relations.

## Abstract

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.