Construction of Dynamically Stable Solutions in Differential Network Games

Abstract

This paper presents a novel measure of the worth of coalitions—named as a cooperative-trajectory characteristic function—to generate a time-consistent Shapley value solution in a class of network differential games. This new class of characteristic function is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under minmax confrontation or Nash non-cooperative stance. The resultant time-consistent Shapley value calibrates the marginal contributions of individual players to the grand coalition payoff based on their cooperative actions/strategy. The cooperative-trajectory characteristic function is also time consistent and yields a new cooperative solution in network differential games.

Appendix: Proof of Proposition 5.1.

Using (5.7), we have

$$\begin{aligned} V(S{}_{1}^{} \cup S_{2}^{} ;x_{0}^{} ,T-t_{0}^{} )=\sum _{i\in S_{1}^{} \bigcup S_{2}^{} }^{} \; \sum _{j\in K(i)\cap (S_{1}^{} \cup S_{2}^{} )}^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) \end{aligned}$$

$$\begin{aligned} =\sum _{i\in S_{1}^{} }^{} \sum _{j\in K(i)\cap S_{1}^{} }^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} )+ \sum _{i\in S_{2}^{} }^{} \sum _{j\in K(i)\cap S_{2}^{} }^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) \end{aligned}$$

$$\begin{aligned} -\sum _{i\in S_{1}^{} \cap S_{2}^{} }^{} \; \sum _{j\in K(i)\cap (S_{1}^{} \cap S_{2}^{} )}^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) \end{aligned}$$

$$\begin{aligned} +\sum _{i\in S_{1}^{} }^{} \sum _{j\in K(i)\cap S_{2}^{} }^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) +\sum _{i\in S_{2}^{} }^{} \sum _{j\in K(i)\cap S_{1}^{} }^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) \end{aligned}$$

$$\begin{aligned} \ge \sum _{i\in S_{1}^{} }^{} \sum _{j\in K(i)\cap S_{1}^{} }^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} )+ \sum _{i\in S_{2}^{} }^{} \sum _{j\in K(i)\cap S_{2}^{} }^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) \end{aligned}$$

$$\begin{aligned} -\sum _{i\in S_{1}^{} \cap S_{2}^{} }^{} \; \sum _{j\in K(i)\cap (S_{1}^{} \cap S_{2}^{} )}^{} \alpha _{ij}^{} (x_{0}^{} ,T-t_{0}^{} ) \end{aligned}$$

$$\begin{aligned} =V(S{}_{1}^{} ;x_{0}^{} ,T-t_{0}^{} )+V(S{}_{2}^{} ;x_{0}^{} ,T-t_{0}^{} )-V(S{}_{1}^{} \cap S_{2}^{} ;x_{0}^{} ,T-t_{0}^{} ). \end{aligned}$$

(5.25)

Hence Proposition 5.1 follows.