A Double-Sided Jamming Game with Resource Constraints

Abstract

In this article, we study the problem of power allocation in teams of mobile agents in a conflict situation. Each team consists of two agents who try to split their available power between the tasks of communication and jamming the nodes of the other team. The agents have constraints on their total energy and instantaneous power usage. The cost function is the difference between the rates of erroneously transmitted bits of each team. We present a 2-level game formulation: At the higher level, the agents solve a continuous-kernel power allocation game at each instant. Based on the communications model, we present sufficient conditions on the physical parameters of the agents for the existence of a Pure Strategy Nash Equilibrium for the continuous-kernel power allocation game. At the lower level, we have a zero-sum differential game between the two teams and use Isaacs’ approach to obtain necessary conditions for the optimal trajectories. The optimal power allocation scheme obtained at the upper level is used to solve the lower level differential game. This gives rise to a games-in-games scenario which is one of the first such phenomena documented in the literature.

Appendix

We include here the expressions for the constant parameters used in the statement of Theorem 3.

$$\displaystyle\begin{array}{rcl} a_{2}& =& \frac{1} { \frac{\sigma }{P_{ max}\rho (1+d^{12})^{-\alpha }} +\delta _{ 1}^{1}\left (\frac{1+d_{1}^{1}} {1+d^{12}} \right )^{-\alpha } +\delta _{ 2}^{1}\left (\frac{1+d_{2}^{1}} {1+d^{12}} \right )^{-\alpha }},\quad b_{2} =\delta _{21}\left (\frac{1 + d_{12}} {1 + d_{1}^{2}}\right )^{-\alpha }, {}\\ c_{2}& =& \frac{\sigma } {P_{max}\rho (1 + d_{1}^{2})^{-\alpha }} +\gamma _{ 1}^{1}\left (\frac{1 + d_{1}^{1}} {1 + d_{1}^{2}}\right )^{-\alpha },d_{ 2} =\delta _{12}\left (\frac{1 + d_{12}} {1 + d_{2}^{2}}\right )^{-\alpha }, {}\\ e_{2}& =& \frac{\sigma } {P_{max}\rho (1 + d_{2}^{2})^{-\alpha }} +\gamma _{ 2}^{1}\left (\frac{1 + d_{2}^{1}} {1 + d_{2}^{2}}\right )^{-\alpha }, {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} k_{1}& =& \frac{1} { \frac{\sigma }{P_{ max}\rho (1+d_{12})^{-\alpha }} +\gamma _{ 2}^{1}\left (\frac{1+d_{2}^{1}} {1+d_{12}} \right )^{-\alpha } +\gamma _{ 2}^{2}\left (\frac{1+d_{2}^{2}} {1+d_{12}} \right )^{-\alpha }},\quad l_{1} =\gamma ^{21}\left (\frac{1 + d^{12}} {1 + d_{1}^{1}}\right )^{-\alpha } {}\\ m_{1}& =& \frac{\sigma } {P_{max}\rho (1 + d_{1}^{1})^{-\alpha }} +\delta _{ 2}^{1}\left (\frac{1 + d_{2}^{1}} {1 + d_{1}^{1}}\right )^{-\alpha },n_{ 1} =\gamma ^{12}\left (\frac{1 + d^{12}} {1 + d_{1}^{2}}\right )^{-\alpha }, {}\\ o_{1}& =& \frac{\sigma } {P_{max}\rho (1 + d_{1}^{2})^{-\alpha }} +\delta _{ 2}^{2}\left (\frac{1 + d_{2}^{2}} {1 + d_{1}^{2}}\right )^{-\alpha }, {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} k_{2}& =& \frac{1} { \frac{\sigma }{P_{ max}\rho (1+d_{12})^{-\alpha }} +\gamma _{ 1}^{1}\left (\frac{1+d_{1}^{1}} {1+d_{12}} \right )^{-\alpha } +\gamma _{ 1}^{2}\left (\frac{1+d_{1}^{2}} {1+d_{12}} \right )^{-\alpha }},l_{2} =\gamma ^{21}\left (\frac{1 + d^{12}} {1 + d_{2}^{1}}\right )^{-\alpha } {}\\ m_{2}& =& \frac{\sigma } {P_{max}\rho (1 + d_{2}^{1})^{-\alpha }} +\delta _{ 1}^{1}\left (\frac{1 + d_{1}^{1}} {1 + d_{2}^{1}}\right )^{-\alpha },n_{ 2} =\gamma ^{12}\left (\frac{1 + d^{12}} {1 + d_{2}^{2}}\right )^{-\alpha }, {}\\ o_{2}& =& \frac{\sigma } {P_{max}\rho (1 + d_{2}^{2})^{-\alpha }} +\delta _{ 1}^{2}\left (\frac{1 + d_{1}^{2}} {1 + d_{2}^{2}}\right )^{-\alpha }, {}\\ \end{array}$$