An Anti-jamming Game When None Player Knows Rival’s Channel Gain

Abstract

We consider a user’s communication with a receiver in presence of a jammer, in the most competitive situation for user and jammer when they do not have access to complete information on channel gains of each other although they could have access to exact information on their own channel gains. The problem is modeled as a Bayesian power control game between user and jammer as players. Incomplete information is modeled as statistical data over possible channel gains (also referred as channel states). Since channel gain is a function on the distance to the receiver, this also covers scenarios where the user and jammer could know its own location via global positioning system (GPS), but none of them know the exact location of the other. A novel approach is suggested to derive equilibrium of such problems in closed form for two communication metrics: signal-to-interference-plus-noise ratio (SINR) metric, reflecting regular data transmission, and latency metric, reflecting emergency data transmission. In particular, it is shown that the user’s equilibrium strategies corresponding to the latency metric is more sensitive to the a priori statistical information, as compared to the SINR metric. This reflects an advantage of implementing latency metric in case of availability of exact information on network parameters, and an advantage of implementing SINR metric in case of lack of such its availability.

Appendix

Proof of Proposition 1: Since $v_{J,m}(\boldsymbol{P},J_m)$ is concave in $J_m$, and $v_{U,k}(P_k,\boldsymbol{J})$ is linear in $P_k$, and set of feasible strategies of each player is compact, the result follows from Nash theorem [4].$\blacksquare$

Proof of Proposition 2: By (3), $v_{U,k}(P_k,\boldsymbol{J})$ is linear in $P_k$. Then, (12) follows from (3) and (7). By (5), we have that

$$\begin{aligned} \frac{\partial v_{J,m}(\boldsymbol{P},J_m)}{\partial J_m} = \frac{g_m H(\boldsymbol{P})}{(N+g_m J_m)^2} -C_J. \end{aligned}$$

(49)

Since $v_{J,m}(\boldsymbol{P},J_m)$ is concave in $J_m$, by (8) and (49), we have that

$$\begin{aligned} \, &\text{ BR}_{J,m}(\boldsymbol{P})={\left\{ \begin{array}{ll} 0,&{} \displaystyle \frac{\partial v_{J,m}(\boldsymbol{P},0)}{\partial J_m}\le 0,\\ \displaystyle \frac{\partial v_{J,m}(\boldsymbol{P},J_m)}{\partial J_m}=0,&{} \displaystyle \frac{\partial v_{J,m}(\boldsymbol{P},\overline{J})}{\partial J_m}<0< \frac{\partial v_{J,m}(\boldsymbol{P},0)}{\partial J_m}, \\ \overline{J},&{} \displaystyle 0\le \frac{\partial v_{J,m}(\boldsymbol{P},\overline{J})}{\partial J_m}. \end{array}\right. } \end{aligned}$$

(50)

Substituting (49) into (50) and taking into account (6) and (14) imply (13).$\blacksquare$

Proof of Proposition 3: By (1) and (12), for a fixed $\boldsymbol{J}$ there exists an unique $\tilde{k}(\boldsymbol{J})\in \{0,1,\ldots ,K,K+1\}$ such that

$$\begin{aligned} \, \text{ BR}_{U,k}(\boldsymbol{J}) = {\left\{ \begin{array}{ll} 0,&{}k< \tilde{k}(\boldsymbol{J}),\\ \in [0,\overline{P}],&{}k= \tilde{k}(\boldsymbol{J}),\\ \overline{P}&{} k>\tilde{k}(\boldsymbol{J}) \end{array}\right. } \text{ or } \text{ BR}_{U,k}(\boldsymbol{J}) = {\left\{ \begin{array}{ll} 0,&{}k< \tilde{k}(\boldsymbol{J}),\\ \overline{P}&{} k\ge \tilde{k}(\boldsymbol{J}). \end{array}\right. } \end{aligned}$$

(51)

Thus, user’s types equilibrium strategies have to have the form $\boldsymbol{P}(x)$ given by (18) with an $x\in [\tilde{k}(\boldsymbol{J})\overline{P}, (\tilde{k}(\boldsymbol{J})+1)\overline{P}).$ To derive an equation to find such x,  note that, by (9), $\boldsymbol{P}(x)$ is user’s types equilibrium strategies if and only if

$$\begin{aligned} \boldsymbol{P}(x)=\text{ BR}_{U}(\text{ BR}_J(\boldsymbol{P}(x)). \end{aligned}$$

(52)

By (4), (13) and (20), we have that

$$\begin{aligned} T\left( \text{ BR}_J(\boldsymbol{P}(x))\right) =\mathcal {T}(x), \end{aligned}$$

(53)

with $I_0(x)\triangleq \left\{ m\in \mathcal {M}:\mathcal {H}(x)\le A_m \right\}$, $I(x)\triangleq \left\{ m\in \mathcal {M}: A_m<\mathcal {H}(x)< B_m\right\}$ and $\overline{I}(x)\triangleq \left\{ m\in \mathcal {M}: B_m\le \mathcal {H}(x)\right\} .$ By (25) and (26) these sets $I_0(x)$, I(x) and $\overline{I}(x)$ can be present in equivalent form given by (22)–(24). Then substituting (12) and (53) into right side of fixed point equation (52), and (18) into its right side imply (19), and the result follows. $\blacksquare$

Proof of Theorem 1: Let $\boldsymbol{P}(x)$ be an equilibrium. Then, by Proposition 3, there are $\tilde{k}$ and $x\in [\tilde{k}\overline{P},(\tilde{k}+1)\overline{P})$ such that

$$\begin{aligned} P_k(x) = {\left\{ \begin{array}{ll} 0,&{}k< \tilde{k},\\ \in [0,\overline{P}],&{}k= \tilde{k},\\ \overline{P}&{} k>\tilde{k} \end{array}\right. } \text{ or } P_k(x) = {\left\{ \begin{array}{ll} 0,&{}k< \tilde{k},\\ \overline{P}&{} k\ge \tilde{k} \end{array}\right. } \end{aligned}$$

(54)

with

$$\begin{aligned} P_{\tilde{k}}(x)=(\tilde{k}+1)\overline{P}-x \end{aligned}$$

(55)

and

$$\begin{aligned} P_k(x)= {\left\{ \begin{array}{ll} 0,&{}\mathcal {T}(x)<C_P/h_k,\\ \in [0,\overline{P}],&{}\mathcal {T}(x)=C_P/h_k,\\ \overline{P},&{}\mathcal {T}(x)>C_P/h_k, \end{array}\right. } \text{ for } k\in \mathcal {K}. \end{aligned}$$

(56)

Note that, by (20),

$$\begin{aligned} \mathcal {T}(x) \text{ is } \text{ non-decreasing } \text{ continuous } \text{ function. } \end{aligned}$$

(57)

Thus, by (56), if (27) holds then $x=0,$ and (a) follows. If (28) holds then $x=K\overline{P},$ and (b) follows. Let (27) and (28) do not hold. Let user’s types equilibrium strategies are given by the right formula in (54). Then, by (55), $x=\tilde{k}\overline{P},$ and, thus, by (56), (29) holds, and (c-i) follows.

Let user’s types equilibrium strategies are given by the left formula in (54). Then, by (55), $\tilde{k}\overline{P}<x<(\tilde{k}+1)\overline{P}$, and by (56), $\mathcal {T}(x)=C_P/h_{\tilde{k}}.$ This equation has a root in $[\tilde{k}\overline{P},(\tilde{k}+1)\overline{P}]$ if and only if (30) holds, and (c-ii) follows. $\blacksquare$

Proof of Proposition 4: It is clear that $v^L_{U,k}(P_k,\boldsymbol{J})$ is concave in $P_k$ and $v^L_{J,m}(\boldsymbol{P},J_m)$ is linear in $J_m$. Thus, by Nash theorem [4], equilibrium exists.$\blacksquare$

Proof of Proposition 5: By (34), $v^L_{J,m}(\boldsymbol{P},J_m)$ is linear in $J_m$. Then, (37) follows from (34). Further, by (34), we have that

$$\begin{aligned} \frac{\partial v^L_{U,k}(P_k,\boldsymbol{J})}{\partial P_k}= \frac{W(\boldsymbol{J})}{h_k P^2_k}-C_P. \end{aligned}$$

(58)

Since $v^L_{U,k}(P_k,\boldsymbol{J})$ is concave in $P_k$ and $v^L_{U,k}(0,\boldsymbol{J})=-\infty$ we have that

$$\begin{aligned} \, &\text{ BR}^L_{U,k}(\boldsymbol{J})={\left\{ \begin{array}{ll} \displaystyle \frac{\partial v^L_{U,k}(P_k,\boldsymbol{J})}{\partial P_k}=0,&{} \displaystyle \frac{\partial v^L_{U,k}(\overline{P},\boldsymbol{J})}{\partial P_k}<0, \\ \overline{P},&{} \displaystyle 0\le \frac{\partial v^L_{U,k}(\overline{P},\boldsymbol{J})}{\partial P_k}. \end{array}\right. } \end{aligned}$$

(59)

Substituting (58) into (59) implies (36). By (34), $v^L_{J,m}(\boldsymbol{P},J_m)$ is linear in $J_m$, and, then, (37) immediately follows from (34). $\blacksquare$

Proof of Proposition 6: By (2) and (37), for a fixed $\boldsymbol{P}$ there exists an unique $\tilde{m}(\boldsymbol{P})\in \{0,1,\ldots ,M,M+1\}$ such that

$$\begin{aligned} \text{ BR}^L_{J,m}(\boldsymbol{P}) = {\left\{ \begin{array}{ll} 0,&{}m< \tilde{m}(\boldsymbol{P}),\\ \in [0,\overline{J}],&{}m= \tilde{m}(\boldsymbol{P}),\\ \overline{J}&{} m>\tilde{m}(\boldsymbol{P}) \end{array}\right. } \text{ or } \text{ BR}^L_{J,m}(\boldsymbol{P}) = {\left\{ \begin{array}{ll} 0,&{}m< \tilde{m}(\boldsymbol{P}),\\ \overline{J}&{} m\ge \tilde{m}(\boldsymbol{P}). \end{array}\right. } \end{aligned}$$

(60)

Thus, jammer’s types equilibrium strategies have to have the form $\boldsymbol{J}(y)$ given by (39) with $y\in [\tilde{m}(\boldsymbol{P})\overline{J}, (\tilde{m}(\boldsymbol{P})+1)\overline{J}).$ To derive an equation to find such y,  note that, $\boldsymbol{J}(y)$ is jammer’s types equilibrium strategies if and only if

$$\begin{aligned} \boldsymbol{J}(y)=\text{ BR}^L_{J}(\text{ BR}^L_U(\boldsymbol{J}(y)). \end{aligned}$$

(61)

By (35), (36) and (41)–(43), we have that $R\left( \text{ BR}_U(\boldsymbol{J}(y))\right) =\mathcal {R}(y).$ Then this, (37), (39) and (61) and (39) imply (40), and the result follows.$\blacksquare$

Proof of Theorem 2: By (33) and (39), $W(\boldsymbol{J}(y))$ is strictly decreasing in $[0,M\overline{J}],$ by (41)–(43), $\mathcal {R}(y)$ is either constant or strictly increasing in $[0,M\overline{J}].$ Then the result follows from Proposition 6.$\blacksquare$