Keywords

1 Introduction

In many Unmanned Aerial Vehicle (UAV) applications, a Ground Control Station (GCS) communicates with a group of UAVs to send instructions to control each UAV’s mission. However, when such active communication faces the threat of hostile interference, the result can be delay or even interruption in getting such instructions. Larger periods of delay or interruption lead to reduced freshness of received instructions, and can decrease the probability of mission success. Thus we model the probability of mission success as a function of the age of the received information. Such considerations have been gaining prominence in the research literature lately, as reflected by interest in the age of information (AoI), a system delay performance metric that has been widely employed in different applications [1, 13, 23, 25, 26].

The pioneering paper, on the impact of hostile interference on age of information is [17]. Our work is based the model of [17] suggesting a relationship between SINR at the receiver and AoI of update packets. We employ this approach to model a scenario where a group of UAVs is cooperating to perform a mission. A transmitter GCS employs directional antennas to control the UAVs while jammer radiates a spherically symmetric interference intended to cause the mission to fail. We model this problem via a game-theoretical approach. An interesting feature of this problem is that the rivals have different structures for their strategies. Specifically, the GCS’s strategy is power allocation individually between the UAVs, while the interferer’s strategy involves assigning power to jam the whole UAV group. While for most jamming games studied in literature rivals strategies have the same structure: either power allocation for both rivals [16, 20, 24] or assigning power level [7, 12, 17, 22] for both rivals.

2 Model

We consider a group of n UAVs that, following their route/mission to a target in a protected zone, must communicate with the GCS to get/verify position and mission data. The GCS is equipped with n antennas (or n separate antenna beams) to communicate with these UAVs. This communication can be damaged by active interference that might lead to loss of navigation commands and failure of the mission. An interferer, located in the protected zone, is a source of spherically symmetric interference intended to cause the UAV mission to fail. As a metric for data updating in this paper, we consider AoI which reflects the time that has passed since the last update. We assume that the probability \(\pi _S\) of mission success is a function of the average age of information A, and this function is decreasing with A such that: (a) \(\pi _S(0)=1\), i.e., if the data is up-to-date the mission succeeds with certainty; and (b) \(\pi _S(A)\downarrow 0\) for \(A\uparrow \infty \), i.e., if data is never updated, then the mission fails with certainty. To model the probability of mission success, we will use the ratio form contest success function. This is commonly used to translate involved resources into probability of winning or losing, and has been widely applied in different economic and attack-defense problems in the literature; see, for example, [5, 9, 11, 19, 21]. In our scenario, the metric that dictates whether the mission is successful or fails, is age of information. Specifically, in terms of positive constants a and b, the probability of mission success is

$$\begin{aligned} \pi _S(A)=\frac{b}{b+aA}. \end{aligned}$$
(1)

2.1 Age of Information

To model age of information, we will employ a generalization of the model introduced in [17]. For convenience of the readers, we give a brief description:

(i) :

The GCS can transmit at a rate that is proportional to the signal to interference plus noise ratio (SINR) at the receiver. Following [17], when \(p_i\) and q are the powers of the transmitting signal by GCS to UAV i and interfering signal, and \(h_i\) and \(g_i\) are the corresponding channel gains to the UAV, the packet transmission rate associated with power profile \((p_i,q)\) is

$$\begin{aligned} \begin{aligned} \mu _i(p_i,q)=z_i\text{ SINR }(p_i,q)=z_i\frac{g_ip_i}{N_i+h_iq}, \end{aligned} \end{aligned}$$
(2)

where \(N_i\) is background noise power and \(z_i\) is a positive constant.

(ii) :

Depending on the model for how update packets are delivered to the UAV, the age of information metric \(A_i\) takes on the form

$$\begin{aligned} \begin{aligned} A_i(p_i,q)=\frac{c}{\lambda _i}+\frac{d}{\mu _i(p_i,q)} \end{aligned} \end{aligned}$$
(3)

for packet arrival rate \(\lambda _i\) and constants \(c\ge 0\) and \(d>0\). In particular, when \((c,d)=(1,2)\) and fresh update packets are generated at the UAV as a rate \(\lambda \) Poisson process, the age metric A(p, q) corresponds to the average peak age of an M/G/1/1 queue [3, 10]. This is the age metric employed in [17].

We note that various other age metrics can be modeled by specifying (c, d) in (3). For example, with \((c,d)=(1,1)\), \(A_i(p_i,q)\) is the average AoI of an M/M/1/1 server supporting preemption in service [15]. Furthermore, with \((c,d)=(0,2)\) and just-in-time arrivals (i.e., a fresh update goes into service precisely when the server would become idle) at a rate \(\mu _i(p_i,q)\) memoryless server, \(A_i(p_i,q)\) is again the average AoI [14]. Finally, with \((c,d)=(0,3/2)\), \(A_i(p_i,q)\) corresponds to just-in-time updates transmitted with deterministic service times at rate \(1/\mu _i(p_i,q)\) [14]. In the following, we refer to \(A_i(p_i,q)\) as the AoI for any \(c\ge 0\) and \(d>0\).

2.2 Formulation of the Game

To define game we have to describe: (a) the set of players, (b) the set of feasible strategies of each player, and (c) the player’s payoff [4]. In our scenario, there are two players: the interferer and the GCS. A strategy of the GCS is a non-negative power vector \({\varvec{p}}=(p_1,\ldots ,p_n)\), where \(p_i\) is the power employed to communicate with UAV i, and \(\sum _{i=1}^n p_i={\overline{p}}\) is the total power. Let \(\varPi _{\text {GCS}}\) be the set of all feasible GCS strategies. A strategy of the interferer is a power level q of the jamming signal. Let \(\varPi _I={\mathbb {R}}_+\) be the set of all feasible interferer’s strategies. Note that the probability of mission success for UAV i is

$$\begin{aligned} \pi _S(A_i(p_i,q))&=\frac{b}{b+a A(p_i,q_i)}=\frac{b z_ig_ip_i}{(b+a c/\lambda _i)z_ig_ip_i+ d a (N_i+h_iq)}. \end{aligned}$$
(4)

We now introduce the auxiliary notations:

$$\begin{aligned} \alpha _i=bg_iz_i,\,\, \beta _i=(b+ac/\lambda _i)g_iz_i,\,\, \gamma _i=dah_i, \delta _{i}=daN_i, \text{ and } \varGamma _i(q)=\gamma _i q+\delta _i. \end{aligned}$$
(5)

With this notation, (4) becomes

$$\begin{aligned} \begin{aligned} \pi _S(A_i(p_i,q))=\frac{\alpha _i p_i}{\beta _ip_i+ \varGamma _i(q)}. \end{aligned} \end{aligned}$$
(6)

As criteria for mission success we consider proportional fairness criteria [6, 18] for mission success of each UAV, i.e.,

$$\begin{aligned} \begin{aligned} v_{\text {GCS}}({\varvec{p}},q)=\sum _{i=1}^n \ln (\pi _S(A_i(p_i,q))). \end{aligned} \end{aligned}$$
(7)

This utility is the payoff for the GCS. For the interferer, the cost function is the sum of the GCS payoff and the involved cost of the effort, i.e.,

$$\begin{aligned} v_I({\varvec{p}},q)=v_{\text {GCS}}({\varvec{p}},q) +C_I q, \end{aligned}$$
(8)

where \(C_I\) is the cost per unit of jamming power. The GCS wants to maximize its payoff, while the interferer aims to minimize its cost function. So, \(-v_I\) is the payoff to the interferer. We are looking for Nash equilibrium. Recall that \(({\varvec{p}},q)\) is a Nash equilibrium [4] if and only if:

$$\begin{aligned} v_{\text {GCS}}({\tilde{{\varvec{p}}}},q)&\le v_{\text {GCS}}({\varvec{p}},q),\nonumber \\ v_I({\varvec{p}},q)&\le v_I({\varvec{p}},{\tilde{q}}) \text{ for } \text{ any } ({\tilde{{\varvec{p}}}},{\tilde{q}})\in \varPi _{\text {GCS}}\times \varPi _I. \end{aligned}$$
(9)

We denote this game by \(\varGamma =\varGamma (v_{\text {GCS}},\varPi _{\text {GCS}};-v_I,\varPi _I).\)

Lemma 1

\(v_{\text {GCS}}({\varvec{p}},q)\) is concave in \({\varvec{p}},\) and \(v_I({\varvec{p}},q)\) is convex in q.

Proof

Note that

$$\begin{aligned} \frac{\partial ^2 v_{\text {GCS}}({\varvec{p}},q)}{\partial p^2_i }&= -\frac{\varGamma _i(q)(2\beta _ip_i+\varGamma _i(q)) }{p^2_i(\beta _ip_i+\varGamma _i(q))^2}<0,\\ \frac{\partial ^2 v_I({\varvec{p}},q)}{\partial q^2 }&= \sum _{i=1}^n \frac{ \gamma ^2_i }{(\beta _ip_i+\varGamma _i(q))^2}>0, \end{aligned}$$

and the result follows. \(\blacksquare \)

Lemma 1 and the Nash theorem [4] imply the following result.

Theorem 1

In the game \(\varGamma \) there exists at least one equilibrium.

3 Solution of the Game

In this section we design equilibrium strategies of the game \(\varGamma \). By (9), \({\varvec{p}}\) and q are equilibrium strategies if and only if each of them is the best response to the other, i.e., they are solutions of the equations:

$$\begin{aligned} {\varvec{p}}&= \text{ BR }_{\text {GCS}}(q)= \hbox {argmax}\{v_{\text {GCS}}({\varvec{p}},q):{\varvec{p}}\in \varPi _{\text {GCS}} \}, \end{aligned}$$
(10)
$$\begin{aligned} q&= \text{ BR }_I({\varvec{p}})= \hbox {argmin}\{v_I({\varvec{p}},q):q\in \varPi _I \}. \end{aligned}$$
(11)

To solve these best response equations we will employ a constructive approach.

3.1 Best Response Strategies

In this section we derive the best response strategies.

Theorem 2

The best response strategy \({\varvec{p}}\) of the GCS to jamming power q is unique and given as follows:

$$\begin{aligned} p_i=P_i(\varOmega (q),q) \text{ for } i=1,\ldots ,n, \end{aligned}$$
(12)

where for each fixed q, \(\varOmega (q)=\omega \) is the unique positive root of the equation

$$\begin{aligned} S_P(\omega ,q)={\overline{p}}, \end{aligned}$$
(13)

with

$$\begin{aligned} S_P(\omega ,q)&\triangleq \sum _{i=1}^n P_i(\omega ,q), \end{aligned}$$
(14)
$$\begin{aligned} P_i(\omega ,q)&\triangleq \frac{\varGamma _i(q)}{2\beta _i}\left( \sqrt{1+\frac{\displaystyle 4\beta _i}{\displaystyle \varGamma _i(q)\omega }} -1\right) . \end{aligned}$$
(15)

Proof

Since, by Lemma 1, (10) is a concave NLP problem, to find the best response strategy \({\varvec{p}}\) to q we have to introduce a Lagrangian depending on a Lagrange multiplier \(\omega \) as follows: \( L_{\omega }({\varvec{p}})= v_{\text {GCS}}({\varvec{p}},q)+\omega \left( {\overline{p}}-\sum _{i=1}^np_i\right) . \) Then, KKT Theorem implies that \({\varvec{p}}\in \varPi _{\text {GCS}}\) is the best response strategy to q if and only if the following condition holds:

$$\begin{aligned} \frac{\partial L_{\omega }}{\partial p_i }= \frac{\varGamma _i(q) }{p_i(\beta _ip_i+\varGamma _i(q))}-\omega {\left\{ \begin{array}{ll} =0,&{}p_i>0,\\ \le 0,&{}p_i=0. \end{array}\right. } \end{aligned}$$
(16)

By (16), we have that \(p_i>0\) for any i. Thus, also \(\omega >0\), and

$$\begin{aligned} \frac{\varGamma _i(q) }{p_i(\beta _i p_i+\varGamma _i(q))}=\omega \text{ for } \text{ any } i. \end{aligned}$$
(17)

Solving this equation in \(p_i\) implies \(p_i=P_i(\omega ,q)\) as given by (15).

Since \({\varvec{p}}\in \varPi _{\text {GCS}}\) the \(\omega \) is defined by the condition that the total power resource has to be utilized by the GCS, i.e., by Eq. (13).

Note that \(P_i(\omega ,q)\) given by (15) has the following properties:

(i) :

\(P_i(\omega ,q)\) is differentiable in \(\omega \) and q

(ii) :

\(P_i(\omega ,q)\) is decreasing in \(\omega \) from infinity for \(\omega \downarrow 0\) to zero for \(\omega \uparrow \infty \).

(iii) :

\(P_i(\omega ,q)\) is increasing in q to \(1/\omega \) for \(q\uparrow \infty \).

Note that (i) and (ii) straightforwardly follow from (15). By (15), for a fixed \(\omega >0\)

$$\begin{aligned} \lim _{q\uparrow \infty }P_i(\omega ,q)=1/{\overline{\omega }} \end{aligned}$$
(18)

Also, \(P_i(\omega ,q)=f_i(\varGamma _i(q))/(2\beta _i),\) where \(f_i(x)=x(\sqrt{1+m/x}-1)\) with \(m=4\beta _i/\omega \). Since \( \frac{df_i(x)}{dx}=\frac{2x+m}{2\sqrt{x^2+mx}}-1>0,\) \(P_i(\omega ,q)\) increases with q. This and (18) implies (iii). Then, (i) and (ii) yield existence of the unique root \(\omega =\varOmega (q)\) for Eq. (13). While (i)–(iii) and (18) imply that \(\varOmega (q)\) increases with q to \(n/{\overline{p}}\). \(\blacksquare \)

Note that, \(\varOmega (q)\) can be found via bisection method and \(\varOmega (q)\) is differentiable for \(q\ge 0\) and increasing from \(\omega _0\) for \(q=0\) to \(n/{\overline{p}}\) for \(q\uparrow \infty \), where \(\omega _0\) is the unique positive root of the equation:

$$\begin{aligned} S_P(\omega _0,0) =\sum _{i=1}^n \frac{\delta _i}{2\beta _i}\left( \sqrt{1+\frac{\displaystyle 4\beta _i}{\displaystyle \delta _i \omega _0}}-1\right) ={\overline{p}}. \end{aligned}$$
(19)

Theorem 2 straightforwardly implies the following result.

Corollary 1

The inverse function \(Q(\omega )=\varOmega ^{-1}(q)\) to \(\varOmega (q)\) is defined for \(\omega \in [\omega _0,n/{\overline{p}})\) and increases from \(Q(\omega _0)=0\) to \(\lim _{\omega \uparrow (n/{\overline{p}})} Q(\omega )=\infty \). Moreover, \(S_P(\omega ,Q(\omega ))={\overline{p}}.\)

Theorem 3

The best response strategy q of the interferer to \({\varvec{p}}\) is unique and given as follows:

figure a

such that, when (20b) holds, \(q_+\) is the unique positive root of

$$\begin{aligned} \sum _{i=1}^n \frac{ \gamma _i }{\beta _i p_i+\varGamma _i(q_+)}=C_I. \end{aligned}$$
(21)

Proof

Since, by Lemma 1, \(v_I({\varvec{p}},q)\) is a convex in q, by (11), q is the best response strategy to \({\varvec{p}}\) if and only if the following condition holds:

$$\begin{aligned} \frac{\partial v_I({\varvec{p}},q)}{\partial q}&=-\sum _{i=1}^n \frac{ \gamma _i }{\beta _i p_i+\varGamma _i(q)}+C_I{\left\{ \begin{array}{ll} =0,&{}q>0,\\ \ge 0,&{}q=0. \end{array}\right. } \end{aligned}$$
(22)

Since \(\gamma _i/(\beta _i p_i+\varGamma _i(q))\) is decreasing in q, the result straightforward follows from (22). \(\blacksquare \)

3.2 Equilibrium

In this section we establish threshold value of the jamming cost for the interferer to be active, derive the form the equilibrium has to have and design a fixed point algorithm to find the equilibrium.

Theorem 4

(a) If

$$\begin{aligned} \sum _{i=1}^n \frac{ \gamma _i }{\delta _i \left( 1+\sqrt{1+4\beta _i/(\delta _i\omega _0)}\right) }\le \frac{C_I}{2}. \end{aligned}$$
(23)

then \(({\varvec{p}},q)=({\varvec{P}}(\omega _0,0),0)\) is the unique equilibrium where \(\omega _0\) and \({\varvec{P}}\) given by (15) and (19) correspondingly.

(b) If (23) does not hold then \(({\varvec{p}},q)=({\varvec{P}}(\varOmega (q),q),q)\) is the equilibrium where \({\varvec{P}}\) given by (15) and q is the positive root of the equation

$$\begin{aligned} F(q)=C_I/2, \end{aligned}$$
(24)

where

$$\begin{aligned} F(q)\triangleq \sum _{i=1}^n \frac{ \gamma _i }{\varGamma _i(q)\left( 1+\sqrt{1+4\beta _i/(\varOmega (q) \varGamma _i(q))}\right) }. \end{aligned}$$
(25)

Proof

Let \(({\varvec{p}},q)\) be an equilibrium. By Theorem 2, \({\varvec{p}}>0\). Thus, only two cases arise to consider: (a) \(q=0\) and (a) \(q>0\).

  1. (a)

    Let \(q=0\). Then, by Theorem 2, \({\varvec{p}}={\varvec{P}}(\omega _0,q)\). Substituting this \({\varvec{p}}\) into (20a) implies (20a).

  2. (b)

    Let \(q>0\). Then, by Theorem 2, \({\varvec{p}}={\varvec{P}}(\varOmega (q),q)\). Substituting this \({\varvec{p}}\) into (21) implies (24) and (25). By (21), q is decreasing in \(C_I\). Thus, by (24) and (25), F also decreasing in \(C_I\), and the result follows. \(\blacksquare \)

By Theorem 2, we have that \(\lim _{q\uparrow \infty } F(q)=0\). Moreover, if (23) does not hold then \(F(0)>C_I/2.\) Also, note that (23) establishes the threshold on the jamming cost for the interferer to be active (i.e., for \(q>0\) to be an equilibrium) or non-active (i.e., for \(q=0\) to be an equilibrium). While the GCS is always active in communication with each of the UAVS. This remarkably differs with OFDM jamming problem where some of sub-subcarriers could be not involved in transmission [8] and network security problem where some not might be not protected [2].

Interestingly, the equilibrium q can be found using fixed point algorithm. To do so, note that, by Theorem 2 and Corollary 1, there is one-to-one correspondence between \(\omega \) and q. That is why first in the following proposition we derive an equation for \(\omega \), and, then, in Theorem 5, we prove convergence of the fixed point algorithm to find the \(\omega \).

Proposition 1

Equation (24) is equivalent to

$$\begin{aligned} G(\omega )=\omega , \end{aligned}$$
(26)

with \(q=Q(\omega )\), where

$$\begin{aligned} G(\omega )\triangleq \frac{2 C_I}{\displaystyle \sum _{i=1}^n \gamma _i \left( \sqrt{1+4\beta _i/(\omega \varGamma _i(Q(\omega )))}-1\right) /\beta _i}. \end{aligned}$$
(27)

Proof

Note that

$$\begin{aligned}&\frac{ \gamma _i }{\varGamma _i(q)\left( 1+\sqrt{1+4\beta _i/(\omega \varGamma _i(q))}\right) }=\frac{ \gamma _i \left( \sqrt{1+4\beta _i/(\omega \varGamma _i(q))}-1\right) }{ 4 \beta _i/\omega }. \end{aligned}$$

Substituting this into (24) and (25) imply the result. \(\blacksquare \)

The following theorem shows that Eq. (26) can be solved by fixed point algorithm.

Theorem 5

\(G(\omega )\) has the following properties:

(i) :

\(G(\omega _0)<\omega _0\);

(ii) :

\(G(\omega )\) is continuous and increasing on \(\omega \);

(iii) :

There is \(\omega _*\) such that \(G(\omega )<\omega \) for \(\omega <\omega _*\) and

$$\begin{aligned} G(\omega _*)=\omega _*; \end{aligned}$$
(28)
(iv) :

The fixed point \(\omega _*\) of (28) can be found via fixed point algorithm:

$$\begin{aligned} \omega ^m=G^{-1}(\omega ^{m-1}) \text{ for } m=1,2,\ldots \text{ with } \omega ^0 \text{ is } \text{ fixed. } \end{aligned}$$

The algorithm converges to \(\omega _*\) for any \(\omega ^0\in (\omega _0,\omega _*)\).

Proof

(i) follows from (20b). (ii) follows from Corollary 1 and (27). (iii) follows from (i), (ii), Theorems 1, 4(b) and Proposition 1.

Since \(\omega ^0<\omega _*\), by (ii) and (iii), \(G(\omega ^0)<\omega ^0.\) Then, (28) implies that there is the unique \(\omega ^1\in (\omega ^0,\omega _*)\) such that \(G(\omega ^1)=\omega ^0.\) Thus, \(\omega ^1=G^{-1}(\omega ^0).\) Similarly, there is the unique \(\omega ^2\in (\omega ^1,\omega _*)\) such that \(G(\omega ^2)=\omega ^1,\) and so on, i.e., there is the unique \(\omega ^{m}\in (\omega ^{m-1},\omega _*)\) with \(m\ge 1\) such that \(G(\omega ^{m})=\omega ^{m-1}.\) Thus, \(\omega ^m\) is increasing and upper-bounded. Thus, there exists \(\lim _{m\uparrow \infty }\omega ^m\), and, this limit is equal to \(\omega _*\). \(\blacksquare \)

Note that for boundary cases of the jamming cost and total transmission power the equilibrium strategies can be obtained in closed form:

Proposition 2

(a) Let \(C_I\) be small. Then

$$\begin{aligned} q\approx n/C_I \text{ and } p_i\approx {\overline{p}}/n \text{ for } i=1,\ldots , n. \end{aligned}$$
(29)

(b) Let \({\overline{p}}\) be small. Then \(p_i\approx {\overline{p}}/n, i=1,\ldots , n\) and

figure b

where \(q_*\) is the unique positive root of the equation

$$\begin{aligned} \sum _{i=1}^n 1/(q_*+\delta _i/\gamma _i)=C_I. \end{aligned}$$
(31)

(c) Let \({\overline{p}}\) be large. Then \(q=0\) and

$$\begin{aligned} p_i\approx \frac{\sqrt{\delta _i/\beta _i}}{\sum _{j=1}^n \sqrt{\delta _j/\beta _i}}{\overline{p}} \text{ for } i=1,\ldots , n. \end{aligned}$$
(32)

Proof

Let \(C_I\) be small. Then, by (20b), \(q=q_+\). While, by (21), \(q_+\) is large. Then, Eq. (21) can be approximated by \(n/q_+\approx C_I\). Thus, \(q=n/C_I.\) Substituting this q into (15) implies that \(P_i(\omega ,q)\approx 1/\omega \), and (a) follows. Let \({\overline{p}}\) be small. Then \(p_i\) also is small for any i. Substituting these \(p_i\) into (20a), (20b) and (21) and taking into account that \({\varvec{p}}\in \varPi _{\text {GCS}}\) implies imply (b). Let \({\overline{p}}\) be large. Then \(p_i\) is large for at least one i. Then, by (20a), \(q=0\). Then, by (13) and (15), \(\omega \) is small, and \( p_i=P_i(\omega ,0)\approx \sqrt{\delta _i/\beta _i}/\sqrt{\omega }. \) Then, since \({\varvec{p}}\in \varPi _{\text {GCS}}\), (32) follows. \(\blacksquare \)

If background noise can be neglected, then equilibrium strategies also can be found in closed form.

Proposition 3

If \(\delta _i=0\) for all i,  then,

  1. (a)

    if

    $$\begin{aligned} \sum _{i=1}^n \sqrt{\gamma _i/\beta _i}\le \sqrt{{\overline{p}}C_I} \end{aligned}$$
    (33)

    then \(q=0\) is the unique interferer strategy, while there is a continuum of the GCS equilibrium strategies, namely, any strategy \({\varvec{p}}\in \varPi _{\text {GCS}}\) such that:

    $$\begin{aligned} \sum _{i=1}^n \gamma _i/(p_i\beta _i)\le C_I. \end{aligned}$$
    (34)
  2. (b)

    if

    $$\begin{aligned} \sum _{i=1}^n \sqrt{\gamma _i/\beta _i}> \sqrt{{\overline{p}}C_I} \end{aligned}$$
    (35)

    then q and \({\varvec{p}}\) are uniquely defined as follows:

    $$\begin{aligned} p_i= & {} P_i(\omega )=\frac{\sqrt{\left( {\overline{p}}\omega / C_I\right) ^2+4{\overline{p}}\beta _i/(\gamma _iC_I)}-{\overline{p}}\omega / C_I}{2\beta _i/\gamma _i} \text{ for } i=1,\ldots ,n,\end{aligned}$$
    (36)
    $$\begin{aligned} q= & {} \omega {\overline{p}}/C_I, \end{aligned}$$
    (37)

    where \(\omega \) is the unique positive root of the equation: \(\sum \limits _{i=1}^nP_i(\omega ) ={\overline{p}}.\)

Fig. 1.
figure 1

(a) The equilibrium q for \({\overline{p}}\in \{0.1,5,10\}\), (b) the equilibrium \({\varvec{p}}\) for \({\bar{p}}=0.1\), (c) the equilibrium \({\varvec{p}}\) for \({\bar{p}}=5\) and (d) the equilibrium \({\varvec{p}}\) for \({\bar{p}}=10.\)

Full size image

Proof

Since \(\delta _i=0\) for all i, if \(q = 0\) then \({\varvec{p}}\) is any feasible strategy such that \(\sum _{i=1}^n \gamma _i/(\beta _ip_i)\le C_I.\) Such an equilibrium strategy exists if and only if

$$\begin{aligned} \min _{{\varvec{p}}\in \varPi _{\text {GCS}}}\sum _{i=1}^n \gamma _i/(\beta _ip_i)\le C_I. \end{aligned}$$
(38)

It is clear that left-side of (38) is a convex NLP problem, and straightforward applying the KKT theorem implies that its solution is

$$\begin{aligned} p_i=({\overline{p}}\sqrt{\gamma _i/\beta _i})/\sum _{j=1}^n \sqrt{\gamma _j/\beta _i}. \end{aligned}$$

Substituting this strategy into (38) implies (33), and (a) follows. While, if \(q>0\) then by (21) and (33) we have that \(\sum _{i=1}^n \omega p_i/q=C_I.\) This and the fact that \({\varvec{p}}\in \varPi _{\text {GCS}}\) implies (37). Substituting (37) implies that \({\varvec{p}}\) is given by (36). Note that \(\varphi (\omega )=\sum _{i=1}^nP_i(\omega )\) decreases with \(\omega \) and tends to zero for \(\omega \uparrow \infty \). Then, equation \(\varphi (\omega )={\overline{p}}\) has the positive root if and only if \(\varphi (0)>{\overline{p}}\), and this condition is equivalent to (35). \(\blacksquare \)

Figure 1(a) illustrates a decrease in applied jamming power with an increase in jamming power cost and the total transmission power. Figure 1(b) illustrates that for \({\bar{p}}\) the GCS tends to serve the UAV uniformly. While an increase in \({\bar{p}}\) allows the GCS to serve in more individual form according to non-uniform power allocation (32). This makes the problem remarkably distinguish from OFDM transmission where uniform strategy arise for large total power resource [8]. This is caused by the fact that OFDM utility can be approximated by a superposition of logarithm and linear function of transmission power for large applied power while proportional fairness utility of the considered game can be approximated similar way for small applied power.

4 Conclusions

The problem to maintain freshness of the commands received by a group of UAVs to succeed a mission under hostile interference was modeled as non-zero game. Proportional fairness in mission success by each of the UAVs is considered as criteria for the GCS. The problem is formulated and solved as non-zero some game The considered game differs remarkably from the conventional jamming games considered in literature [16, 20, 24] because the structure of the rivals’ strategies differ from from each other. In particular, the GCS’s strategy is power allocation between the UAVs, while the interferer’s strategy is a common power level assignment to jam the whole UAV’s group. Moreover, in OFDM jamming game with throughput as transmitter’s payoff [8], transmitter’s equilibrium strategy is uniform power allocation for large total transmitting power, while, in the considered game, GCS’s equilibrium strategy is uniform power allocation for small total transmitting power.