Jamming in FixedRate Wireless Systems with Power Constraints  Part I: Fast Fading Channels
Abstract
This is the first part of a twopart paper that studies the problem of jamming in a fixedrate transmission system with fading, under the general assumption that the jammer has no knowledge about either the codebook used by the legitimate communication terminals, or the source’s output. Both transmitter and jammer are subject to power constraints which can be enforced over each codeword (shortterm / peak) or over all codewords (longterm / average), hence generating different scenarios. All our jamming problems are formulated as zerosum games, having the probability of outage as payoff function and power control functions as strategies. The paper aims at providing a comprehensive coverage of these problems, under fast and slow fading, peak and average power constraints, pure and mixed strategies, with and without channel state information (CSI) feedback. In this first part we study the fast fading scenario. We first assume full CSI to be available to all parties. For peak power constraints, a Nash equilibrium of pure strategies is found. For average power constraints, both pure and mixed strategies are investigated. With pure strategies, we derive the optimal power control functions for both intraframe and interframe power allocation. Maximin and minimax solutions are found and shown to be different, which implies the nonexistence of a saddle point. In addition we provide alternative perspectives in obtaining the optimal intraframe power control functions under the longterm power constraints. With mixed strategies, the Nash equilibrium is found by solving the generalized form of an older problem dating back to Bell and Cover [1]. Finally, for comparison purposes, we derive a Nash equilibrium of the game in which no CSI is fed back from the receiver. We show that full channel state information brings only a very slight improvement in the system’s performance.
Keywords: Fast fading channels, fixed rate, capacity, jamming, zerosum game, outage probability, power control.
I Introduction.
The importance of designing antijamming strategies cannot be overstated, due to the extremely wide deployment of wireless networks, the very essence of which makes them vulnerable to attacks. Although the bases of jamming and antijamming strategies have been set in the 80’s and 90’s [2, 3, 4], new interest has been recently generated by the increasing demand for wireless security. Jamming and antijamming strategies were developed for the broadcast channel [5], the multiple access channel [6], and even studied from the perspective of an arbitrarily varying channel [7]. Under all scenarios, the jamming problem is formulated as a twoplayer, zerosum game. The corresponding objective functions are the sumrate [5], the ergodic capacity [6] or the capacity [7]. Although most often the jammer is assumed to have access to either the transmitter’s output or input [2, 4, 8] and consequently is able to produce correlated jamming signals, the correlation assumption can only be accurate for repeater protocols, or other situations where the jammer gets the chance to jam a signal about which it has already obtained some information from eavesdropping previous transmissions.
The approach of [7] is quite relevant to our work. The jamming problem is viewed as a special case of an arbitrarily varying channel (AVC). Constraints are placed either on the power invested in each codeword (peak power constraints), or on the power averaged over all codewords (average power constraints). The capacity, which is used to evaluate system performance, is defined as the maximum transmission rate that guarantees a probability of codeword error less than , under random coding. It is shown that when peak power constraints are imposed on both transmitter and jammer, the capacity is constant for , and therefore is the same as the channel capacity. No fading is assumed in [7], and consequently no power control strategies are necessary. However, fading channels are often the more practical models for wireless applications.
Traditionally, fast fading channels are characterized by their ergodic capacity, which is completely determined by the probability distribution of the channel coefficient and the transmitter power constraints. The physical interpretation of this measure of channel quality is related to the capabilities of channel codes. In the fast fading scenario, the codewords are assumed long enough to reveal the longterm statistical properties of the fading coefficient (in practical systems, this requirement may be satisfied by the use of interleaving [9]). Implicitly, power constraints are imposed over each codeword. Therefore, for achieving asymptotic error free communication, all codewords need to be transmitted at the same rate not exceeding the channel’s ergodic capacity.
However, applications like video streams in multimedia often require fixed data rates that could exceed the channel’s ergodic capacity, but can tolerate nonzero codeword error probabilities. Therefore, in situations when the transmitter’s available power is not sufficient for supporting a certain rate for each codeword in the traditional framework, the transmitter can choose to concentrate its power on transmitting only a subset of the codewords, while dropping the others. This maneuver ensures error free decoding of the transmitted messages, at the cost of a nonzero probability of message decoding error, which is feasible when power constraints are imposed over the ensemble of all codewords, instead of over each single codeword. This justifies the evaluation of fixed rate systems in fast fading channels by a quantity that is best known to characterize slow fading channels: the outage probability. Note that unlike the case of slow fading, in fast fading channels, due to the large codeword length, the channel conditions affecting the transmission of different codewords are asymptotically identical.
In this paper, we consider a fast fading AWGN channel where codewords (we denote the span of a codeword by the term frame) are considered long enough to reveal the longterm statistical properties of the fading coefficient. Our channel model is depicted in Figure 1. It was shown in [10] that the ergodic capacity of the fast fading AWGN channel can be achieved by a constantrate, constantpower Gaussian codebook, provided that when the fading coefficients are available at the transmitter, the transmitter employs a dynamic scaling of the code symbols, by the appropriate power allocation function. For this reason we assume in out model that the transmitter uses a capacityachieving complex Gaussian codebook. The jammer is assumed to have no knowledge about this codebook or the actual output of the transmitter, and hence its most harmful strategy is to transmit white complex Gaussian noise [11].
The channel coefficient is a complex number, the squared absolute value of which will be denoted throughout this paper by . The average powers invested by the transmitter and jammer in transmitting and jamming a codeword, respectively, are denoted by and . The transmitter and the jammer are subject to either peak power constraints (over each frame, or codeword) of the form and , or average power constraints (over all frames) of the form and , where the expectation is taken with respect to the players’ strategies of allocating the powers and between frames.
A codeword is decoded with strictly positive probability of error (i.e. outage) if the ergodic capacity calculated over the frame is below the fixed rate . The probability of this event (the equivalent of in [7]) will be denoted as the probability of outage . The transmitter aims at minimizing the probability of outage for a fixed rate , while the jammer attempts to maximize it. Our contributions can be summarized as below:

We first investigate the scenario where full channel state information (CSI) is available to all parties. For this case we show that peak power constraints are not efficient for high rate transmissions or large jammer power;

We formulate the scenario of average transmitter/jammer power constraints as a twoperson, zerosum game with the probability of outage as the payoff function.

Under average power constraints, we first investigate pure strategies and find the maximin and minimax solutions, as a result of two levels of power control: one within frames and one concerning the additional randomization introduced by the transmitter. Optimal strategies are derived for both levels, and it is shown that a Nash equilibrium of pure strategies does not exist in general.

Finally, for comparison purposes, we find the optimal transmitter and jammer mixed strategies for the case when the receiver does not feed back the CSI. Our results show that CSI feedback only brings slight improvements in the overall transmission quality.
One comment is in order. Note that Nash equilibria of mixed strategies are not always the best approach to practical jamming situations. An equilibrium of mixed strategies usually assumes that none of the two players knows exactly when or with what power the other player is going to transmit. While this may generally be true for the legitimate transmitter, a smart jammer might constantly eavesdrop the channel and detect both the legitimate transmitter’s presence and its power level. Therefore, many real jamming scenarios might be more accurately characterized by the solutions of the maximin problem formulation with pure strategies when the jammer tries to minimize and the transmitter tries to maximize the objective, and the solutions of the minimax problem formulation with pure strategies when the jammer tries to maximize and the transmitter tries to minimize the objective (the latter case applies to the present paper). At worst, these solutions provide a valid lower bound on system performance.
The paper is organized as follows. Section II formalizes the peak power constrained problem when full CSI is available to all parties. It turns out that this problem has an intuitive solution. Under the same full CSI assumption, Section III studies the problem of average power constraints and pure strategies, and is divided into three subsections. The first one presents the optimal strategies for allocating power over one frame. Using the results therein, the maximin and minimax solutions are derived in Subsection IIIB. Some numerical results are shown in Subsection IIIC. Section IV investigates the problem of full CSI, average power constraints and mixed strategies and provides the Nash equilibrium point. The scenario when the channel coefficients are only known to the receiver is investigated in Section V. Finally, conclusions are drawn in Section VI.
Ii CSI Available to All Parties. Jamming Game with Peak Power Constraints.
This game represents a more general version of the game discussed in Section IV.B of [6], and its solution relies on the results therein. The transmitter’s goal is to:
(1) 
while the jammer’s goal is to:
(2) 
where
is the ergodic capacity, which is completely determined by the p.d.f. of the channel coefficient and the transmitter/jammer power control strategies and . The expectation is defined as .
We prove that this game is closely related to the two player, zerosum game of [6], which has the mutual information between Tx and Rx as cost/reward function:
(3) 
(4) 
This latter game is characterized by the following proposition, proved in Section IV.B of [6]:
Proposition 1
(5) 
(6) 
where and are constants that can be determined from the power constraints and .
The connection between the two games above is made clear in the following theorem, the proof of which follows in the footsteps of [12] and is given in Appendix A.
Theorem 1
Let and denote the Nash equilibrium solutions of the game described by (3) and (4). Then the original game of (1), (2) has a Nash equilibrium point, which is given by the following pair of strategies:
(7) 
(8) 
where and are some arbitrary power allocations satisfying the respective power constraints. (Note that no particular improvements are obtained by setting , since only peak power constraints are in effect.)
The results are intuitive: if the ergodic capacity under the optimal jammer/transmitter strategies is larger than the fixed rate , reliable communication can be established over each frame, and hence the probability of outage is . In this case, the actual power allocation of the jammer does not matter anymore, since the jammer has already lost the game.
On the other hand, if the ergodic capacity is less than , outage occurs on all frames (), and the actual transmitter strategy makes no difference. As will be shown in the next section, enforcing average power constraints in this case gives the transmitter more freedom, and results in a smaller outage probability.
Iii CSI Available to All Parties. Jamming Game with Average Power Constraints: Pure Strategies.
In this section power constraints are imposed over a large number of frames rather than on each frame. The transmitter and jammer may increase their transmission and jamming powers over any frame from to , and from to , respectively. To satisfy the average power constraints imposed by and , less power has to be allocated to other frames. We shall prove that for both players, the optimal way to control the power allocation between frames is to employ ON/OFF strategies. Since all frames are equivalent from the point of view of their corresponding channel realizations, the manner in which the “discarded” codewords are picked is somewhat random. However, note that this type of randomization only aims at ensuring that a possibly larger or is obtained. We don’t consider mixing strategies in this section [13]. Although each player picks up a frame randomly, we assume this is known by its opponent when considering the maxmin and minimax problems as formulated below. That is, the maximin scenario assumes the transmitter has perfect noncausal access to the jammer’s strategy (we say the jammer “plays first”), while the minimax case assumes the jammer has perfect, noncausal access to the transmitter’s strategy (we say the transmitter “plays first”). The first player in the minimax or maxmin cases is always more vulnerable in the sense that the follower has the freedom to adapt its strategy such that it minimizes the first player’s payoff.
The minimax scenario is the more practical one. In addition to being pessimistic from the system designer’s point of view, it accurately models the situation where the jammer (who is not interested in exchanging any information of its own) listens to the feedback carrying the channel coefficients and senses the transmitter’s presence and power level, hence estimating the transmitter’s strategy. The maximin scenario is not of less importance, since it is required for determining the nonexistence of a Nash equilibrium and for comparison with the minimax approach.
An important remark should be made here. We shall prove in the sequel that under both the pure strategies and the mixed strategies scenarios, the optimal power allocation over a frame is done similarly. Therefore, the major difference between the two cases is in the strategies of allocating power to different frames. We should note that it is easier for one of the players to detect the presence of the other player over a frame, than to estimate the other player’s transmission power. Under the minimax solution of pure strategies, the jammer only needs to detect the presence of the transmitter (the optimal strategies are of ON/OFF type) to have complete information about the transmitter’s behavior. However, if the transmitter chose to use mixed strategies, a complete characterization of its behavior would require not only knowledge about its presence, but also about the power it decided to allocate to that frame.
The average power constrained jamming game can be formulated as:
(9) 
(10) 
where and are defined as in (1), (2), the expectation is taken over all frames with respect to the power allocation strategies introduced by the transmitter and jammer, and and are the upperbounds on average transmission power of the source and jammer, respectively.
Iiia Power Allocation within a Frame
The game between transmitter and jammer has two levels. The first (coarser) level is about power allocation between frames, and has the probability of outage as a cost/reward function. The probability of outage is determined by the number of frames over which the transmitter is not present or the jammer is successful in inducing outage. This set is established in the first level of power control which is investigated in detail in the next two subsections, but which cannot be derived before the second level strategies are available.
The second (finer) level is that of power allocation within a frame. In this subsection we derive the optimal second level of power allocation strategies for both maximin and minimax problems, and show they are connected by a special kind of duality.
Note that decomposing the problem into several (two or three) levels and solving each one separately does not restrict the generality of our solution. Our proofs are of a contradictory type. Instead of directly deriving each optimal strategy, we assume an optimal solution has already been reached and show it has to satisfy a set of properties. We first assume these properties are not satisfied, and then show that under this assumption there is room for improvement. Thus we prove that any solution not satisfying our set of properties cannot be optimal (i.e. the properties are necessary). We pick the properties in such a manner that they are sufficient for the complete characterization of the optimal solution. That is, we make sure that the system of necessary properties has a unique solution.
In the maximin case (when jammer plays first), assume that the jammer has already allocated some power to a given frame. Depending on the value of , and its own power constraints, the transmitter decides whether it wants to achieve reliable communication over that frame. If it decides to transmit, it needs to spend as little power as possible (the transmitter will be able to use the saved power for achieving reliable communication over another set of frames, and thus to decrease the probability of outage). Therefore, the transmitter’s objective is to minimize the power spent for achieving reliable communication over each frame. Note that if the jammer is present over a frame, the value of required to achieve reliable communication over that frame is a function of . However, the transmitter should attempt to minimize the required even when the jammer is absent. The jammer’s objective is then to allocate the given power over the frame such that the required is maximized.
In the minimax scenario (when transmitter plays first) the jammer’s objective is to minimize the power used for jamming the transmission over a given frame. The jammer will only transmit if the transmitter is present with some . The transmitter’s objective is to distribute within a frame such that the power required for jamming is maximized.
The two problems can be formulated as follows:
Problem 1 (for the maximin solution  jammer plays first)
(11) 
Problem 2 (for the minimax solution  transmitter plays first)
(12) 
Let denote the probability measure introduced by the probability density function (p.d.f.) of , i.e., for a set , we have . Denote . Note that the expectation is defined as . Similarly, we define .
Solution of Problem 1
The transmitter’s optimization problem:
(13) 
has linear cost function and convex constraints. Write the Lagrangian as:
(14) 
With the notation , the resulting KKT conditions yield the unique solution [14]:
(15) 
where
(16) 
and is the set of channel coefficients over which , and . We say the transmitter is “nonabsent” over , and “absent” on .
The following proposition, the proof of which is given in Appendix BA, states that the jammer should only be present where the transmitter is nonabsent.
Proposition 2
The jammer should only transmit where the transmitter is ”nonabsent”. Otherwise, if and for in some set , the jammer can decrease over and maintain the same required transmitter power over the frame.
Since the set depends on the jammer power allocation , solving the optimization problem above analytically is difficult. This is why we next provide an alternative method for finding the solution. Our method examines the properties of the sets over which the transmitter is present and over which the jammer is present, as well as those of the optimal transmitter/jammer strategies.
Fixing , the Lagrangian for the jammer’s optimization problem can be written as
(19) 
This yields the new KKT conditions:
(20) 
(21) 
(22) 
where is the set of channel coefficients on which the jammer transmits nonzero power.
For fixed and , the jammer’s optimal strategy has to satisfy these KKT conditions. The resulting optimal strategy is
(23) 
The expression above states that for any two channel realizations with coefficients belonging to , we have
(24) 
Note that for any two channel realizations (i.e. ) we also have
(25) 
The following proposition brings more insight into the optimal jamming strategy. Its proof is deferred to Appendix BB.
Proposition 3
The optimal jamming strategy is such that is a continuous decreasing function of over all of , and is of the form . Moreover, this implies that is of the form .
The optimal transmitter/jammer strategies for allocating power over a frame are described in Figure 2.
Substituting (23) into (16), we get a new expression for :
(26) 
which together with (15) yields
(27) 
An interesting remark which supports the results of the next subsection is that, for the optimal solution of Problem 1, has to be strictly greater than zero, hence eliminating the possibility that the jammer allocates positive power to frames where the transmitter, although “nonabsent”, could allocate zero power. In Appendix BB it is shown how this remark follows from Proposition 3.
The second level power allocation solution for the maximin problem is thus completely determined by the triple , or equivalently by . By Proposition 3 above, (by continuity in ), and . Rearranging these two relations, along with (IIIA) in a more convenient form, we obtain the following system of equations, which has to hold for any solution to our problem:
(30) 
(31) 
(32) 
The equations above lead to the following result:
Proposition 4
The solution of the maximin second level power allocation problem is unique.
It is easy to see that the right hand side of (31) is a strictly decreasing function of , for fixed , and a strictly decreasing function of , for fixed , while being equal to a constant. Hence, for given , (31) yields as a strictly decreasing function of .
Similarly, the right hand side of (IIIA) is a strictly decreasing function of , for fixed , and a strictly increasing function of , for fixed , while being equal to a constant. Hence, (IIIA) yields as a strictly increasing function of .
Since (31) and (IIIA) have to be satisfied simultaneously by any solution, the solution has to be unique.
Another insightful remark that follows from (30)–(IIIA) is that as increases, both and should be decreasing.
The following proposition, characterizing the function, is necessary for deriving the optimal power allocation between frames in the next section. The proof is deferred to Appendix BC.
Proposition 5
Under the optimal maximin second level power control strategies, the “required” transmitter power over a frame is a strictly increasing, unbounded and concave function of the power that the jammer invests in that frame.
Throughout the remainder of this paper, we shall denote by the function that characterizes the “required” transmitter power over a frame where the jammer invests power , in the maximin case.
Solution of Problem 2
To solve the minimax intraframe power allocation problem by using the same techniques as in Problem 1 turns out to be more difficult. Instead we use the above solution of Problem 1 and show that for both problems, the second level power allocation follows the same rules.
Theorem 2
If is the value used for the second constraint in Problem 1 above, and is the resulting value of the cost/reward function, then solving Problem 2 with yields the cost/reward function . Moreover, any pair of second level power allocation strategies that makes an optimal solution of Problem 1, should also make an optimal solution of Problem 2, and this also holds conversely.
The result is a direct consequence of Theorem 8 in Appendix BD, if we denote , , , and . We shall denote by the function that characterizes the “required” jamming power over a frame where the transmitter invests power , in the minimax case. By Theorem 2, we have that and .
Further comments on the power control within frames
Although the second level optimal power allocation strategies for the maximin and minimax problems coincide, this result should not be associated to the notion of Nash equilibrium, since the two problems solved above do not form a zerosum game, while for the game of (9) and (10), first level power control strategies are yet to be investigated.
Instead, the result should be interpreted as a form of duality. In fact, a much stronger result can be observed as a consequence of Theorem 8. Namely, a similar “duality” property links Problem 1 and Problem 2 above to the auxiliary problem of (3) and (4) appearing in the peak power constraints scenario. This explains the resemblance between the solution of the peak power constraints auxiliary problem (6) and the solution of Problem1 (26), (27).
Also, this common solution implies that over the set of channel realizations where both jammer and transmitter are present. Although the transmitter is also active over the set of nonzero measure as in Figure 2, under practical conditions the measure of this set is relatively small. This is the reason why the curve appears to be linear (although it is not) in Figure 3 of the numerical results section.
IiiB Power Allocation between Frames
The Maximin Solution
In this subsection we present the first level optimal power allocation strategies for the maximin problem. Recall that all frames are equivalent in the sense that they are all characterized by the same channel realizations (although not necessarily occurring in the same chronological order).
The maximin scenario assumes that the transmitter is completely aware of the jammer’s power control strategy (only pure strategies are considered in this section). Given a jammer’s strategy that allocates different jamming powers to different frames, the optimal way of allocating the transmitter’s power is always to ensure that reliable communication is obtained on the frames that require the least amount of transmitter power. The jammer’s optimal strategy (which is based solely on this knowledge about the transmitter’s strategy) is presented in the following theorem.
Theorem 3
Under the maximin scenario it is optimal for the jammer to allocate the same amount of power to all frames.
The proof relies on the concavity of . Consider the optimal maximin interframe power allocation strategies. Let denote the sets of frames over which the transmitter and the jammer are present, respectively. Note that the jammer can itself compute the optimal transmitter strategy in response to its own, and hence is fully informed of the transmitter’s response.
We first look at the set of frames where the transmitter is active. Denote the power invested by the jammer in this set by . Note that is the average “required” transmitter power over .
If the two players’ strategies are both optimal, then by modifying the allocation of over the frames of , the new average “required” transmitter power over can only be less than or equal to . In other words, if we denote by the generic power level allocated by the jammer to a frame in , then
(33) 
subject to
(34) 
By writing the KKT conditions for the maximization problem in (33) and (34) above, it is straightforward to see that, at an optimum, should be constant all over . Taking into account the fact that is concave, we have that a uniform jamming power allocation of over achieves this optimum.
We next look at the set of frames where the transmitter cannot afford to be active. This means that the “required” transmitter power over is greater than or equal to , or equivalently, the power invested by the jammer is greater than or equal to . But since the jammer already knows the transmitter’s strategy, investing more than in any of the frames of would be a waste.
Therefore, under the optimal maximin interframe power allocation strategies, the jammer can invest the same amount of power into all the frames of (which means ).
But since the transmitter decides to match the required transmitter power on , there can be no frames where the jammer is not active, and hence is the set of all frames.
The jamming power allocated to each frame is . In this case the transmitter faces an indifferent choice space. The power required for the transmitter to achieve reliable communication is . Hence, the transmitter’s optimal strategy is to randomly pick as many frames as possible and allocate power to each of them. This is equivalent to saying the transmitter is present over a frame with probability , given by . The resulting probability of outage is now .
Note that if , the probability of outage can be reduced to zero. This corresponds to the case when the ergodic capacity of the channel, computed in the conventional way, with peak power constraints, is larger than the rate .
The Minimax Solution
Theorem 2 showed that for the minimax problem the power allocation within a frame, as well as the relationship between the total powers used by transmitter and receiver over a particular frame, are identical to the maximin problem. Hence, by rotating the plane, we get the characteristic curve for the minimax problem.
The minimax scenario assumes that the jammer knows exactly when and with what power level the transmitter transmits. Given a transmitter’s strategy that allocates different powers to different (equivalent) frames, the optimal way of allocating the jammer’s power is such that outage is first induced on the frames that require the least amount of jamming power.
Under these conditions, the transmitter’s optimal strategy is presented in the following theorem.
Theorem 4
Under the minimax scenario it is optimal for the transmitter to transmit over a maximum number of frames, with the same power that minimizes the probability of outage.
The proof relies on the convexity of . Consider the optimal minimax interframe power allocation strategies, and let denote the sets of frames over which the transmitter and the jammer are present, respectively. It is clear in this scenario that .
We first look at the set of frames where the jammer is active. Denote the power invested by the jammer in this set by , and the power invested by the transmitter by . Note that is the average “required” jamming power over .
If the two players’ strategies are both optimal, then by modifying the allocation of over the frames of , the new average “required” jamming power over can only be less than or equal to . In other words, if we denote by the generic power level allocated by the transmitter to a frame in , then
(35) 
subject to
(36) 
From the KKT conditions for the maximization problem in (35) and (36) above, we see that, at an optimum, should be constant all over . Taking into account the fact that is convex, we have that a uniform transmitter power allocation of over achieves this optimum.
We should emphasize here that the above arguments hold under the assumption that the jammer is active over the whole set , i.e. when over . Of course, the overall required jamming power is increased by increasing the transmitter power over some frames of , while neglecting the others. But this action modifies the set itself, and thus the initial assumptions.
We next look at the set of frames where the jammer cannot afford to be active. This means that the “required” jamming power over is greater than or equal to , or equivalently, the power invested by the transmitter is greater than or equal to . But since the transmitter already knows the jammer’s strategy, investing more than in any of the frames of would be a waste.
Therefore, under the optimal maximin interframe power allocation strategies, the transmitter can invest the same amount of power into all the frames of .
The frames over which the transmitter allocates the optimal can be chosen at random. This is equivalent to the transmitter being active over a frame with probability given by . Searching for the optimal is equivalent to searching for the optimal .
The jammer’s strategy is to attack as many of the frames where the transmitter is present as possible. In order to induce outage over these frames, the jammer needs to allocate to each of them. This is equivalent to the jammer transmitting on a frame on which the transmitter is present, with probability given by . Note that represents the conditional probability that the jammer transmits over a frame, given that the transmitter is present over that frame. Outage over a frame occurs in two circumstances: either the transmitter (and consequently also the jammer) decides to ignore the frame, or the transmitter attempts to transmit the corresponding codeword, but the jammer is present (and since this is the minimax scenario, it is also successful).
The resulting probability of outage is or, only as a function of :
(37) 
The transmitter finds the optimal value of as the argument that minimizes above. A numerical approach should perform exhaustive search with the desired resolution in the interval , where can be set such that we have for a fixed . Since as independently of the curve, such a finite bound exists for any .
Note that if the curve is strictly concave, the jammer can never achieve an outage probability . This is because the transmitter can invest all its power over a small enough set of frames, such that the jamming power required to jam all the frames in this set exceeds the jammer’s power budget. If however the probability measure is chosen such that is an affine function of the form , and furthermore if , then for all values of , and the probability of outage becomes .
IiiC Some Numerical Results
An example of the curve is given in Figure 3 for a fixed rate , noise power and a channel coefficient distributed exponentially, with parameter .
For the same parameters used to generate Figure 3, the probability of outage was computed for a jammer power constraint and different values of the transmitter power constraint . The results were plotted in Figure 4. For comparison, the same figure shows for the case when the jammer does not use any power control strategy (nonintelligent jammer). Since the jammer’s first level of power control for the maximin scenario reduces to uniformly distributing the available power to all frames, the only difference between the maximin scenario and the nonintelligent jammer scenario is in the power allocation within frames. However, as seen from Figure 4, this difference is almost negligible.
Figure 5 shows how the outage probability varies with the rate , for fixed power constraints and . The curves delimitate the achievable capacity vs. outage regions for both peak power constraints and average power constraints (minimax and maximin cases).
Note that even for the minimax solution of the average power constraints problem, there exist values of (Figure 4), or of the rate (Figure 5) for which the outage probability is less than that achievable under peak power constraints.
Also note that the maximin curve coincides with the peak power constraints curve at large transmitter power (in Figure 4) or at small rates (in Figure 5). Recall that the jammer’s strategy in the maximin scenario is the same as in the peak power constraints scenario (i.e. the jammer allocates the same amount of power to each frame). Due to the favorable conditions in the regions characterized by large or small , the transmitter can also spread its power uniformly over all frames (just like in the peak power constraints scenario), overcoming the jammer completely (hence the resulting zero probability of outage).
Iv CSI Available to All Parties. Jamming Game with Average Power Constraints: Mixed Strategies.
In the previous section we studied the maximin and minimax solutions of the jamming game when only pure strategies were allowed. Implicitly, we assumed that the power control strategies employed by the first player are perfectly known to the second player, even if they include a form of ON/OFF randomization. We made a case that such a situation as the minimax case can emerge when the jammer does not transmit unless it senses that the transmitter is on (and it can always serve as a pessimistic scenario for the transmitter).
However,our previous assumption may sometimes be inappropriate from a practical point of view. For example, if the transmitter does not stick with the optimal minimax solution, the jammer may have a hard time following the transmitter’s behavior. The reason for this is that, as we have already mentioned, the jammer would find it much harder to correctly estimate the amount of power that the transmitter invests in a given frame, than to just detect the presence of the transmitter.
In this section we investigate the jamming game with average power constraints when mixed (probabilistic) strategies are considered. Similarly to the pure strategies scenario of the previous section, this game is played on two levels, with the first (coarser) level dealing with power allocation between frames. Its cost/reward function is the probability of outage. We assume that the jammer’s and transmitter’s randomized strategies consist of picking the power values to be invested over a frame in a random manner. In our previous notation, and are now random variables, and each frame is characterized by a realization of the pair .
Given this realization, each player has to distribute its power over the frame in an optimal way. This is the purpose of the second (finer) level of power control. The objective of each player at this level is to make the best of the available resources (i.e. the powers ). This means maximizing (or minimizing, respectively) the average rate supported by the frame, in the hope that the resulting average rate will be above (or below, respectively) the system’s fixed rate .
Iva Power allocation within a frame
We can formulate the second level of power control similarly to the twoplayer, zerosum game of (3) and (4) having the ergodic capacity calculated over a frame as cost function. The difference is that under the current scenario, none of the players knows the other player’s constraints, because is a random event. Theorem 5 below provides the optimal transmitter/jammer strategies for power allocation within a frame.
Theorem 5
Given a realization of , let denote the solution of Problem 1 in Section III with , and denote the solution of Problem 2 in Section III with .
The transmitter’s optimal strategy is the solution of the game in (3) and (4), where the jammer is constrained to and the transmitter is constrained to . The jammer’s optimal strategy is the solution of the game in (3) and (4), where the transmitter is constrained to and the jammer is constrained to .
Note that each of the two players deploys the strategy that results from the most pessimistic scenario that it can handle successfully.
Denote the solution of the game in (3) and (4), where the jammer is constrained to and the transmitter is constrained to by , and the solution of the game in (3) and (4), where the transmitter is constrained to and the jammer is constrained to by .
Denote the solution of the game in (3) and (4), where the jammer is constrained to and the transmitter is constrained to by ..
We will show that (i) even if mixed strategies are considered for the game in (3) and (4), any Nash equilibrium has the same value as the Nash equilibrium of pure strategies; (ii) even if the jammer’s power is different from , the transmitter’s strategy is still optimal; (iii) even if the transmitter’s power is different from , the jammer’s strategy is still optimal.
(i): Since the game of (3) and (4) is a twoperson zerosum game, all Nash equilibria of mixed strategies yield the same value of the cost/reward function [13]. Moreover, the two players are indifferent between all equilibria. It was shown in [6] that this game has a Nash equilibrium of pure strategies. But any equilibrium of pure strategies is also an equilibrium of mixed strategies [13] and hence it is enough to consider the equilibrium of pure strategies found in [6].
(ii),(iii): Assume the transmitter plays the strategy given by .
If , it is clear that the optimal solution for both transmitter and jammer is the solution of the game in (3) and (4), where the jammer is constrained to and the transmitter is constrained to . In this case, it is as if each player knows the other player’s power constraint.
If , then by Lemma 4 in Appendix BC we have that . Since is a strictly decreasing function of (under the order relation defined in Appendix BD), this implies that . Note that is the jammer’s strategy when the jammer knows the transmitter’s power constraint . Thus we have shown that when the transmitter plays and , the jammer cannot induce outage over the frame even if it knew the value of .
The condition is equivalent to (by Theorem 8). In this case, since the jammer plays the strategy given by , a similar argument as above (but this time applied to the transmitter’s strategy) shows that the transmitter cannot achieve reliable communication over the frame even if it knew the exact value of .
This accomplishes the proof and shows that is a Bayes equilibrium [13] for the game with incomplete information describing the power allocation within a frame.
IvB Power allocation between frames
Due to the form of the optimal second level power allocation strategies described in the previous subsection, the outage probability can be expressed as
(38) 
where is the strictly increasing, unbounded and concave function of Proposition 5. The optimal mixed strategies for power allocation between frames are presented in the following theorem.
Theorem 6
The unique Nash equilibrium of mixed strategies of the twoplayer, zerosum game with average power constraints described in (9) and (10) is attained by the pair of strategies satisfying:
(39) 
(40) 
where denotes the CDF of a uniform distribution over the interval , and denotes the CDF of a Dirac distribution (i.e. a step function), and the parameters and are uniquely determined from the following steps:

Find the unique value which satisfies:
(41) 
Compute .

If , then is the unique solution of
(42) (43) and
(44) 
If then , .

If , then is the unique solution of
(45) (46) and
(47)
IvC Numerical results
For the same parameters as in subsection IIIC we evaluated numerically the optimal probabilistic power control strategies. Figure 6 shows the probability of outage obtained under the mixed strategies Nash equilibrium, versus the transmitter power constraint , for a fixed rate , noise power , a jammer power constraint and a channel coefficient distributed exponentially, with parameter . All the previously obtained curves are shown for comparison.
Figure 7 shows the same probability of outage when and the system rate is varied.
In both figures it can be seen that the system performance under the Nash equilibrium of mixed strategies is better (from the transmitter’s point of view) than the minimax and worse than the maximin solutions of the pure strategies game. This is expected since the pure strategies solutions assume that the second player (the “follower”) is constantly at a disadvantage with the first player (the “leader”).
V CSI Available to Receiver Only. Jamming Game with Average Power Constraints: Mixed Strategies
In this section we investigate the scenario when the receiver does not feed back any channel state information. Since we have already shown that the long term power constraints problem is the more interesting and challenging one, we further focus only on the scenario of average power constraints and mixed strategies. As in the previous sections, we have to discuss two levels of power control: within a frame and between frames.
Va Power allocation within a frame
The jammer and transmitter powers allocated to each frame will be established in the next subsection. For now we are concerned with the optimal power allocation within a frame, given the amounts of power invested in that frame by each one of the players. For a given frame, denote these powers by and , to be consistent with our previous notation. Both the transmitter and the jammer will choose a probability distribution for the randomly variable power levels and , respectively, such that and , where the notations and denote the expectations with respect to these probability distributions. For the generic channel use, the channel coefficient , the transmitter’s power and the jammer’s power are all independent random variables, which yield the randomly variable instantaneous mutual information . For a frame, this results in the ergodic capacity , where denotes expectation with respect to the channel coefficient.
The transmitter’s purpose is to use the allocated power in an attempt to make this ergodic capacity larger than the rate . Similarly, the jammer is concerned with using for making the ergodic capacity fall below . The problem of allocating the power within the frame can be written as:
(48) 
Denote and let us observe that
(49) 
(50) 
(51) 
(52) 
which implies that is a strictly increasing, concave function of for fixed , and a strictly decreasing, convex function of for fixed .
Thus, we can write
(53) 
and hence the uniform distribution of and over the frame achieves a Nash equilibrium. A frame to which the transmitter allocates power