A Prospect Theoretic Look at a JointRadar and Communication System

Andrey Garnaev1,2(B), Wade Trappe2, and Athina Petropulu3

1 Saint Petersburg State University, Saint Petersburg, Russiagarnaev@yahoo.com

2 WINLAB, Rutgers University, North Brunswick, USAtrappe@winlab.rutgers.edu

3 ECE, Rutgers University, Piscataway, USAathinap@rutgers.edu

Abstract. In this paper, we consider the problem of finding how a jointradar and communication system should divide its effort between sup-porting the radar and communication objectives when the system oper-ates in an environment with hostile interference. Our model exploresthe uncertainty of the jammer’s location by assuming the joint systemknows only the a priori probabilities of jammer’s positions. The underly-ing problem is formulated and solved as a Bayesian game involving thejoint radar/communication system and a jammer. We then explore howirrational behavior by the rivals can affect the equilibrium strategies byusing prospect theory (PT). It is shown that the PT system strategyis not sensitive to the jammer’s probability weighting parameter, whilejammer’s strategy is sensitive to probability weighted parameters of bothrivals.

Keywords: Communication · Radar · Bayesian gameProspect theory

1 Introduction

Recently, there has been interest in enabling radar and communication systemsto co-exist in the same frequency bands in order to allow spectrum to be utilizedmore efficiently [20]. This has given rise to a significant amount of research onmethods for spectrum sharing between the two systems. One approach to achievethis is to formulate waveform design using OFDM signals and then optimallyallocating the subcarriers [14,24]. Radar waveform design for controlled interfer-ence is considered in [2,3], while the cooperative design of the two systems wasexplored in [4,17].

In this paper, we consider a dual purpose communication-radar system thatemploys OFDM style waveforms and explore the complementary aspect of findingthe optimal frequency of performance for radar and communication objectiveswhen the joint system faces with hostile interference. Moreover, we considerc© Springer Nature Switzerland AG 2018O. Galinina et al. (Eds.): NEW2AN 2018/ruSMART 2018, LNCS 11118, pp. 483–495, 2018.https://doi.org/10.1007/978-3-030-01168-0_43

484 A. Garnaev et al.

that the system might know only a priori probabilities about jammer’s position.The problem is formulated and solved as a Bayesian game. To address the riskthat the rivals’ behaviour might be irrational, equilibrium strategies are foundusing prospect theory. We note that, although prospect theory originally wasdesigned to take into account possibility of the risk of irrational rivals behaviourin economic problems [15,16], it has been applied in engineering applications[18,22,25].

The organization of this paper is as follows: in Sect. 2, we present the basicsystem model, and then we incorporate a jammer in the model in Sect. 3. InSect. 4, a basic zero-sum game for selecting the mode of attack and transmissionfor a fixed jammer position is formulated, while, in Sect. 5, its PT solution isfound. In Sect. 6, the basic game is generalized to a Bayesian game where thesystem knows only a priori probabilities associated with the jammer’s possiblelocation, while, in Sect. 7, the PT solution for this Bayesian game is found.Finally, in Sect. 8, discussion of the results and conclusions are given.

2 Basic Model

We begin our formulation by considering an operational scenario involving anRF transceiver that is attempting to support two different objectives: commu-nication with a communication receiver that is distant and separate from thetransmitter, while also supporting the tracking of a radar target through thereflections witnessed at the RF transmitter. In order to support these two differ-ent objectives, the transmitter uses a spectrum band that is modeled as consist-ing of n adjacent sub-channels, which may be associated with n different subcar-riers. In this paper we employ a transmission scheme like OFDM, as consideredin [12,13] for designing a bargaining strategy for a dual radar and communica-tion system in the absence of hostile interference. With each of these n differentsubcarriers, two different (fading) channel gains are associated. Specifically, welet hR,i correspond to the i-th radar channel gain associated with the round-tripeffect of the transmitted signal, reflected off the radar target, and received atthe RF transceiver, while hC,i denotes the i-th channel gain associated with thei-th communication subcarrier between the transmitter and the communicationrecipient. Although there are two different objectives, there is nonetheless a sin-gle transmitter responsible for deciding how to allocate power across the differentsubcarriers. We consider a power-allocation strategy for the transmitter to bethe power vector P = (P1, . . . , Pn) where Pi is the power assigned for transmittingon subcarrier i, and

∑ni=1 Pi = P where P is the total power budget allocated

for transmission. We assume that the system, to avoid mutual interference ofthe signals, can work in one of two modes: (i) communication mode for per-forming only the communication task, and (ii) radar mode for performing onlythe radar task. In order to unify the examination of radar and communicationmetrics, we note that radar detection/tracking and communication throughputare both closely related to the associated signal-to-interference-plus-noise ratio(SINR) as witnessed at the appropriate recipient. Let the radar SINR be given

A Prospect Theoretic Look 485

byn∑

i=1

hR,iiPi/σ2R and the communication SINR by

n∑

i=1

hC,iPi/σ2C , where σ2

R and

σ2C are corresponding background noises. For the communication objective, the

SINR is used as the payoff function for two reasons: first, it is easily linearized;and, second, for a low SINR regime, SINR in an approximation to throughput.For the radar objective, SINR is used as the payoff function since it is closelyrelated to the associated detection metrics [19].

3 The Jammer

Now, suppose there is an adversary present in the environment who seeks tointroduce hostile interference to disrupt the functionality of either the radar orcommunication system. His strategy is a power vector J = (J1, . . . , Jn) whereJi is the power assigned for jamming subcarrier i, and

∑ni=1 Ji = J where J

is the total power budget allocated for jamming. Under a jamming attack, thecommunication and radar SINRs are given by: SINRR(P ,J) =

n∑

i=1

hR,iPi/(σ2R +

gR,iJi) and SINRC(P ,J) =n∑

i=1

hC,iPi/(σ2C + gC,iJi), where gR,i are fading channels

gains between the jammer and the radar, and gC,i are fading gains between thejammer and the communication receiver.

When the system works in a particular mode the corresponding SINR is thepayoff to the system, i.e., vm(P ,J) = SINRm(P ,J) for m ∈ {C, R}. Let us assumethe jammer knows what mode the system is in, then the system payoff can beconsidered as the cost function for the jammer. Thus, here we deal with a zero-sum game-theoretical scenario [5]. Recall that a pair of strategies (Pm,Jm) is a(Nash) equilibrium for the game with payoff function vm for m ∈ {C, R} if andonly if, for any strategies (P ,J), the following inequalities hold:

vm(P ,Jm) ≤ vm(Pm,Jm) ≤ vm(Pm,J). (1)

This game can be solved following [1]. See, also, [6,8,9,11,23,26], as examplesof other jamming games.

4 Mode Selection Game

In this section, we consider the scenario where the jammer does not know whichthe mode the system is in, while the system does not know which mode thejammer’s effort is optimized against. The system can choose its mode to use,while the jammer can choose the mode he wants to optimize his effort against.If a mode is selected, each of the rivals allocates power according to the optimalstrategy for this mode. This problem can be described by the following 2 × 2payoff matrix

M =(R C

R A aC b B

), (2)

486 A. Garnaev et al.

with A = vR(P R,JR), a = vR(P R,JC), B = vC(P C ,JC), b = vC(P C ,JR). By (1),

we can assume that

A < a, B < b. (3)

In matrix M , the rows correspond to the system’s strategies, i.e. signal trans-mission according to one of the two modes; and the columns correspond to thejammer’s strategies, i.e. jamming against a particular system mode.

Let x = (x, 1 − x)T and y = (y, 1 − y)T , be randomized (mixed) strategies [5]for the system and the jammer, i.e., x and 1−x (y and 1−y) be the probabilitiesfor employing (pure) strategies “R” and “C” by the system (respectively, theadversary). Then, the expected payoff to the system is given:

v(x,y) = xT My = Axy + ax(1 − y) + b(1 − x)y + B(1 − x)(1 − y). (4)

while for the jammer v is cost function. We look for a (Nash) equilibrium [5],i.e., for a pair of strategies (x∗,y∗) such that the following inequalities hold:

v(x,y∗) ≤ v(x∗,y∗) ≤ v(x∗,y) for all (x,y) (5)

This is a classical 2 × 2 matrix zero sum game [5], which has a closed formsolution.

Theorem 1. The game has the unique equilibrium (x,y). Namely,

x =

⎧⎪⎨

⎪⎩

ξ, B < a & A < b,

1, A ≥ b,

0, B ≥ a,

y =

⎧⎪⎨

⎪⎩

Ξ, B < a & A < b,

1, A ≥ b,

0, B ≥ a,

(6)

with ξ := (b − B)/(a + b − A − B) and Ξ := (a − B)/(a + b − A − B).

Note that two inequalities A ≥ b and B ≥ a cannot hold simultaneouslysince otherwise summing them up implies A + B ≥ a + b, and this contradictsassumption (3). Thus, (6) defines (x,y) uniquely.

5 PT Solution for Mode Selection Game

In the previous section, we assumed that rivals’ decisions were according tothe expected utility theory. To describe decisions under the risk that the agents’behaviour might be irrational occur, prospect theory was developed [15]. Accord-ing to [15], agents use their subjective probabilities w rather than objectiveprobabilities p to weight the values of possible outcomes. Moreover, agents tendto over-weight low probability outcomes and under-weight moderate and highprobability outcomes. This feature is captured by weighting the probability dis-tribution by an S-shaped function, the so-called weighting function. The originalexample of weighting function [16] is given by

A Prospect Theoretic Look 487

wγ(p) = pγ/(pγ + (1 − p)γ)1/γ , (7)

with γ ∈ [1/2, 1] the probability weighting parameter, where the lower bound onγ comes from [21]. In particular, this weighting function infinitely overweightsinfinitesimal probabilities and infinitely underweights near-one probabilities, i.e.wγ(p)/p tends to infinity for p ↓ 0 and (1−wγ(p))/(1−p) tends to infinity for p ↑ 1.

Denote by α and β these parameters for the system and the jammer, respec-tively. Then, the PT-utilities for the rivals in the matrix game (2) are given asfollows:

uPTS (x, y) := x (Awβ(y) + awβ(1 − y)) + (1 − x) (bwβ(y) + Bwβ(1 − y)) ,

uPTJ (x, y) := y (Awα(x) + bwα(1 − x)) + (1 − y) (awα(x) + Bwα(1 − x)) .

(8)

Then, the PT-equilibrium is given as the solution of the best response equations:

x = BRPTS (y) := argmax

x∈[0,1]

uPTS (x, y), y = BRPT

J (x) := argminy∈[0,1]

uPTJ (x, y). (9)

Theorem 2. The game has the unique PT equilibrium (x,y). Namely,

x =

⎧⎪⎨⎪⎩

ξ(α), B < a&A < b,

1, A ≥ b,

0, B ≥ a,

y =

⎧⎪⎨⎪⎩

Ξ(β), B < a&A < b,

1, A ≥ b,

0, B ≥ a,

(10)

with ξ(α) := (b − B)1/α/((a − A)1/α + (b − B)1/α) and Ξ(β) := (a − B)1/β/((a −B)1/β + (b − A)1/β).

Proof: Since uPTS (x, y) is linear on x while uPT

J (x, y) is linear on y solving theoptimization problems (7) we find the best response strategies as follows:

BRPTS (y) =

⎧⎪⎨⎪⎩

0, (b − A)wβ(y) > (a − B)wβ(1 − y),any in [0, 1], (b − A)wβ(y) = (a − B)wβ(1 − y),1, (b − A)wβ(y) < (a − B)wβ(1 − y),

(11)

BRPTJ (x) =

⎧⎪⎨⎪⎩

1, (a − A)wα(x) > (b − B)wα(1 − x),any in [0, 1], (a − A)wα(x) = (b − B)wα(1 − x),0, (a − A)wα(x) < (b − B)wα(1 − x).

(12)

Substituting (7) with γ = α into (12) implies

BRPTJ (x) =

⎧⎪⎨⎪⎩

1, x > ξ(α),any in [0, 1], x = ξ(α),0, x < ξ(α).

(13)

488 A. Garnaev et al.

Fig. 1. (a) Probability x and (b) probability y for PT equilibrium.

Let b ≤ A. Then, by (3), a > B. Thus, (11) implies that BRPTS (y) ≡ 1. Then,

by (13), BRPTJ (x) ≡ 1.

Let a ≤ B. Then, by (3), b > A. Thus, (11) implies that BRPTS (y) ≡ 0. Then,

by (13), BRPTJ (x) ≡ 0.

Let b > A and b > A. Then, substituting (7) with γ = β into (11) yields that

BRPTS (y) =

⎧⎪⎨⎪⎩

0, y > Ξ(β),any in [0, 1], y = Ξ(β),1, y < Ξ(β).

(14)

Thus, by (13) and (14), x ∈ (0, 1) and y ∈ (0, 1) is the PT equilibrium if andonly if x = ξ(α) and y = Ξ(β), and the result follows. �

It is clear that the PT equilibrium in mixed strategies (10) coincides withthe Nash equilibrium (10) for α = β = 1. The threshold condition for switchingthe PT equilibrium from mixed to pure strategies is stable with respect to theprobability weighting parameters (i.e. it does not depend on them). Both mixedstrategies ξ(α) and Ξ(β) are monotonic with respect to the probability weightingparameters. Namely, if a + B > b + A then ξ(α) is increasing and less than theNE strategy x. If a + B < b + A then ξ(α) is decreasing and always greater thanthe NE strategy x. While if b + B > a + A then Ξ(β) is increasing and less thanthe NE strategy y. If b + B < a + A then Ξ(β) is decreasing and less than theNE strategy y. Fig. 1 illustrates these monotonic properties by an example n = 5,

σ2C = 1, σ2

R = 1, hC = (0.9, 0.7, 0.8, 1, 0.9), gC = (1, 2, 3, 4, 5), hR = (0.8, 0.9, 1, 0.7, 0.8),

gR = (5, 4, 3, 2, 1) and P = 1. Moreover, Fig. 1(a) illustrates that variation innetwork parameters (in this case, total jamming power) can change the type ofmonotonicity.

6 Uncertainty About the Jammer’s Position

The channel gains depend on the positioning of the jammer and the receiver [27].In this section we assume that the system does not know the jammer’s positionwith certainty, but knows only a priori probabilities about jammer’s position.Namely, let the jammer be at point (Xt, Yt) with a priori probability γt, and letthe corresponding channel gains be {hC,i,t, gC,i,t, hR,i,t, gC,i,t}. Let Pm,t and Jm,t

be the equilibrium strategies when the system is in mode m and the jammer acts

A Prospect Theoretic Look 489

versus this mode. We assign the type, namely, type t, to the jammer associatedwith his location. Let Mt be matrix of payoffs for the system facing a jammer oftype t designed based on strategies Pm,t and Jm,t, while At, at, Bt, bt be entriesof this matrix. Following (3), we can assume that

At < at and Bt < at. (15)

Let yt = (yt, 1 − yt)T be the strategy of the jammer with type t, and Y =

(y1, . . . ,yT ). Let x = (x, 1 − x)T be the strategy of the system. Then, the(expected) payoff to the system is given as follows v(x, Y) =

∑Tt=1 γtx

T Mtyt,

while the cost function for the jammer of type t is given by vt(x,yt) = xT Mtyt.

The system wants to maximize its expected payoff while each of type of thejammer wants to minimize its cost function. Thus, here we deal with a Bayesiangame. Recall that (x∗, Y∗) is the Bayesian equilibrium if and only if for any (x, Y)

the following inequalities hold:

v(x,Y∗) ≤ v(x∗,Y∗) and vt(x∗,yt∗) ≤ vt(x∗,yt) for t = 1, . . . , T. (16)

Let us introduce the following axillary notations:

Θt :=t∑

τ=1

γτ (aτ + bτ − Aτ − Bτ ) for t = 1, . . . , T and Θ0 = 0,

ξt := (bt − Bt)/(at + bt − At − Bt) for t = 1, . . . , T and ξ0 = 0,

Θ :=T∑

t=1

γt(at − Bt).

(17)

By (15), 0 < ξt < 1. To avoid bulkiness in formulas we assume that ξt �= ξτ

for any t �= τ . Then, without loss of generality we can assume that

0 = ξ0 < ξ1 < ξ2 < . . . < ξT < 1. (18)

Theorem 3

(a) Let Θ ≤ 0. Then the equilibrium (x, Y) is unique and x = 0, yt = 0, t = 1, . . . , T .(b) Let ΘT < Θ. Then the equilibrium is unique and x = 1, yt = 1, t = 1, . . . , T .

(c) Let Θt∗−1 < Θ < Θt∗ for a t∗ ∈ {1, . . . , T}. Then the equilibrium is uniqueand

x = ξt∗ and yt =

⎧⎪⎨⎪⎩

1, t ≤ t∗ − 1,

(Θ − Θt∗−1)/(at∗ + bt∗ − At∗ − Bt∗), t = t∗,0, t ≥ t∗ + 1.

(19)

(d) Let Θ = Θt∗ for a t∗ ∈ {1, . . . , T}. Then each x such that ξt∗−1 < x < ξt∗ is anequilibrium strategy for the system while the jammer equilibrium strategy is uniqueand given as:

490 A. Garnaev et al.

yt =

{1, t ≤ t∗ − 1,

0, t ≥ t∗.(20)

Note that although a continuum of equilibria might arise for the system theyare equivalent since they return the same payoff.

Proof. By (16), (x, Y) is an equilibrium if they are best response strategy toeach other, i.e., solution of the following equations:

x = BRS(Y) := argmaxx

v(x,Y), yt = BRJ,t(x) := argminyt

vt(x,yt), t = 1, . . . , T.

(21)

Since v(x, Y) is linear on x while vt(x,yt) is linear on yt solving the optimizationproblems (21) we find the best response strategies as:

BRS(Y) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

1,T∑

t=1γt(at + bt − At − Bt)yt <

T∑t=1

γt(at − Bt),

any in [0, 1],T∑

t=1γt(at + bt − At − Bt)yt =

T∑t=1

γt(at − Bt),

0,T∑

t=1γt(at + bt − At − Bt)yt >

T∑t=1

γt(at − Bt),

(22)

BRJ,t(x) =

⎧⎪⎨⎪⎩

0, (at + bt − At − Bt)x < bt − Bt,

any in [0, 1], (at + bt − At − Bt)x = bt − Bt,

1, (at + bt − At − Bt)x > bt − Bt.

(23)

Then, (3), (17) and (23) imply

BRJ,t(x) =

⎧⎪⎨⎪⎩

1, x > ξt,

any in [0, 1], x = ξt,

0, x < ξt.

(24)

Let x = 0. Substituting into (24) implies that yt = BRJ,t((0, 1)) = 0, t =

1, . . . , T . Finally substituting x = 0, yt = 0.t = 1, . . . , T into (22) implies thatΘ ≤ 0, and (a) follows.

Let x = 1. Substituting into (24) implies that yt = BRJ,t((1, 0)) = 1, t =

1, . . . , T . Finally substituting x = 1, yt = 1, t = 1, . . . , T into (22) implies thatΘ ≤ Θ, and (b) follows.

Thus, only the case 0 < x < 1 is left to consider. Two subcases arise: (I)there is t+ such x = ξt+ and (II) there is t+ such ξt+−1 < x < ξt+ .

(I) Let there exist t+ such x = ξt+ . Then, by (18) and (24),

yt = BRJ,t(x) ≡

⎧⎪⎨⎪⎩

1, t < t+,

any in [0, 1], t = t+,

0, x > t+.

(25)

A Prospect Theoretic Look 491

Substituting these {yt} into (22) implies

x = BRS(Y) =

⎧⎪⎨⎪⎩

1, Lt+(yt+) < 0,

any in [0, 1], Lt+(yt+) = 0,0, Lt+(yt+) > 0,

(26)

where

Lt+(yt+) :=t+−1∑t=1

γt(bt − At) −n∑

t=t+

γt(at − Bt) + γt+(at+ + bt+ − At+ − Bt+)yt+ .

(27)By (27), x = ξt+ if and only if Lt+(yt+) = 0. Since Lt+(yt+) is linear and increasingin y+, this equation has a root y+ ∈ (0, 1) (and it is unique) if and only ifLt+(yt+) < 0 and Lt+(yt+) > 0. This is equivalent to t+ = t∗ where t∗ is given bycondition (a-iii). Then, solving Lt∗(yt∗) = 0 by yt∗ implies (19), and (c) follows.

(II) Let there exist t+ such ξt+−1 < x < ξt+ . Then, by (18) and (24),

yt = BRJ,t(x) ≡{

1, t < t+,

0, t ≥ t+.(28)

Substituting these {yt} into (22) implies

x = BRS(Y) =

⎧⎪⎨⎪⎩

1, Lt+(0) < 0,

any in [0, 1], Lt+(0) = 0,0, Lt+(0) > 0.

(29)

By (29), ξt+−1 < x < ξt+ if and only if Lt+(0) = 0, i.e., t+ = t∗ where t∗ is givenby condition of (b). This and (28) imply (d), and the result follows. �

7 PT Solution for Bayesion Game

Denote the probability weighting parameter for the system by α. We assumethat the jammer’s weighting parameter β does not depend on jammer’s position.Then, the PT-utilities for the rivals in the Bayesian game are given as:

uPTS (x, Y) := x

T∑

t=1

(Atwβ(yt) + atwβ(1 − yt)) + (1 − x)T∑

t=1

(btwβ(y) + Btwβ(1 − yt)),

uPTJ,t (x, yt) := yt(Atwα(x) + btwα(1 − x)) + (1 − yt)(atwα(x) + Btwα(1 − x)).

(30)

Then, the PT-equilibrium is given as the solution of the best response equations:

x = BRPTS (Y) := argmax

xuPT

S (x, Y), yt = BRPTJ,t (x) := argmin

yt

uPTJ,t (x, yt), t = 1, . . . , T.

492 A. Garnaev et al.

Let us introduce the following auxiliary notation: ξt(α) = (bt − Bt)1/α/((at −

At)1/α + (bt − Bt)

1/α) for t = 1, . . . , T and ξ0(α) = 0. It is clear that 0 < ξt(α) < 1.

To avoid bulkiness in formulas we assume that ξt(α) �= ξτ (α) for any t �= τ . Then,without loss of generality we can assume that

0 = ξ0(α) < ξ1(α) < ξ2(α) < . . . < ξT (α) < 1. (31)

Theorem 4. The PT equilibrium coincides with the Bayesian equilibrium given byTheorem 3 except for the case where equilibrium strategies of both rivals are mixed, i.e.,the case (c). In this case, i.e., if Θt∗−1 < Θ < Θt∗ then

x = ξt∗(α) and yt =

⎧⎪⎨

⎪⎩

1, t ≤ t∗ − 1,

Ξt∗(β), t = t∗,

0, t ≥ t∗ + 1,

(32)

where Ξ = Ξt∗(β) is the root in (0, 1) of the equation:

Lβ,t∗(Ξ) :=

t∗−1∑

t=1

γt(bt − At) −T∑

t∗+1

γt(at − Bt)

+ γt∗ (bt∗ − At∗) wβ (Ξ) − γt∗ (at∗ − Bt∗) wβ (1 − Ξ) = 0.

(33)

Proof: Since uPTS is linear on x while uPT

J,t is linear on yt the best responsestrategies are given as follows:

BRPTS (Y)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

= 0,T∑

t=1γt(bt − At)wβ(yt) >

T∑t=1

γt(at − Bt)wβ(1 − yt),

∈ [0, 1],T∑

t=1γt(bt − At)wβ(yt) =

T∑t=1

γt(at − Bt)wβ(1 − yt),

= 1,T∑

t=1γt(bt − At)wβ(yt) <

T∑t=1

γt(at − Bt)wβ(1 − yt),

(34)

BRPTJ,t (x)

⎧⎪⎨⎪⎩

= 1, (at − At)wα(x) > (bt − Bt)wα(1 − x),∈ [0, 1], (at − At)wα(x) = (bt − Bt)wα(1 − x),= 0, (at − At)wα(x) < (bt − Bt)wα(1 − x).

(35)

Substituting (7) with γ = α into (35) implies

BRPTJ,t (x)

⎧⎪⎨⎪⎩

= 1, x > ξt(α),∈ [0, 1], x = ξt(α),= 0, x < ξt(α).

(36)

Then, by (7), (31), (34) and (36) all of the cases besides (c) can be provedsimilarly to Theorem 3. Thus, we have to consider only the cases where x = ξt∗(α)

for a t∗ such that Θt∗−1 < Θ < Θt∗ . By (34), x = ξt∗ if and only if Lβ,t∗(yt∗) = 0.

Note that Lβ,t∗(0) = Θt∗−1 − Θ < 0 and Lβ,t∗(1) = Θt∗ − Θ > 0. Thus, the rootexists and the result follows. �

A Prospect Theoretic Look 493

Fig. 2. (a) Probability x, (b) probability y1 and (c) probability y2 with T = 2, q1 ∈[0, 1], α = β = 1/2 q2 = 1 − q1, A1 ∈ (0, a1) and M1 = ((A1, 1), (1, 0.3)), M2 =((0.3, 0.9), (0.8, 0.4)).

8 Discussion of the Results and Conclusions

By Theorems 3 and 4, the PT system strategy is not sensitive to the probabilityweighted parameter for the jammer, and it depends only on the probabilityweighted parameter of the system. On the other hand, the PT strategy for thejammer depends on both of these parameters. Although the system’s strategy isany probability vector, i.e., the feasibility set consists of a continuum of elements,Theorems 3 and 4 shows that there is only a finite set Γ := {ξt(α) : t = 0, . . . , T}∪{1} of strategies given in closed form containing all of the equilibrium. Whileelements of this set depend on weighted parameter of the system, i.e., the PTsystem strategy is sensitive to this parameter, the selection rule to identify theequilibrium is stable with respect to this parameter. Figure 2 illustrates that thePT system strategy can be sensitive to a priori probabilities (namely, q1) whilethe jammer’s strategy is sensitive to the system’s parameters (namely, A1). Thisis similar to what can be observed in bandwidth protection games [7,10], wherean agent sometimes has to react sharply to small variations in the environmentparameters. Finally, we note that, in this paper, although we have employed theoriginal probability weighting function (7) given in [16], the obtained result canbe generalized for any such S-shaped weighting function. A goal for our futurework is to develop solutions for the joint radar and communication system designusing cumulative prospect theory [16], which is a variant of prospect theory thatallows one to take into account risk aversion of the rivals.

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