Page 121 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 1

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 1

4.4 Equilibrium and its uniqueness mated as follows:

In this section we ind the equilibrium and establish its ℎ

uniqueness. ≈ ℎ − 2 , (33)
+
ℎ
Theorem 2 In the game Γ( , , ), Nash equilibrium , ≤ ,
2

( , ) is unique. Moreover, this Nash equilibrium is ≈ ℎ ℎ (34)

( ( , ), ( , )) given by (14) and (15), where = ∗ √ , > ,
2
uniquely given by (29) and = Ω( ) uniquely given by ℎ
∗
(20). where is the unique positive root of the equation:

The proof can be found in Appendix 9.5. − ℎ = (35)
ℎ 2
∈ +
5. THE BOUNDARY CASES
and
In this section we ind the equilibrium strategies in closed
form for boundary cases of network parameters such = + √ . (36)
as the TCC, the total jamming/transmission power re‑ 2 ℎ 2 ℎ
sources and background noise at the receivers. ≤ ℎ / > ℎ /
The proof can be found in Appendix 9.7.
5.1 Negligible background noise at the re‑
ceivers Proposition 6 Let the total transmission power be
small. Then the Nash equilibrium ( , ) can be approxi‑
In this section we consider the scenario with negligible mated as follows:
background noise at the receivers.
/ℎ

≈ , (37)

Proposition 4 Let the background noise at the receivers ∑ ∈ /ℎ

be negligible, i.e.,
≈ / for ∈ . (38)

= 0 for ∈ . (30) The proof can be found in Appendix 9.8.

In particular, Proposition 5 and Proposition 6 imply that
Then the unique Nash equilibrium ( , ) is given as follows: for large or small total transmission power the transmit‑
ter’s strategies and the jammer strategy are insensitive to
/( + ℎ ) the TCC.

= , (31)

∑ ∈ /( + ℎ )

5.3 Large or small total jamming power
/( + ℎ )

= , ∈ . (32) In this section we consider the cases where the total jam‑

∑ /( + ℎ )
∈ ming power is either large or small.
The proof can be found in Appendix 9.6. Proposition 7 Let the total jamming power be large.
Proposition 4 implies that, for negligible background Then, the Nash equilibrium ( , ) can be approximated as
noise at the receivers, the equilibrium strategies of the follows:
transmitter and the jammer are proportional to ratio / .
1
Note that, in the SLCC problem solved in [14, 15] for ≈ (Ω , 0) = 1 + 1 + 4 Ω , (39)
0
0

negligible background noise at the receiver, equilibrium 2Ω 0 ∈ ℎ
strategies are given in closed form. Proposition 4 also
supplies the equilibrium strategies in closed form for the ≈ / for ∈ , (40)

MLCC problem. Thus, an increase in the number of com‑ where Ω is given by (21).
munication links does not lead to an increase in the com‑ 0
plexity involved in designing the equilibrium strategies. The proof can be found in Appendix 9.9.
Proposition 8 Let the total jamming power be small.
5.2 Large and small total transmission power Then, the Nash equilibrium ( , ) can be approximated as
follows:
In this section we consider the cases where total trans‑
mission power is either large or small. /ℎ
≈ for ∈ (41)

∑ ∈ /ℎ

Proposition 5 Let the total transmission power be
large. Then the Nash equilibrium ( , ) can be approxi‑ ∈ such that supp( ) ⊂ ℐ, (42)
© International Telecommunication Union, 2021 105