Page 119 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

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

• By (4), the PCC with the receiver is given as follows: 4. SOLUTION OF THE GAME
In this section we ind equilibrium strategies in closed
− form using a constructive approach via inding all solu‑
ℎ

( , ) = , (7) tions of the best response equations. Recall that, by (11),

1 + ( , ) is a Nash equilibrium if and only if each of these
ℎ strategies is the best response to the other, i.e., ( , ) is

where ℎ and are channel gains and is the back‑ a solution of the best response equations:

ground noise.
= argmax{ ( , ) ∶ ∈ }, (12)
• The PCC with all the receivers is given as follows: = argmin{ ( , ) ∶ ∈ }. (13)

− Note that (12) and (13) are Non‑Linear Programming

ℎ (NLP) problems.
( , ) = ( , ) = . (8)

∈ ∈ 1 +
ℎ 4.1 Explicitformfortheequilibriumstrategies

The goal of the transmitter is to maximize this PCC, while In this section we ind in closed form all of the possible
the jammer wants to minimize this probability. Such solutions of the best response equations, i.e., equilibrium
a problem could arise in military operations where one strategies, as functions of two auxiliary parameters and
radio station (say, GCS) must transmit data to mili‑ (Lagrange multipliers for the NLP problems (12) and
tary units (e.g., drones) under hostile interference. Thus, (13) correspondingly).
( , ) is the payoff to the transmitter, while for the jam‑ Proposition 1 Each equilibrium ( , ) of the game
mer ( , ) is the cost function. Thus, here we deal with a Γ( , , ) has to have the following form
zero sum game. We look for the Nash equilibrium [2]. Re‑
call that ( , ) is a Nash equilibrium in a zero‑sum game
∗ ∗ = ( , )

if and only if the following inequalities hold:
2
2
⎧ , ≤ ,
( , ) ≤ ( , ) ≤ ( , ) for all ( , ) ∈ × . ⎪ ℎ ℎ

∗
∗
∗
∗
(9) ⎪ ℎ
Let ≜ 1 + 1 + 4 +
⎨ ℎ 2 2
( , ) = ln( ( , )) ⎪ ℎ , ℎ >
⎪ 2 +
ℎ ⎩

= ln − . (10)
ℎ + ℎ (14)

∈ ∈
and
Since ln(⋅) is an increasing function, the problem to ind
the Nash equilibrium with payoff ( , ) to the transmit‑ = ( , )
ter is equivalent to inding the Nash equilibrium with pay‑
off ( , ) to the transmitter, i.e., such ( , ) that ℎ
∗ ∗ ⎢ ⎥
⎢ 1 + 1 + 4 ℎ + ⎥
( , ) ≤ ( , ) ≤ ( , ) for all ( , ) ∈ × . ⎢ 1 ℎ ⎥
∗
∗
∗
∗
(11) ≜ ⎢ − ℎ ⎥ , (15)
Denote this game by Γ = Γ( , , ). ⎢ 2 + ⎥
Note that the transmitter’s equilibrium strategy also re‑ ⎣ ⎦ +
lects the most fair power resource allocation to main‑
tain communication with all the receiversunder the worst where ∈ and ⌊ ⌋ ≜ max{ , 0}. Thus, ⌊ ⌋ = if ≥ 0
+
+
hostile interference since the utility given by (10) can and ⌊ ⌋ = 0 otherwise. Moreover, > 0 and > 0 are
+
also be considered as a proportional fairness utility [20, solutions of the following equations:
21].
( , ) ≜ ( , ) = , (16)

Theorem 1 The game Γ( , , ) has at least one Nash ∈
equilibrium.
( , ) ≜ ( , ) = . (17)

The proof can be found in Appendix 9.1. ∈
Note that, generally in resource allocation problems even
if the payoffs are concave the game might have multiple The proof can be found in Appendix 9.2.
equilibria (see, for example, [22]). In this paper we es‑ Note that, by Theorem 1 and Proposition 1, the non‑linear
tablish uniqueness of the equilibrium as a side effect of equations (16) and (17) have at least one solution. To ind
solving the best response equations associated with (11). this solution and to establish its uniqueness we cannot
© International Telecommunication Union, 2021 103