Networked Life-Homework 1 Distributed Power Control and Zero Sum Game Solved

Exercise 1. Distributed power control 

a    Consider three pairs of transmitters and receivers in a cell, with the following channel gain matrix G and noise of 0.1 mW for all the receivers. The target SIRs are shown below. 

 

                                                                               1     0.1    0.3                1

                                                                𝐺 = (0.2      1       0.3) , 𝛾 = (1.5) 

                                                                             0.2    0.2      1                 1

With an initialization of all transmit powers at 1 mW, run DPC for ten iterations and plot the evolution of transmit powers and received SIRs. You can use any programming language, or even write the steps out by hand. 

b    Now suppose the power levels for logical links 1, 2 and 3 have converged to the equilibrium in (a). A new pair of transmitter and receiver, labeled as logical link 4, shows up in the same cell, with an initial transmit power of 1 mW and demands a target SIR of 1. The new channel gain matrix is shown below. 

 

                                                                                       1       0.1       0.3     1

                                                                         𝐺 = (0.2        1       0.3      0.1)

                                                                                     0.2     0.2       1       0.1

                                                                                     0.1     0.1     0.1       1

Similarly to what you did in (a), show what happens in the next ten timeslots. What happens at the new equilibrium? 

Exercise 2.Power control infeasibility 

Consider a three-link cell with the link gains 𝐺𝑖𝑗 shown below. The receivers request 𝛾1 = 1, 𝛾2 = 1 and 𝛾3 = 1.The noise 𝑛𝑖 = 0.1 for all i . 

 

                                                                                             1       0.5     0.5

                                                                               𝐺 = (0.5        1       0.5)

                                                                                           0.5     0.5       1

 

Show this set of target SIRs is infeasible. 

Exercise 3.Zero sum game 

In the following two-user game, the payoffs of users Alice and Bob are exactly negative of each other in all the combinations of strategies (a,a), (a,b), (b,a) and (b,b). This models an extreme case of competition and is called a zero-sum game. Is there any pure strategy equilibrium? How many are there? 

                                                                                              𝑎          𝑏 

                                                                                   𝑎   2, −2     3, −3

                                                                                       (                           ) 

                                                                                   𝑏    3, −3     4, −4

 

 

 

 

 

 

Exercise 4: DPC as a game. 

We have two pairs of transmitter-receivers as in the figure below: 

 

 

 

 

Assume 𝐺11 = 𝐺22 = 2, 𝐺12 = 𝐺21 = 0.5, 𝛾1 = 4, 𝛾2 = 2 and 𝑛1 = 𝑛2 = 0.3. 

a    Draw the feasible region of (𝑝1, 𝑝2). 

b    Suppose each player is a pair of transmitter-receiver choosing its transmit power 𝑝𝑖. The player can choose its 𝑝𝑖 ∈ {1, 2, 3}. Complete the following payoff matrix: 

 

                                                                                        1        2         3

1             ? , ?         ? , ?         ? , ? 2(? , ?               ? , ?         ? , ?) 3   ? , ?         ? , ?               ? , ?

c. Find the Nash equilibrium of the previous game.