Repeated Game (we consider a special version that we can study from the view of static games) Consider two wireless transmitters, which can transmit at high power (H) or low power (L). The transmission rate that they can achieve depends on both their transmission power and on the other player's decision, according to the following payoff matrix: Table 4: Wireless Transmission L H L (4, 4) (0, 6) H (6, 0) (2, 2) They both want to maximize their transmission rate, and they have to transmit T packets. They can decide the transmission power for each packet, and they have to make their decision simultaneously, without knowing what the other transmitter will do. Therefore, the strategy space for each player is T1 = {H, L} x {H, L} X... X {H, L}, which consists of T times {H, L} (4) and each strategy (action) is a sequence of T symbols such as s;= (H, L,..., H). The payoff for a strategy is the sum of the payoffs for transmission of the packets in that strategy. 1. How many possible pure strategies (actions) does each player have? (1 point) Note that we are considering a static game. 2. Draw the matrix

12:33 Note that (Az"); is the i-th row of (Az"). Also, the support of the probability distribution y" is the set of indices corresponding to the set of non-zero elements of y" (i.e., the set of pure strategies with positive probabilities). 3. If we know the support of a Nash equilibrium (y", z"), can we compute the Nash equilibrium? Briefly justify your answer (i.e., (steps) how to compute the NE). (1 point) 5 Problem 5 (5 points) Repeated Game (we consider a special version that we can study from the view of static games) Consider two wireless transmitters, which can transmit at high power (H) or low power (L). The transmission rate that they can achieve depends on both their transmission power and on the other player's decision, according to the following payoff matrix: Table 4: Wireless Transmission L H L (4, 4) (0, 6) H (6, 0) (2, 2) They both want to maximize their transmission rate, and they have to transmit T packets. They can decide the transmission power for each packet, and they have to make their decision simultaneously, without knowing what the other transmitter will do. Therefore, the strategy space for each player is T = (H, L} x {H, L} x ... x {H, LY, which consists of T times {H, L} (4) and each strategy (action) is a sequence of T symbols such as s, = (H, L, ..., H). The payoff for a strategy is the sum of the payoffs for transmission of the packets in that strategy. 1. How many possible pure strategies (actions) does each player have? (1 point) Note that we are considering a static game. 2. Draw the matrix representing the game for T = 2 packets. Find all Nash equilibria of this game with T = 2. (2 points) 3. What is (are) the NE for T = 100? (1 point) 4. Now assume each player can revise its action after seeing the outcome of the game at the previous stage. Consider the trigger strategy: a player i playing the trigger strategy starts by playing L, and keeps playing L. However, if the other player plays H, then player i changes to H in the next stage, and keeps playing # for the rest of the game. What would be the outcome of the game under the trigger strategy? Is this a Nash equilibrium strategy? Justify your answer. (1 point) X