Ü B U N G S A U F G A B E N. S p i e l t h e o r i e

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1 T E C H N I S C H E U N I V E R S I T Ä T D R E S D E N F A K U L T Ä T E L E K T R O T E C H N I K U N D I N F O R M A T I O N S T E C H N I K Ü B U N G S A U F G A B E N S p i e l t h e o r i e by Alessio Zappone Prof. Eduard A. Jorswieck Lehrstuhl Theoretische Nachrichtentechnik

2 Fakultät Elektrotechnik und Informationstechnik Institut für Nachrichtentechnik Lehrstuhl Theoretische Nachrichtentechnik Prof. Eduard A. Jorswieck, Alessio Zappone Übung zur Lehrveranstaltung Spieltheorie Exercise 1: Answer the following questions: 1. What is a game in strategic form? 2. What is a non-cooperative game? 3. What is a static game? 4. What is a game with complete information? 5. What is a pure strategy and a mixed strategy Nash equilibrium? Exercise 2: For the two-player, non-cooperative games represented by the following payoff matrices, determine pure-strategy and mixed-strategy Nash equilibria. a) Player 2 D C D (4,4) (10,2) Player 1 C (2,10) (8,8) b) Player 2 D C D (0,0) (10,2) Player 1 C (2,10) (6,6) c) Player 2 R P S R (0,0) (-1,1) (1,-1) Player 1 P (1,-1) (0,0) (-1,1) S (-1,1) (1,-1) (0,0) d) Player 2 P Q R S P (0,7) (2,5) (7,0) (0,1) Player 1 Q (5,2) (3,3) (5,2) (0,1) R (0,0) (2,5) (0,7) (0,1) S (7,0) (0,-2) (0,0) (10,-1)

3 Exercise 3: Consider one Base Station (BS) and two mobiles which communicate with the BS in two-phases. The channel coefficients are h 1 = 1 und h 2 = 0.5 and the noise power is σ 2. In the first phase (MAC pahse), MS 1 and MS 2 tune the transmit power p [0; 1] and q [0; 1], respectively. The corresponding achievable rates for MS 1 and MS 2 are ) R1 MAC p = log 2 (1 + σ q and ( R2 MAC = log q ) σ 2. + p In the second phase, the BS transmits the powers (1 p) und (1 q) to serve MS 1 and MS 2, respectively. The corresponding achievable rates are ) R1 BC 1 p = log 2 (1 + σ 2 + (1 q) and ( ) R2 BC 0.5(1 q) = log σ (1 p) The two mobiles do not cooperate with each other and their goal is to choose p and q to maximize their utilities, defined as the achievable sum-rate in the two phases. Formulate the problem as a non-cooperative game and compute the NE of the game.

4 Exercise 4: Consider two mobile stations MS 1 and MS 2 which communicate with a base station BS. The transmit for the mobiles are p 1 [0; P max ] and p 2 [0; P max ], with P max the maximum feasible transmit power for both mobiles. The noise power for each communication link is 1 and the channel power gains are h 1 and h 2. Mobiles MS 1 and MS 2 consume a hardware power of P c,1 and P c,2 to operate the devices. Each mobile is interested in maximizing its energy efficiency, defined as the ratio of the achievable rate over the consumed power. ( ) log 1 + p 1h 1 1+p 2 h 2 EE 1 = p 1 + P c,1 ( ) log 1 + p 2h 2 1+p 1 h 1 EE 2 = p 2 + P c,2 1. Formulate the problem as a non-cooperative game in normal form. 2. Establish whether the game admits a Nash equilibrium. Exercise 5: Give the definitions of: 1. Potential game. 2. Supermodular game. Exercise 6: Consider the same communication system as in Exercise 4, but assume now that the utility function of p mobile i is u i = f( i h i 1+p i h i ) cp i, with i = {1, 2}, and c a positive parameter. Formulate the problem as a non-cooperative game in normal form and find conditions on the function f such that the game is supermodular.

5 Exercise 7: Given the following two games with perfect information in extensive form 1. Write down the components of the game. 2. Write the game in strategic form. 3. Find the Nash equilibria. 4. Calculate the subgame perfect equilibria. Exercise 8: Consider two mobile stations MS 1 and MS 2, which communicate to a base station BS. Each MS i can choose whether to transmit a power p i = P or to not transmit, i.e. p i = 0. The BS implements a successive interference cancellation receiver and chooses whether to decode MS 1 first (1 2) or MS 2 first (2 1). The utility of MS i is: i j. The utility of the BS is with µ 0. { log(1 + pi ) µp i if j i u MSi = log(1 + p i 1+p j ) µp i if i j, u BS = µ(p 1 + p 2 ), 1. Formulate the game in extensive form. 2. Find the subgame perfect equilibria of the game.

6 Exercise 9: Define the following axioms 1. Linearity 2. Symmetry 3. Pareto Efficiency 4. Independence of Irrelevant Alternatives Exercise 10: Given a convex and compact payoff region U R 2 and d U, describe the Nash bargaining solution f(u, d). Exercise 11: Consider two communication pairs each composed of a transmitter (BS) and a receiver (MS), and both sharing two parallel frequency bands. The channel coefficients are α 11 = α 22 = β 11 = β 22 = 1, α 12 = α 21 = β 12 = β 21 = 0.5. The noise power in both bands is 1. Transmitter 1 transmits with power π 1 [0, 1] in the first band and a with power 1 π 1 [0, 1] in the second band. Transmitter 2 transmits with power π 2 [0, 1] in the first band and a with power 1 π 2 [0, 1] in the second band. The utility function for each communication system is the maximum achievable rate treating interference as noise. ( R 1 (π 1, π 2 ) = log π ) ( 1α 11 + log (1 π ) 1)β π 2 α 21 ( R 2 (π 1, π 2 ) = log π 2α π 1 α Find the Nash equilibrium of the game 1 + (1 π 2 )β 21 ) ( + log (1 π ) 2)β (1 π 1 )β Calculate the Nash bargaining solution of the game using the Nash equilibrium as a conflict point.

7 Exercise 12: Answer the following questions 1. What are coalitional games? 2. What is the difference between transferable utility and non-transferable utility? Exercise 13: Define the following concepts. 1. Core. 2. Nucleolus. Exercise 14: Each of n factories draws water from a lake and discharges waste into the same lake. Each factory requires clean water and pays kc to clean its water supply, where k is the number of factories that do not treat their waste before discharging it into the lake. The cost of treating the waste is b, with c b nc. 1. Model this situation as a coalitional game under the assumption that the worth v(s) of a coalition S is the highest payoff that v(s) can guarantee. 2. Find the conditions under which the game has a nonempty core and conditions under which the core is a singleton.

8 Exercise 15: Compute the Shapley value for the game {1, 2, 3, 4}, v wherein v({1, 2, 3, 4}) = 3 v(s) = 0 if S contains at most one player in {1, 2, 3} v(s) = 2 otherwise Exercise 16: Consider three men M = {m 1, m 2, m 3 } and three women W = {w 1, w 2, w 3 }. The preferences of men and women are given as follows P (m 1 ) = w 1, w 2, w 3 P (w 1 ) = m 2, m 1, m 3 P (m 2 ) = w 2, w 1, w 3 P (w 2 ) = m 1, m 2, m 3 P (m 3 ) = w 1, w 2, w 3 P (w 3 ) = m 1, m 2, m 3 Is the matching [(m 1, w 3 ), (m 2, w 2 ), (m 3, w 1 )] stable? Is the matching [(m 1, w 1 ), (m 2, w 2 ), (m 3, w 3 )] stable? Find two stable matchings.