Problem #3: Signaling Games (12 pts) Consider the game of imperfect information depicted in the game...

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Problem #3: Signaling Games (12 pts) Consider the game of imperfect information depicted in the game tree below: Quiche Fight Walk 0.1 5.0 Cowboy is Coward (Prob.p) Beer Fight Walk -5,1 0.0 34 Cowboy is Gunslinger (Prob = 1.p) 0.5 EA -1,-1 O Quiche Fight Walk 0.0 Beer Fight Wak 0-1 1,0 In this game, the players are a Cowboy who wanders into a town in the Wild West, and an Outlaw who terrorizes the town. The Cowboy may be either a Coward or a Gunslinger; the probability p of each state is 0.5. The Cowboy knows which type they are but the Outlaw does not. The Cowboy wants breakfast, but the hotel serves only two things for breakfast: Quiche and Beer. The Outlaw enjoys picking on the weak, and would like to pick a fight with a Cowboy who is a Coward, but is not actually a good shot and would lose a fight to a Gunslinger. The Outlaw can observe what the Cowboy orders for breakfast before deciding whether to start a Fight or Walk away. 4 a) List all of the strategies belonging to each player. Note that each player has two information sets. Comboy: 1c (Continued on next page) Problem #3: Signaling Games (12 pts) Consider the game of imperfect information depicted in the game tree below: Quiche Fight Walk 0.1 5.0 Cowboy is Coward (Prob.p) Beer Fight Walk -5,1 0.0 34 Cowboy is Gunslinger (Prob = 1.p) 0.5 EA -1,-1 O Quiche Fight Walk 0.0 Beer Fight Wak 0-1 1,0 In this game, the players are a Cowboy who wanders into a town in the Wild West, and an Outlaw who terrorizes the town. The Cowboy may be either a Coward or a Gunslinger; the probability p of each state is 0.5. The Cowboy knows which type they are but the Outlaw does not. The Cowboy wants breakfast, but the hotel serves only two things for breakfast: Quiche and Beer. The Outlaw enjoys picking on the weak, and would like to pick a fight with a Cowboy who is a Coward, but is not actually a good shot and would lose a fight to a Gunslinger. The Outlaw can observe what the Cowboy orders for breakfast before deciding whether to start a Fight or Walk away. 4 a) List all of the strategies belonging to each player. Note that each player has two information sets. Comboy: 1c (Continued on next page)