Econ109-SummerI2016-Midterm-sols - Economics 109 Game...

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Economics 109: Game Theory Midterm Solutions: Summer I 2016 1. (15 pts) Bob, Lauren and Ebert are playing the following game. First Bob can choose to either wear a hat ( W ) or not ( N ). He then walks into a room where Lauren is waiting. She can observe his choice of action. If he is wearing the hat Lauren can then choose to remove it ( R ) or to leave it on ( L ). If he is not wearing the hat she can choose to put a hat on him ( P ) or to leave it off ( LO ). Bob then walks into the next room where Ebert is waiting. Ebert can only observe whether or not Bob is wearing a hat. Ebert then guesses who put a hat on Bob, either Bob ( B ), Lauren ( LA ) or neither ( NE ). At this point the game ends and some payoffs result. (Don’t worry about the payoffs.) a. Represent this game in the extensive form. That is graph the game tree. If necessary attach primes ( ','' , etc.) to distinguish the different strategies. b. How many (pure) strategies does Bob have? How many does Lauren have? How many does Ebert have? Bob has 2. Laurent has 2 x 2 = 4. Ebert has 3 x 3 = 9. B L E W N R L P LO B LA NE B LA NE B LA NE B LA NE
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c. Clearly describe one specific strategy profile for this game. Anything from { W , N } x { RP , R ( LO ), LP , L ( LO )} x { BB ’, B ( LA ’), B ( NE ’), ( LA ) B ’, ( LA )( LA ’), ( LA )( NE ’), ( NE ) B ’, ( NE )( LA ’), ( NE )( NE ’)}. One example is ( W , RP , BB ’).
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2. (20 pts) Consider the following normal form game. 2 1 L R 2 1 L R U 25, 9, 1 50, 6, 2 U 25, 9, 3 0, 6, 0 D 75, 6, 1 75, 3, 1 D 150, 0, 0 75, 3, 3 A B 3 a. Evaluate u 1 ( σ 1 , R , A ) for σ 1 = 3 7 , 10 10 . (You will receive full credit if you
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