C__DOCUME~1_IPEKHZ~1_LOCALS~1_Temp_plugtmp-63_plugin-Econ%20S-1050%20Problem%20Set%201%20Answers

C__DOCUME~1_IPEKHZ~1_LOCALS~1_Temp_plugtmp-63_plugin-Econ%20S-1050%20Problem%20Set%201%20Answers

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1 E CONOMICS S-1050 H ARVARD U NIVERSITY S UMMER 2009 S UMMER S CHOOL P ROBLEM S ET 1 A NSWERS due July 2 , in class Show all calculations, explain all answers (up to 5 lines), and fully label graphs. Also: (a) staple sheets, (b) acknowledge assistance, (c) highlight final answers, and (d) write neatly! 1. Consider the following game [payoffs are in the form: (Ann, Bob, Carol)]: - 3.0 - a) List each player’s actions and strategies. When the extensive form “tree” is drawn sideways, from left to right (as above), it is customary to list nodes, actions, etc. from left to right then top to bottom. (If drawn vertically, from top to bottom, then list them top to bottom then left to right.) Use regular set notation for all answers, as shown below. Ann’s actions = {U, D} (need not mention node 1 as she has only one node) Ann’s strategies = {U, D} Bob’s actions @ node 1 = {L, R} Bob’s actions @ node 2 = {L, M, R} Bob’s strategies = {LL, LM, LR, RL, RM, RR} Carol’s actions @ node 1 = {N, S} Carol’s actions @ node 2 = {N, S} Carol’s strategies = {NN, NS, SN, SS} - 0.1 × 10 - Bob L R U D L M R (5,-2,3) (1, 2, 2) (3, 5, 5) Ann Bob Carol Carol (1,2,-3) (2,-4,6) (4,7,-1) (-2,1,0) N S N S
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2 b) If Ann “buys” Carol’s position in the game (i.e., she moves for Carol and collects Carol’s payoffs in addition to her own), what are her new strategies? Replace Carol with Ann: Ann’s actions @ node 1 = {U, D} & Ann’s actions @ node 2 = {N, S} Ann’s actions @ node 3 = {N, S} Ann’s new strategies = {UNN, UNS, USN, USS, DNN, DNS, DSN, DSS} - 0.5 - c) The game is represented above in what form? Represent the new game in the normal (strategic) form. The game is represented above in the extensive form - 0.2 - Normal (strategic) form representation of the new game: - 0.5 - LL LM LR RL RM RR UNN -2,2 -2,2 -2,2 3,7 3,7 3,7 UNS -2,2 -2,2 -2,2 -2,1 -2,1 -2,1 USN 8,-4 8,-4 8,-4 3,7 3,7 3,7 USS 8,-4 8,-4 8,-4 -2,1 -2,1 -2,1 DNN 8,-2 3, 2 8, 5 8,-2 3, 2 8, 5 DNS 8,-2 3, 2 8, 5 8,-2 3, 2 8, 5 DSN 8,-2 3, 2 8, 5 8,-2 3, 2 8, 5 DSS 8,-2 3, 2 8, 5 8,-2 3, 2 8, 5 Bob Ann
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3 d) If Ann cannot observe Bob’s actions, draw this game in the two standard forms. Extensive form representation of the revised game: - 0.3 - Normal (strategic) form representation of the revised game: - 0.5 - LL LM LR RL RM RR UN -2,2 -2,2 -2,2 3,7 3,7 3,7 US 8,-4 8,-4 8,-4 -2,1 -2,1 -2,1 DN 8,-2 3, 2 8, 5 8,-2 3, 2 8, 5 DS 8,-2 3, 2 8, 5 8,-2 3, 2 8, 5 Bob Ann Bob L R U D L M R Ann N S N S (-2,2) (8,-4) (3, 7) (-2,1) (8,-2) (3, 2) (8, 5) Ann
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4 2. Consider the following game: - 2.0 - L M R U 3 -1 2 M 1 1 -4 D -2 0 3 a) Does either player have a dominant strategy? Find the best-response of each player to the other’s strategies: L M R U 3 -1 2 M 1 1 -4 D -2 0 3 row player = player-1 , having strategies = {U,M,D}, denoted by red circles column player = player-2 , having strategies = {L,M,R}, denoted by blue diamonds (These terms and depictions will be used in every assignment in this course.) As only one payoff per strategy pair (“cell”) is given, this is a
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C__DOCUME~1_IPEKHZ~1_LOCALS~1_Temp_plugtmp-63_plugin-Econ%20S-1050%20Problem%20Set%201%20Answers

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