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Unformatted text preview: E CONOMICS S-1050 H ARVARD U NIVERSITY S UMMER 2008 S UMMER S CHOOL S TRATEGY , C ONFLICT & C OOPERATION • M IDTERM E XAMINATION A NSWERS NAME : You have 120 minutes to finish the exam. This exam has 12 pages (check that you have all). Answer all questions on the exam itself. Closed books & notes, but calculator allowed. You must answer all 4 questions , each worth 25 points , for a total of 100 points . You should spend about 25 minutes on each question, so that you have time to re-check . Show all calculations, explain all answers, fully label graphs, and highlight all final answers. 2 1. Consider the following game: L C R U 4 2 M 1 3 3 D 4 5 2 a) What is the saddlepoint? What is the value of the game? _ 10 _ Reduce the game by eliminating the dominated strategies U and C: L C R U 4 2 M 1 3 3 D 4 5 2 L R M 1 3 p D 4 2 q Note that the pure prudent strategies, D for Player 1 and R for Player 2, does not yield a saddlepoint, as maximin [= 2 at (D,R)] ≠ minimax [= 3 at (M,R)]. So look for mixed : Let: p = Player 1’s probability of playing M, q = Player 2’s probability of playing L ∴ EP 1 (L | p) = 1 p + 4 (1 – p) = -3 p + 4 EP 1 (R | p) = 3 p + 2 (1 – p) = 1 p + 2 ⇒-3 p* + 4 = 1 p* + 2 ⇒ p* = ½ EP 2 (M | q) = -1 q + -3 (1 – q) = 2 q – 3 EP 2 (D | q) = -4 q + -2 (1 – q) = -2 q – 2 ⇒ 2 q* – 3 = -2 q* – 2 ⇒ q* = ¼ ∴ saddlepoint = {( ½, ¼ )} _ 5 _ ∴ EP 1 (M | p*=½) = -3 × ½ + 4 = 2.5 ⇒ value = 2.5 _ 5 _ b) What is the Nash equilibrium of the game? _ 5 _ NE = {( ½, ¼ )} which is the same as the saddlepoint, as this is a zero-sum game. 3...
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