pracfinal - ‘ 74v 4 Ti Cé M 7" L OR&IE 435 Final...

Unformatted text preview: ‘ 74v 4 Ti Cé M 7": L. OR&IE 435 Final Exam Fall ’04 2 1/2, hours. Game Theory 3 sheets of reminders There are five questions, on both sides of the paper. Give brief justifications for your answers. ' 1. (20 points: 4, 2, 4, 3, 2, 2, 3) Consider the following 2—person game. First, player I chooses L or R, and announces his choice to player 11. Then player II chooses l or r. If I chooses L and II chooses l, the outcome is (W1, 732); if L and 7‘, (£1,£2); if R and l, (£1,£2), and if R and r, (D1,W2). Here W1 >1 D1 >1 £1 and W2 >2 D2 >2 £2. (a) What is the extensive form of the game? Be sure to mark all edges, outcomes, and information sets. (b) How many pure strategies does I have? What are the four pure strategies of II? (c) What is the strategic form of this game? (d) Find all pure-strategy Nash equilibria in the game from (0). Which of these are subgame—perfect? ' (6) Now suppose player 11 is deaf and cannot hear player I announce his move, and I , knows that II is deaf. How does this change your answer to (a)? What property of the game in (a) is lost in this variation? ~ ' 2(f) How many strategies does II have now? What is the new strategic form of the l game? I (g) Explain briefly why it might be important in the game in (e) and (f) to assume I that the players’ preferences satisfy the axioms of von Neumann and Morgen- stern. 2. (25 points: 5,6,3,5,6) (a) Consider the following 3-person cooperative game. Player 1 has one cup of black tea and one cup of black coffee. His preferred beverage is teagwith milk; he doesn’t like coffee, but he will drink black tea if necessary. He is indifferent between a cup of back tea for sure and a 50-50 chance between a Cup of tea with milk and nothing. Player 2 has some sugar. She likes coffee with sugar, but nothing else. Player 3 has some milk, but he only likes black coffee. Each 1 player assigns a utility of 1 to his or her preferred beverage, and O to having no beverage or an undesired beverage. Suppose utility is transferable. Calculate the characteristic function of this garne, where 12(3) is the total utility coalition S can achieve by sharing its resources. ’ i (b) Consider the n-person cooperative game (N, v) where N = {1,2, 3} and «2(1) '= 1/2, 11(2) = 0, 11(3) = 0, . “({2: = 0a “({173}) = 2: “({1,2}) = 3/23 (1) “({1,2,3}) = 2- i. Is this game essential? ii. Is it constant-sum? iii. What is the (O,1)-normalized game strategically equivalent to (N, v)? (c) Draw the set of imputations for the game in (b). (For slightly less credit, yo can use the (GD-normalized game in (b(iii)).) ; n (d) Find the core for the game in (For slightly less credit, you can use the (0-1)-norma.lized game in (b(iii)).) (e) Formulate a linear programming problem whose solution would give a point in the core of the game in (b) if there is one. Do Lot solve. 3. (10 points: 4,6) (a) In the 2—person 0—sum game II find all security strategies for player I. (b) Use complementary slackness to find all Security strategies for player II. 4. (30 points: 4,4,6,6,4,6) Consider the bimatrix game (A, B) Where 3 2 0 —-2 0 2 A: 2 0 —1 , B: 6 5 2 2 0 2 1 4 ——2 Parts (a)-(e) show how to compute a Nash equilibrium for this game without using pivoting or drawing response curves. (3.) Find an i such that pi : 0 in any Nash equilibrium (11(7). Then find a j be that (jj = O in any such ", (j). (b) Now consider the smaller bimatrix game (21,1?) with A ’20 a__ 0 2 A“[o 2], 3"(4 —2]' By considering the best response of each player to each pure strategy of the other, show that there are no Nash equilibria (g6, (j) with either 13 or (2 (or both) . a pure strategy. h . A (c) You now know thatfievery Nash equilibrium (13,97) fqr (AB) has 131 0 add 152 > 0. Argue that Alcj = Agé. (Ai is the ith row of A.) Hence compute q. | (d) Similarly compute 13. (e) Show that (p = (3/4,O,1/4)T;(j == (0,1/2,1/2)T) is a Nash equilibrium for the original game (A, B). _ (f) Now suppose you are going to compute a Nash equilibrium for (A, B) using the complementary pivoting algorithm. Write down the linear Systems of equations and inequalities so that any nontrivial complementary solution yields a Nash equilibrium. Suppose you choose mm = 0 as the complementarity condition to relax. Which variable enters the basis and which leaves at the first iteration of the complementary pivoting algorithm? Do not perform the pivot! Which variable enters and which leaves at the second iteration? Again, do not perform the pivot. ' ' 5. (15 points: 6, 5, 4) (a) State what is meant by two Nash equilibria of a bimatrix game being equivalent. Answer the same question for interchangeable. For what games are all Nash equilibria equivalent and interchangeable? (b) Give a game where, in every Nash equilibrium, player 11 uses a pure strategr and player I a mixed strategy with positive probabilities on each of his pure strategies. (Hint: you don’t have to start from scratch to find the game.) (c) For the bargaining problem with cooperative payoff region X = {(331,932)T : 931 + 332 S 6} and disagreement point d = (2, 2)T, someone sug- gests the solution :7: = (4, 2)T. Which of the axioms defining the Nash bargaining solution is violated by this solution? ...
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