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1. A normal form Game is depicted below. Player 1 chooses the row (T or B), Player 2 chooses the column (L.M. or R), and

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1. A normal form Game is depicted below. Player 1 chooses the row (T or B), Player 2 chooses the column (L.M. or R), and Player 3 chooses the matrix (W,X,Y, or Z). W X L M R L M R T 1,1,10 1,0,10 5.-5,0 T 1,1,-10 1,0, 10 5,-5,0 B 0, 1, 10 0,0,-10 5,-5,0 B 0, 1, 10 0,0, 10 5,-5,0 L M R L M R 20,15,2 12,20,2 5,-5,40 8,8,-20 = T 8,8,-20 5,40,-20 B 16,20,2 20,15,2 5,-5,40 B 8,8,-20 8,8,-20 5,40,-20 Y Z (a) (5 points) Write a strategic form game tree for this game, and indicate the payoffs on any two terminal nodes of your choice. You don't need to write the payoffs at any other terminal nodes. (b) (5 points) Find all pure strategy Nash Equilibria for this game. (c) (5 points) Find a mixed strategy equilibrium of the two-player game (between Players 1 and 2) that results if Player 3 is forced to play Y. (d) (10 points) Find all of the rationalizable strategies in the full 3 player game. Show your reasoning.(a) (REQUIRED; 15 points) If Bob, Sue and Mary are rational voters with strict preferences given in the table to the right, with top being better, and all this is common knowledge, what outcome do you expect the binary agenda at left to produce? Bob Sue May X2 X1 XO X3 X3 X2 Xo Xa Strictly Worse X3 X2 (b) (CHOOSE (b) OR (c);10 points) What, if anything, is wrong with the following pattern of choices? (If you don't have a calulator and want to know: 5/6 = 0.83.) . Choice 1: 0.5[$100] + 0.5[$0] = p > q = 0.6[$80] + 0.4[$0]. . Choice 2: 1[$80] = r > > = (5/6) [$100] + (1/6)[$0].(10 points) (c) (If you already answered (b) don't do this - we won't grade it!) Consider a Judicial Settlement problem: . At each date t = 1, 2, ...n the Plaintiff makes a settlement offer s. The Defendent can either accept or reject each offer. (Note that the same player is making offers each period.) . If the Plaintiff accepts at date t, the "game" ends with the Defendent paying s, to the Plaintiff, and the Defendent and Plaintiff paying top and top to their respective lawyers. . If the Plaintiff rejects at all dates, the case goes to court. The Plaintiff will lose and have to pay / to the Defendent. The Plaintif and Defendent will also have to pay lawyer's fees (n + 1)cp and (n + 1)cp respectively.3. In this question you are asked to compute the rationalizable strategies in a linear Bertrand-duopoly with discrete prices and fixed "startup" costs. We consider a world where the prices must be an odd multiple of 10 cents, i.e., P = {0.1, 0.3, 0.5, ..., 0.1 + 0.2n, ...} is the set of feasible prices. For each price p, the demand is: Q(p) = max {1 - p. 0}. We have two firms N = {1, 2}, each with 0 marginal cost, but each with a fixed "start- up" cost k. That is, if the firm produces a positive amount, it must bear the cost k. If it produces 0, it does not have to pay k. Simultaneously, each firm sets a price p; E P. Observing prices p, and pa, consumers buy from the firm with the lowest price. When prices are equal, they divide the demand equally between the two firms. Each firm i wishes to maximize its profit. p:Q(p;) - k if pi 0) wi(p. p) = (pQ(p)/2-k pi =p; and Q(p;) >0 0 otherwise (a) If k = 0.1 : 1. (5 points) Show that p; =0.1 is strictly dominated. 2. (5 points) Show that there are prices greater than the monopoly price (p = 0.5) that are not strictly dominated. 3. (15 points) Iteratively eliminate all strictly dominated strategies to find the set of rationalizable strategies. Explain your reasoning.4. There are three "dates", t = 1, 2,3, and two players: Government and Worker. . At f = 1, Worker expends effort to build K e [0, co) units of capital. . At t = 2, Government sets tax rates Tx 6 0, 1] and T. E [0, 1] on capital-holdings and on labor income. . At t = 3, Worker chooses effort ez ( [0, co) to produce output Kez. The payoffs of Government and Worker are: UG = TRK+T.Key and Uw = (1 - Te)Ken + (1 - TK)K - K2/2 -

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