5. In the lecture, we have shown the following: Theorem: Suppose strategy s, (in a finite game) is strictly dominated by strat- egy ŝ; when opponents are restricted to using pure strategies. That is, Then strategy s, will never be in the support of a best response. That is, for all 0-i o; €BR;(G-;) = 0;(s})=0. (a) Use this result to find all the Nash equilibria (including mixed strategy ones) of the following game. player 2 L R T 10,5 4,1 player 1 M 6,1 8,8 B 5,0 0,10

5. In the lecture, we have shown the following: Theorem: Suppose strategy s, (in a finite game) is strictly dominated by strat- egy ŝ; when opponents are restricted to using pure strategies. That is, Then strategy s, will never be in the support of a best response. That is, for all 0-i o; €BR;(G-;) = 0;(s})=0. (a) Use this result to find all the Nash equilibria (including mixed strategy ones) of the following game. player 2 L R T 10,5 4,1 player 1 M 6,1 8,8 B 5,0 0,10

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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Answer all questions, show all working. Consider the following game in strategic or normal form. A2 B2 C2 A1 1,0 1,2 -2,1 B1 6,2 0,3 2,3 C1 2,2 -2,1 2,3 Use the iterative elimination of strictly dominated strategies to reduce the game as much as possible. What is the set of rationalizable strategies for each player? What is/are Nash equilibrium(s) in this game? Explain the differences between cooperative and non- cooperative games. What are the fundamental hypotheses about the behavior of the persons in Game theory?
1 True or False : Every finite extensive - form game of imperfect information admits at least one pure - strategy Nash equilibrium . Justify if true or give a counter - example if not
5 Suppose there are only two electric vehicle producers–Tesla and BMW. Using game theory in microeconomics, explain whether BMWand Tesla willdesign the same charge plug or different plug. What factors should the firms take into consideration?
5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:A writes down either number 1 or number 2, and B must guess which one.If the number that A has written down is i and B has guessed correctly, B receives i units from A.If B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.Suppose B guesses 1 with probability p and 2 with probability 1 −p.Let X1 denote B’s gain (or loss) in a game where A has written down 1.Let X2 denote B’s gain (or loss) in a game where A has written down 2.(a) Find the pmf of X1 and X2(b) Find B’s expected gain for these two cases, E[X1] and E[X2].(c) What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player ASuppose that A writes down 1 with probability q and 2 with probability 1 −q.Let Y1 be A’s loss (or gain) if B chooses number 1.Let Y2 be A’s loss (or gain) if B…
In the coin game, two players begin with 13 coins. Players take turns to remove coins. At each turn each player is allowed to remove one, two, or three coins. The winner is the last player to remove coin(s). Suppose there are 9 coins remaining and it is your turn to remove coin(s). Your best strategy is to remove how many coins? a. 0 b. 1 c. 2 d. 3
1) Suppose that Player A can take two actions, either Up or Down. Player A is thinking to choose Up 50 percent of the time, and Down 50 percent of the time. This type of strategy is called a ____ ? 2) Consider a payoff matrix of a game shown below. In each cell, the number on the left is a payoff for Player A and the number on the right is a payoff for Player B. In order for (Down, Right) to be a unique pure strategy Nash equilibrium, a must be (greater, or smaller) than 3 and b must be (greater, or smaller) than 3. refer to image
(a) Assuming that each fishery chooses fi ∈ (0,F), to maximize its payoff function, derive the players’ best response functions and find a Nash equilibrium. (b) Is the equilibrium you found in (a) unique or not? What are equilibrium payoffs? (c) Suppose that a benevolent social planner wants maximize the util- ity of both fisheries. In other words, the social planner solves the following problem: max w(f1, f2) = u1(f1, f2) + u2(f1, f2) (f1 ,f2 )= 2ln(f1)+2ln(f2)+2ln(F −f1 −f2). Find the social planner’s solution. (d) What are the fisheries’ payoffs if the quantities of fish they catch are solutions to the social planner’s problem? What can you say about the Nash equilibrium quantities of fish being caught as compared to the social planner’s solution? (e) If fishery j decides to follow the recommendation of the social planner, how much fish will firm i catch?
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5) Suppose that the letter grade earned on a test for each student in a class depends upon how well he/she does relative to other students in the class. This exhibit shows a prisoner’s dilemma setting for two representative students in the class, George and Gina. What is the expected final Nash Equilibrium? a) Study 4 hours for both Gina and George. b) Study 2 hours for both Gina and George. c) George studies for 2 hours, and Gina studies for 4 hours. d) George studies for 4 hours, and Gina studies for 2 hours.
12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s <w. If an army attacks but its rival does not, no costs are borne by either side. Identify all 12.7 Exercises • 267 the pure-strategy Bayesian Nash equilibria of this game for the following two cases, and briefly describe…
1. Suppose Company A is about to play a game with Company B. The following facts are known about the two players. The first Company (Company A) is a row player and uses three different strategies i.e. (Strategy X, Strategy Y and Strategy Z). Whereas the column player (Company B) has two different strategies i.e. (M and N) that can be used accordingly. The payoff matrix is given in the table below. Player B Player A M N X -5 3 Y 3 -7 Z -6 5 Answer the following questions based on the information given above a. Determine the strategies of each firm using graphical technique b. Compute the value of the game.
on 8.1 Consider the following game: Player 1 A C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1, 1 F 0,0 1,1 4,4 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.