Question 2 (Subgame-Perfect Nash Equilibrium Consider the following sequential game of imperfect information. E 5,5 P2 (c) soning. T 0,6 P1 L P2 B ( 6,0 R 2,0 P2 I P1 T e 1,1 5,1 b 0,2 P2 P2 A' R 0,0 P1 B' { 0,0 P2 1,5 (a) How many information sets does this game have? What constitutes a strategy in a sequential game of imperfect information? (b) Find all the pure strategy subgame-perfect Nash equilibria of this game. Remember to write them down properly. Could this game support a SPNE in mixed strategies? Explain your rea- (d) Suppose this game were a game of perfect information. What would the game tree look like? What would be the SPNE outcome?

Question 2 (Subgame-Perfect Nash Equilibrium Consider the following sequential game of imperfect information. E 5,5 P2 (c) soning. T 0,6 P1 L P2 B ( 6,0 R 2,0 P2 I P1 T e 1,1 5,1 b 0,2 P2 P2 A' R 0,0 P1 B' { 0,0 P2 1,5 (a) How many information sets does this game have? What constitutes a strategy in a sequential game of imperfect information? (b) Find all the pure strategy subgame-perfect Nash equilibria of this game. Remember to write them down properly. Could this game support a SPNE in mixed strategies? Explain your rea- (d) Suppose this game were a game of perfect information. What would the game tree look like? What would be the SPNE outcome?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.1P

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Aedri Quick Lesson in Game Theory A Nash Equilibrium is an outcome in which neither player is better off by changing their strategy. 2.7 Is a Dominant Strategy equilibrium also a Nash equilibrium? a) Yes b) No esc The Ice Cream Guys $3.99 $4.99 + Chucky's Chunky CCT: $20,000 $3.99 ICG: $20,000 CCT: $60,000 ICG: $10,000 tab Treats CCT: $10,000 $4.99 ICG: $60,000 CCT: $40,000 ICG: $40,000 The table above is the payoff matrix for the annual profit of the only two ice-cream-truck firms operating in Beach City. They are deciding the price of an ice cream cone. %3D caps lo 2.8 What is Chucky's dominant strategy? a) $3.99 b) $4.99 c) Not enough information hift 2.9 What is the dominant strategy equilibrium in this situation? a) Both charge $3.99 b) Both charge $4.99 c) CCT charges $3.99 and ICG charges $4.99 d) CCT charges $4.99 and ICG charges $3.99 2.10 Suppose these two firms engaged in collusion (which, of course, totally doesn't happen because it is against the law). Which outcome would…
4. Boeing and Airbus, change in profits ($m) from the development of a super jumbo jet.AirbusDevelop Do not developBoeing Develop -50, -50 -30, 80Do not develop 80, -30 0, 0a) Identify the key assumption of Game theoryb) What important lesson does game theory reveal about coordination inan oligopoly market?c) In the game above, what is the Nash equilibrium?
Question 9 Player 1 Y Player 2 3,2 2 -0,1 1,3 -0,1 Which of the following is the Subgame Perfect Nash Equilbrium in this game?
rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
5. Aaron and Betty play the following one-shot game. Aaron Up Middle Down Left 10,0 5,10 1,1 Betty Right 0,5 1,0 2,2 a. Does the game above have any (pure strategy) Nash equilibria? Find any equilibria or explain why an equilibrium doesn't exist. b. Suppose instead that Aaron moves first and Betty moves second. Represent this situation with a game tree and find the subgame perfect Nash equilibrium. Are Aaron and Betty better off playing a simultaneous game or a sequential game in which Aaron moves first? Explain. c.
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…
3 Use the following payoff matrix for a one-shot game to answer the accompanying questions. 20 oints eBook Player 1 Strategy A B Player 2 X Y 12, 12 -30, 6 6, -30 30, 30 a. Determine the Nash equilibrium outcomes that arise if the players make decisions independently, simultaneously, and without any communication. Instructions: In order to receive full credit, you must make a selection for each option. For correct answer(s), click the box once to place a check mark. For incorrect answer(s), click twice to empty the box. ? (-30, 6) (12.12) ? (6.-30) ? (30, 30) Which of these outcomes would you consider most likely? O (30,30) O(-30, 6) O (12, 12) O(6,-30) b. Suppose player 1 is permitted to "communicate" by uttering one syllable before the players simultaneously and independently make their decisions. What should player 1 utter? OA OB What outcome do you think would occur as a result? O (12, 12) O (-30, 6) O (30, 30) 0 (6,-30) c. Suppose player 2 can choose its strategy before player 1,…
What is the Nash Equilibrium of the following game? |0, 0 Up Down 3, 2 |2, 1 3, 1 2, 6 4, 2 5,5 10, 0 a. (Down, D) b. (Down, C) c. (Up, A) d. (Up, B)
Solve for the Nash equilibrium (or equilibria) in each of the following games. (a) The following two-by-two game is a little harder to solve since firm 2’spreferred strategy depends of what firm 1 does. But firm 1 has a dominantstrategy so this game has one Nash equilibrium. Firm 2 Launch Don’tFirm 1 Launch 60, -10 100, 0 Don’t 80, 30 120, 0 What is the Nash equilibrium of this simultaneous-move game? (b) What would the outcome of this game be if instead firm 1 moved first and then, after seeing what firm 1 chose, firm 2 chose it strategy? In this case firm 1 doesn’t necessarily need to choose a best response, but firm 2 must choose a best response since it moves second.
Consider the following price game: Firm 1 Firm 2 High Low High 20, 20 12, 24 Low 24, 12 14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?
12) What is a Nash equilibrium? Why do we generally think that Nash equilibria will be likely outcomes of games?
1. Use the following one-shot, normal form game to answer the questions below. Strategy A 100, 125 300, 250 200, 100 250, 0 500, 500 750, 400 400, 300 -100, 350 0, -100 (a) Find each player's dominant strategy, if it exists. (b) Find the Nash equilibrium.