4. Consider the following model of a bank run. Depositors may either Hold or Withdraw their funds from a bank. Both players hold deposits of 6, which earn interest of 1 if kept in the bank. If either depositor withdraws then the bank fails. If both try to withdraw then each depositor can only recover half of their deposits. The payoffs for a two-player version of this game are given below: Player 1 Hold Withdraw Hold 7,7 6,0 Player 2 Withdraw 0,6 3,3 (a) Find any pure-strategy Nash equilibria of this game. (b) Find any mixed-strategy Nash equilibria of this game. (c) How do the concepts of 'payoff dominance' and 'risk dominance' apply in this context? Do they help us to predict which equilibrium is more likely to occur?

4. Consider the following model of a bank run. Depositors may either Hold or Withdraw their funds from a bank. Both players hold deposits of 6, which earn interest of 1 if kept in the bank. If either depositor withdraws then the bank fails. If both try to withdraw then each depositor can only recover half of their deposits. The payoffs for a two-player version of this game are given below: Player 1 Hold Withdraw Hold 7,7 6,0 Player 2 Withdraw 0,6 3,3 (a) Find any pure-strategy Nash equilibria of this game. (b) Find any mixed-strategy Nash equilibria of this game. (c) How do the concepts of 'payoff dominance' and 'risk dominance' apply in this context? Do they help us to predict which equilibrium is more likely to occur?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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3. The following is an interpretation of the rivalry between the United States (USA) and the Soviet Ünion (USSR) during the cold war. Each side has the choice of two strategies: Aggressive and Restrained. The payoff table is given as follows: USSR Restrained Aggressiveness Restrained 4,3 1,4 USA Aggressiveness 3,1 2,2 a) Consider this game when the two countries move simultaneously. Find all pure strategy Nash equilibria. b) Next consider three alternative ways in which the game could be played with sequential moves: (i) The USA moves first and the USSR moves second. (i) the USSR moves first and the USA moves second. (i) The USSR moves first, and the USA moves second, but the USSR has a further move after the USA moves. For each case, draw the game tree and find the subgame-perfect Nash equilibrium. c) What are the key strategic issues (commitment, credibility and so on) for the two countries. (Note: Be concise. Your answer should not exceed 300 words].
7. Solving for dominant strategies and the Nash equilibrium Suppose Carlos and Deborah are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Carlos chooses Right and Deborah chooses Right, Carlos will receive a payoff of 5 and Deborah will receive a payoff of 1. Carlos Deborah Left Left 4,4 Right 2,4 Right 6,7 5,1 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Carlos chooses and Deborah chooses
1. Consider the following game: There are two companies: Nike and Adidas. They both need to make a choice: to increase the marketing costs or not change it. If both companies increase, they both will have $100 million profit. If neither company change, each will have a profit of $50 million. If only one company increases marketing costs, its profit will be $120 million and the other airline's profit will be $40 million. Draw the payoff matrix. Find the Nash equilibrium. Explain why this is the Nash equilibrium.
Consider the following game played between an incumbent (I) a potential entrant (E): Incumbent: 80, Entrant: 40 Enter High Price Stay Out Incumbent: 120, Entrant: 0 High-Cost I, Enter Incumbent: 60, Entrant: 40 Nature Low Price Stay Out Incumbent: 100, Entrant: 0 Low-Cost Enter Incumbent: 150, Entrant: -10 Low Price Es Stay Out Incumbent: 200, Entrant: 0 Suppose the entrant observes a low price but is unsure whether they face a low-cost or high-cost incumbent, i.e. they are unsure whether they are at decision node E4 or E5. What is the maximum probability the incumbent is low cost such that the potential entrant will enter the market? Оа. 0.9 O b. 0.5 O c. 0.8 O d. 0.2
4. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff and the second entry is Animal B's payoff: Animal A Hawk (rows)/Animal B (columns) Hawk Dove (-10,-10) (0,20) Dove (20,0) (8,8) An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why they are equilibria.
5. Consider the following Extended Form game where P1 stands for Player one and P2 stands for player two with A, B and A', B' being their respective strategies. A' (1,1) P2 B' (2,3) P1 A' B (4,5) P2 What is the set of Nash Equilibria? Clearly explain your answer. Full-screen Snip B' (6,7)
In the following games, all payoffs are listed with the row player's payoffs first and the column player's payoffs second. GAME 33 Player A Player B B1 10, 12 A1 A2 9,3 A3 8, 10 In Game 33 above, B2 B3 8,8 12, 10 7,6 11, 1 9,4 14, 3 B1 is a dominated strategy for Player B B2 is a dominated strategy for Player B B3 is a dominated strategy for Player B Player B has no dominated strategies.
QUESTION 3 Up Down In the game above, what is/are the sub-game perfect Nash equilibrium? Ⓒ (up,up) (up,down) Player 1 (down, up) (down, down) No equilibrium exists Up Down Up Down Player 2 P1 gets $25 P2 gets $25 P1 gets $7 P2 gets $30 P1 gets $13 P2 gets $9 P1 gets $8 P2 gets $6
7. Solving for dominant strategies and the Nash equilibrium Suppose Antonio and Caroline are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Antonio chooses Right and Caroline chooses Right, Antonio will receive a payoff of 9 and Caroline will receive a payoff of 8. Caroline Left Right Left 8, 5 8. 7 Antonio Right 3, 6 9, 8 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Antonio chooses and Caroline chooses
Problem 4 Consider the following two-person game Player 2 R 0,1 2, 3 Player 1 M 5, 9 7, 3 1, 7 5, 5 10, 10 1, 5 a) Find the dominant strategies for both players. b) Explain why (B, L) is not a Nash Equilibrium. c) Find the Nash equilibrium of the above game
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7. Solving for dominant strategies and the Nash equilibrium Suppose Gabriel and Nia are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Gabriel chooses Up and Nia chooses Down, Gabriel will receive a payoff of 6 and Nia will receive a payoff of 4. Gabriel Up Down Up 6,3 3,3 Nia Down 6, 4 7,4 In this game, the only dominant strategy is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Gabriel chooses and Nia chooses