5. Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either 1, 2, 3, or 4 matchsticks. The player to remove the last matchstick wins the game. 1. Suppose there are only 6 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? Explain your reasoning. 2. Suppose there are 12 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? (Hint: Use your answer to part (a) and roll back.) 3. Now start from the beginning of the game. If both players play optimally, who will win?

5. Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either 1, 2, 3, or 4 matchsticks. The player to remove the last matchstick wins the game. 1. Suppose there are only 6 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? Explain your reasoning. 2. Suppose there are 12 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? (Hint: Use your answer to part (a) and roll back.) 3. Now start from the beginning of the game. If both players play optimally, who will win?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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4. Consider a two player game with Fred and Barney, who tal turns removing matchsticks from a pile. They start with 33 matchsticks, and Fred goes first. On each turn, ecach player may remove either one, two, three, four, or five matchsticks. The player to remove the last matchstick wins the game. What are the optimal strategies for each player? Who will win? b. Suppose now that they can remove up to six matchsticks, how will the optimal strategies change for- each player? a.
5. The Stag Hunt is a famous coordination game where two hunters must work together to hunt a stag but each one has the option to go solo and hunt a hare. The following payoff matrix depicts such a game: a. b. Hunter A Stag Hare Stag 5,5 4,0 Hunter B Hare 0,4 4,4 List any pure strategy Nash equilibria if they exist. Does the game have a mixed strategy Nash equilibrium? If so, solve for it.
Use the following game table to answer the question: Player 2 Y A 10, 5 6, 3 0,0 Player 1 B 2,6 1, 3 0,0 3,5 3, 4 3, 3 One of these players has a strictly dominant strategy. Which player, and what strategy is it? Player 2; Y Player 1; A Player 2; X O Player 1; B O Player 1; X
Table 8. This table shows a game played between two players, A and B. The payoffs are given in the table as (Payoff to A, Payoff to B). A Select one: Up Middle Down Left (4,2) a. Middle-Center O b. Down-Center (3, 1) (1,3) B Center (2,5) (5, 3) (4,4) Refer to Table 8. This table shows a game played between two players, A and B. The payoffs in the table are shown as (Payoff to A, Payoff to B). Which of the following outcomes represents a Nash equilibrium in the game? Right (3, 3) (5,2) (6, 1) O c. Up-Left O d. More than one of the above is a Nash equilibrium in this game.
2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.
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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
11. The following payoff table involves two players each of whom has three possible moves. L M N L 30,15 45,35 55,25 M 25,35 20,20 30,50 N 15,35 35,40 15,15 a. Are there any dominant strategies in this game?
3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…
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1. Consider a game of chicken: two players approach a narrow bridge with room only for one, and each chooses either to go through and hope the other guy backs off (call this action H, for "hawk"), or backs off to let the other go through (call this action D, for "dove"). If both players are normal, then each gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H against D, 0 if they choose D (regardless of what the other player chooses). Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent type", who gets 1 from playing H regardless of Pl's action, and -1 from playing D, again regardless of P1's action. (a) Illustrate the game tree corresponding to the Harsanyi transformation of this game (i.e. the dynamic game of imperfect information in which nature moves first). (b) Illustrate the extended game (i.e. the normal form game corresponding to your game tree in part (a)). (c) Find all values of p for which there is a BNE in which P1…