6. Consider the following game in extensive form: At stage 1, player 1 chooses between R and D. - If he chooses D, player 1 gets 1 and player 2 gets 0; - If he chooses R, the game moves to the second round. At stage 2, player 2 chooses between r and d. - If he chooses d, player 1 gets 0 and player 2 gets 2:- If he chooses r, the game moves to the second round. At stage 3, player 1 chooses between R - If he chooses D, player 1 gets 3 and player 2 gets 1; - If he chooses R, the game moves to the second round. At stage 4, player 2 chooses between r and d. - If he chooses d, player 1 gets 2 and player 2 gets 4; - If he chooses R, both players get 3. a. Draw the tree representation of the game.

6. Consider the following game in extensive form: At stage 1, player 1 chooses between R and D. - If he chooses D, player 1 gets 1 and player 2 gets 0; - If he chooses R, the game moves to the second round. At stage 2, player 2 chooses between r and d. - If he chooses d, player 1 gets 0 and player 2 gets 2:- If he chooses r, the game moves to the second round. At stage 3, player 1 chooses between R - If he chooses D, player 1 gets 3 and player 2 gets 1; - If he chooses R, the game moves to the second round. At stage 4, player 2 chooses between r and d. - If he chooses d, player 1 gets 2 and player 2 gets 4; - If he chooses R, both players get 3. a. Draw the tree representation of the game.

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.
4 Consider an extensive game where player 1 starts with choosing of two actions, A or B. Player 2 observes player 1’s move and makes her move; if the move by player 1 is A, then player 2 can take three actions, X, Y or Z, if the move by player 1 is B, then player 2 can take of of two actions, U or V. Write down all teminal histories, proper subhistories, the player function and strategies of players in this game.
2. Consider a two-stage game of complete but imperfect information where players 1 and 2 simultaneously choose actions a₁ and a₂ in stage 1 and players 3 and 4 simultaneously choose actions a3 and a in stage 2. Provide the notation for the Nash equilibrium in stage 2. How come players 1 and 2 can anticipate this outcome in stage 2?
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For each of the following games, identify the backwards induction equilibrium and the equilibrium strategy for each player. 5. b. (1,1) (0,2) (2.2) (0,1) (1,) (1.0)
7. Which of the following is not true of the “Battle of the Sexes" Game? a. The predicted outcome demonstrates how independent and rational play by players can quickly reach an agreement or a convergence of expectations concerning which strategy will be played by each player in equilibrium. b. The Battle of the Sexes Game can be used to depict a variety of situations where the players' interests do not precisely coincide, except for the fact that they both want to avoid the outcomes that would result from choosing different actions from one another. c. Players do not have a dominant strategy but playing “Boxing" say, is a best response to the other playing "Boxing" even if it doesn't give the player the payoff associated with his or her preferred alternative. d. Played only once, the outcome where each goes to their preferred venue cannot be Nash Equilibrium.
Use the following game table to answer the question: Player 2 Y 6, 3 1, 3 A 10, 5 0,0 Player 1 B 2,6 0,0 3,5 3, 4 3, 3 Is the Nash equilibrium of this game for Player 1 to choose A and Player 2 to choose X? Why or why not? O It is; if Player 2 chose B or C they would get less than 10, and if Player 1 chose Y or Z they would get less than 5. It is not; Player 2 could get 6, instead of 5, if they chose B. O It is; if Player 1 chose B or C they would get less than 10, and if Player 2 chose Y or Z they would get less than 5. It is; if Player 2 chose B or C they would get less than 5, and if Player 1 chose Y or Z they would get less than 10. It is not; Player 1 could get 6, instead of 5, if they chose B.
3. Consider a two-player, sequential-move game where each player can choose to play right or left. Player 1 moves first. Player 2 observes player 1's actual move and then decides to move right or left. If player 1 moves right, player 1 receives £20 and player 2 receives £45. If both move left, player 1 receives £15 and player 2 receives £30. If player 1 moves left and player 2 moves right, player 1 receives £40 and player 2 receives £40. a. Draw the above situation in the form of an extensive form game. b. Find the sub-game perfect Nash equilibrium of the extensive form 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.
5. Instead of choosing Game A or Game B in the first stage, player 1 chooses either game B or directly which action to play in game A. Game U D L (10,5) (5,10) R (0,0) (0,0) :Game B: L R U (6,6) (0,0) D (0,0) (0,0) (a) Write the game in extensive form. (b) Solve for all SPNEs of the modified game. (c) Do all the SPNEs give intuitively reasonable predictions? Why or why not? (Hint: think about what the modification of the payoff to game A entail for player 2 in that game and how the predictions contrast with the case where the game is chosen first, then the action.)
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