Exercise 1.3. Alice (Player 1), Bob (Player 2), and Charlie (Player 3) play the following simultaneous game. They are sitting in different rooms facing a keyboard with only one key and each has to decide whether or not to press the key. Alice wins if the number of people who press the key is odd (that is, all three of them or only Alice or only Bob or only Charlie), Bob wins if exactly two people (he may be one of them) press the key and Charlie wins if nobody presses the key. (a) Represent this situation as a game-frame. Note that we can represent a three- player game with a set of tables. Player 1 chooses the row, Player 2 chooses the column and Player 3 chooses the table (that is, we label the rows with Player 1's strategies, the columns with Player 2's strategies and the tables with Player 3's strategies). (b) Using the game-frame of part (a) obtain a reduced-form game by adding the information that each player prefers winning to not winning and is indifferent between any two outcomes where he/she does not win. For each player use a utility function with values from the set {0,1}. (c) Using the game-frame of part (a) obtain a reduced-form game by adding the information that (1) each player prefers winning to not winning, (2) Alice is indifferent between any two outcomes where she does not win, (3) conditional on not winning, Bob prefers if Charlie wins rather than Alice, (4) conditional on not winning, Charlie prefers if Bob wins rather than Alice. For each player use a utility function with values from the set {0,1,2}. 1.E.2. Exercises for Section 1.2: Strict and weak dominance The answers to the following exercises are in Appendix S at the end of this chapter. Exercise 1.4. There are two players. Each player is given an unmarked envelope and asked to put in it either nothing or $300 of his own money or $600. A referee collects the envelopes, opens them, gathers all the money, then adds 50% of that amount (using his own money) and divides the total into two equal parts which he then distributes to the players. (a) Represent this game frame with two alternative tables: the first table showing in each cell the amount of money distributed to Player 1 and the amount of money distributed to Player 2, the second table showing the change in wealth of each player (money received minus contribution). (b) Suppose that Player 1 has some animosity towards the referee and ranks the outcomes in terms of how much money the referee loses (the more, the better), while Player 2 is selfish and greedy and ranks the outcomes in terms of her own net gain. Represent the corresponding game using a table. (c) Is there a strict dominant-strategy equilibrium? Problem 3. Val and Earl are at work on a very hot afternoon. There is a job waiting for them. It takes only one person to do the job. The one who does it must crawl through a mucky culvert to connect a pipe. Once the pipe is connected, they can both go to a comfortable tavern for a beer. The game tree is shown below, where at the terminal nodes, the top number is Val's payoff and the bottom number is Earl's payoff. Val Do it Stall V 5 E 10 Earl Do it Stall 9 Val 4 Do it Stall 38 Earl Do it Stall 7 Val Do it Stall Earl 110 Do it Leave it 5 0 X X At the beginning of play, Earl is working on another job, but Val could do the job and if he does it right away, they can both go out for beer. Val could choose to stall, waiting for Earl to come back. If Val decides to stall, then when Earl gets back, he could do the job, or he could stall. If Earl chooses to stall, Val could either do the job, or stall again, and so on... If Val and Earl both stall until Earl's final decision node at the end of the workday, Earl can either do the job, or they can both leave the job undone. The payoffs to both from leaving the job undone are given by the variable X. C) Find the subgame perfect Nash equilibrium profile if X < 0. In this equilibrium, who if anyone, does the job? Explain your answer. D) Find the subgame perfect Nash equilibrium profile if 0 < x < 1. In this equilibrium, who if anyone, does the job? Explain your answer. E) Find the subgame perfect Nash equilibrium profile if 1 < X < 2. In this equilibrium, who if anyone, does the job? Explain your answer.