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- 1Y You have to decide whether to invest $100 in a friend's enterprise, where in a year's time the money will increase to $150. You have agreed that your friend will split the profits evenly each year with you. Your alternative is to keep at the bank at an interest rate of r. Suppose this game is played repeatedly infinitely often. The rate of interest between any two successive periods is r, the same as that in the bank and the same for you and your friend. What is the cut-off interest rate which will make cooperation possible?4. Two investors have each deposited 5 (all amounts are in $10,000s) with a bank. The bank has invested these deposits in a long-term project. If both investors make withdrawals at date 1 then each receives 3 and the game ends. If only one investor makes a withdrawal at date 1 then that investor receives 5, the other receives 1, and the game ends. Finally, if neither investor makes a withdrawal at date 1 then the project matures and the investors make withdrawal decisions at date 2. If both investors make withdrawals at date 2 then each receives 7 and the game ends. If only one investor makes a withdrawal at date 2 then that investor receives 9, the other receives 5, and the game ends. Finally, if neither investor makes a withdrawal at date 2 then the bank returns 7 to each investor and the game ends. Show the bi-matrix for dates 1 and 2.3. 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…
- Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.First Player can invest $1.00 with Second Player. Second Player chooses whether tc cooperate and make the promised investment, or appropriate and keep First Player'a $1.00. Based on the payoff matrix shown below (first player payoffs on bottom left of cell, second player payoffa on upper right of cell), what will be First Player's payoff if contracis are not enforceable? Second Player Cooperate Appropriate Invest 1.5 1.0 First Player 1.5 -1.0 Don't Invest
- 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? 4. What are the optimal (complete) strategies for each player?9. Find all of the Nash equilibria for the three-player game here. Player 1 a b Player 3: A Player 2 X y Z 1,1,0 2,0,0 2,0,0 3,2,1 1,2,3 0,1,2 2,0,0 0,2,3 3,1,1 Player 1 a b C Player 1 Player 3: C X Player 2 y 2,0,0 0,1,2 0,1,1 1,2,1 3,1,2 0,1,2 a b C Player 3: B Player 2 X 2,0,0 1,2,0 0,1,2 Z 0,1,2 0,1,2 1,1,2 y 0,0,1 1,2,1 2,2,1 Z 2,1,2 1,2,1 2,1,0Costco Spend $1B on advertising Spend $1.5B on advertising Spend $1B on advertising $3.5B; $2.7B $4.2; $2.2B Target Spend $1.5B on advertising $3.2B; $3.4B $4B; $3B
- 1.1. What is a game in normal form? What is a strictly dominated strategy in a normal-form game? What is a pure-strategy Nash equilibrium in a normal-form game? 1.2. In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibria? L C R T 2.0 1,1 4,2 M 3,4 1,2 2,3 B 1,3 0,2 3,03. Repeated Game Consider the game in which the stage game depicted below is infinitely repeated and in which both players discount future payoffs with discount factor 6 € [0, 1]. Player 1 E S Player 2 E 3,3 4, -1 S -1.4 1,1 (a) Suppose that both players play the tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player copies the action chosen by his opponent in the previous period (i.e if Player 2 shirks (plays S) in period 2, then Player 1 plays S in period 3). Then, for what values of d can tit-for-tat be supported on the equilibrium path of play? (b) Suppose that both players play the perfect tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player plays E unless the actions are disagreed in the previous period. Then, for what values of d can perfect tit-for-tat be supported on the equilibrium path of play?2. Consider the following simultaneous move stage game. In each cell, player l's payoff is shown first. Player 2 L C R T 3, 1 0 ,0 5 , 0 Player 1 M 2, 1 I, 2 3 , 1 В 1, 0 , 1 4, 4 This game is played twice, without discounting of the second stage payoffs. Both players observe the outcome of the first stage prior to making their second stage choices. Determine whether or not (4,4) can be the first stage payoffs from a pure strategy subgame perfect Nash equilibrium. Explain your answer carefully.