4. Consider the following grenade game. First, player 1 chooses between giving player 2 $500 and giving player 2 nothing. Second, player 2 observes player 1's move and then chooses whether or not to explode a grenade that will kill both players. What is the subgame-perfect Nash equilibria?
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- Type out the correct answer ASAP with proper explanation of it In the Ultimatum Game, player 1 is given some money (e.g. $10; this is public knowledge), and may give some or all of this to player 2. In turn, player 2 may accept player 1’s offer, in which case the game is over; or player 2 may reject player 1’s offer, in which case neither player gets any money, and the game is over. a. If you are player 2 and strictly rational, explain why you would accept any positive offer from player 1. b. In reality, many players reject offers from player 1 that are significantly below 50%. WhyQ3. Two ice cream trucks operate on a beach and play a simultaneous pricing game. If one of them prices low and the other high it gets all the customers and a pay off of 12 while the other gets zero. If both price high each gets 6, and if both price low each get 5. The best strategy is for both to price high. True/False/Uncertain. Explain.Return to the game between Monica and Nancy in Exercise U10 in Chapter 5. Assume that Monica and Nancy choose their effort levels sequentially instead of simultaneously. Monica commits to her choice of effort level first. On observing this decision, Nancy commits to her own effort level. What is the subgame - perfect equilibrium of the game where the joint profits are 5m + 4n+ mn, the costs of their efforts to Monica and Nancy are m2 and n2, respectively, and Monica commits to an effort level first? Compare the payoffs to Monica and Nancy with those found in Exercise U10 in Chapter 5. Does this game have a first-mover or second - mover advantage? Using the same joint profit function as in part (a), find the subgame - perfect equilibrium for the game where Nancy must commit first to an effort level. U10. Return to the game between Monica and Nancy in Exercise U10 in Chapter 5. Assume that Monica and Nancy choose their effort levels sequentially instead of simultaneously. Monica commits…
- 2. Consider the two-player strategic form game represented in the following payoff table: t 1 m b 7,2 1,2 3,7 2 M 2,4 3,4 1,6 R 1,3 7,1 4,2 Player 1 chooses between rows, S₁ = {t, m, b), and receives the first listed payoff in each cell. Player 2 chooses between columns, S₂ = {L, M, R), and receives the second listed payoff in each cell. a. Does either player have a strictly dominant strategy? Explain. b. Explaining your steps carefully, identify the set of rationalizable strategies in this game. c. Making sure to explain your solution method, solve for all pure strategy Nash equilibria in this game.1Find the SPNE for the centipede game. Centipede game: Two players, 1 and 2, take turns choosing one of two actions each time, continue or stop. They both start with $1 in their respective piles, and each time i says continue, $1 is taken away from his pile, and $2 are added to the other player's pile. The game automatically stops when both players have $100 in their respective piles.
- 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.2The chicken game has often been used to model crises. Recall that in this game, the two players drive straight at each other. They can choose to swerve or keep going straight. If one swerves, and the other goes straight, assume that the one that swerves gets -10 utility and the one that goes straight gets 10 utility, since the one that swerves is deemed the loser. If both swerve, both get 0 utility. If both go straight, they crash and get -50 utility. Assume both players have a discount rate of 0.9 Draw the stage game of date night List all pure strategy Nash equilibria of the single stage game Consider an infinite horizon version of Chicken. Can you get an SPNE in which the both players swerve using a grim trigger type strategy? Consider the following strategies: both players swerve, as long as neither ever went straight. If one player ever plays straight, in all subsequent rounds the player that swerved goes straight and the player that went straight swerves. Can you think…
- 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 chooses3. 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…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…