1. The table below shows expected payoffs (ir $billions) for GM and the United Auto Workers (UAW) as they enter contract negotiations. Strike GM UAW Concede Bargain Hard -20, -10 10, -15 Concede -10, -2 15, 0 a. Using game theory for a one-shot, simultaneous decision game, find all the Nash equilibria. b. Each side enters negotiations threatening to take a hard line (Bargain Hard or Strike). How credible are each of these threats and why (briefly)?
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- Consider the following price game: Firm 1 Firm 2 High Low High 20, 20 12, 24 Low 24, 12 14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?Solve for the Nash equilibrium (or equilibria) in each of the following games. (a) The following two-by-two game is a little harder to solve since firm 2’spreferred strategy depends of what firm 1 does. But firm 1 has a dominantstrategy so this game has one Nash equilibrium. Firm 2 Launch Don’tFirm 1 Launch 60, -10 100, 0 Don’t 80, 30 120, 0 What is the Nash equilibrium of this simultaneous-move game? (b) What would the outcome of this game be if instead firm 1 moved first and then, after seeing what firm 1 chose, firm 2 chose it strategy? In this case firm 1 doesn’t necessarily need to choose a best response, but firm 2 must choose a best response since it moves second.4. This problem is based on after chapter question 11. If you are not familiar with the rule of baseball game, just use the figure below. a. Find the subgame perfect Nash equilibrium. Note: like many sports games we discussed, the equilibrium is often about mixed-strategies. b. What are the equilibrium payoffs of two players? FIGURE PR9.11 The Suicide Squeeze Game Torre Suicide squeeze Torre 10 LaRussa 0 Pitchout No suicide squeeze 2 ∞0 N LaRussa 8 Suicide squeeze No pitchout Suicide squeeze 9 Pitchout No suicide squeeze 4 6 No suicide squeeze Torre LaRussa No pitchout Suicide squeeze 37 No suicide squeeze 5 55 5
- a. The president of a Japanese electronics company, Takashi Hashiyama, famously chose the auction housefor his company’s art collection by having representatives from two auction houses, Christie’s and Sotheby’s,play a round of rock-paper-scissors. At stake were millions of dollars in commissions. Show the payoff matrixfor a game where the payoff is 1 if you win, 0 if you tie, and -1 if you lose. I’m sure you know this already, butjust in case: rock breaks scissors, scissors cut paper, and paper smothers rock. Here’s an article about the actualgame if you’re interested. b. If you were in a high stakes game…1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).Consider the following game: Player 2 In Out Player 1 In -2,-2 2, 0 Out 0, 2 0, 0 (a) What is the Nash equilibrium of this game, or what are the Nash equilibriaof this game? (b) Does either firm have a dominate strategy (a strategy that is always abest response)? Which? (c) Suppose Player 1 could move before Player 2 and Player 2 could observe Player 1’s move. What do you think would happen?
- 1. For the following game Player B Player A Left 7,17 10,5 4,4 Middle Right 14,11 4,3 10,25 21,21 Тop Middle 14,4 Bottom 7,3 a) Determine the Nash equilibrium is any? b) Represent the game sequentially starting with player 1 and determine the sub-game perfect equilibrium c) Represent the game sequentially starting with player 2 and determine the sub-game perfect equilibrium d) Find the mixed strategy Nash equilibrium of the following game e) Firm 1 Firm 2 High price Low price High price 4,3 2,4 Low price 2,4 3,320.38 C 1 Search il 4G O Microeconom... 1. In the sequential games such as the sequential Battle of the Sexes, why does the Nash equilibrium allow for outcomes with noncredible threats? 2. Left Right Up 1, 3 3, 1 Down 2, 2 4, 4 a. Find the Nash equilibrium or equilibria b. Which player, if any, has a dominant strategy? c. Does A benefit from changing the game by reducing his or her payoff in this way? 3. Noah and Xaviera love to play "Hide and Seek." It is a simple game, but it continues to amuse. It goes like this. Xaviera hides upstairs or downstairs. Noah can look upstairs or downstairs but not in both places. If he finds Xaviera, Noah gets one scoop of ice cream and Xaviera gets none. If he does not find Xaviera, Xaviera gets one scoop of ice cream and Noah gets none. a. Fill in the payoffs in the matrix below! Is this a zero-sum game? What are the Nash equilibria in pure strategies? Hide and Seek Xaviera Upstairs Downstairs Upstairs Noah Downstairs b. Find a Nash equilibrium in…5. Aaron and Betty play the following one-shot game. Aaron Up Middle Down Left 10,0 5,10 1,1 Betty Right 0,5 1,0 2,2 a. Does the game above have any (pure strategy) Nash equilibria? Find any equilibria or explain why an equilibrium doesn't exist. b. Suppose instead that Aaron moves first and Betty moves second. Represent this situation with a game tree and find the subgame perfect Nash equilibrium. Are Aaron and Betty better off playing a simultaneous game or a sequential game in which Aaron moves first? Explain. c.
- 1. Static and Dynamic Game. Consider the following 2-by-2 game: C D 1A (1,10) (1,1) B (2, a) (0,1) (a) erwise, find the pure-strategy Nash equilibria, if any of this game. (There is no need to look for equilibria in mixed strategies) Assume for now that a = 2. Using dominant strategies or oth- (b) Now, consider the dynamic game in which player 1 moves before player 2, and the payoffs remain unchanged. i.: possible strategies for each player? Recall that a strategy profile for a player not at the initial node of the game tree must specify an action for the player at every node. Draw the game tree for this dynamic game. What are the ii "Find the backward-induction solution(s) to this game. Calculate the equilibrium payoffs for each player. Com- pared to the simultaneous game, is there a first-mover advantage or a ii. second-mover advantage? (c) Now, assume that a = 0 in the original static game. i. Find all the Nash equilibria in pure and mixed strategies. Denoting p the probability…Which of the following is FLASE for an extensive form game with perfect information? A. All Nash equilibria has the property of non-credible threat B. Subgame perfect equilibrium eliminates non-credible threats O C. In a Nash equlibrium strategy, each agent's strategy is a contingent plan1. Consider the following sequential game: Left (3,5) Mary Go (6,4) Up Right John Stop (5.7) John (a) Draw the game tree of the game. John Go Down (4,1) Left Mary Stop (2,5) Right (6,-1) (John as player 1 and Mary as player 2) By backward induction, find a Nash equilibrium and the corresponding payoffs. 2. Eve and Noa start with $10 in each of their piles. They take turns choosing one of two actions, continue or stop, with Eve choosing first. Each time a player says continue, $10 will be removed from her pile, and $20 will be added to the other player's pile. The game automatically stops when the total amount in their piles reaches $60. (Eve as player 1 and Noa as player 2) (b) By backward induction, find a Nash equilibrium and the corresponding payoffs.