(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 8 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 6 can perfect tit-for-tat be supported on the equilibrium path of play?
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- 3. Player 1 and Player 2 are going to play the following stage game twice: Player 1 Top Bottom Left 4,3 0,0 Player 2 Middle 0,0 2,1 Right 1,4 0,0 There is no discounting in this problem and so a player's payoff in this repeated game is the sum of her payoffs in the two plays of the stage game. (a) Find the Nash equilibria of the stage game. Is (Top, Left) a Nash of the stage game? (b) Find a subgame perfect Nash equilibrium of the repeated game where the first time they play the stage game Player 1 chooses Top and Player 2 chooses Left.3. 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.
- 1. Consider the game where initially She chooses between "Stay Home" and "Go Out". If She chooses "Stay Home" then She gets 2 and He gets 0. If She chooses "Go Out" then they each simultaneously choose "Movie" or "Concert" where the payoffs are 0,1 or 3 as in the Battle of the Sexes Game. What are the subgame perfect Nash Equilibria of this game ?Question 4. Zeynep and Mehmet will eventually play the following game. Mehmet L R U 3,1 0,0 Zeynep D 0,3 1,3 In a preliminary stage, Zeynep has already asked Mehmet to allow her to move first and proposed to pay him 1 unit of her own payoff in exchange. So Mehmet has to options: • If he accepts Zeynep's offer: they will play the sequential move version of the above game in which Zeynep moves first. Mehmet will receive 1 unit of utility more, and Zeynep will receive 1 unit of utility less (in any outcome of the game) compared to the payoffs given in the bimatrix. 2 • If he rejects Zeynep's offer: they will play the simultaneous move game. a. Represent this strategic interaction in a game tree. b. How many information sets does each player have? c. Characterize the set of pure strategies for both players. d. Present this game as a normal-form game, and characterize the set of pure strategy Nash equi- libria.2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
- 14. You have baked a cake, but your two dear daughters won't stop fighting on who gets the biggest slice. To settle the dispute, to ask your dear daughter one (DD1) to cut the cake and your dear daughter two (DD2) to choose which piece she wants. (a) Draw the extensive form of the game. Let dear daughter one's strategies be "Cut Evenly" or "Cut Unevenly"; depending on what is on the platter, dear daughter two's strategies might in- clude "Take Big Slice", "Take Small Slice", or "Take Equal Slice". Assign payoffs to dear daughter one and dear daughter two that grow with the size of the slice that they receive. (b) Use backward induction to find the equilibrium outcome of this game. (c) Is the promise to take a small slice by DD2, if DD1 cuts unevenly, credible? Explain carefully. (d) After the rules are announced, dear daughter two says "It is not fair! I want to be the one who gets to cut the cake, not the one who chooses the slice!". Is dear daughter two's complaint valid? You are…し(5,3) b I(2,2) も(0,0) (4,12) a (12.4) (0,0) i). List all subgame pertect Nash equilibria and name one Nash eqvilibrivm that is not subgame pertect i). How many strategies does playot and player 2 have?Consider the following simultaneous game: Player 1 U D Player 2 L 30,20 -10, -10 R -10, -10 20,30 Please indicate whether each of the following statements is true or false. Player 1 has a dominant strategy. This game has two Nash equilibria in pure strategies. Player 1's payoff in each of the Nash equilibria is 30.
- Consider the following simultaneous game: Player 1 U D Player 2 L 30,10 -10, 20 R 10, 20 5,-10 Please indicate whether each of the following statements is true or false. Player 1 has a dominant strategy. This game has a Nash equilibrium. This game has a Nash equilibrium in pure strategies. Player 1's best response is D if player 2 plays R.Two business partners jointly own a firm and share equally the revenues. They individually and simultaneously decide how much effort to put into the firm. Let s₁ and s2 denote the effort choices of partner 1 and partner 2, respectively. Assume si € [0, 4]. The cost of effort is given by s? for i E {1, 2}. The firm's revenue is given by 4(81 +82 + bs182) where 0 ≤ b ≤ 1. (Note that the parameter 6 reflects the synergies between the effort levels. b> 0 implies that the more one partner works, the more productive the other partner is.) The payoffs for partners 1 and 2 are: u₁ (81, 82) u2 (81, 82) - = 1 [4(81 +82 +68182)] − 8² - 1 [4(81 +82 + bs182)] – $²4. Suppose that Anne and Bob must simultaneously name a number in the set {1,2, ..., 10}. If they name the same number, the each get a payoff of 1; if they name different num- bers, they each get a payoff of 0. (a) Find all (pure and mixed strategy) Nash equilibria of this game. (b) How many are there? Explain why. (Hint: there are a lot of them!)