Consider the Nash Bargaining game. There are 2 players who have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive sį and s2, with 0 < s1, s2 < 1. If s + s2 < 1, then the players receive the shares they named and if si + 82 > 1, then both players fail to achieve an agreement and receive zero. Find a strictly dominant strategy. Find a pure strategy Nash equilibrium?
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- GAME 5 Player B B2 B1 Player A A1 7, 3 5, 10 A2 3, 8 9, 6 In Game 5 above, O there are no Nash equilibria in pure strategies. Player A choosing A1 and Player B choosing B2 is a Nash equilibrium. O Player A choosing A1 and Player B choosing B1 is a Nash equilibrium. O Player A choosing A2 and Player B choosing B1 is a Nash equilibrium.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.GAME UUU B1 Player B B2 A1 7,13 5, 10 A2 3,8 9,16 Player A A3 5,8 4,7 In Game UUU (see table above), assuming players move simultaneously, Player A choosing A1 and Player B choosing B3 is a Nash equilibrium. Player A choosing A3 and Player B choosing B2 is a Nash equilibrium. Both Player A choosing A1 and Player B choosing B1 and Player A choosing A2 and player B choosing B2 are Nash equilibria in pure strategies Player A choosing A1 and Player B choosing B2 is a Nash equilibrium.
- A Nash equilibrium occurs when O no player has an incentive to unilaterally change strategies. each player has an incentive to unilaterally change strategies. O both players can cooperate to increase their payoffs. O no player can earn a higher payoff from any other strategy.Consider a repeated game with the following stage game: Player 1 O O O Cooperate Not Cooperate This game is repeated T-120 times, and t each period both players must choose their actions simultaneously. The players are deciding whether or not to cooperate specifically, whether or not to follow this cooperation strategy: r≤ 3 S|S|AW|N N|H "Each player will play cooperate as long as no one has played Not Cooperate before. If someone has played Not Cooperate before, then each player will play Not Cooperate forever." rs 5 rs Player 2 In this case, cooperation will be possible if the interest rate r is Not Cooperate 5,15 10, 10 None of the above Cooperate 12, 12 15,5Consider the following sequential game: Player 1 plays first and can choose between L and R. Player 2 plays second and can choose between U and D. • If player 1 chooses L, player 2 does not get to play and both players get £1. If player 1 plays R, player 2 gets to play. If player 2 plays U then both players get £2. If player 2 plays D then player 2 gets £3 and player 1 gets £0. What the the outcome of the subgame perfect equilibrium? O a. £3-£0 O b. £1-£1 O c. This game does not have a subgame-perfect Nash-equilibrium. O d. £0-£3 O e. £2-£2
- Two companies are playing an entry game. In the first stage of the game the incumbent firm decides whether to invest in new robotic equipment, which will lower its marginal cost of production. In the second stage, a potential rival decides whether to enter the market. o If the incumbent firm chooses to invest and the rival firm does not enter the market, the incumbent makes 8 million in profits. o If the incumbent firm chooses to invest and the rival firm enters the market, the incumbent makes 2 million in profits, and the rival firm looses 1 million. o If the incumbent firm chooses NOT to invest and the rival firm does not enter the market, the incumbent makes 10 million in profits. o If the incumbent firm chooses NOT to invest and the rival firm enters the market, both the incumbent firm and the rival firm make 4 million each. (a) Draw a decision tree for this game. (b) Use backward induction to solve the game. What is the rival best strategy? What is the incumbent best strategy? (C)…6. There are 2 players who have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive $₁ and $2, with 0 ≤ $1, S2 ≤ 1. If $1 + $2 ≤ 1, then the players receive the shares they named and if 8₁ + 82 > 1, then both players fail to achieve an agreement and receive zero. Which of the following strategy profiles is a pure strategy Nash equilibrium? O a) (0.3, 0.7); Ob) (0.5, 0.5); O c) (1.0, 1.0); Od) All of the aboveBob - Don't Confess Confess В: 10 years J: 10 years В: 20 years J: 1 year Confess Joe В: 1 year J: 20 years В: 2 years J: 2 years Don't Confess The tal above shows the payoff matrix for a prisoners' dilemma. In the Nash equilibrium O Joe will serve 20 years, and Bill will serve 1 year. Bill will serve 20 years, and Joe will serve 1 year. O Both prisoners get 2 years in jail. Both prisoners get 10 years in jail.
- The following table contains the possible actions and payoffs of players 1 and 2. Player 1 U M D L 2,10 2,5 8,7 Player 2 C 5,2 11,8 8,4 R 5,4 2,9 8,3 This is a simultaneous move game. In a pure strategy Nash equilibrium of this game, Player 1 receives a payoff of ✓and player 2 receives a payoff of If, instead of playing simultaneously, player 2 moves first, then in the Nash equilibrium player 1 receives a payoff of ✓and player 2 receives a payoff of1. 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 ?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.