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(a) Describe Bi. Explain.
(b) Find Ri. Explain.
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- Consider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?please solve it in 30 minute please Consider an infinitely repeated game where the stage game is the game of chicken: Swerve Straight Swerve 0,0 -1,1 Straight 1, –1 -10, -10 Assume that the discount factor is very close to one. For concreteness, you can assume that d = 0.99. (a) Is there an SPE in which player l's payoff is 1/(1 – 6)? Explain. (b) Is there an SPE in which each player gets a payoff of 0? Explain. (c) Is there an SPE in which player l's payoff is -2/(1 – 8)? Explain.6. Two players each pick a positive integer between 1 and 100. If the numbers are identical, noone wins. If the numbers differ by 1, the one with the lower number pays 1 to the opponent. If the difference is at least 2, the one with the higher number pays 2 to the opponent. What is the NE? Hint: Draw a sketch of the game matrix (while you can't consider all 100 strategies, at least consider the lowest 5-6 numbers) and think about dominated strategies.
- 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.Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.Two workers are on a production line. They each have two actions: exert effort, E, or shirk, S. Effort costs a worker e > 0 and shirking costs them nothing. If two workers do action E a lot of output is produced and the workers earn £3 each. If only one worker chooses action E less output is produced and they both earn £1. The workers earn nothing if they both shirk. (i) Describe this situation as a strategic form game (assuming the workers do not observe each other's effort choice when making their own decision). (ii) For what values of e does this game have strictly dominant strategies? (iii) Describe the Nash equilibria of this game for e = 0,1, 2, 3. (iv) The workers now are re-arranged into a production line. First worker 1 moves and then worker 2 moves. Worker 2 can now see worker l's effort level before they choose their effort. Draw this extensive form game. (v) Find the subgame perfect equilibria of the production-line game for c = 1/2 and c = 3/2.
- Subgames • • • At each node where a player makes a decision, we can think about the "subgame" starting at that node A subgame is the portion of the tree following a particular node How many subgames does our first sequential move example have? • How many subgames does our entry game have? • How many subgames does the game on the right have? • What are its Nash equilibria? Player 1 (1,4) T Player 2 U B (5,2) (3,3) р Player 1 D L 9 (2, 0) Player 2 R (6, 2)rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.0 E F 3 2 FIGURE 2 23 D G 2 H 0 1 (2) Two friends, Albert (A) and Berta (B), have to decide on going to one of three bars, Xenia's (X), Yara's (Y), or Zana's (Z). They decide to select a bar by alternatively vetoing bars until one remains. First Albert vetoes a bar. If at least two bars remain, then Berta vetoes another bar. That process continues until a single bar remains unvetoed. Suppose Albert prefers Xenia's to Yara's to Zana's and Berta prefers Zana's to Yara's to Xenia's. Assume that, for each of the two, going to their most preferred bar gives a utility of 2, going to their next preferred bar gives a utility of 1, and going to their least preferred bar gives a utility of 0. Model this as an extensive form game and find its Nash equilibria. Which of the Nash equilibria are subgame perfect?
- Example 1: Zero Sum Game Game Table 2 Game Table 1 Player 2 Player 2 S1 S2 Si S2 RI (3,-3) (-5,5) Player 1 R1 3 -5 RI (0,0) (7,-7) R1 7 Player 1 All players payoff shown =) Fnd Nasn ewwliLtlun> zero sum shorthandParamter y = 0 If ⟨a, d⟩ is played in the first period and ⟨b, e⟩ is played in the second period, whatis the resulting (repeated game) payoff for the row player?Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.