5. Consider the following game: 1 L 3,1 2,0 C 1,-1 D 0,-1 ABCD A 2 R -2,-1 1,4 2,0 4,0 a. Find all pure NE. b. There is a unique MNE. Find it and show why there is no other one.
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- 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.11. What are the subgame perfect equilibria of the following game? Out In T B 7,21 L R L/R 9,3 1,1 1,1 3,9 (a) (InT, L), (OutB, R) and (OutT+OutB, L + R). (b) (InT, L) and (OutB, R). (c) Out and (InT, L). (d) (InT, L), (OutB, R) and (OutT+OutB, L + R). (e) (InT, L). 2Exercise 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.
- 1. Consider a repeated game in which the following stage game is played in each of two periods and there is no discounting. U Player 1 M D L 5,5 0,0 0,12 Player 2 C 0,0 2,2 0,0 R 12,0 0,0 10, 10 a) Find the Nash equilibrium that maximises the players' final payoffs, i.e. the sum of their payoffs from each period. I b) Discuss the intuition of this equilibrium.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.5) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.
- see image Use the following setting for questions 1-3. Consider the following static game: C R L 3,1 0,0 5,0 T M 2,1 1,2 3,1 B 1.2 0,1 4,4 Suppose this game is played for two periods. At the beginning of period 2, the players can observe the outcome from period 1. There is no discounting. 1. First, assume only pure-strategy SPNE such that either player's second period strategy DOES NOT depend on the first period outcome. How many such SPNE are in this game? (a) 1 (b) 2 (c) 3 (d) 4 (e) none 2. Consider only SPNE from the previous question. What is the maximum sum of the players' payoffs that can be achieved in the first period? (a) 0 (b) 3 (c) 4 (d) 5 (e) 8 3. Now consider all possible pure-strategy SPNE such that (B,R) is played in the first period. What must be the equilibrium outcome in the second period of such SPNE? (a) (T,L) (b) (M,C) (c) (B,R) (d) (T,R) (e) There are no such SPNE3. 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.4. Consider a normal form game with N= {1,2} and S= {A,B,C,D} x {W,X,Y,Z}. Assume that L1 allows to eliminate D and X, L2 allows to eliminate B and C, and L3 allows to eliminate Z. Provide payoffs that lead to this outcome. State the rationality assumptions explicitly.
- 8.1 Consider the following game: Player 1 A B C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1.1 E 0,0 1,1 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game..4. Consider the following game matrix, where a and c are some fixed numbers strictly between -1 and 1: 1 -2 -1 -2 -1 C 2 2 -1 -1 1 1 1 -2 a Argue that an optimal mixed strategy of the column player is y* = (1/2,0, 0, 1/2,0), and write down an optimal mixed strategy of the row player with exactly 3 nonzero entries (probabilities). 2.5. Consider a general two-stage game of complete but imperfect information. How do we denote the subgame-perfect outcome? a. (a₁, až, až (α₁, A₂), A† (A₁, A2)) * b. (a₁, až, az (a₁, až), α₁ (a₁, až)) c. (a₁, a₂, a3 (a₁, A₂), α₁ (α₁, α₂)) d. * * * (a₁, až‚ až (aƒ‚ a₂), a‡ (aƒ‚ a²))
