. Consider a marriage market with three men and three women, all of whom prefer any match to being unmatched. Preferences over matches are as follows: m₁w₁ W2 > W3 m₂ : W₁ W2 > W3 m3 w2w3 > W₁ (a) Find a stable matching. w₁m1 m2 > m3 W₂ m₂ > m₁ > M3 w3m₁ m2 > m3 (b) Is there a matching that is Pareto efficient but not stable?
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- Determine the optimum strategies and the value of the game with the followingpayoff matrix of player A where A1, A2 are the strategies for player A and B1, B2 are for player B.B1 B2A1 5 1A2 3 4Find the value of q for which player 2 is indifferent between L and R. (Note: this will be player 1's equilibrium mixed strategy.) 1/8 O 1/4 2/5 3/8 1/2 3/5 5/8 О3/4P1| P2 Left Middle Right Left 4,2 3,3 1,2 Middle 3,3 5,5 2,6 Right 2,1 6,2 3,3 Consider the simultaneous move game represented by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted, or the discount factor is equal to 1. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. a) (Left, Left) is played in both periods. b) (Right, Right) is played in both periods. c) (Middle, Middle) is played in both periods. d) (Middle, Middle) is played in the first period, followed by (Left, Left). e) (Middle, Middle) is played in the first period, followed by (Right, Right).
- Which of the following conditions correctly describes a Nash equilibrium when two firms are in the market? Multiple Choice n91s1. $2)2 (51. $2) for all s1 and 12(51, $2)212(51. 52) for all s2. aqls1, s2)22(s1, s2) for all s1 and a2(s1, 52) 2 n1ls1.52) for all s2. Rylsi. s2)2 als1. 52) for all s as1. 52)22(1. s2) for all sy and a2(s. s2)2 a1(s1, s2) for all s2Exercise 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.There are 3 workers W1,W2,W3. Neither of these workers can be employed alone so that their value for remaining single is 0. The pair W1 and W2 if employed jointly can earn 100. Likewise the pair W1 and W3. Say the pair W2 and W3 if employed jointly can earn z. The three workers cannot be employed together. What is the maximum value for z such that there is a stable matching in this game ?
- Suppose there are two players playing a game with east or west and south and nerth ways. Find the expected Nash equilibrium by using the concept of probabilities. Player X Left[L) Right|R) Player Y Up(U) (5,6) (0,8) (4,6) Down[D) (0,9)Paramter 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?(b) Consider the simultaneous-move game below with two players, 1 and 2. Each player has two pure strategies. If a player plays both strategies with strictly positive probability, we call it a strictly mixed strategy for that player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. Player 2 b₁ b₂ Player 1 a₁ 3,0 0,1 a2 2,1 2,1
- Player 1 B L 8 9 Player 2 8 0 R 0 1 9 1 Collusion 5. Consider the following game and suppose that it is repeated an infinite number of times. Players have a discount value of 8. (a) For what values of 8 can collusion on (T,L) be sustained under the following grim trigger strategy: • Play (T,L) if (T,L) has been played in all previous periods . Otherwise play (B,R).Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.Consider the following normal form game Player 2 L C Player 1 U 5,4 4,5 A 3,8 UMD 8,3 8,0 8.8 R 5,5 6,3 0,8