L R 1,2,3 3,1,1 D 3,1,0 2,5,2 L T 0,2,0 4,1,4 4,1,1 0,5,2 Consider both pure and mixed strategies, if probability p assigns to Tof player 1. q assigns to L of player 2. r assigns to L of player 3. Does this have a nash equilibrium if r =1 or 0?
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- Section 1: Two Romantics Ann and Bob had their first date. Each either felt romantic chemistry (C) or no chemistry (NC) with the other person. Each person knows his/her own feeling but does not know the feeling of the other person. Assume a common prior belief that the other person felt chemistry with probability Pr(C) = p and no chemistry with probability Pr(NC) = 1-p. Ann and Bob are old-fashioned romantics and they made the following rule after the first date: No texts/calls/DMs. Instead, they can choose whether to appear (A) or not appear (NA) under the USyd Quadrangle clock tower at sunset on the next day. Their payoffs are given as follows: (From a first-person perspective) If I felt chemistry (C) and I appear (A) under the clock tower, my payoff is 100 if the other person also appears (A) and -100 if the other person doesn't (NA). If I felt chemistry (C) and I choose not to appear (NA) under the clock tower, my payoff is -30 regardless of the other person's action (because I…Consider the game with the payoffs below. Which of the possible outcomes are MORE efficient than the Nash Equilibrium (NE)? Note, they do NOT need to be Nash equilibria themselves, they just need to be more efficient than the NE. Multiple answers are possible, but not necessary. You need to check ALL correct answers for full credit. JILL High Medium LowMAGGIE Left 3,4 2,3 2,2Center 4,8 9,7 8,7Right 7,6 8,5 9,4Group of answer choices (Left, Low) There is no strategy combination that is more efficient than the Nash equilibrium for this game. (Right, Medium) (Left, High) (Center, Medium) (Center, High) (Center, Low) (Left, Medium) (Right, Low) (Right, High)3. Consider the game below. С1 C2 C3 R1 1, 1 4, 6 8, 5 1, 2 5, 4 R2 R3 2, 6 2, 7 7, 6 0, 7 3.1. Does the game have any pure strategy NEs? 3.2. Check whether a mixed strategy NE exists in which A is mixing R1 and R2 with positive probabilities, playing R3 with zero probability, while B is mixing C1 and C3 with positive probabilities while playing C2 with zero probability. [Let (p1,P2, P3) be the probabilities with which A plays (R1, R2,R3) and let (q1,92, 93) be the probabilities with which B plays (C1, C2,C3). Make use of the following NE test: m* is a NE if for every player i, u;(mị , m²¿) = u;(Si, m²¡) for every si E S¡|m¡(sji) > 0 and u¡(m¡ ,m²¡) > u¡(s¡,m;) for every si E S¡ |m¡ (s¡) = 0. Hint: Each player must be indifferent between those of her pure strategies that are used (with positive probability) in her mixed strategy, and unused strategies must not yield a payoff that is higher than the payoff a player gets with her NE (mixed) strategy.] %3D
- Here is a Bayesian BOS game. Man has a special preference t1 on boxing while woman has a special preference t2 on opera. Both are in [0,2]. The preference information is private. i.e. Each knows his (her) own preference but does not know the other's preference. The below is the payoff. M/W Boxing Opera Воxing 3+t1,1 0,0 Opera 0,0 1,3+t2 We are checking whether the below strategy profile is a Bayesian Nash Equilibrium. "Man chooses boxing if t1>p while woman chooses opera if t2>g." 1) Given t1, man will choose boxing if t1> /g- 2) Given t2, woman will choose opera if t2> /p- 4) By shoving the simultaneous equations, we get p=q=sqrt(6)-Consider the following two-player game with three options for each player. (Payouts are listed for the row player first, then the column player.) player Y layery 3,3 A 1,5 4,4 6,2 K 8,1 3,7 5,2 0,6 1,1 Find a mixed Nash equilibrium for this game. Solution suggestion: Use two variables per player. If p and are the probabilities of selecting the first two strategies, then 1-p-q is the probability of selecting the third strategy. You will need to solve a system of equations.The count is three balls and two strikes, and the bases are empty. The batter wants to maximize the probability of getting a hit or a walk, while the pitcher wants to minimize this probability. The pitcher has to decide whether to throw a fast ball or a curve ball, while the batter has to decide whether to prepare for a fast ball or a curve ball. The strategic form of this game is shown here. Find all Nash equilibria in mixed strategies.
- Player 1 and Player 2 will play one round of the Rock-Paper-Scissors game. The winner receives a payoff +1, the loser receives a payoff -1. If it is a tie, then each player receives a payoff zero. Suppose Player 2 is using a non-optimal strategy: he/she plays Rock with probability 35%; Paper with probability 45%; and Scissors with probability 20%. Given that Player 1 knows Player 2 is using this strategy, the best response of Player 1 is to play O a mixed strategy with probability 1/3 each (Rock, Paper and Scissors) O Scissors with probability 100% O any strategy (any strategy is a best response for Player 1) O Rock with probability 100% O Paper with probability 100%Problem 2. Consider the partnership-game we discussed in Lecture 3 (pages 81-87 of the textbook). Now change the setup of the game so that player 1 chooses x = [0, 4], and after observing the choice of x, player 2 chooses y ≤ [0, 4]. The payoffs are the same as before. (a) Find all SPNE (subgame perfect Nash equilibria) in pure strategies. (b) Can you find a Nash equilibrium, with player 1 choosing x = 1, that is not subgame perfect? Explain.Finding Nash Equilibria Consider the following two player, normal form game: Player 1 Player 2 C L (2,1) U M (-2,-2) D R (2, -1) (1,2) (-1, 1) (0,0) (3,1) (0,0) (-1,-1) Find all pure and mixed strategy Nash equilibria. Calculate each player's expected payoffs at each equilibrium.
- Consider the following infinitely repeated game: 1/2 C D с 2,2 -3,3 D 3,-3-2,-2 a. Start with C and continue playing C as long as neither player deviates. If any player deviates, play D from that point onwards. For what value of the discount factor will this strategy be subgame perfect. b. Start with C, D and alternate between C,D and D,C. In case of deviation, play D,D forever. Can this strategy be subgame perfect?Suppose that you are a manager. You are considering whether or not to monitor employees with the payoffs in the normal-form accompanying game. Worker Work Shirk Manager Monitor -1,1 1,-1 Don't Monitor 1,-1 -1,1 Which of the following pairs of strategies constitutes a Nash equilibrium? Multiple Choice Manager monitors and worker shirks. Manager does not monitor and worker works. Manager monitors and worker works. O None of the answers is correct.Mixed Nash Equilibrium Player 2 C D A 5,1 1,3 B 2,6 4,2 PLAYER 1 a) calculate the probability In equilibrium Player 1 chooses A with probability a= [?] and player 2 chooses M with probability B = [ ?] b) calculate payoff for mixed strategy nash equilibrium Player 1 gets payoff of [?] Player 2 gets payoff of [?] c) Suppose the payoff for (B,C) would change to (0,8) Would probability a increase,decrease, stay the same? Would probability B increase,decrease, stay the same?