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- 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.6. Players 1 and 2 are leaving the house and need to decide whether or not to bring an umbrella. They know there's a 75% chance that it will rain (thanks AccuWeather). If they don't bring an umbrella and it rains they receive a payoff of -7. If they do bring an umbrella and it rains, they receive a payoff of -4. If it doesn't rain, but they bring an umbrella they receive a payoff of -2. Finally, if it doesn't rain and they don't bring an umbrella they receive a payoff of 2. Player 1 learns the weather before they leave the house, but player 2 does not. Player 2 can, however, observe player 1s choice. (a) Represent this game in the extensive form. i. Use backwards induction to solve for the Nash equilibrium (b) Represent this game in the normal form i. Solve for the Nash equilibrium (c) Are the equilibrium strategy profiles the same in a) and b)? Explain why or why not.Question 5 ave Answer Player 1 has two options: L and R. If Player 1 chooses L, then Player 2 has two options; she can choose A or B. If Player 2 chooses A payoffs are (2; 1) and if she chooses B payoffs are (1; 2). If Player 1 chooses R, then Player 2 has two options; she can choose Y or N. If Player 2 shows Y payoffs are (0; 3) and if she chooses N payoffs are (1; 2). Which of the following statements is true? O A. (L; B,Y ) is a subgame perfect equilibrium of this game O B. (R; B,N ) is a subgame perfect equilibrium of this game O C.(L; A,Y) is a subgame perfect equilibrium of this game O D. (R; B,Y ) is a subgame perfect equilibrium of this game
- 1. 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 ?1. Each of two players simultaneously announces either Rock, Paper, or Scissors. Paper beats Rock, Rock beats Scissors, and Scissors beats Paper. The player who names the winning object receives $1 from her opponent. If both players make the same choice, no payment is made. Each players's preferences are represented by by the expected amount of money she receives. a) Formulate the situation as a strategic game and find all its mixed strategy equilibria. Make sure that you consider all different supports. b) Find all the mixed strategy equilibria of the modified game in which player 2 is prohibited from announcing Rock.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.
- 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.For the following simultaneous move game, we want to find a mixed strategy Nash equilibrium. ROWENA Safe 4*q + 4*(1-q) = Risky Safe 1 4,4 1,4 COLIN Risky 4,1 We denote Colin's mixture probability on Safe by q.. To find q in Colin's mixed strategy Nash equilibrium, we apply indifference property to player Rowena 's expected payoffs from pure strategies in mixture: 6,6 +q+6 *(1-q)3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…
- 3. Repeated Game Consider the game in which the stage game depicted below is infinitely repeated and in which both players discount future payoffs with discount factor 6 € [0, 1]. Player 1 E S Player 2 E 3,3 4, -1 S -1.4 1,1 (a) Suppose that both players play the tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player copies the action chosen by his opponent in the previous period (i.e if Player 2 shirks (plays S) in period 2, then Player 1 plays S in period 3). Then, for what values of d can tit-for-tat be supported on the equilibrium path of play? (b) Suppose that both players play the perfect tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player plays E unless the actions are disagreed in the previous period. Then, for what values of d can perfect tit-for-tat be supported on the equilibrium path of play?i. ii. QUESTION ONE A. A Nash equilibrium is a strategy profile such that every player's strategy is the best response to all the other players. It requires that each player makes a best response and that expectations regarding the play of other players are correct. Below is the table showing strategies and payoff for Player 1 and Player 2. PLAYER 1 R1 R2 R3 R4 C1 0,7 5,2 7,0 6,6 C2 2,5 3,3 2,5 2,2 PLAYER 2 C3 7,0 5,2 0,7 4,4 CA 6,6 2,2 4,4 10,4 REQUIRED; Transform the normal form game above into an imperfect extensive game form Find the Nash equilibrium for the game above using iterative deletion of strictly dominated strategies. Find the Nash equilibrium using brute force or cell by cell inspection.on 8.1 Consider the following game: Player 1 A C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1, 1 F 0,0 1,1 4,4 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.
