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.
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- 4. Two investors have each deposited 5 (all amounts are in $10,000s) with a bank. The bank has invested these deposits in a long-term project. If both investors make withdrawals at date 1 then each receives 3 and the game ends. If only one investor makes a withdrawal at date 1 then that investor receives 5, the other receives 1, and the game ends. Finally, if neither investor makes a withdrawal at date 1 then the project matures and the investors make withdrawal decisions at date 2. If both investors make withdrawals at date 2 then each receives 7 and the game ends. If only one investor makes a withdrawal at date 2 then that investor receives 9, the other receives 5, and the game ends. Finally, if neither investor makes a withdrawal at date 2 then the bank returns 7 to each investor and the game ends. Show the bi-matrix for dates 1 and 2.Emma and Mia are invited to two events A and B. They want to go to both events but are not on speaking terms. Going to event A results in a payoff of 5 while going to B results in a payoff of 4. When they go to the same event, they each get a payoff of -8. 6. Write the payoff table for this game. 7. Find the maximin solution. (Show the steps leading to the solution.) 8. Find the Nash Equilibrium. (Show the steps leading to the NE.)5. Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either 1, 2, 3, or 4 matchsticks. The player to remove the last matchstick wins the game. 1. Suppose there are only 6 matchsticks left, and it is Barney' s turn. What move should Barney make to guarantee himself victory? Explain your reasoning. 2. Suppose there are 12 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? (Hint: Use your answer to part (a) and roll back.) 3. Now start from the beginning of the game. If both players play optimally, who will win? 4. What are the optimal (complete) strategies for each player?
- 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 game38. Consider the following game. There are ten churches in town: nine are Catholic and one is Eastern Orthodox. All churches conduct their divine services at 10am on Sunday morning. You and your friend, without communication, agree to meet at a church service. If you fail to meet at the same church, then you are sad and become a Doomer. What is this game's focal point? Remember that you cannot communicate about this. (A) You meet at any of the Catholic churches. (B) You meet at any of the churches – denomination does not matter. We're all the same bro. (C) You meet at the Orthodox church. (D) There is no focal point. 39. Consider a fractional-reserve banking system in which an initial deposit of $1,000 can generate up to $4,000 in money. What is the reserve ratio? (A) 4 percent. (B) 5 percent. (C) 25 percent. (D) 75 percent.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.
- Game theory Consider a simultaneous move game with two players. Player 1 has three possible actions (A, B, or C) and Player 2 has two possible actions (D or E.) In the payoff matrix below, each cell contains the payoff for Player 1 followed by the payoff for Player 2. Player 2 7. Player 1 A B C D -3, -3 0, -11 -4, 3 -11, E 0 -7, -7 -12, 0 (a) Identify any dominated strategies in this game. If there are none, state this clearly. (b) Identify any pure strategy Nash Equilibria in this game. If there are none, state this clearly1. Players 1 and 2 must decide whether or not to carry an umbrella when leaving home. They know that there is a 50-50 chance of rain. Each player's payoff is -5 if he does not carry an umbrella and it rains, -2 if he carries an umbrella and it rains, −1 if he carries an umbrella and is sunny, and 1 if he doesn't carry an umbrella and it is sunny. Player 1 learns the weather before leaving home; player 2 does not, but he can observe player 1's action before choosing his own. Give the extensive and strategic forms of the game and find its Nash equilibria. Is it dominance solvable?Consider a situation of after-match penalty shoot-out. The striker can target either East or West side of the goal. If he targets West, with 80% chance he shoots on target. If he shoots East, he is accurate with 75%. The goalkeeper has to choose the corner to jump to. If he does not guess the corner correctly and the shot is on target, then the striker scores. If the striker shoots West, the shot is on target and the goalkeeperjumps West, then with 75% chance he saves the goal. If the striker shoots East, the shot is on target and the goalkeeper jumps East, then with chance of 2/3 he saves the goal. Suppose, that it is a zero-sum game and if the striker scores his payoff is 1, otherwise it is 0.1. Formulate this situation as a strategic game and Find all Nash equilibria of the game.
- A and B are a couple and they want to go into a concert. A comes from Bonn and loves the music of Ludwig van Beethoven. B comes from Salzburg and loves the music of Wolfgang Amadeus Mozart. There are two concerts in town: one with the music of Beethoven and one with the music of Mozart. Both (A and B) prefer to go to a concert together. If both go to a concert of Beethoven, A has a pay-off of 4 and B has a pay-off of 2. If both go to a concert of Mozart, B has a pay-off of 4 and A has a pay-off of 2. If A goes to a concert of Mozart and B to a concert of Beethoven, they are both miserable and get a pay-off of 0. But, if B goes to a concert of Mozart and A to a concert of Beethoven, they are both a little better off with a pay-off of 1. Both A and B have to make a simultaneous decision and cannot communicate prior to the decision. (a) Construct the pay-off matrix for this game. (b) Identify the Nash equilibrium or equilibria of the game. (c) Which off the allocations are…Consider the below payoff matrix for a game of chicken. Two players drive their cars down the center of the road directly at each other. Each player chooses SWERVE or STAY. Staying wins you more payoff only if the other player swerves. Swerving loses face if the other player stays. However, the worst outcome is when both players stay. Stay Swerve Stay -0,-0 -6,6 Swerve 6,-6 9,9 Player 1 chooses rows and Player 2 chooses columns. Denote the probability of "Stay" for Player 1 as p and for Player 2 as 9. What is the value of p so that Player 2 is indifferent between Stay and Swerve? Write your answer as a decimal number with 2 decimal places (e.g. 0.05)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…