1. What is the number of pure strategies that each player has: 1 Left Right 2 2 Right Left, 1,0 2,2 Left 2,2 a) Both have 2 strategies. b) Both have 4 strategies. c) Player 1 has 2, and player 2 has 4. d) Player 1 has 3, and player 2 has 4. Left 2,2 Right 1 Right 0,0
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- Player 2 Left Center 30, 40 6, 20 20, 20 Right 21, 24 32, 35 10, 5 Up Middle 5, 5 40, 11 Player 1 Down 38, 42 Use the information in the game matrix above to answer questions #3 and # 4. 3. If both Players move simultaneously, then a. Player 1 has a dominant strategy, and Player 2 does not. b. Player 2 has a dominant strategy, and Player 1 does not. c. There is more than one pure strategy Nash Equilibrium. d. Both players have a dominant strategy. e. There are no Nash Equilibria in pure strategies. 4. If Player 1 moves first, then is the subgame perfect equilibrium. a. (Up, Left) b. (Middle, Center) c. (Middle, Right) d. (Down, Left) e. (Down, Center)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?1. A row of minus signs is written on a whiteboard (can be any number). Two players take turns in replacing either a single minus sign by a plus sign or two adjacent minus signs by two plus signs. The player who makes the last move wins. Discuss the possible strategy for the first mover to force a win.
- 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 game6. 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.Consider the game below: Player C Strategy C1 50, 50 Strategy C2 Strategy R1 Strategy R2 Player R 800, 800 1000, 1000 100, 100 Playing the game sequentially would result in C getting: 100 800 It depends if R plays first or second 50 1,000
- GAME 5 Player B B1 B2 Player A A1 7,3 | 5, 10 A2 3, 8| 9, 6 In Game 5 above, O Neither player has a dominant strategy. O Player B has a dominant strategy. O Player A has a dominant strategy. O Both players have dominant strategies.15) Find all Nash equilibria (both pure and mixed strategy) in the following game Player 2 Left Right Player 1 Up 10,8 0,5 Down 4,0 3,6Q5 A and B are sharing a room with a TV and a DVD player. DVD's can be rented for the weekend for a cost of $5. They would each get $4 worth of enjoyment from a weekend movie, so if both of them rent a movie a on a particular weekend they can each get $8 worth of enjoyment at a cost of $5 rental. (a) Model this as a strategic form game with both players having the strategies Rent (R) and No Rent (N), (b) What is the equilibrium if they share the room for one weekend only? (e) Draw the extensive form of the game if they stay together for two weeks, and identify the subgame perfect equilibria. (d) Explain whether or not the outcome of the game will change if they stay together for 10 weeks.
- 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.Cameron and Luke are playing a game called ”Race to 10”. Cameron goes first, and the players take turns choosing either 1 or 2. In each turn, they add the new number to a running total. The player who brings the total to exactly 10 wins the game. a) If both Cameron and Luke play optimally, who will win the game? Does the game have a first-mover advantage or a second-mover advantage? b) Suppose the game is modified to ”Race to 11” (i.e, the player who reaches 11 first wins). Who will win the game if both players play their optimal strategies? What if the game is ”Race to 12”? Does the result change? c) Consider the general version of the game called ”Race to n,” where n is a positive integer greater than 0. What are the conditions on n such that the game has a first mover advantage? What are the conditions on n such that the game has a second mover advantage?Consider the game below. How many best responses does Player 1 have to strategy L of Player 2? 1) 0 2)1 3) 2 4)3 Player 1 T M B L 2,0 2, 2 2,3 Player 2 с 2,1 3,2 1, 2 R 3,2 0,1 0,3