Consider the following two player game. In each cell the first number refers to the payoff to Player 1 while the second number refers to the payoff to Player 2. Suppose the two players move simultaneously (at the same time). Which one of the following statements is CORRECT? Player #1 Player #2 Top Bottom Left 7.7 4.0 Right 0.4 4,4

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?

2. Consider two players playing a simultaneous moves game. These players can be one of the two states of the world, state 1 or state 2, while playing the game. The payoffs-matrix for the two state games are as follows: State 1 P1 U D P2 L R 1,4 1,0 1,6 2,16 0,0 0,24 M State 2 P1 U D P2 L M R 1,0 1,4 1,6 2,16 0,24 0,0 Each players believes that the two states are equally likely, Derive the BNE of the game.

11) In the following game, list any dominant or dominated strategies. Then, solve the game Player 2 Left Center Right Player 1 Up 1,2 4,6 5,5 Down 6,3 2,4 3,1

Use the following extensive-form game to answer the questions below. A B W Z (60, 120) (50, 50) (0, 0) (100, 150) Use backward induction to solve for the equilibrium. What are the equilibrium strategies? Player 1 plays B, and Player 2 plays W if A and Z if B. Player 1 plays B, and Player 2 plays W if A and Y if B. O Player 1 plays B, and Player 2 plays X if A and Z if B. Player 1 plays A, and Player 2 plays X if A and Z if B. Player 1 plays A, and Player 2 plays W if A and Y if B.

Please explain how to solve the question and arrive at the provided answer. Thank you.

5. A game of Chicken is played by two lorry drivers who speed toward a narrow bridge. If one driver stops to let the other pass, they waste time, whereas the one who does not stop gets to their destination quicker. However, if both keep driving straight on then they crash. The game is played once, and drivers move simultaneously. Payoffs are given below (assume 0 < x < 10): Drive Stop Drive -10,-10 -x, 8 Stop 8,-1 0,0 (a) Find any pure strategy Nash equilibria of this game. (b) Find the mixed strategy Nash equilibrium of this game. (c) What happens to the column player's equilibrium mixed strategy as x increases? Explain why this happens. (d) Let x = 1. What are the rationalizable outcomes of this game? Do these offer a better prediction than the Nash equilibria?

The following normal form representation of a racquetball game implies both players choose their strategies simultaneously. Nimah hits the ball, and Raina is moving at the same time to try and position herself where the ball will be. (please refer to figure provided)    If the game is slowed down a little, giving Raina a chance to see what direction Nimah has hit to, what should happen?That is, what outcome would we predict for this as a sequential game when Nimah moves first?Choose one: A.  Raina always wins the point. B.  Nimah wins the point about half the time, and Raina wins about half the time. C.  Nimah wins the points to the right, and Raina wins the points to the left. D.  Nimah always wins the point.

!

In the following games, all payoffs are listed with the row player's payoffs first and the column player's payoffs second. GAME 33 Player A Player B B1 10, 12 A1 A2 9,3 A3 8, 10 In Game 33 above, B2 B3 8,8 12, 10 7,6 11, 1 9,4 14, 3 B1 is a dominated strategy for Player B B2 is a dominated strategy for Player B B3 is a dominated strategy for Player B Player B has no dominated strategies.