Two hunters are on a stag hunt. They split up in the forest and each have two strategies: hunt for a stag (S), or give up the stag hunt and instead hunt for rabbit (R). If they both hunt for a stag, they will succeed and each earn a payoff of 9. If one hunts for stag and the other gives up and hunts for rabbit, the stag hunter receives 0 and the rabbit hunter 8. If both hunt for rabbit then each receives 7. Compute all Nash equilibria for this game, called 'The Stag Hunt', depicted below. Which of these equilibria do you think is most likely to be played? Why? S R S 9,9 8,0 R 0,8 7,7
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- Alice chooses action a or action b, and her choice is observed by Bob. If Alice chooses action a, then Alice receives a payoff of 5 and Bob receives a payoff of 7. If Alice chooses action b, then Bob chooses action cor action d. If Bob chooses action c, then Alice receives a payoff of 10 and Bob receives a payoff of 5. If Bob chooses action d, then Alice receives a payoff of 0 and Bob receives a payoff of 3. Which of the following are correct statements about the game described in the previous paragraph? (Mark all that are correct.) O This game has imperfect information. O Alice's backward induction payoff is 5. O This is a promise game. O Alice's backward induction payoff is 0. O This is a prisoners' dilemma. O Bob's backward induction payoff is 5. O Bob's backward induction payoff is 3. O This is a threat game.CLASSES OF GAMES First, Alice chooses either action al or action a2. If Alice chooses action al, then she receives a payoff of 2 and Bob receives a payoff of 8. If Alice chooses action a2, then Bob chooses either action b1 or action b2. If Bob chooses action b1, then Alice receives a payoff of 0 and Bob receives a payoff of 2. If Bob chooses action b2, then Alice receives a payoff of 4 and Bob receives a payoff of 4. Which of the following are correct statements about the game described in the previous paragraph? (Mark all that are correct.) This game has perfect information. The strategy profile (a2,b1) is a Nash equilibrium of this game. Bob's backward induction payoff is 8. The strategy profile (al,b1) is a Nash equilibrium of this game. The strategy profile (a1,b2) is a Nash equilibrium of this game. The strategy profile (a2,b2) is a Nash equilibrium of this game. This is a bargaining game. This is a promise game. This is a threat game. Alice's backward induction payoff is 2.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)Suppose Edison and Hilary are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that Edison chooses Right and Hilary chooses Right, Edison will receive a payoff of 3 and Hilary will receive a payoff of 7. Hilary Left Right Left 4, 6 6, 8 Edison Right 7, 5 3, 7 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Edison chooses and Hilary choosesAlice and Julia want to spend time together but have different preferences for their activities. They can choose between two options: going to a soccer game or watching a football game. Alice prefers football and Julia prefers the football game, but they both prefer being together over being apart. If they both choose to attend a soccer game then Julia receives a payoff of 2 and Alice a payoff of 1. If they both choose to attend a football game, then Julia receive a payoff of 1 and Alice a payoff of 2. If one chooses a football game and the other a soccer game then they both receive a payoff of O. The payout matrix below illustrates these payoffs. Which of the following statements are true: I. There is a dominant strategy II. A nash equilibrium exists where Alice attends a football game and Julia attends a soccer game III. A nash equilibrium exists where both friends attend a soccer game IV. Both attending a football game is a stable outcome. Alice Soccer Football Julia Soccer Football…
- Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.Consider the following sequential game. Wally first chooses L or H. Having observed Wally's choice, Elizabeth chooses between A and F. The payoffs are as follows. If Wally chose L and Elizabeth chose A, the payoffs are 30 to Wally and 20 to Elizabeth. If Wally chose L and Elizabeth F, the payoffs are 40 to Wally and 10 and to Elizabeth. If Wally decides to opt for H and Elizabeth A, the payoffs are 10 and 2 to Wally and Elizabeth, respectively. Finally, if Wally opts for H and Elizabeth F, the payoffs are 35 toWally and 5 to Elizabeth. What is the outcome in the subgame perfect equilibrium of this game? (L,F) (H,A) (H,F) and (L,F) (L,A) and (H,F) (H,F)Consider a sequential game where there are two players, Jake and Sydney. Jake really likes Sydney and is hoping to run in to her at a party this weekend. Sydney can't stand Jake. There are two parties going on this weekend and each player's payoffs are a function of whether they see one another at the party. The payoff matrix is as follows: Sydney Party 1 Party 2 Party 1 6, 18 18, 6 Jake Party 2 24,8 0,24 a) Does this game have a pure strategy Nash Equilibrium? b) What is the mixed strategy Nash Equilibrium? c) Now suppose Sydney decides what party she is going to first. Her roommate is friends with Jake and will call him to tell him which party they go to. Write the extensive form of this game (game tree). d) What is the subgame perfect Nash equilibrium from part c?
- Two roommates John and Joe are playing a simultaneous game of cleaning the apartment. If neither of them clean, the apartment gets filthy and both get a utility of 2. If John cleans and Joe doesn't, John gets a utility of 1 and Joe gets a utility of 4. If Joe cleans and John doesn't, Joe gets a utility of 1 and John gets a utility of 4 and if both clean up the apartment, they each get a utility of 3. Is the equilibrium efficient?The following table shows the payoff matrix that represents the strategic interaction between two classmates, Amelie and Brianna. They are deciding which Scottish island to visit on a day trip. Brianna Kerrera Cumbrae Amelie Kerrera 5, 2 1, 1 Cumbrae 3, 3 3, 5 a. Amelie likes Kerrera more than Cumbrae, conditional on both choosing the same island b. In a sequential game where Brianna can move first, Amelie would choose to visit Cumbrae c. In a one-shot, simultaneous game, both Amelie and Brianna will choose to visit Kerrera since their total happiness would be the highest d. There are two Nash equilibriaCameron 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?