Betty and John are opening a restaurant together. To proceed with the project two choices must be made: the type of restaurant and the location. Since Betty and John don't agree on where/what store to open, they've divided the decision into two parts. Betty will choose the type of restaurant and John the location. The payoff matrix is presented below. |Strip Mall 1,0 2, 1 |0, 4 Arts District |4, 4 5, 0 3, 5 Business District Steak house Italian Thai |7, 3 |1, 0 6, 9 What is the Nash-Equilibrium? a. (Italian, Strip Mall) b. (Tahi, Arts District) c. (Thai, Business District) d. (Steak house, Business District)
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- Two athletes of equal ability are competing for a prize of $10,000. Each is deciding whether to take a dangerous performance enhancing drug. If one athlete takes the drug, and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of X dollars. a) Draw a 2×2 payoff matrix describing the decisions the athletes face. b) For what X is taking the drug the Nash equilibrium? c) Does making the drug safer (that is, lowering X) make the athletes better or worse off? Explain.Two athletes of equal ability are competing for a prize of $12,000. Each is deciding whether to take a dangerous performance-enhancing drug. If one athlete takes the drug and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of XX dollars. Complete the following payoff matrix describing the decisions the athletes face. Enter Player One's payoff on the left in each situation, Player Two's on the right. Player Two's Decision Take Drug Don't Take Drug Player One's Decision Take Drug , , Don't Take Drug , , True or False: The Nash equilibrium is taking the drug if X is greater than $6,000. True False Suppose there was a way to make the drug safer (that is, have lower XX). Which of the following statements are true about the effects of making the drug safer? Check all that…There is a city, which looks like chopped isosceles triangle, as shown below. Citizens live uniformly distributed all over the city. Two ice-cream vendors, A and B, must independently set up stores in the city. Each citizen buys from the vendor closest to their location and when equidistant from both vendors they choose by coin toss. Each vendor’s aim is to maximize the expected number of customers. A choice of location by the two vendors is a Nash equilibrium if no vendor can do better by deviating unilaterally. Does this game have a Nash equilibrium? If so, describe it. If not, explain why not
- Two neighboring homeowners, i = 1,2, simultaneously choose how many hours to spend maintaining a lawn. The AVERAGE benefit per hour for i is (e.g., it is for homeowner 1) And the (opportunity) cost per hour for each homeowner is 4. (a) Give each homeowner’s (net) payoff as a function of and . (b) Compute the Nash equilibrium.Two friends are deciding where to go for dinner. There are three choices, which we label A, B, and C. Max prefers A to B to C. Sally prefers B to A to C. To decide which restaurant to go to, the friends adopt the following procedure: First, Max eliminates one of three choices. Then, Sally decides among the two remaining choices. Thus, Max has three strategies (eliminate A, eliminate B, and eliminate C). For each of those strategies, Sally has two choices (choose among the two remaining). a.Write down the extensive form (game tree) to represent this game. b.If Max acts non-strategically, and makes a decision in the first period to eliminate his least desirable choice, what will the final decision be? c.What is the subgame-perfect equilibrium of the above game? d. Does your answer in b. differ from your answer in c.? Explain why or why not. Only typed AnswerConsider the following game played by four individuals, players 1, 2, 3, and 4. Each individual has $10,000. Each player can donate between $0 and $10,000 to build a public park that costs $20,000. If they collect enough money, they construct the park, which is worth $9,000 to each of them. However, if they collect less than $20,000, they cannot build a park. Furthermore, regardless of whether the park is built or not, individuals lose any donations that they make. a) Describe the Nash equilibria for a simultaneous game. What makes them equilibria? Hint: There are many equilibria, so you may want to use a mathematical expression! b) Suppose that players 1, 2, and 3, each donate $4,000 for the park. How much will player 4 donate and why. What are the resulting payoffs for the players? c) Suppose instead that player 1 donated first, player 2 second, player 3 third, and player 4 last. Furthermore, players could only donate in intervals of 1,000 (0, $1,000, $2,000, etc.). How much will…
- Rachel, Monica, and Phoebe are roommates; each has 10 hours of free time you could spend cleaning your apartment. You all dislike cleaning, but you all like having a clean apartment: each person’s payoff is the total hours spent (by everyone) cleaning, minus a number 1/2 times the hours spent (individually) cleaning.That is, ui(s1, s2, s3) = s1 + s2 + s3 -1/2si Assume everyone chooses simultaneously how much time to spend cleaning. a. Find the Nash equilibrium. b. Find the Nash if the payoff for each player is: ui(s1, s2, s3) = s1 + s2 + s3 − 3si Is the Nash equilibrium Pareto efficient? If not, can you find an outcome in which everyone is better off than in the Nash equilibrium outcome?**Practice*** Amy and Bob are playing the following board game:(I) Amy starts. She has three possible actions: Pass, Attack, or Defend.(II) Bob observes what Amy chose, and then chooses between three actions with the same names: Pass, Attack, or Defend.(III) If either player passes, or one attacks and the other defends, then the game ends. But if either both players attack, or if both players defend, then Amy has to choose between two actions: Respond or Not Respond. The payoffs are as follows:- If both players pass, both players get a payoff of 0.- If a player attacks and the other player defends, the player that attacks gets a payoff of 1, while the player that defended gets a payoff of 2.- If a player passes but the other player attacks or defends, the player who passes gets a payoff of -1, and the player who attacked or defended gets a payoff of 3.- If both players attack or both players defend:– If Amy responds, she gets a payoff of 4, and Bob gets a payoff of 0.– If Amy does…You are given the payoff matrix below. B1 B2 B3 A1 1 Q 6 A2 P 5 10 A3 6 2 3 a. Determine the range of values of P and Q in order for Player A to choose strategy A2 and PlayerB to choose strategy B2.b. Solve the problem in the perspective of the opponent. Are the ranges the same?c. What are the values of the game for (a) and (b)?
- 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.Someone at a party pulls out a $100 bill and announces that he is going to auction it off. There are n=10 people at the partywho are potential bidders. The owner of the $100 bill puts forth the following procedure: All bidders simultaneously submit a written bid. Only the highest bidders pay their bid (assuming that the highest bid is positive). If m people submit the highest bid, then each receives 1/m of the $100. Each person’s strategy set is {0,1,2,...,1000}{0,1,2,...,1000} so bidding can go as high as $1,000. The payoff of a player bidding bi is:0 if bi < max{b1,b2,…,bn},and 100/m − bi if bi = max {b1,b2,…,bn}where,m is the number of bidders whose bid equals max{b1,...,bn}. How many pure-strategy Nash equilibria does this game have? 1) 0 2) 1 3) 4 4) More than 4.Suppose two bidders compete for a single indivisible item (e.g., a used car, a piece of art, etc.). We assume that bidder 1 values the item at $v1, and bidder 2 values the item at $v2. We assume that v1 > v2. In this problem we study a second price auction, which proceeds as follows. Each player i = 1, 2 simultaneously chooses a bid bi ≥ 0. The higher of the two bidders wins, and pays the second highest bid (in this case, the other player’s bid). In case of a tie, suppose the item goes to bidder 1. If a bidder does not win, their payoff is zero; if the bidder wins, their payoff is their value minus the second highest bid. a) Now suppose that player 1 bids b1 = v2 and player 2 bids b2 = v1, i.e., they both bid the value of the other player. (Note that in this case, player 2 is bidding above their value!) Show that this is a pure NE of the second price auction. (Note that in this pure NE the player with the lower value wins, while in the weak dominant strategy equilibrium where both…