Three friends Linda, Stella, and Lea-are deciding where to go together for vacation. They all agree that they should go one of the three places: Greece, Mexico, or Hawaii. They also agree that they will have two pairwise votes to determine where to go on vacation, with the majority determining the outcome on each vote. The first, second, and third choices for each person are as indicated in the following table. First choice Second choice Third choice Linda Greece Mexico Hawaii Stella Mexico Hawaii Greece Lea Hawaii Greece Mexico Refer to Table 22-1. Depending on the order of the pairwise voting, the friends could go to either

Three friends Linda, Stella, and Lea-are deciding where to go together for vacation. They all agree that they should go one of the three places: Greece, Mexico, or Hawaii. They also agree that they will have two pairwise votes to determine where to go on vacation, with the majority determining the outcome on each vote. The first, second, and third choices for each person are as indicated in the following table. First choice Second choice Third choice Linda Greece Mexico Hawaii Stella Mexico Hawaii Greece Lea Hawaii Greece Mexico Refer to Table 22-1. Depending on the order of the pairwise voting, the friends could go to either

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Male antelopes (lopes) fight for the right to mate. Whenever two lopes encounter each other, they first test each other out by sparring. If they both back off, there’s an 80% chance that they will go on and mate. If they fight to the death, there’s a 40% chance for each that they will survive to mate. If one wins and the other runs off, the winner has a 100% chance of mating and the one who leaves has a 70% chance of mating. a. What are the Nash equilibria in this game? b. Which equilibria are important for the species to continue to propagate?
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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?
A crime is witnessed by 3 citizens. Every citizen would like the police to be informed about the crime, but prefers that someone else reports it (filing a report is a hassle). Each of the three citizens chooses simultaneously (and independently) whether to call the police or not. When no one makes a call, every citizen receives payoff 0. If at least one citizen calls the police, citizens who call get payoff 5, and those who don't call get payoff 9. (a) Find all Nash equilibria in pure strategies. (b) Compute a symmetric Nash equilibrium in mixed strategies (i.e., an equilibrium in which every citizen calls with the same probability p E (0, 1)). (c) Does the game have a Nash equilibrium in pure or mixed strategies different from those you identified in (a) and (b)? If yes, construct one. If not, argue why not.
Rita is playing a game of chance in which she tosses a dart into a rotating dartboard with 8 equal-sized slices numbered 1 through 8. The dart lands on a numbered slice at random. This game is this: Rita tosses the dart once. She wins $1 if the dart lands in slice 1, $2 if the dart lands in slice 2, $5 if the dart lands in slice 3, and $8 if the dart lands in slice 4. She loses $3 if the dart lands in slices 5, 6, 7, or 8. (If necessary, consult a list of formulas.) (a) Find the expected value of playing the game. | dollars (b) What can Rita expect in the long run, after playing the game many times? O Rita can expect to gain money. She can expect to win dollars per toss. Rita can expect to lose money. She can expect to lose dollars per toss. O Rita can expect to break even (neither gain nor lose money).
Consider 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…
39 In the following sequential game, person 1 chooses first and person 2 chooses second. Using backward induction, what choice should person 1 make? chooses no if person 2 chooses no chooses yes if person 2 chooses yes chooses yes chooses no
From a certain place K, 2000 people set off for work in the distant place S in the morning by car. Each of the drivers must decide whether to come to place S via point G or via point D (see the directed graph below). The travel time on segments G-S and K-D is not dependent on the number of drivers: each driver traverses the G-S segment in 60 minutes, and the K-D segment in 45 minutes, regardless of the number of drivers. However, the travel time on segments K-G and D-S depends on the number of drivers: if x drivers travel on the K-G segment, each driver on this segment consumes 0.005x minutes; if x drivers travel on the D-S segment, each driver on this segment consumes 0.01x minutes (see the directed graph below). Assume that all people start simultaneously and do not know the decisions of other people. Find all pure Nash equilibria of the game.
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If the players play pure strategies, the game has no Nash equilibrium. But what if they choose their moves randomly? Let each player instead opt for a mixed strategy instead of a pure strategy. The first will play action Z with probability p, and the second will play action L with probability q. At which pair (p, q) are the mixed strategies of the players in equilibrium? At which pair (p, q) does neither player want to change strategy? When are both strategies simultaneously the best response?
We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does Jerry have any strictly dominant strategy? Does Elaine?
Economics Two players are bargaining over a three period bargaining model as discussed in class with player 1 making offers in rounds 1 and 3. Player 2 makes an offer in round 2 only. Each player has a common discount factor delta. The two players are bargaining to split $20. They have three time periods available to them for their bargaining game. At the end of round 3, if no agreement has been reached then player 1 receives $2 and player 2 receives $1 and the rest of the money is destroyed. Find the subgame perfect Nash equilibrium outcome in the finite horizon model in which the game ends after period 3.