Ex. 4 Strength Can Be WeaknessA three-person committee has to choose a winner for a prize. After some debate, thereare three candidates still under consideration. Let's call these candidates a, b and c, andcall those committee members 1, 2 and 3. The committee members only care about whichcandidate wins the prize, and their preferences as follows: member 1 prefers a to b andb to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. Therules of the competition say that the committee should first apply majority vote (secretballot, one member one vote) and the candidate with the most votes wins. If the vote istied, that is, the majority rule select a unique winning candidate, then the winner willbe the candidate for whom member 1 voted. Thus, it might seem that member 1 has anadvantage.(1) Write down the strategic form of this voting game. [You may assign any numberto the payoff of each voter, as long as it is consistent with her preference order.](2) For each voter, which strategies are weakly or strictly dominated? [You may wantto pay extra attention to voter 1.](3) Now consider the reduced game in which all weakly and strictly dominated strate-gies have been removed from the original game. For each voter, which strategies arenow weakly or strictly dominated? What is the outcome predicted by the processof iterated elimination of dominated strategies? Compare this outcome to voter l'spreferences and comment.

Dominated Strategy: In game theory, a strategy is said to be dominated if there exists another…

Ex. 4 Strength Can Be Weakness A three-person committee has to choose a winner for a prize. After some debate, there are three candidates still under consideration. Let's call these candidates a, b and c, and call those committee members 1, 2 and 3. The committee members only care about which candidate wins the prize, and their preferences as follows: member 1 prefers a to b and b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The rules of the competition say that the committee should first apply majority vote (secret ballot, one member one vote) and the candidate with the most votes wins. If the vote is

Ex. 4 Strength Can Be Weakness A three-person committee has to choose a winner for a prize. After some debate, there are three candidates still under consideration. Let's call these candidates a, b and c, and call those committee members 1, 2 and 3. The committee members only care about which candidate wins the prize, and their preferences as follows: member 1 prefers a to b and b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The rules of the competition say that the committee should first apply majority vote (secret ballot, one member one vote) and the candidate with the most votes wins. If the vote is

Principles of Economics, 7th Edition (MindTap Course List)

7th Edition

ISBN:9781285165875

Author:N. Gregory Mankiw

Publisher:N. Gregory Mankiw

Chapter22: Frontiers Of Microeconomics

Section: Chapter Questions

Problem 7PA

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Ex. 4 Strength Can Be Weakness A three-person committee has to choose a winner for a prize. After some debate, there are three candidates still under consideration. Let's call these candidates a, b and c, and call those committee members 1, 2 and 3. The committee members only care about which candidate wins the prize, and their preferences as follows: member 1 prefers a to b and b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The rules of the competition say that the committee should first apply majority vote (secret ballot, one member one vote) and the candidate with the most votes wins. If the vote is tied, that is, the majority rule select a unique winning candidate, then the winner will be the candidate for whom member 1 voted. Thus, it might seem that member 1 has an advantage. (1) Write down the strategic form of this voting game. [You may assign any number to the payoff of each voter, as long as it is consistent with her preference order.] (2)…
6 Two people will select a policy that affects both of them by applying a "veto" in a sequential and alternate manner, that is: person 1 begins to veto a policy and then person 2 exercises his "veto" with the remaining policies; the process repeats until only one policy remains. Assume that there are 3 policies: X,Y,Z, and that person 1 prefers X to Y to Z and person 2 prefers Z to Y to X. a. Represents the game extensively b. Give the number of subgames C. Indicate the total strategies of the players d. find all subgame perfect nash equilibria e. Find all Nash Equilibriums.
2. Individual Problems 15-2 Mr. and Mrs. Ward typically vote oppositely in elections and so their votes "cancel each other out." They each gain 20 units of utility from a vote for their positions (and lose 20 units of utility from a vote against their positions). However, the bother of actually voting costs each 10 units of utility. The following matrix summarizes the strategies for both Mr. Ward and Mrs. Ward. Mr. Ward Vote Vote Mr. Ward: -10, Mrs. Ward: -10 Don't Vote Mr. Ward: -20, Mrs. Ward: 10 The Nash equilibrium for this game is for Mr. Ward to payoff of O True Mrs. Ward O False Don't Vote Mr. Ward: 10, Mrs. Ward: -20 Mr. Ward: 0, Mrs. Ward: 0 units of utility and Mrs. Ward receives a payoff of Suppose Mr. and Mrs. Ward agreed not to vote in tomorrow's election. This agreement not to vote True or False: This agreement would increase utility for each spouse, compared to the Nash equilibrium from the previous part of the question. and for Mrs. Ward to units of utility. a Nash…
Using backward induction, solve the Vote Payoffs Game for Two Candidates. A challenger runs against an incumbent in a political party nomination. Each candidate can choose to either follow the party platform or deviate from the party platform. The incumbent chooses first. o If they both "follow" or both "deviate", the incumbent wins. Otherwise, the challenger wins. o The payoff for winning is 10. o A candidate that "deviates" gets a deduction of 5 in payoff. Draw the extensive form and solve by backward induction.
0 E F 3 2 FIGURE 2 23 D G 2 H 0 1 (2) Two friends, Albert (A) and Berta (B), have to decide on going to one of three bars, Xenia's (X), Yara's (Y), or Zana's (Z). They decide to select a bar by alternatively vetoing bars until one remains. First Albert vetoes a bar. If at least two bars remain, then Berta vetoes another bar. That process continues until a single bar remains unvetoed. Suppose Albert prefers Xenia's to Yara's to Zana's and Berta prefers Zana's to Yara's to Xenia's. Assume that, for each of the two, going to their most preferred bar gives a utility of 2, going to their next preferred bar gives a utility of 1, and going to their least preferred bar gives a utility of 0. Model this as an extensive form game and find its Nash equilibria. Which of the Nash equilibria are subgame perfect?
part G H I 4.7. The instant recall method works as follows. If there is a candidatewith the majority of the first-place votes, that candidate is declaredthe unique winner. Otherwise, the candidate with the plurality of thefirst-place votes is eliminated from the slate and the process is repeatedamong the remaining candidates. (Ignore the possibility that there maybe a tie for plurality winner.) The method is inspired by the recall elections that are permitted for California gubernatorial elections, which ineffect leave a system where plurality winners lacking majority supportare recallable, and so the ultimate holder of the governor’s office, if theelectorate so chooses, is the first candidate able to withstand a recall(the first candidate to have majority support) after prior plurality winners are eliminated (i.e., recalled). Arnold Schwarzenegger took office inCalifornia via a recall election.(a) Does instant recall satisfy anonymity?(b) Does instant recall satisfy neutrality?(c)…
In game theory, a dominant strategy is the best strategy to pick, no matter which moves are chosen by the other player. O to make the exact same move that was made by the other player. the choice that causes the payoff for the other player to be minimized, regardless of the payoff it earns the best strategy to pick, assuming the other player makes his or her best possible choice. to allocate all personnel resources towards defensive talent in order to dominate opposing offenses. 46°F
For the simple game depicted below, choose the corresponding voting rule that it represents: 100 BLOCKING WINNING 50 LOSING BLOCKING 250 500 Lower House A bill requires a 3/4 majority in both houses. O The Constitution can be amended with 3/4 of the LH votes and 1/2 of the Senate votes O A bill requires a majority in the LH and a majority in the Senate. O The Constitution can be amended with 3/4 of the Senate and 1/2 of the LH votes. Senate
2. Consider a game that game theory people refer to as the “ultimatum game.”We will refer to our two players as the “offerer” and the “decider”. How the gameworks is that the offerer proposes a way to split $1000 between the two players.While this could be done in a variety of ways, we will assume that the offerersonly has two possible proposals: Either a 50-50 split, or she offers the decider$50 and keeps the rest. The decider can either accept or reject the offer. If the offer is accepted, the money is split as proposed. If the offer is rejected, themoney spontaneously combusts and nobody gets anything. a) List the strategies for each player and write an extensive form version of thegame with payouts. b) List all the Nash equilibria of this game. c) Explain which, if any of the Nash equilibrium are not sub-game perfect. d) Write the game out in normal form and find the pure strategy Nashequilibrium. Explain how this matches with your answers to (b) and (c) . Alsoexplain why there…
Two politicians from the same party decide whether to support or oppose the gov- ernment's environmental policy. Politician A prefers that they both support, but Politician B prefers that they both oppose. Both politicians agree that the party should have a coherent position and the worst outcome is that one supports and the other opposes the policy. (a) Formulate this situation as a strategic game - specify the players, actions, and payoffs (you can set-up a table of the actions and payoffs). (b) What are the Nash equilibria of this game? Explain your answer.
3. Suppose that the various Balls of basketball or internet fame Lavar, Lamelo, Liangelo, Lonzo,and Spalding are considering playing in a 1-on-1 tournament. They get a payoff of 0 if theydecide not to participate. The following table illustrates their payoffs across the variousscenarios. 1 Player 2 Players 3 Players 4 Players 5 Players Lavar 10 4 2 -2 -5 Lamelo 10 7 5 2 -1 Liangelo 10 7 5 2 -1 Lonzo 10 8 7 6 5 Spalding 10 9 8 7 6 a. Find and describe all pure strategy Nash Equilibria.b. Suppose now Spalding, since it is in fact a literal basketball, gets a payoff of 0 in anycircumstance, being an inanimate object. The table is now as follows: 1 Player 2 Players 3 Players 4Players 5 Players Lavar 10 4 2 -2 -5 Lamelo 10 7 5 2 -1 Liangelo 10 7 5 2 -1 Lonzo 10 8 7 6 5 Spalding 0 0 0 0 0 Find and describe all pure strategy Nash Equilibria.
All societies must take decisions on collective resources should be allo- cated. In democratic societies such decisions are often taken by majority voting, as a matter of fact democracy is often associated with majority voting. However, it has for a long time been known that this way of taking collective decisions is associated with certain problems. (a) Assume that a small city should decide whether they should build a new public library (x), a new public swimming pool (y), or improving the street lightning along the city's main street (z). Only one of these alternative can be started during the next budget year. The city population can be divided into three equally large groups, the members of each group has the following preference ordering for these alternatives: 1:x > y > z, 2: y > z> x, 3:z > x> y.