Suppose Tom, Mary and Jill have apartments in the same building, and installing a security system for their building costs $75O. Tom is willing to pay $100 for the security system, Mary is willing to pay $300 and Jill is willing to pay $300. If the three of them hold a vote on whether to install the system and split the cost equally, how will the vote go? O Nobody would vote yes Only one would vote yes Only two would vote yes All three would vote yes
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- You and your friend will divide $4. You have agreed to use the following procedure.Each of you will name a number of dollars, either $0, $1, $2, $3, or $4. You will chooseyour numbers simultaneously. If the sum of the amounts is less than or equal to $4, theneach of you receives the amount you named and the rest of the money is thrown away.If the sum of the amounts is greater than $4 and the amounts named are different, thenthe person who named the smaller amount receives that amount and the other personreceives the remaining money. If the sum of the amounts is greater than $4 and theamounts named are the same, then each receives $2. (a)Draw the payoff matrix of this game. Let “you” be the row player and “yourfriend” be the column player.(b) Derive the best reply functions of all players.(c) Find the Nash equilibrium (or all of the equilibria) of this game using thebest reply functions you found in part (a).8. Two states, A and B, have signed an arms-control agreement. This agreementcommits them to refrain from building certain types of weapons. The agreement is supposed tohold for an indefinite length of time. However, A and B remain potential enemies who wouldprefer to be able to cheat and build more weapons than the other. The payoff table for A (player1, the row player) and B (player 2, the column player) in each period after signing thisagreement is below. a) First assume that each state uses Tit-for-Tat (TFT) as a strategy in this repeated game.The rate of return is r. For what values of r would it be worth it for player A to cheat bybuilding additional weapons just once against TFT? b) For what values of r would it be worth deviating from the agreement forever to buildweapons? c) Convert both values you found in parts a and b to the equivalent discount factor dusing the formula given in lecture and section. d) Use the answers you find to discuss the relationship between d and r:…Five roommates arc planning to spend the weekendin their dorm room watching movies, and they arcdebating how many movies to watch. Here is theirwillingness to pay:Quentin Spike Ridley Martin StevenFirst film $14 $10 $8 $4 $2Second film 12 8 4 2 0Third film 10 6 2 0 0Fourth film 6 2 0 0 0Filth film 2 0 0 0 0Buying a DVD costs SIS, which the roommates splitequally, so each pays $3 per movie.a. What is the efficient number of movies to watch(that is, the number that maximizes total surplus)?b. from the standpoint of each roommate, what isthe preferred number of movies?c. What is the preference of the median roommate?d. (f the roommates held a vote on the efficient out~come versus the median voter's preference, howwould each person vote? Which outcome wouldg~ a majority?c. lf one of the roommates proposed a differentnumber of movies, could his proposal beat thewinner from part (d) in a vote?f. Can majority rule be counted on to reach efficientoutcomes in the provision of public go<Xis?
- 53. The Indiana University basketball team trails by twopoints with eight seconds to go and has the ball. Shouldit attempt a two-point shot or a three-point shot?Assume that the Indiana shot will end the game and thatno foul will occur on the shot. Assume that a three-pointshot has a 30% chance of success, and atwo-point shot has a 55% chance of success. Finally,assume that Indiana has a 55% chance of winning inovertime.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…1.4. Suppose you are against one of two alternatives but 90% of theelectorate disagrees with your position and favors that option. Is there avoting method that is anonymous, neutral, and monotone that preventsthat option from being selected as the winning alternative?
- you and a friend decide to run a three mile race. If you agree to run together, you keep up with himfor the first mile, but you overexert yourself and run the last two miles at slower paces on your own. Tomake up for lost time, your friend runs the last two miles at a faster pace. Your mile times are 6:30, 7:00,and 7:30. Your friend’s times are 6:30, 6:00, and 6:00. If you both agree to run on your own, you run aconstant pace of 7:05 while your friend runs at a constant pace of 6:05. If you want to run together butyour friend wants to run solo, he runs his constant pace of 6:05. You, on the other hand, want to showhim that you can run faster, but you end up overexerting yourself after the first mile. You run times of6:20, 7:05, and 7:30. If he wants to run together but you do not, you both run at your pace of 7:05. Thissituation can be turned into an economic game, with the payoffs the overall race times. You each wantto run the fastest time you possibly can.(a) Who are the players in…V4. Assume there are only two oil producing countries in the world, C1 and C2. Each can export either 2 million or 4 million barrels of oil. If a total of 4 million barrels of oil are exported (both countries combined) then each barrel sells at $25. If there are 6 million barrels exported between them, each barrel sells at $15 and if there are 8 million barrels exported between them, then each barrel sells at $10. a) Write the “pay-off” matrix/table (i.e., the table or matrix indicating the strategies for each country, i.e., the amount of barrels they export and the revenue they make). b) Determine the Nash equilibrium for the game. c) Is there a strategy that results in larger revenue for both the countries? If so, which is that? d) For what wegitage for future payoff, δ, will both countries agree to use the strategy that benefits both of them, better?Consider a situation of after-match penalty shoot-out. The striker can target either East or West side of the goal. If he targets West, with 80% chance he shoots on target. If he shoots East, he is accurate with 75%. The goalkeeper has to choose the corner to jump to. If he does not guess the corner correctly and the shot is on target, then the striker scores. If the striker shoots West, the shot is on target and the goalkeeperjumps West, then with 75% chance he saves the goal. If the striker shoots East, the shot is on target and the goalkeeper jumps East, then with chance of 2/3 he saves the goal. Suppose, that it is a zero-sum game and if the striker scores his payoff is 1, otherwise it is 0.1. Formulate this situation as a strategic game and Find all Nash equilibria of the game.
- Five roommates are planning to spend the weekendin their apartment watching movies, and they aredebating how many movies to watch. The tablebelow shows each roommate’s willingness to pay foreach of the movies:Ava Ridley Spike Kathryn QuentinFirst film $14 $10 $8 $4 $2Second film 12 8 4 2 0Third film 10 6 2 0 0Fourth film 6 2 0 0 0Fifth film 2 0 0 0 0A movie on their streaming service costs $15, which theroommates split equally, so each pays $3 per movie.a. What is the efficient number of movies to watch(that is, the number that maximizes total surplus)?b. For each roommate, what is the preferred numberof movies to watch?The New York City Council is ready to vote on two billsthat authorize the construction of new roads in Manhattanand Brooklyn. If the two boroughs join forces, they can passboth bills, but neither borough by itself has enough powerto pass a bill. If a bill is passed, then it will cost the taxpayersof each borough $1 million, but if roads are built in aborough, the benefits to the borough are estimated to be $10million. The council votes on both bills simultaneously, andeach councilperson must vote on the bills without knowinghow anybody else will vote. Assuming that each boroughsupports its own bill, determine whether this game has anyequilibrium points. Is this game analogous to the Prisoner’sDilemma? Explain why or why not.64. (This problem assumes knowledge of the basic rulesof baseball.) George Lindsey (1959) looked at boxscores of more than 1000 baseball games and foundthe expected number of runs scored in an inning foreach on-base and out situation to be as listed in the fileP09_64.xlsx. For example, if a team has a man on firstbase with one out, it scores 0.5 run on average untilthe end of the inning. You can assume throughout thisproblem that the team batting wants to maximize theexpected number of runs scored in the inning.a. Use this data to explain why, in most cases,bunting with a man on first base and no outs isa bad decision. In what situation might buntingwith a man on first base and no outs be a gooddecision?b. Assume there is a man on first base with one out.What probability of stealing second makes an attempted steal a good idea?