1. The Dean of Columbia College is looking for a tenured professor to teach in the Core Curriculum. Monetary incentives are needed to get someone interested, but how much? The Dean decides to use an auction to do the job. Two professors, equally qualified, applied for the position. The two professors are invited to covertly submit their bids to the Dean. The Dean will give the position to the professor who submits the lower bid (if there is a tie, the job is assigned randomly). The professor who gets the job will be paid his/her own bid. Each professor's reservation value for teaching the course is his/her private information. It is common knowledge that their reservation values are independently and uniformly distributed over [0, 100]. So if a professor with a reservation value of 50 wins with a bid of 60, his payoff is 60 – 50 = 10.' (a) Find a Bayesian Nash equilibrium of the bidding game. (b) Suppose the two professors' reservation values are 60 and 70, respec- tively. What are their bids in the Bayesian Nash equilibrium you computed in part (a)? Who is the winner? What's the payoff of the

1. The Dean of Columbia College is looking for a tenured professor to teach in the Core Curriculum. Monetary incentives are needed to get someone interested, but how much? The Dean decides to use an auction to do the job. Two professors, equally qualified, applied for the position. The two professors are invited to covertly submit their bids to the Dean. The Dean will give the position to the professor who submits the lower bid (if there is a tie, the job is assigned randomly). The professor who gets the job will be paid his/her own bid. Each professor's reservation value for teaching the course is his/her private information. It is common knowledge that their reservation values are independently and uniformly distributed over [0, 100]. So if a professor with a reservation value of 50 wins with a bid of 60, his payoff is 60 – 50 = 10.' (a) Find a Bayesian Nash equilibrium of the bidding game. (b) Suppose the two professors' reservation values are 60 and 70, respec- tively. What are their bids in the Bayesian Nash equilibrium you computed in part (a)? Who is the winner? What's the payoff of the

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.11P

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The Dean of a College is looking for a tenured professor to teachin the Core Curriculum. Monetary incentives are needed to get someoneinterested, but how much? The Dean decides to use an auction to do thejob. Two professors, equally qualiÖed, applied for the position. The twoprofessors are invited to covertly submit their bids to the Dean. The Deanwill give the position to the professor who submits the lower bid (if thereis a tie, the job is assigned randomly). The professor who gets the job willbe paid his/her own bid. Each professorís reservation value for teachingthe course is his/her private information. It is common knowledge thattheir reservation values are independently and uniformly distributed over[0;100]: So if a professor with a reservation value of 60 wins with a bid of50, his payoff is 60- 50 = 10: (A) Find a Bayesian Nash equilibrium of the bidding game.(B) Suppose the two professorsíreservation values are 60 and 70, respectively. What are their bids in the Bayesian Nash…
Choice under uncertainty. Consider a coin-toss game in which the player gets $30 if they win, and $5 if they lose. The probability of winning is 50%. (a) Alan is (just) willing to pay $15 to play this game. What is Alan’s attitude to risk? Show your work. (b) Assume a market with many identical Alans, who are all forced to pay $15 to play this coin-toss game. An insurer offers an insurance policy to protect the Alans from the risk. What would be the fair (zero profit) premium on this policy? can you help me for par (b) plase?
Choice under uncertainty. Consider a coin-toss game in which the player gets $30 if they win, and $5 if they lose. The probability of winning is 50%. (a) Alan is (just) willing to pay $15 to play this game. What is Alan’s attitude to risk? Show your work.(b) Assume a market with many identical Alans, who are all forced to pay $15 to play this coin-toss game. An insurer offers an insurance policy to protect the Alans from the risk. What would be the fair (zero profit) premium on this policy? i need help with question B please.
5) Three legislators are set to vote on a bill to raise the salary of legislators. The majority wins, so all three will receive the raise if at least two of them vote in favor of the bill. The raise is valued at R by each legislator. Voting in favor of the bill comes with political backlash from constituents, though, even if the bill fails. Let C be the cost of backlash for anyone voting in favor of the bill. Finally, suppose that 0 < C < R. There are four possible payoffs for each legislator: 0: if they vote against the bill and at least one other legislator votes against it (so the bill fails) R: if they vote against the bill and the others vote for the bill (so the bill passes) -C: if they vote for the bill and no one else votes for the bill (so the bill fails) R-C: if they vote for the bill and at least one other legislator votes for it (so the bill passes). The three legislators are named X, Y, and Z, and voting happens sequentially and orally. So X announces their vote (to…
10 Use the expected value information to illustrate how having more bidders in an oral auction will likely result in a higher winning bid.
Suppose that the University of Alabama and Clemson are making spending decisions for theupcoming year. Assume that Alabama is currently spending $15 million on their recruiting andfacilities, and Clemson is spending $10 million. Each team has an additional $5 million to spendor keep as profits. If they both choose to not spend the additional $5 million then Alabama hasa 60% chance of getting the highest quality quarterback recruit to commit to them (getting thecommitment of the player is the goal). However, if they both choose to spend the additional $5million then there is a 57% chance that Alabama gets the high quality quarterback to commit. IfAlabama spends the additional $5 million but Clemson doesn’t then there is a 67% chanceAlabama gets the recruit. However, if Alabama does NOT spend the additional $5million butClemson does then there is a 50% change either team gets the recruit’s commitment. Setup thepayoff matrix and label the players, their strategies, and their payoffs, and…
Micro Nash game theory Consider the game matching pennies with an external option u for player 1. Suppose that u in ]0, 1[. a. Calculate in a very complete form the best response apps for each player. b. Give the set of Nash equilibria for the game. please I need a formal answer, not the one already posted.
Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.
5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:A writes down either number 1 or number 2, and B must guess which one.If the number that A has written down is i and B has guessed correctly, B receives i units from A.If B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.Suppose B guesses 1 with probability p and 2 with probability 1 −p.Let X1 denote B’s gain (or loss) in a game where A has written down 1.Let X2 denote B’s gain (or loss) in a game where A has written down 2.(a) Find the pmf of X1 and X2(b) Find B’s expected gain for these two cases, E[X1] and E[X2].(c) What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player ASuppose that A writes down 1 with probability q and 2 with probability 1 −q.Let Y1 be A’s loss (or gain) if B chooses number 1.Let Y2 be A’s loss (or gain) if B…
please discuss the separation eqilibrium in details 1. Assume that two players play the Tinder game. There are two types of potential boyfriends: the gentleman (25% of the population) and the douchebag (75% of the population), and they can decide on the different level of effort: low, medium, and high. For the first type, the respective costs are 30, 50, and 80. For the second type, the respective costs are 50, 75, and 120. For the potential boyfriend, the payoff from being in relationship is 130, and the payoff associated with being dumped is 70. The girl can get 60 if she stays single, 100 if she is in relationship with the gentleman, and 25 if she dates the douchebag (credits to Owen Sims) a) Present the game in the extensive form b) Calculate the expected pay-off for the girl. Should she date at all? c) Discuss whether the game has a separating equilibrium
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…
Consider a game where each player picks a number from 0 to 60. The guess that is closest to half of the average of the chosen numbers wins a prize. If several people are equally close, then they share the prize. The game theory implies that (A) all players have dominant strategies to choose 0 (B) all players have dominant strategies to choose 30 (C) there is a Nash equilibrium where all players pick 0 (D) there is a Nash equilibrium where all players pick positive numbers Behavioral data in such games suggests that (A) most subjects choose 0; (B) most subjects choose 30;(C) common answers include 30, 15, 7.5, and 0; (D) most subjects use randomization.