Three voters vote over two candidates (A and B), and each voter has two pure strategies: vote for A and vote for B. When A wins, voter 1 gets a payoff of 1, and 2 and 3 get payoffs of 0; when B wins, 1 gets 0 and 2 and 3 get 1. Thus, 1 prefers A, and 2 and 3 prefer B. The candidate getting 2 or more votes is the winner (majority rule). Find a weakly dominant strategy. Compute for the pure strategy Nash equilibrium.

Three voters vote over two candidates (A and B), and each voter has two pure strategies: vote for A and vote for B. When A wins, voter 1 gets a payoff of 1, and 2 and 3 get payoffs of 0; when B wins, 1 gets 0 and 2 and 3 get 1. Thus, 1 prefers A, and 2 and 3 prefer B. The candidate getting 2 or more votes is the winner (majority rule). Find a weakly dominant strategy. Compute for the pure strategy Nash equilibrium.

Managerial Economics: A Problem Solving Approach

5th Edition

ISBN:9781337106665

Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Chapter18: Auctions

Section: Chapter Questions

Problem 9MC

See similar textbooks

Similar questions

Consider the following voting game. There are three players, 1, 2 and 3. And there are three alternatives: A, B and C. Players vote simultaneously for an alternative. Abstaining is not allowed. Thus, the strategy space for each player is {A, B, C}. The alternative with the most votes wins. If no alternative receives a majority, then alternative A is selected. Denote ui(d) the utility obtained by player i if alternave d {A, B, C} is selected. The payoff functions are, u1 (A) = u2 (B) = u3 (C) = 2 u1 (B) = u2 (C) = u3 (A) = 1 u1 (C) = u2 (A) = u3 (B) = 0 a. Let us denote by (i, j, k) a profile of pure strategies where player 1’s strategy is (to vote for) i, player 2’s strategy is j and player 3’s strategy is k. Show that the pure strategy profiles (A,A,A) and (A,B,A) are both Nash equilibria. b. Is (A,A,B) a Nash equilibrium? Comment.
5.Each of Player 1 and Player 2 chooses an integer from the set {1, 2, ..., K}. If they choose the same integer, P1 gets +1 and P2 gets -1; if they choose different integers, P1 gets -1 and P2 gets +1. (a) Show that it is a NE for each player to choose every integer in {1, 2, ..., K} with equal probability, K1 . (b) Show that there are no NE besides the one you found in (a).
We’ll now show how a college degree can get you a better job even if itdoesn’t make you a better worker. Consider a two-player game between aprospective employee, whom we’ll refer to as the applicant, and an employer. The applicant’s type is her intellect, which may be low, moderate,or high, with probability 1/3 , 1/2 , and 1/6 , respectively. After the applicantlearns her type, she decides whether or not to go to college. The personalcost in gaining a college degree is higher when the applicant is less intelligent, because a less smart student has to work harder if she is to graduate. Assume that the cost of gaining a college degree is 2, 4, and 6 for an applicant who is of high, moderate, and low intelligence, respectively.The employer decides whether to offer the applicant a job as a manageror as a clerk. The applicant’s payoff to being hired as a manager is 15,while the payoff to being a clerk is 10. These payoffs are independent ofthe applicant’s type. The employer’s payoff from…
Two bidders compete in a second price auction (i.e., the winning bidder pays the losing bidder’s bid, and the losing bidder does not pay anything). They submit sealed bids, and the one with the highest bid wins the contract and pays the other bidder’s bid. Each bidder i’s private valuation is vi and is distributed independently and uniformly between 0 and 50. 1. For any given bidder, prove that he has a dominant strategy bid and show what it is. 2. Assuming each bidder bids his dominant strategy noted above, if a bidder with vi = 40 wins, what price does he expect to pay?
True/False a. Consider a strategic game, in which player i has two actions, a and b. Let s−i be some strategy profile of her opponents. If a IS a best response to s−i, then b is NOT a best response to s−i. b. Consider the same game in (a). If a IS NOT a best response to s−i, then a does NOT weakly dominates b. c. Consider the same game in (a). If a mixed strategy of i that assigns probabilities 13 and 23 to a and b, respectively, IS a best response to s−i, SO IS a mixed strategy that assigns probabilities 32 and 13 to a and b, respectively. d. Consider the same game in (a). If a mixed strategy of i that assigns probabilities 13 and 23 to a and b, respectively, is NOT a best response to some strategy profile of her opponents, s−i, NEITHER is a mixed strategy that assigns probabilities 32 and 13 to a and b, respectively. e. Consider the same game in (a). If a IS a best response to s−i, SO IS any mixed strategy that assigns positive probability to a. f. Consider the same game in (a). If a…
Consider two bidders – Alice and Bob who are bidding for a second-hand car. Each of them knows the private value she/he assigns to the car, but does not know the exact value of others. It is common knowledge that the value of other bidders is randomly drawn from a uniform distribution between 0 and $10000. Assume that Alice values the car at $8500 and Bob values the car at $4500. a) If Alice and Bob participated in the second-price sealed bid auction, what would they bid and what would be the result of the auction? Explain your answer. b) If they participate instead in a first-price sealed bid auction, what would they bid and what would be the result of the auction? Explain your answer. c) Calculate and compare the revenue of the seller in the above situations. Which type of auction should the seller use? Explain your answer
There is an election with three candidates: Lady Left (LL). Care Central (CC), and Robert Right (RR). The following are the outcomes in various two- and three-way races: CC. 100%. RR, 0% /LL, 0%. CC/ 100% RR, 51%, LL, 49% RR, 35%. LU 33% CC 329 The voting system is such that first there is a three way race. Then, if none of the three candidates wins more than 50% of the votes, there is a two-way runoff election between the two can Is this voting system consistent? Explain which property of consistency is violated, if any. Short answer
Mohamed and Kate each pick an integer number between 1 and 3 (inclusive). They make their choices sequentially.Mohamed is the first player and Kate the second player. If they pick the same number each receives a payoff equal to the number they named. If they pick a different number, they get nothing. What is the SPE of the game? a. Mohamed chooses 3 and Kate is indifferent between 1, 2 and 3. b. Mohamed chooses 3 and Kate chooses 1 if Mohamed chooses 1, 2 if Mohamed chooses 2, and 3 if Mohamed chooses 3. c. Mohamed chooses 1 and Kate chooses 1 if Mohamed chooses 1, 2 if Mohamed chooses 2 and 3 if Mohamed chooses 3. d. Mohamed chooses 3 and Kate chooses 3.
There are N>=2 collectors who engage in the auction of an antique. The collectorshave a common valuation of the antique, denoted by v, which is known to all. Thecollectors make a simultaneous bid. Let pn denote the bid by collector n = 1,....,N. The one with the highest bid wins the antique. The winner receives payoff v-pi.The other(s) receive zero payoff. If more than one collectors make the same highestbid, then they have an equal chance of winning the item. Prove that: A) It is not a Nash Equilibrium (NE) if the highest bid is v and onlyone collector bids this price.(b) It is not a NE if the highest bid is less than v.(c) It is a NE that the highest bid is v and more than one collector bidsthis price
You and a rival are engaged in a game in which there are three possible outcomes: you win, your rival wins (you lose), or the two of you tie. You get a payoff of 50 if you win, a payoff of 20 if you tie, and a payoff of 0 if you lose. What is your expected payoff in each of the following situations? (a) There is a 50% chance that the game ends in a tie, but only a 10% chance that you win. (There is thus a 40% chance that you lose.) (b) There is a 50–50 chance that you win or lose. There are no ties. (c) There is an 80% chance that you lose, a 10% chance that you win, and a 10% chance that you tie.
Consider the following ‘war of attrition’. Two animals are in a stand off for a prey. Theyindependently decide when to give up. Waiting is costly, but the animal giving up last winsthe prey (they each get nothing if they walk away at the exact same time). Getting the preygives a benefit of 80 while waiting costs 2 per unit of time. Formally payoffs are given asfollows:u1(t1, t2) =(−2t1 if t1 ≤ t280 − 2t2 if t1 > t2u2(t1, t2) =(80 − 2t1 if t1 < t2−2t2 if t1 ≥ t2,where ti is the amount of time animal i decided to wait. Assuming that animals aim tomaximize payoffs (consciously or not), figure out the Nash equilibria of this game by answeringthe following questions (similar to how we proceeded to solve the Bertrand game).(a) Show that there is no Nash equilibrium where both animals wait a strictly positiveamount of time. For this, consider two subcases: (i) both wait the same amount oftime, or (ii) one gives in earlier than the other.(b) Assume now that one animal, say the first one,…
Each of the two players independently (and simultaneously with the other) decides whether to go to a play or a concert. Each would rather go with the other to a concert than with them to a play, but prefers this to not being together, in which case they don't care where they go alone. Additionally, each is indifferent between attending the play together and participating in a lottery where both go to the concert with a probability of ¾ and to different events with a probability of ¼. Describe the game in matrix form and find all its equilibria under the assumption that the players have von Neumann-Morgenstern preferences.