Consider an extensive game. First, a firm from City 1 (Player 2) makes Betty (Player 1) a job offer. The offer promises an income y1. Then Betty decides whether to accept the offer. If the o§er is accepted, the payffs to Betty and the firm are (y1 - x1; 1- y1), where x1 is the house price in City 1. While Betty is contemplating over this o§er, she receives another job offer from a firm in City 2. This outside option promises an income of y2 and a house price x2. If Betty rejects Player 2ís offer and accepts the outside option, the payoffs to the two players are (y2 - x2; 0). If Betty rejects both offers, then the payo§s are (0; 0). Assume y1 > x1, y2 > x2, 0 < y1 < 1 and y2 - x2 + x1 <=1 .assume that Betty will accept an offer if she is indifferent from accepting and rejecting it. Do the following: (a) Draw the game tree. (b) Find the subgame perfect equilibrium (SPE) by specifying strategies used. (c) What is Bettyís payoff in the SPE? How does this payoff change respectively with x1, y2, and x2?

Consider an extensive game. First, a firm from City 1 (Player 2) makes Betty (Player 1) a job offer. The offer promises an income y1. Then Betty decides whether to accept the offer. If the o§er is accepted, the payffs to Betty and the firm are (y1 - x1; 1- y1), where x1 is the house price in City 1. While Betty is contemplating over this o§er, she receives another job offer from a firm in City 2. This outside option promises an income of y2 and a house price x2. If Betty rejects Player 2ís offer and accepts the outside option, the payoffs to the two players are (y2 - x2; 0). If Betty rejects both offers, then the payo§s are (0; 0). Assume y1 > x1, y2 > x2, 0 < y1 < 1 and y2 - x2 + x1 <=1 .assume that Betty will accept an offer if she is indifferent from accepting and rejecting it. Do the following: (a) Draw the game tree. (b) Find the subgame perfect equilibrium (SPE) by specifying strategies used. (c) What is Bettyís payoff in the SPE? How does this payoff change respectively with x1, y2, and x2?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

See similar textbooks

Similar questions

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:…
Prove that in the variation on the centipede game given in figure 14.5(b) the unique sequential equilibrium described is, in fact, the unique Nash equilibrium. (Hint: Take some presumed Nash equilibrium and suppose information set 2n+ 1 [for player 2] is the first unreached information set. Derive an immediate contradiction. Then suppose that node (2n) t is the first unreached information set and derive a contradiction that is one degree removed from immediate.)
For the operating systems game, let us now assume the intrinsic superiorityof Mac is not as great and that network effects are stronger for Windows.These modifications are reflected in different payoffs. Now, the payoff fromadopting Windows is 50 X w and from adopting Mac is 15 + 5 X m;n consumers are simultaneously deciding between Windows and Mac.a. Find all Nash equilibria.b. With these new payoffs, let us now suppose that a third option exists,which is to not buy either operating system; it has a payoff of 1,000.Consumers simultaneously decide among Windows, Mac, and nooperating system. Find all Nash equilibria.
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.
What is the payoff for player 1 in the normal form game below: O. 3 O. 1 O. 0 O. 4
Consider the location game we covered in Lecture 3. Now assume there arethree players (vendors). As we assumed in the lecture, consumers in each area choosethe closest vendor and if there are multiple closest vendors then these vendors receiveequal share of consumers in the area. Notice Si = {1, 2, 3, ...., 9} for i = 1, 2, 3. Here aresome examples of payoffs: u1(1, 1, 1) = 3, u1(1, 1, 9) = u2(1, 1, 9) = 2.25, u3(1, 1, 9) =4.5, u1(1, 5, 9) = u3(1, 5, 9) = 2.5 and u2(1, 5, 9) = 4. (a) Is s′1 = 1 strictly dominated by s′′1 = 2 for player 1?(b) Is s′1 = 1 weakly dominated by s′′1 = 2 for player 1?(c) Can you find a Nash equilibrium in pure strategies?
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…
4 Consider an extensive game where player 1 starts with choosing of two actions, A or B. Player 2 observes player 1’s move and makes her move; if the move by player 1 is A, then player 2 can take three actions, X, Y or Z, if the move by player 1 is B, then player 2 can take of of two actions, U or V. Write down all teminal histories, proper subhistories, the player function and strategies of players in this game.
Consider the strategic voting game discussed at the endof this chapter, where we saw that the strategy profile (Bustamante, Schwarzenegger,Schwarzenegger) is a Nash equilibrium of the game. Show that (Bustamante, Schwarzeneg-ger, Schwarzenegger) is, in fact , the only rationalizable strategy profile. Do this by firstconsidering the dominated strategies of player L. (Basically, the question is asking youto find the outcome of the iterative elimination of strictly dominated strategies)
Consider a modified Traveler’s Dilemma. In terms of strategy options that the players have and the dollars they earn, it is like the standard Traveler’s Dilemma, but the players do not have endless appetite for money. Up to 100 dollars, each dollar feels like a dollar. But any moneybeyond 100 is psychologically like 100 dollars. Assuming that players are maximizers of ‘psychological’ dollars instead of real dollars, describe all the Nash equilibria of this modified Traveler’s Dilemma.
Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.
answer the ff: Suppose that each company cancharge either a high price for tickets or a low price. Ifone company charges $300, it earns low profit if theother company also charges $300 and high profit ifthe other company charges $600. On the other hand,if the company charges $600, it earns very low profit ifthe other company charges $300 and medium profitif the other company also charges $600.a. Draw the decision box for this game.b. What is the Nash equilibrium in this game?Explain.c. Is there an outcome that would be better than theNash equilibrium for both airlines? How could itbe achieved? Who would lose if it were achieved?