Two individuals are in negotiations to terminate a contract. Person A, was the signatory to the control that she had to execute in partnership with Person B and is in possession of the pay- out. The contract is worth R1 000. The mediator asks Person A to choose how much of this contract value she is willing to offer person B. Person B, will then be asked to decide whether to accept the offer or reject it. If Person B rejects the offer, neither player receives any money. Using table or decision tree illustrate the possible payoffs. Determine if Nash equilibrium exist.
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- 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:…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.A game is played as follows: First Player 1 decides (Y or N) whether or not to play.If she chooses N, the game ends. If she chooses Y, then Player 2 decides (Y or N) whetheror not to play. If he chooses N the game ends. If he chooses Y, then they go ahead and playanother game with the payoffs shown below. A player who opts out by choosing N gets 2 andthe other player gets 0. Draw the tree of this game and then find the two subgame-perfect Nashequilibria.
- 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…The first player can choose either U or D. If he chooses U, the second player has a choice of two strategies: L and R. If the second player moves L he obtains 1 and the first player gets 5. If the second player chooses R he obtains 2 units of payoff while the first player receives 1. Following a move D by the first player, both players engage in a simultaneous-move “Bach or Stravinsky” game (as it was described in class). Find the SPE of this game and write it down in a mixed and behavior form.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
- 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.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?** Please be advsed that this is practice only from previous yeasr *** Answers: (a) There are no Nash equilibria.(b) There are two pure strategy Nash equilibra, one with (H,H) and another with (L,L), and no mixed strategy Nash equilibria.(c) There are two pure strategy Nash equilibra, one with (H,H) and another with (L,L), and one mixed strategy Nash equilibria with p = 1/2 and q = 1/2.(d) There are two pure strategy Nash equilibra, one with (H,H) and another with (L,L), and one mixed strategy Nash equilibria with p = 1/2 and q = 3/4.(e) There are two pure strategy Nash equilibra, one with (H,H) and another with (L,L), and one mixed strategy Nash equilibria with p = 3/4 and q = 1/2.
- Look again at Figure 21.12. It appears that the investor in panel b can't lose and the investor in panel a can' t win. Is that correct? Note: The solution should not be hand written.Consider the “trust game” discussed in class. The first player starts with a $100 endowment and chooses how much to give to the second player. The gift triples in value (i.e. if $20 is given, the second player receives $60). The second player then chooses how much to give back. The first player receives exactly how much is returned (i.e. if $40 is returned, the first player receives $40). The Nash equilibrium of the game is: Group of answer choices: -First player gives $100, second player returns nothing. -First player gives $50, second player returns $50. -First player gives $100, second player returns $300. -There is no Nash equilibrium of this game. -First player gives nothing, second player returns nothing.Find all NE of the stage game.(b) Consider a two-period game without discounting in which the stage game is played ineach period. Find all pure strategy SPNE.(c) What’s the min-max payoff of each player?(c1) Consider pure strategies only.(c2) Consider all strategies, including the mixed ones.(d) Now suppose the stage game is repeated infinitely many times. Use the Fudenberg-Maskin Folk theorem to find all possible values of payoff that can be supported as aSPNE.