Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak; she assigns probability a to person 2's being strong. Person 2 is fully informed. Each person can either fight or yield. Each person's preferences are represented by the expected value of a Bernoulli payoff function that assigns the payoff of 0 if she yields (regardless of the other person's action) and a payoff of 1 if she fights and her opponent yields; if both people fight then their payoffs are (-1, 1) if person 2 is strong and (1, -1) if person 2 is weak. Formulate this situation as a Bayesian game and find its Nash equilibria if a < 1/2 and if a > 1/2. "
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- Charles is participating in an experiment. His payoff in the experiment is tied to his effort e doing a mundane task. There is also some risk involved by design-there is a chance p that Charles is going to get a fixed payment L regardless of his effort. Charles' payoff is thus: with probability p w.e with probability 1- p Charles has to pay a cost C, which increases with his effort. First, let us assume that Charles' utility is the expected payoff net of this cost: U(e) = pL + (1 – p)we – c(e) Derive the first order condition with respect to e. b. How doesp affect Charles' effort e? c. How does L affect e?Consider the St. Petersburg Paradox problem first discussed by Daniel Bernoulli in 1738. The game consists of tossing a coin. The player gets a payoff of 2^n where n is the number of times the coin is tossed to get the first head. So, if the sequence of tosses yields TTTH, you get a payoff of 2^4 this payoff occurs with probability (1/2^4). Compute the expected value of playing this game. Next, assume that utility U is a function of wealth X given by U = X.5 and that X = $1,000,000. In this part of the question, assume that the game ends if the first head has not occurred after 40 tosses of the coin. In that case, the payoff is 240 and the game is over. What is the expected payout of this game? Finally, what is the most you would pay to play the game if you require that your expected utility after playing the game must be equal to your utility before playing the game? Use the Goal Seek function (found in Data, What-If Analysis) in Excel.In the signaling game represented below, there are two types of Player 1, smart and dumb, the probabilities of which are 0.4 and 0.6, respectively. Player 1 is in college and can either ((Q)uit or (G)raduate. Player 2 is a prospective employer and can either (N)ot hire or (H)ire Player 1. Player 2's payoff does not depend upon l's education, only her intelligence. Player 1's payoff depends partly on her education: both types benefit from completing their education, but the smart type gets more out of it. Player l's payoff also depends on 2's hiring decision: the smart type wants a job but the weak type does not. 0,0 1, 1 2, 1 0,0 N H N H 2 Q Q .4 C .6 18 G G N H H 2,0 3,1 3, 1 1,0 (a) Find a separating PBE. (b) Find a pooling PBE. (c) , Find an equilibrium in which one type of player 1 mixes, playing both Q and G with positive probability.
- 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…The Dean is looking for a tenured professor to teacha course. Monetary incentives are needed to get someoneinterested, but how much? The Dean decides to use an auction to do thejob. Two professors, equally qualified, 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 50 wins witha bid of 60, his payoff is 60 - 50 = 10. Please answer part b). (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…Boris and Priti have to work together on a project as agents. Ursula, the principal, will get a payoff of 12 if they complete it successfully but she can only observe the outcome not how much effort Boris and Priti are putting in. Boris and Priti can choose either to work or shirk where working costs 4 and shirking nothing. The project will be successful only if both Boris and Priti work, otherwise it fails. The payoff that Ursula offers for successful project completion is R, which is split equally between Boris and Priti. (a) Why does it matter that Ursula cannot observe the effort put in by Boris and Priti? What outcome do you expect to be reached if Ursuala cannot observe what Boris and Priti do? (b) Write down the payoff matrix to represent how Boris and Priti decide whether to work or shirk and derive the pure strategy Nash equilibria. (c) Compare the outcome when R = 2, 4 or 9. Does this mean that higher rewards do not always elicit more effort? (d) Do you think that it would…
- Professor can give a TA scholarship for a maximum of 2 years. At the beginning of each year professor Hahn decides whether he will give a scholarship to Gong Yi or not. Gong Yi can get a scholarship in t=2, only if he gets it in t=1. Basically, the professor and TA will play the following game twice. TA can be a Hardworking type with probably 0.3 and can be a Lazy type with a probability of 0.7. Professor does not know TA's type. If TA is hard working, it will be X=5 and TA will always work if he gets a scholarship. If TA is lazy, it will be X= 1. There is no time discount for t=2. Find out a Perfect Bayesian Equilibrium of the game.You and a coworker are assigned a team project on which your likelihood or a promotion will be decidedon. It is now the night before the project is due and neither has yet to start it. You both want toreceive a promotion next year, but you both also want to go to your company’s holiday party that night.Each of you wants to maximize his or her own happiness (likelihood of a promotion and mingling withyour colleagues “on the company’s dime”). If you both work, you deliver an outstanding presentation.If you both go to the party, your presentation is mediocre. If one parties and the other works, yourpresentation is above average. Partying increases happiness by 25 units. Working on the project addszero units to happiness. Happiness is also affected by your chance of a promotion, which is depends on howgood your project is. An outstanding presentation gives 40 units of happiness to each of you; an aboveaverage presentation gives 30 units of happiness; a mediocre presentation gives 10 units…First Player can invest $1.00 with Second Player (low reliance) or $2.00 (high reliance). Based on the payoffs shown below, what is the probability of performance that makes High Reliance optimal? Write your answer as a two digit integer. E.g., if the answer is 33%, write 33. Second Player Perform Breach Invest & Low Reliance 0.25 1.0 First Player 0.25 -1.0 Invest & High Reliance 0.5 1.0 0.75 -2.0
- Consider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs: Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00 a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they existA Bank has foreclosed on a home mortgage and is selling the house at auction. There are two bidders for the house, Zeke and Heidi. The bank does not know the willingness to pay of these three bidders for the house, but on the basis of its previous experience, the bank believes that each of these bidders has a probability of 1/3 of valuing it at $800,000, a probability of 1/3 of valuing at $600,000, and a probability of 1/3 of valuing it at $300,000. The bank believes that these probabilities are independent among buyers. If the bank sells the house by means of a second- bidder, sealed-bid auction, what will be the bank’s expected revenue from the sale? The answer is 455, 556. Please show the steps in details thank you!A risk-eutral plaintiff in a lawsuit must decide whether to settle a claim or go to trial. The defendants offer $70,000 to settle now. If the plaintiff does not settle, the plaintiff believes that the probability of winning at trial is 40%. If the plaintiff wins, the amount awarded to the plaintiff is X. Will the plaintiff settle if X is $87,500? What if X= $280,000? What is the critical value of X that would make the plaintiff indifferent between settling and going to trial? If the plaintiff were risk averse instead of risk neutral, would this critical value of X be higher or lower? If the amount to be awarded at trial with a win (X) were $87,500, then the plaintiff would settle If the amount to be awarded at trial with a win (X) were $280,000, then the plaintiff would not settle The critical value of X that would make the plaintiff indifferent between settling and going to trial is $. (Enter your response using rounded to two decimal places.)