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. "

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. "

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 18.1IP

See similar textbooks

Similar questions

Consider the following variation to the Rock (R), Paper (P), Scissors (S) game:• Suppose that the Player 1 (row player) has a single type, Normal.• Player 2 (column player) has two types Normal and Simple.• A player of Normal type plays this zero-sum game as we studied in class whereas a player of type Simple always play P.• Player 2 knows whether he is Normal or Simple, but player 1does not.a) Suppose player 2 is of type Normal with probability 1/3 and of type Simple with probability (2/3). Find all pure strategy Bayesian Nash Equilibria.b) Suppose player 2 is of type Normal with probability 2/3 and of type Simple with probability (1/3). Find all pure strategy Bayesian Nash Equilibria.
Matthew is playing snooker (more difficult variant of pool) with his friend. He is not sure which strategy to choose for his next shot. He can try and pot a relatively difficult red ball (strategy R1), which he will pot with probability 0.4. If he pots it, he will have to play the black ball, which he will pot with probability 0.3. His second option (strategy R2) is to try and pot a relatively easy red, which he will pot with probability 0.7. If he pots it, he will have to play the blue ball, which he will pot with probability 0.6. His third option, (strategy R3) is to play safe, meaning not trying to pot any ball and give a difficult shot for his opponent to then make a foul, which will give Matthew 4 points with probability 0.5. If potted, the red balls are worth 1 point each, while the blue ball is worth 5 points, and the black ball 7 points. If he does not pot any ball, he gets 0 point. By using the EMV rule, which strategy should Matthew choose? And what is his expected…
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…
A risk-averse agent, Andy, has power utility of consumption with riskaversion coefficient γ = 0.5. While standing in line at the conveniencestore, Andy hears that the odds of winning the jackpot in a new statelottery game are 1 in 250. A lottery ticket costs $1. Assume his income isIt = $100. You can assume that there is only one jackpot prize awarded,and there is no chance it will be shared with another player. The lotterywill be drawn shortly after Andy buys the ticket, so you can ignore therole of discounting for time value. For simplicity, assume that ct+1 = 100even if Andy buys the ticket How large would the jackpot have to be in order for Andy to play thelottery? b) What is the fair (expected) value of the lottery with the jackpot youfound in (a)? What is the dollar amount of the risk premium that Andyrequires to play the lottery? Solve for the optimal number of lottery tickets that Andy would buyif the jackpot value were $10,000 (the ticket price, the odds of winning,and Andy’s…
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.
[Adverse Selection] Each of the two players receives an envelope, in which there is anamount of money that is equally distributed from $0, $1, $2, ..., $100. The amounts in twoenvelopes are independent. After receiving the envelope, each individual can check exactlyhow much money is put in his/her own envelope. Then each player has the option to exchangehis/her envelope for the other individual's prize. The decisions are made simultaneously. Ifboth individuals agree to exchange, then the envelopes are exchanged; otherwise, if at leastone player chooses not to exchange, each individual keeps his/her own envelope and receivesits attached sum of money.a. Model this game as a static Bayesian game (write the normal formrepresentation) and find the Bayesian Nash equilibrium.b. Consider a new game where the probability distribution of money in eachenvelope is changed. The amount is equal to $100 with probability 90%, and is equalto each number in $0, $1, $2, ... ,$99 with probability 0.1%.…
Utility functions incorporate a decision maker’s attitude towards risk. Let’s assume that the following utilities were assessed for Danica Wary. x u(x) -$2,000 0 -$500 62 $0 75 $400 80 $5,000 100 Would a risk neutral decision maker be willing to take the following deal: 30% chance of winning $5,000, 40% chance of winning $400 and a 30% chance of losing $2,000? Using the utilities given in the table above, determine whether Danica would be willing to take the deal described in part a? Is Danica risk averse or is she a risk taker? What is her risk premium for this deal?
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.
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.
You're a contestant on a TV game show. In the final round of the game, if contestants answer a question correctly, they will increase their current winnings of $1 million to $3 million. If they are wrong, their prize is decreased to $750,000. You believe you have a 25% chance of answering the question correctly. Ignoring your current winnings, your expected payoff from playing the final round of the game show is [$ blank]. Given that this is positive [blank (positive/negative)], you should [blank (should/should not)] play the final round of the game. (Hint: Enter a negative sign if the expected payoff is negative.) The lowest probability of a correct guess that would make the guessing in the final round profitable (in expected value) is [blank]. (Hint: At what probability does playing the final round yield an expected value of zero?)
Clancy has $4800. He plans to bet on a boxing match betweenSullivan and Flanagan. He finds that he can buy coupons for $6 thatwill pay off $10 each if Sullivan wins. He also finds in another storesome coupons that will pay off $10 if Flanagan wins. The Flanagantickets cost $4 each. Clancy believes that the two fighters each have aprobability of ½ of winning. Clancy is a risk averter who tries tomaximize the expected value of the natural log of his wealth. Whichof the following strategies would maximize his expected utility? (a) Don’t Gamble (b) Buy 400 S tickets and 600 F tickets(c) Buy exactly as many F tickets and S tickets (d) Buy 200 S and 300 F(e) Buy 200 S and 600 F