2. Consider a cheap talk game in which Nature moves by choosing a sender'stype, where the type space has four elements: −1, 1, 2, and 3, each occur-ring with equal probability of 1½. The sender learns his type and choosesone of three possible messages: bumpy, smooth, and slick. The receiverobserves the sender's message and then chooses one of three actions: 0, 5,and 10. The sender's payoff equals his type multiplied by the receiver'saction. The receiver's payoff equals the sender's type multiplied by twicethe receiver's payoff.a. Find a separating perfect Bayes-Nash equilibrium.b. Find a semiseparating perfect Bayes-Nash equilibrium.

a. Separating Perfect Bayes-Nash Equilibrium: In a separating equilibrium, the sender's type is…

Answered: 2. Consider a cheap talk game in which…

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

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1. A dealer decides to sell a rare book by means of an English auction with a reservation price of 54. There are two bidders. The dealer believes that there are only three possible values, 90, 54, and 45, that each bidder’s willingness to pay might take. Each bidder has a probability of 1/3 of having each of these willingnesses to pay, and the probabilities for each of the two bidders are independent of the other’s valuation. Assuming that the two bidders bid rationally and do not collude, the dealer’s expected revenue is approximately ______. 2. A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of 5 and the other has a buyer value of 10 and with probability 1/2, one has a buyer value of 8 and the other has a buyer value of 15. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must…
In the final round of a TV game show, contestantshave a chance to increase their current winnings of$1 million to $2 million. If they are wrong, theirprize is decreased to $500,000. A contestant thinkshis guess will be right 50% of the time. Should heplay? What is the lowest probability of a correctguess that would make playing profitable?
2. Kier, in The scenario, wants to determine how each of the 3 companies will decide on possible new investments. He was able to determine the new investment pay off for each of the three choices as well as the probability of the two types of market. If a company will launch product 1, it will gain 50,000 if the market is successful and lose 50,000 if the market is a failure. If a company will launch product 2, it will gain 25,000 if the market is successful and lose 25,000 if the market will fail. If a company decides not to launch any of the product, it will not be affected whether the market will succeed or fail. There is a 56% probability that the market will succeed and 44% probability that the market will fail. What will be the companies decision based on EMV? What is the decision of each company based on expected utility value?
. Ayça and Barış are playing a game and following payoff matrix is for the payoffs of Ayça. Answer the questions according to the following payoff matrix. a) What is the probability that the value of the game is 10?
3. Find the saddle point, if it exists, for the following game. (b) Solve the following game by using the principle of dominance and find the probabilities of strategies for each player and the value of the game. Player B Player A II III IV V 3 4 4 II 2 4 III 4 4 IV 4 4 20 2420 8760
2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
In a final round of a MegaMillion TV show, a contestant has won $1 millionand has a chance of doubling the reward. If he loses his winnings drop to$500,000. The contestant thinks his chances of winning are 50%. Should heplay? What is the lowest probability of a correct guess that will make his betprofitable? Show work
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?
7. Suppose the only game in town involves flipping a fair coin (so Heads and Tails are equally likely), with a $x bet. If Heads comes up, the payoff is $0.9x; if Tails comes up, you lose the $x. You have $10,000, and must win at least $5,000 by tomorrow morning to pay off a debt to a mean dude. a. Compute the likelihood of winning at least $5000 by making a single bet of $10,000. b. Compute the likelihood of winning at least $1000 by playing the game 10,0000 times and betting a dollar each time. What is the likelihood of not losing money? Message learned?
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…
A st SW B st 2,1 5, 15 2. Consider the game of chicken above. SW 11,4 3,5 a. Find all Nash equilibrium in pure strategies. b. Say that player A uses a mixed strategy, where she picks st with prob. 0.55 and sw with prob. 0.45. What is B's expected payoffs from choosing pure st and pure sw? c. Player B uses a mixed strategy and wants A to choose sw. What is the minimum prob. p, for which B needs to play st so A will choose sw.
Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.

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