1 Marcia, Marcia, Marcia Greg is deciding whether to ask Marcia o...

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1 Marcia, Marcia, Marcia Greg is deciding whether to ask Marcia out on date. However, Greg isnt sure whether Marcia likes him, and he would rather not ask he expects to be rejected. Whether Marcia likes Greg private information to her. Thus, her preferences regarding Greg constitute her type. Greg does not have any private information. Assume that there is 25% chance that Marcia likes Greg. The Bayesian game is shown here. Should Greg ask Marcia? Find BNE that answers this question. Nature Marcia Marcia doesn' Greg life Greg Probability .25 Probability 75 Greg Invito Do not Do not invite invife invite Marcia Marcia 4 4 Ves No Yes No Greg 10 8 2 Marcia , 1 3 2 Auctions: First Price vs. Second Price Fill in the blanks in the table below. Bidder Value 1st price bid 2nd price bid A 92 B 95 C 31 D 75 E 88 a) Who wins the first price auction and what price do they pay? b) Who wins the second price auction and what price do they pay? 3 NE in Auctions Consider first-price sealed bid auction with risk-neutral bidders (we've been examining risk neutral bidders all along). Each bidder has private value independently drawn from uniform distribution on [0,1]. That is. for each bidder, all values between 0 and are equally likely. The complete strategy of each bidder is bid function that will tell us. for any value v. what amount b(v) that bidder will choose to bid. It is proposed that the equilibrium bid function for is b(v) 0/2 for each of the two bidders. That is. if we have two bidders. each should bid half her value. a) (4 points) Suppose you're bidding against just one opponent whose value is univormly distributed on 10, and always bids half her value. What is the probability that you will win you bid b=.17 If you bid b=.47 If you bid b=.6? b) (4 points) Put together the answers topart a). What is the correct mathematica expression for p(win), the probability that you win. as function of your bid b? c) (4 points) Find an expression for the expected profit you make when your value is and you bid is b. given that your opponent is bidding half her value. Remember that there are two cases: either you win the auction or you lose the auction. You need to average the profit between these cases. d) (4 points) What is the value of that maximizes your expected profit? This should bez function of your value v. e) (4 points) Use your results to argue that it NE for both biddres to follow the same bid function 4 Lemons In the country of Autolandia, used cars all sell for the same price. In Autolandia, the quality of cars is uniformly distributed with sellers valuing the cars from $0 to $1,000 and buyers valuing the cars from $100 to $1 100 (buyers value each individual car at $100 more than the sellers). For example consider a market with 1,001 cars. There will be one worth $0 to seller/$100 to buyer, one worth $1 to a seller/$101 to buyer, etc. up to one worth $1,000 to seller/$1,100 to a buyer. a) (5 points) If neither the buyer nor the seller knows the quality of car. will any cars sell? If so. at what price? b) (5 points) If only the seller knows the quality of car. what will buyer think is the true value of a car with price to the seller of that car? How much would buyer value the average car for sale at prioe p? Give numerical example. c) (5 points) In the version of the market where only the seller knows the true quality of the car, what will be the equilibrium price? d) (5 points) What percentage of the total number of cars get sold? 5 No such thing as a free lunch A local charity has been given grant to serve free meals to the homeless in its community, but it is worried that its program might be exploited by nearby college students who are always interested in a free meal. Both homeless person and college student receive payoff of 10 for free meal. The cost of standing in line for the meal for homeless person and The for college student where m is the number of minutes spent waiting in line. Assume that the staff of the charity cannot observe the true type of people coming for free meals a) (5 points) What the minimum wait time, m. that will achieve separation of types? b) (5 points) After while. the charity finds that it can successfully identify and turn away college students half fo the time. College students who are turned away receive no free meal and also incur a cost of 5 for their time and embarrassment. Will the ability to partially identify college students reduce or increase the answer in part a)? Explain. 6 To Stand or Not to Stand, that is the Question In scene at the end of The Princess Bride (same movie as the poison scene from lesson L01-1) the hero. Wesley confronts the evil Prince Humperdinck. This interaction can be modeled as the following game. Wesley can either be strong or weak, as randomly picked by nature with equal probability. Wesley, knowing if he strong or weak. can then choose to stand or continue lying on the bed. The prince observes this decision but does not know Wesley's type and chooses to fight or surrender. The prince can beat weak Wesley (Wesley was "mostly' dead only few hours earlier) and can really beat up weak Wesley who stays in bed. If weak Wesley stands and the prince fights, then the base payoffs are for the prince and -1 for Wesley. If Wesley stays in bed. then the prince gets and Wesley still gets -1. If Weslev is weak and the prince surrenders regardless of whether Wesley has stood or not the base payoffs are for Wesley and 0 for the prince. strong Wesley, however, will destroy the prince and enjoy doing it should the prince try to fight. It easier for Weslev to beat the Prince he stands. though, so in that case he gets payoff of while the prince gets -2. If Wesley stays in bed and the Prince chooses to fight the payoffs are for Wesley and -1 for the prince. In the event that the prince chooses not to fight. Wesley gets if he stands and if he stays in bed while in either case the prince gets 0. The final complication to the payoffs is that if Wesley is weak, standing costs him some extra amount c, while it costs strong Wesley nothing. a) (5 points) derive the extensive form of the game b) (10 points) derive pooling BNE where q 0,5 represents the prince's initial belief that Wesley is weak. Be sure to include what are the prince's beliefs about Wesley's type if he observes Wesley in bed or observes him standing. Find the range of values for that makes the belief valid. c) (10 points) derive separating BNE for this game as well as the range of e over which it valid

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