1. Consider a game between Joe and Snake. This game takes place in the morning after Joe has had breakfast. Snake can do two things: fight Joe or not. He likes fighting with people who are feeling cowardly and gets a payoff of 1 if he does.
Snake, however, does not like to fight people who are feeling brave. He gets a payoff of -1 if he does. Regardless of how the other person feels Snake gets a payoff of 0 if he doesn't fight. There is a 75% chance that Joe is feeling cowardly and a 25% chance that he is feeling brave. Joe knows whether he is feeling brave or not, but Snake does not. However, Snake does observe what Joe eats for breakfast. Hence, Joe can use his breakfast choice as a signal to Snake. Joe can choose to have either Quiche or Beer for breakfast. Joe prefers to have Quiche for breakfast and he prefers not to fight. Regardless of whether he is feeling brave or cowardly, Joe gets a payoff of 2 by having Quiche and not fighting, a payoff of 1 from having Beer and not fighting, a payoff of 0 from having Quiche and fighting, and a payoff of -1 from having Beer and fighting.
A) Draw the extensive-form version of this game.
B) Find all pure-strategy sub-game perfect Bayes Nash equilibrium.
2. Rumpelstiltskin is having a hard time finding a reliable princess to spin straw into gold for him. On any given day there is a 60% chance his chosen girl will be good and spin 100 ounces of gold and a 40% chance she will be bad and only spin 16 ounces. Rumpy is risk averse and has a utility function of U(g) = g1/2, where g = ounces of gold.
A) What is Rumpelstiltskin's expected utility?
B) How many ounces of gold would Rump be willing to sacrifice on a good day to guarantee the princess spins 36 ounces on a bad day?
C) How many ounces of gold would Rump be willing to pay a princess on a good day if she had an 80% chance of having a good day and only a 20% chance of a bad day? (Assume she takes her payment from the 100 ounces she produces on good days.)
D) How many ounces of gold would Rump be willing to pay a princess who guaranteed him 100 ounces of gold a day?
(Assume she takes her payment from the 100 ounces she produces.)
3. There are currently 500 cars in the used car market. Only 300 of them are good cars. Sellers value good cars at $5000 and bad cars at $2000. Buyers value good cars at $9000 and bad cars at $3000.
A) Will there be failure in the market for good cars? Why or why not?
B) Would sellers of good cars feel inclined to spend money and time providing warranties for their cars? Explain.
C) The government's Cash for Clunker program caused an influx of cars into the used car market. Given the nature of the program, the majority of those cars were bad cars. In the end there were 2000 used cars in the market, 1000 good cars and 1000 bad cars. The program also affected both seller and buyer values for used cars. In particular, sellers now value good used cars at $4000 and bad used cars at $1000. Buyers now value good used cars at $6400 and bad used cars at $1200. What impact would this legislation have had on the desire for sellers to spend resources on things like warranties and guarantees for their good used cars? Explain.
4. Consider the market for soda. Coca Cola and Pepsi Co. must each decide whether to engage in an aggressive advertising campaign (A) or a standard advertising campaign (S). The trouble is, Pepsi Co. has been undergoing some management changes and Coke is currently lacking information about Pepsi Co.'s new corporate structure. There is some chance (p) that Pepsi Co. is still facing very high operating and administrative costs, in which case the profits to the firms will be as shown in game H. There is also some chance (1-p) that Pepsi Co. has fixed their management problems and now faces low operating and administrative costs, in which case the profits to the firms will be as shown in game L. Pepsi Co. knows whether its costs are high or low. Compute all pure-strategy Bayesian Nash equilibrium of this game.
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H
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PepsiCo.
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Coca Cola
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A
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S
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A
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75 ,25
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90 ,30
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S
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25,35
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105 ,85
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L
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PepsiCo.
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Coca Cola
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A
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S
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A
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75 ,75
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90 ,45
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S
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25 ,85
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105 ,105
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5. Two firms face the following market demand: P = 1000 - 0.25*Q. Firm 1 has a marginal cost of $44 and this is common knowledge. Firm 2 has a marginal cost of $60 with 60% probability and a marginal cost of $30 with 40% probability. This is known to both firms. However, only firm 2 actually observes its marginal cost before making its output decision, firm 1 must make its output decision without knowing firm 2's realized marginal cost. Calculate the Bayes Nash equilibrium of this game.