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ps6solutions - Problem Set 6 SOLUTIONS Due Thursday Dec 3rd...

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Problem Set 6 SOLUTIONS Due Thursday, Dec. 3rd, by 10pm 1. Consider a game where a buyer makes a take it or leave it offer, p , to a seller who then decides whether to accept the offer. (a) Suppose the buyer’s payoff is 5 - p if the offer is accepted and 0 if the offer is not accepted. The seller’s payoff is p if the offer is accepted and 1 if the offer is not accepted. Find a subgame perfect Nash equilibrium and and a Nash equilibrium that is not subgame perfect. If we consider the subgame once an offer p has been made, the seller will accept any offer if p 1. With this in mind, the buyer can achieve the highest payoff if he offers p = 1, so the SPNE is: Buyer : { p = 1 } Seller : { Accept if p 1 , Reject if p < 1 } A NE can be generated if the seller makes a non-credible threat, where he says he will reject if p < 2. In this case, the buyer’s best response is to offer p = 2, the the strategies Buyer { p = 2 } ; Seller { Accept if p 2 , Reject if p < 2 } is a NE. However, as stated, this is a non-credible threat; it is not a SPNE. (b) Suppose now that the seller can make an investment which is observed by the buyer before the buyer makes an offer. If the seller makes the investment, this improves the payoff for the buyer if the offer is accepted to 8 - p but lowers the payoff of the seller to p - 1 if the offer is accepted and 0 if the offer is not accepted. What is a subgame perfect Nash equilibrium to this game? Again, considering the final stage of the game, if the seller, does not invest, then he will accept any offer such that p 1 and receive a value of 1. Similarly, if he makes
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