Economic and Game Theory
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Brad (B) wants to buy a ticket for a sports event via mail order. There are two sellers, an honest seller, Harold (H) and a dishonest seller, Donald (D). Brad knows that one of the sellers is honest and one is dishonest but does not know who the honest seller is and who the dishonest seller is. The honest seller prefers to send the ticket; the dishonest seller prefers not to send the ticket. Brad has a willingness to pay for the ticket of 30. Both sellers offer the ticket for 10. Actually sending the ticket costs the honest seller -1 (note that this is a negative cost, because we can assume that he enjoys satisfying the contract and that this more than compensates his costs of production and shipping). For the dishonest seller it costs c > 0 to send the ticket but 0 not to send the ticket. It is commonly known that Brad will also buy one ticket in the next season. His wife, Alice, wants to buy a ticket for this season but not for the next. As a matter of principle, whichever seller Brad chooses this season, she orders her ticket from the other seller. Brad does not care whether Alice gets a ticket or not, but he will learn whether she received her ticket or not. Therefore, when he decides which seller to order his ticket from in the second season, he knows for both sellers whether they have sent the ticket or not. In the first season, he just chooses the seller randomly. In the second season, if he considers both sellers to be equally likely to be honest, he will flip a coin to choose the seller. So the relevant part of the game starts with the simultaneous decision of both sellers whether to ship the ticket or not.
I. Find a value of c such that a separating equilibrium exists. Make sure you describe all strategies of Brad, Harold and Donald, and the beliefs of Brad.
II. Find a value of c such that a pooling equilibrium exists where both sellers ship the ticket in the first season. Make sure you describe all strategies of Brad, Harold and Donald, and the beliefs of Brad, including those following off-equilibrium moves.
III. Is there a further pooling equilibrium? Discuss the plausibility of the off- equilibrium beliefs supporting this equilibrium.
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