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Unformatted text preview: Ec1052: Introduction to Game Theory Handout 2 Harvard University 24 February 2004 Solutions to Problem Set 1 Problem 1. For this exercise it is necessary to assume that all utility is counted in Dollar terms. 1(a) Your roommate can choose between two actions: going to the theater (T) or staying at home (H). We can calculate his utility from both actions: u ( T ) = 20 15 (1) u ( H ) = 0 (2) Your friend faces the same problem when he bought the first and the second ticket. The purchase of the first ticket does not matter for the decision of buying a second ticket because the money for that ticket is lost regardless of whether he buys a second ticket or not. 1(b) Now your friend chooses between two lotteries. If he doesnt buy the theater ticket then he will be able to invite his girlfriend for dinner and get utility G 30 Dollars from the dinner (assuming he gets utility G in Dol lar terms from going for dinner with his girlfriend). His expected utility is therefore u ( H 2) = 1 2 ( G 30). If he buys the theater ticket on the other hand he cannot invite his girlfriend for dinner which causes him a net util ity loss of 100 Dollars. However, he will enjoy a net utility of 5 Dollars from seeing the play. Therefore his expected utility from buying the second ticket is u ( T 2) = 1 2 ( 100) + 5 = 45. He therefore clearly enjoys higher utility from not buying a second ticket. However, he made the correct de cision when he bought the first ticket since his net utility from doing so is u ( T 1) = 1 2 ( G 30) + 5 > u ( H 2) = u ( H 1). 1(c) The net utility from buying the first ticket is 5 Dollars but the net utility from buying the second ticket is far lower because your friend has run down his cash reserves to the extent that he cannot afford to take his girlfriend out for dinner. Therefore the real cost of the second ticket is much greater to him than 15 Dollars. It is 15 Dollars plus the expected loss from being short on liquidity which is 1 2 (100 + G 30). Note, that your friend could afford the second ticket. However, he realizes that this expense might not allow him to spend any further money on emergencies. More generally, EUT tells us that agents should not regret sunk costs since they are paid already regardless of what decision is made in the future. However, sunk costs reduce agents liquidity which can induce them to postpone further spending. Ec1052 Handout 2: Solutions to Problem Set 1 2 Problem 2. * The set of outcomes is A = { a 1 ,a 2 ,..,a n } . Assume that a pref erence relationship over lotteries over these outcomes is consistently represented...
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