1. For each of the following phenomena, briey discuss the difculties that might be encountered trying to explaining the phenomenon using standard economic ideas, and then suggest a possible explanation using ideas from this course. Please explain clearly (but briey) hovt.I the ideas you use apply to the phenomenon. (You are not asked for any mathematical modeling, only to identify relevant concepts and explain how they apply. The quotations are from New York Times articles, supplied on request after the exam.) (a) \"From 1989 to 1992, prices in Boston fell sharply, with condominium prices dropping as much as 40 percent. For a great many of those who bought condominiums during that period, selling could be done only at a signicant loss. And, basically, many people refused to sell....For essentially identical condominiums, people who had bought at the peak and were facing a loss generally listed their properties for signicantly more than those who had bought at a time when prices were lower. . \"Properties listed above the market price just sat there. In the Boston market over all, sellers listed their properties for an average of 35 percent above the expected sale price, and less than 30 percent of the properties sold in fewer than 180 days." (b) \"Genes play a role in Alzheimer's disease, but in most cases the role is not Jlly understood. In...late-onset Alzheimer's, there is no single yest'no gene. Instead, researchers think a combination of genes work together, . . .[and each] gene merely adds to the risk....So far, the strongest inuence comes from a gene called APOE. . _.But APOE is by no means denitive. Many people with [the APOE. gene type that adds to the risk] never become demented, and more titan a third of Alzheimer's patients have [none of that gene type]. ...Because of the uncertainty, the medical profession, the Alzheimer's Association and genetic counselors have for years steadfastly advised against APOE testing, saying that the results are not denitive and if misunderstood could be needlessly upsetting, especially since there is no way to prevent or cure the disease." However, there is now a movement in the medical profession in favor of voluntary APOE testing and revealing the results to patients: \"Not everyone wants to know, but the people who want to know really want to know, and they have their own reasons," Dr. Green said. \"I think it's a little patronizing for the medical establishment to say, 'We could give you that test, but we don't think you can handle it.'\"....\"Feople are eventually going to understand that genetic risk factors are just risk factors, not. determinants," Dr. Green said. \"I think this blanket resistance to APOE exposure is not going to last too much longer.\" (c) \"With the popularity of traditional lotteries waning across the country, many states are turning to instant games priced at $20, $30 and as high as $50 to lure new players and raise revenue. Scratch-off tickets, for example, now account for more than 75 percent of lottery sales in Texas, which this year became the rst state to introduce a $50 scratch-off game. But critics in Texas and elsewhere say games promising this kind of instant gratication are more likely to contribute to the kind of problem gambling that is usually associated with fast-paced casino betting, and they are now trying to limit them. . ..'Scratch-off tickets are to the lottery what crack is to cocaine,' said State Senator Eliot Shapleigh, a Democrat who represents El Paso." 2. A student must do a problem set, but can do it in any one of the three periods t = 0, 1, 2. The immediate utility cost of doing it in period t = 0 is 4; in period t = 1 it is 6; and in period t = 2 it is 9. The student is a hyperbolic discounter with P = 1/2 and 6 = 1. (That is, "self 0"-the student from the point of view of period 0-makes decisions to maximize 0"-period utility plus 1/2 times the (undiscounted) sum of 1"- period and 2"-period utility. Self 1 makes decisions to maximize 1"- period utility plus 1/2 times 2""-period utility. And self 2 makes decisions to maximize 2""-period utility.) For the first two parts of this question, assume that the student cannot make commitments or limit the freedom of choice of future selves in any way. Note however that if the problem set is not done by the start of period 2, self 2 has no choice but to do it. (a) First assume that the student is naive, in the sense that self 0 expects selves 1 and 2 to carry out the period-1 and period-2 parts of self 0's optimal plan, even though selves 1 and 2 have different tradeoffs between periods. Explaining your argument carefully, show that a naive student actually ends up doing the problem set in period 2. (Hint: Start by figuring out what self I will do if the problem set is not done by period 1, and then work backwards to figure out what naive self 0 will do.) (b) Now assume that the student is sophisticated, in the sense that self 0 can correctly predict what selves 1 and 2 will do in whatever situations they find themselves in. Explaining your argument carefully, show that a sophisticated student ends up doing the problem set in period 0. For part (c), assume that self 0 can commit in period 0, completely determining the future decisions of selves 1 and 2 in any way self 0 wishes. (c) When will a sophisticated student who can commit in period 0 to determine the future decisions of selves 1 and 2 end up doing the problem set? When will a naive student who can commit end up doing the problem set?3. In the 3x3 game discussed in class: L C R [l 3 'l' I} 'i' 3 I.- (a) Find each player's strictly dominated strategy or strategies, if any, and each player's strategies that survive iterated deletion of strictly dominated snategies. {b} Find the Nash equilibrium or equilibria. Justify your Nah equilibrium or equilibria as the only possible outcomets} of players' strategic thinking, making whatever assumptions about their rationality andl'or knowledge of each other's rationality you need. For parts {c} and {d}, assume that players are repeatedly paired at random from a large population to play this game, and that they adjust their strategies over time in some way that always reduces the population frequency of each player's pure strategy that has the lowest expected payoff among all of hert'his strategies, given the current mix of strategies in the population. {c} Explain why the population frequency with which Column players play R will decline over time. Show that if it eventually declines close enough to D. then the population frequency with which Row players play B will also start to decline. {d} Can this prmess stop anywhere but with ow playing M and Column playing C with probability one? Do you thin]: it is likely to get there? Explain. 4. Consider a Matching Pennies game but with the ow player's payoff for (Heads, Heads} 2 instead of 1 {and the Column player's payoff for (Heads, Heads} -2 instead of -II so that the payoffs still add to l] for each strategy combination}. The How and lColumn players' payoffs for (Tails. Tails} remain unchanged at l and -1. {a} IWrite the payoff matrix of this 1Itersion of Matching Pennies. {b} Find the Nash equilibrium and players' equilibrium expected payoffs in this game. {c} Compared Iwith the orig-ion] Matching Pennies game, in Iwhich direction does How's equilibrium probability of playing Heads change? ht which direction does Column's equilibrium probability of playing Heads change? Do these directions correspond to your intuitions about the effect of increasing the payoff to matching on Heads on Row's and Column's probabilities of playing Heads? Explain. {d} In which direction do How's and Column's equilibrium expected payoffs change? Do these directions correspond to your intuitions about the effect of increasing the payoff to matching on Heads? Explain. (c) How would you expect most real people, not trained in game theory, to respond to the increased payoff to matching on Heads {relative to standard Matching Pennies} in the Row player's role? In the Column player's role