Universal Studios and Legoland California, among other theme parks, sell day passes that include linecutting privileges for about twice the price of regular admission. Those who do not purchase the linecutting privileges, however, are negatively affected by those who do. Perhaps your school can institute a similar policy. Suppose, for example, that Alan, Ben, and Clara are the only students who want to speak with Professor X during her office hours. All three show up at Professor X's door at the same time and must decide who goes first, second, and third. Alan's value of being first in line is $12, second is $5, and third is $0. Ben's values are $6, $3, and $0. Clara's values are $3, $2, and $0. Being clever, the three design a game to determine the order in which they speak with Professor X. The game has prices for the first two spaces in line: $6 for being first and $2 for being second. They decide to give the proceeds to Professor X. With these prices in place, each person announces, simultaneously with the others, a place in line. If only one person announces a given slot, that person receives the slot. If two or three announce the same slot, then these two or three are randomly assigned, with equal probability, to the desired slot and the unannounced slot(s), each paying the price of his or her randomly assigned slot.

a. What is the Nash equilibrium of this game? Who purchases the right to be first?

b. What is the marginal external cost of the purchase?

c. Are the prices of the line-cutting privileges similar to a tax on the negative externality of line cutting? Explain.

d. What is the sum of each person's value on his or her place in line in the Nash equilibrium? Is there any other line order with a greater sum of values? Explain.