Josiah, a young man, is considering whether or not to ask Lydia to go to prom with him. Lydia is either willing to go with him or not. Josiah surmises that there is a .6 chance that Lydia would say yes. If Josiah does not ask Lydia out, he will find out whether or not she would have accepted his offer afterwards. Let O1 = Josiah asks and Lydia accepts, O2 = Josiah asks and Lydia denies, O3 = Josiah doesn’t ask and Lydia would have accepted, and O4 = Josiah doesn’t ask and Lydia wouldn’t have accepted. Josiah prefers O1 to O4, O4 to O3, and O3 to O2. That is, O1 > O4 > O3 > O2.
A. Assume that Josiah is indifferent between O4 and a lottery with a .9 chance of O1 and a .1 chance of O2, and likewise assume that he is indifferent between O3 and a lottery with a .4 chance of O1 and a .6 chance of O2. Construct an interval utility function for Josiah over the respective outcomes:
Josiah: u(O1) =
u(O2) =
u(O3) =
u(O4) =
B. Use these utilities to construct a 2 X 2 matrix representing the situation. Use the Bayesian decision principle to determine which strategy Josiah would take, were he rational.
C. We treated this game as a game against nature, or a parametric game, rather than a strategic game. Which of these two types of games would be a more realistic model of the described situation? Why or why not?
Sample Solution
