A sister and a brother share the same bedroom. The brother would like to sleep;
ID: 1719820 • Letter: A
Question
A sister and a brother share the same bedroom. The brother would like to sleep; his sister to read. The light bothers the brother when trying to sleep and reduces his sleeping time by s hours. However, her sister benefits from reading through an increase in her performance on the SAT by g points. Their mom would like to optimize the situation, i.e. turn off the light if s > g and leave the light on if g s. However, she does not know the value of g nor s. Only the daughter knows g and only the son knows s. She asks to her son to tell how many hours A of sleep he will lose if the light is on; she asks her daughter how many points B she will forgo if she cannot read. The mom will turn offt he light if and only if A > B .
1. Does each kid have an incentive to lie when reporting a and b?
2. The mom - a bright game theorist - decides to implement the following scheme: if the light left turned
on, the sister needs to give A candies to his mom and her brother receives B candies.
(a) Show that telling the truth (i.e. B = g) is a weakly dominating strategy for the sister;
(b) Show that telling the truth (i.e. A = s) is a weakly dominating strategy for the brother;
(c) Show that with this scheme, if both kids tell the truth and g > s, the mom has a problem.
Explanation / Answer
a)If B is the number of points the daughter forgo if she cannot read and g be the points gained by performance,we conclude that B is greater than g because she gains more points by studying than when she cannot study.
b)If A is number of hours brother loses if light is on and s is the number of hours of sleeping ,we conclude that A will be greater than s therefor he sleeps more than the slep he loses
c)If the daughter tells that she will get more points if she studies well then mother will compromise with sons sleep and let the daughter switch on the light for her to study, if both tell the truth