Sweet tooth Warm Up
Here's the problem with a chunk of the solution from pg 4:
Jeremy will cut the first cake into two pieces, perhaps evenly, perhaps not. After seeing the cut Marie will decide whether she will choose first or allow Jeremy to do so. If she goes first, she will take the larger piece. If she goes second, she can assume Jeremy will take the larger piece.
Next, Jeremy will cut the second cake into two pieces (remember that one of the pieces can be vanishingly small if he so chooses). If Marie had chosen first for the first cake, then Jeremy gets to take the larger piece of the second cake. If Marie had chosen first for the first cake, then Jeremy gets the larger piece of the second cake. If Marie had chosen second for the first cake, then she gets the larger piece of the second cake.
Assuming each person will strive to get the most total cake possible, what is an optimal strategy for Jeremy?
Hint: Assume that Jeremy divides the first cake into fractions f and 1-f where f is at least 1/2. Then explore the consequences if Marie chooses to take the piece of fraction f or if she goes second, so gets the piece having fraction 1-f.
If M takes the fraction f piece, then Jeremy will take the entire second cake. So, Marie will get exactly f and Jeremy will get (1-f) +1. If Marie takes the smaller piece of the first cake (fraction 1-f), Jeremy will do best if he divides the second cake in half. This gives Marie (1-f) = 1/2. Jeremy follows this reasoning, so realizes that the best he can do is to make f = (1-f) +1/2. That is 2f = 1.5 or f = 3/4.
It says "Jeremy follows this reasoning, so realizes that the bes he can do is to make f = (1-f) + 1/2." I understood everything up till this part. More specifically, what I don't understand is how acknowledging that either Marie gets f and John gets (1-f) +1 OR Marie gets (1-f) +1/2 and John gets f + 1/2 leads John to realize that the best he can do is f = (1-f) + 1/2. Why is f = (1- f) + 1/2 the BEST he can do?
Last edited by jsuit; April 6th, 2010 at 11:01 PM.