Question 13.55
25. Suppose that Players 1, 2, and 3 view a cake as follows:
- Provide three drawings to show how each player views a division of the cake by Player 1 into three pieces that he or she considers to be the same size or value. Label the pieces , , and . (We are still assuming that all cuts correspond to vertical lines, so this will require a cut along a vertical center line of some of the squares.)
- Show that neither Player 2 nor Player 3 finds more than one of the three pieces acceptable (with “acceptable” defined as in Exercise 24).
- Identify a single piece that Player 2 and Player 3 agree is not acceptable. (There are actually two such pieces; for definiteness, find the one on the right.)
- Assume that Players 2 and 3 give the piece from part (c) to Player 1, and that they reassemble the rest and Players 2 and 3 divide it between themselves using divide-and-choose (with a single vertical cut). Determine what size piece each of the three players will think he or she received (i) if Player 2 divides and Player 3 chooses, and (ii) if Player 3 divides and Player 2 chooses.
25.
(a) See figures below.
(b) Player 2 finds acceptable (9 square units), but not (5 square units) or (4 square units). Player 3 finds acceptable (12 square units), but not (4 square units) or (2 square units).
(c) Players 2 and 3 both find and unacceptable.
( is on the right.)
(d) (i) Player 1 gets a piece he thinks is 6 square units. Player 2 gets a piece he thinks is 7 square units, and Player 3 gets a piece he thinks is 10 square units.
(ii) Player 1 gets a piece she thinks is 6 square units, Player 2 gets a piece she thinks is square units, and Player 3 gets a piece she thinks is 8 square units.