Solved: Jared, Karla, and Lori are dividing the foot-long half meatballhalf vegetarian

QUESTION:

Jared, Karla, and Lori are dividing the foot-long half meatball–half vegetarian sub shown in Fig. 31 using the lone-chooser method. Jared likes the vegetarian and meatball parts equally well, Karla is a strict vegetarian and does not eat meat at all, and Lori likes the meatball part twice as much as she likes the vegetarian part. Suppose that Karla and Jared are the dividers and Lori is the chooser. In the first division, Karla divides the sub into two shares (a left share \(s_{1}\) and a right share \(s_{2}\)) and Jared picks the share he likes better. In the second division, Jared subdivides the share he picks into three pieces (a “left” piece \(J_{1}\), a “middle” piece \(J_{2}\), and a “right” piece \(J_{3}\)) and Karla subdivides the other share into three pieces (a “left” piece \(K_{1}\), a “middle” piece \(K_{2}\), and a “right” piece \(K_{3}\)). Assume that all cuts are perpendicular to the length of the sub. (You can describe the pieces of sub using the ruler and interval notation, as in [3, 7] for the piece that starts at inch 3 and ends at inch 7.)

a) Describe Karla’s first division into \(s_{1}\) and \(s_{2}\).

(b) Describe which share (\(s_{1}\) or \(s_{2}\)) Jared picks and how he would then subdivide it into the three pieces \(J_{1}\), \(J_{2}\), and \(J_{3}\).

(c) Describe how Karla would subdivide her share into three pieces \(K_{1}\), \(K_{2}\), and \(K_{3}\).

(d) Based on the subdivisions in (a), (b), and (c), describe the final fair division of the sub and give the value of each player’s share (as a percentage of the total value of the sub) in the eyes of the player receiving it.