Chapter 3, Problem 41E

Jackie, Karla, and Lori are dividing the foot-long half meatball—half vegetarian sub shown in $\underset{\_}{\text{Fig}\text{. 3-30}}$ using the lone-chooser method. Jackie 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 Lori are the dividers and Jackie is the chooser. In the first division, Lori divides the sub into two shares (a left share $s_1$ and a right share $s_2$) and Karla picks the share she likes better. In the second division, Karla subdivides the share she picks into three pieces (a “left” piece $K_1$, a “middle” piece $K_2$, and “right” piece $K_3$) and Lori subdivides the other share into three pieces (a “left” piece $L_1$, a “middle” piece $L_2$, and a “right” piece $L_3$). 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 $\left[3,7\right]$ 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 $\left(s_1\text{or}{s}_2\right)$ Jackie picks and how she 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 $\left(a\right)$, $\left(b\right)$ and $\left(c\right)$ 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.