18.4: Fair Division

Last updated
Save as PDF

Page ID: 34292

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

1. Chance values the veggie half at $7.50 and pepperoni half at $2.50.

A full pepperoni slice is $\frac{1}{4}$ of the pepperoni half. Value $\$ 2.50 / 4=\$ 0.625$

A full veggie slice is $\frac{1}{4}$ of the veggie half. Value $\$ 7.50 / 4=\$ 1.875$

A slice that is ½ pepperoni $\frac{1}{2}$ veggie is value $\$ 0.3125+\$ 0.9375=\$ 1.25$

3. Erin: Bowl 1, Catherine: Bowl 2, Shannon: Bowl 3

5. a. 25 Snickers @ $0.01 each, 20 Milky Ways @ $0.05 each, 60 Reese’s @ $0.02 each

Value: $\$ 0.25+\$ 1.00+\$ 1.20=\$ 2.45$

b. No. Dustin values the whole bag at $8, so a fair share would be $4.

c. Lots of possibilities. Here’s a couple:

80 Milky Ways, 0 Snickers, 0 Reese’s

50 Snickers, 50 Milky Ways, 50 Reese’s

7. a. Zoe

b. Maggie: s2, s3. Meredith: s1, s2. Holly: s3

c. Maggie: s2, Meredith: s1, Holly: s3, Zoe: s4

9. a. P5

b. $6.50 (doesn’t need to trim it much since they’re last)

c. P4 would receive it, with value $6.00 (since P4 would trim it)

11. a. $(320+220) / 4=\$ 135$

b. Desk and Vanity both go to A. A pays $\$ 320+\$ 220-\$ 135=\$ 405$ to estate

B gets $95, C gets $125, D gets $110.

c. Surplus of $\$ 405-\$ 95-\$ 125-\$ 110=\$ 75$ gets split, $18.75 each.

A gets desk and vanity, pays $386.25 to estate

B gets $113.75, C gets $143.75, D gets $128.75

13. Fair shares: Abby: 10.333, Ben: 9, Carla: 7.667

Motorcycle to Abby, Car to Ben, Tractor to Abby, Boat to Abby

Initial: Abby pays $10.667, Ben pays $2, Carla gets $7.667

Surplus: $5; $1.667 each

Final: Abby gets Motorcycle, Tractor and Boat, pays $9

Ben gets Car, pays $0.333

Carla gets $9.334

15. Fair shares: Sasha: $135, Megan: $140

Sasha gets: Couch, detail cleaning. Value $80

Megan gets: TV, Stereo, carpets. Value: $260

Initial: Sasha gets $55, Megan pays $120.

Surplus: $65; $32.50 each

Final: Sasha gets Couch and does detail cleaning, gets $87.50

Megan gets TV and stereo, and cleans carpets, pays $87.50

17. a. s3, worth $270

b. s1 and s4 have combined value $440 for Greedy, so piece would be worth $220