18.4: Fair Division
- Page ID
- 34292
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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