Question

homework 9-equitability procedure and envy - free divisions: problem 6
(1 point)
washington, adams, jefferson, and madison bid on a much coveted resource (who will get the presidency).
the statesmen decide that washington should get the presidency. washington will then pay everyone else $152.
(a) what is w, the value of the winning bid in this example?
(b) what is m, the average of all the bids in this example?
(c) what is x_adams in this example?
(d) what is x_jefferson in this example?
(e) what is x_madison in this example?
(f) what is x_washington in this example?
(g) what is adamss fair share?
(h) what is jeffersons fair share?
(i) what is madisons fair share?
(j) what is washingtons fair share?
(k) is the arrangement fair?

Explanation:

Step1: Find the winning bid ($w$)

The winning bid is the highest bid. Looking at the values: Washington - $760$, Adams - $460$, Jefferson - $1120$, Madison - $160$. The highest value is $1120$.

Step2: Calculate the average ($m$) of all bids

The formula for the average of a set of numbers $a,b,c,d$ is $m=\frac{a + b + c + d}{4}$. Here $a = 760$, $b=460$, $c = 1120$, $d = 160$. So $m=\frac{760+460 + 1120+160}{4}=\frac{2500}{4}=625$.

Step3: Calculate $x_{Adams}$

$x_{Adams}$ is the amount Adams receives. Since Washington pays everyone else $152$, $x_{Adams}=152$.

Step4: Calculate $x_{Jefferson}$

$x_{Jefferson}=152$ (same reason as for Adams, Washington pays each of the others $152$).

Step5: Calculate $x_{Madison}$

$x_{Madison}=152$ (Washington pays each of the others $152$).

Step6: Calculate $x_{Washington}$

$x_{Washington}$ is the value of the resource minus the total amount paid out. The value of the resource is $w = 1120$. Washington pays $3\times152=456$. So $x_{Washington}=1120 - 456=664$.

Step7: Calculate fair - share

The fair - share for each person is $\frac{\text{their bid}}{4}$.
For Adams: fair - share $=\frac{460}{4}=115$.
For Jefferson: fair - share $=\frac{1120}{4}=280$.
For Madison: fair - share $=\frac{160}{4}=40$.
For Washington: fair - share $=\frac{760}{4}=190$.

Answer:

(a) $1120$
(b) $625$
(c) $152$
(d) $152$
(e) $152$
(f) $664$
(g) $115$
(h) $280$
(i) $40$
(j) $190$
(k) To check if the arrangement is fair:

• Adams: $x_{Adams}=152>115$
• Jefferson: $x_{Jefferson}=152<280$
• Madison: $x_{Madison}=152 > 40$
• Washington: $x_{Washington}=664>190$

Since Jefferson's received amount ($152$) is less than his fair - share ($280$), the arrangement is not fair.