Question

The Method of Sealed Birds. How much money does Anne pay in the final settlement?

	anne	bessie	cindy
chore 1	65	70	55
chore 2	100	85	95
chore 3	60	50	45
chore 4	75	80	90

I am assuming this is reverse auction

Answer 1

Solution:

Step 1:Find each player’s total valuation of the complete set of amount to be divided. Divide by the number of players to determine each player’s fair share.

	Anne	Bessie	Cindy
Chore 1	65	70	55
Chore 2	100	85	95
Chore 3	60	50	45
Chore 4	75	80	90
Total	300	285	285
Fair share	300/3=100	285/3=95	285/3=95

For example, Anne’s total is 65+100+60+75=300, and her fair share is 300/3 = 100

Step 2: Determine who gets which item by awarding each item to the highest bidder. (It is possible for a player to get multiple items, or to get no items.)

	Anne	Bessie	Cindy
Chore 1	65	70*	55
Chore 2	100*	85	95
Chore 3	60*	50	45
Chore 4	75	80	90*
Total	300	285	285
Fair share	100	95	95

For example, Bessie gets the chore 1 because her bid of $70 was higher than Anne’s $65 or Cindy’s $55.

Step 3: Determine the total value of the items received by each player.

	Anne	Bessie	Cindy
Chore 1	65	70*	55
Chore 2	100*	85	95
Chore 3	60*	50	45
Chore 4	75	80	90*
Total	140	215	195
Fair share	100	95	95
Item value	100+60=160	70	90

For example, Anne got the chore 2 and chore 3, so her item value is 100 + 60= 160

Step 4: Subtract each player’s item value from their fair share to determine the first settlement.

If this number is positive, the player receives cash from the estate(since her item value was less than the fair share.)

If this number is negative, the player must pay cash to the estate(since her item value was more than the fair share.)

	Anne	Bessie	Cindy
Chore 1	65	70*	55
Chore 2	100*	85	95
Chore 3	60*	50	45
Chore 4	75	80	90*
Total	140	215	195
Fair share	100	95	95
Item value	160	70	90
First installment	100-160=-60	95-70=25	95-90=5

Anne’s fair share was $100 but her item value was $160, so in order to receive an exact fair hare she must pay $60. The minus sign indicates that she owes money.

Bessie’s fair share was $95 but she only received $70 worth of items, so in order to make up a fair share she must receive $95−$70 = $25 from the estate.

Cindy’s fair share was $95 but she only received $90 worth of items, so in order to make up a fair share she must receive $95−$90 = $5 from the estate.

Step 5: Add up the first-settlement numbers. The number you get will always be negative or zero (since more cash should have been paid to the estate than taken out of it). Take off the negative sign to get the amount of extra cash on hand. This is the surplus.

In this case the sum of the first-settlement numbers is -60+25+5 = −30 which tells you that the surplus is $30.

	Anne	Bessie	Cindy
Chore 1	65	70*	55
Chore 2	100*	85	95
Chore 3	60*	50	45
Chore 4	75	80	90*
Total	140	215	195
Fair share	100	95	95
Item value	160	70	90
First installment	100-160=-60	95-70=25	95-90=5
Surplus	30

Step 6: Divide the surplus by the number of players to find each player’s share of surplus.

Here, $30 divided by 3 equals $10.

	Anne	Bessie	Cindy
Chore 1	65	70*	55
Chore 2	100*	85	95
Chore 3	60*	50	45
Chore 4	75	80	90*
Total	140	215	195
Fair share	100	95	95
Item value	160	70	90
First installment	100-160=-60	95-70=25	95-90=5
Surplus	30
Share of surplus	10	10	10

Step 7: Add each player’s first settlement to their share of the surplus share to obtain her final settlement.

As before, a plus sign indicates that the player receives money, while a minus sign indicates that the player owes money. If you have done everything correctly, the final settlements should all add up to zero.

	Anne	Bessie	Cindy
Chore 1	65	70*	55
Chore 2	100*	85	95
Chore 3	60*	50	45
Chore 4	75	80	90*
Total	140	215	195
Fair share	100	95	95
Item value	160	70	90
First installment	100-160=-60	95-70=25	95-90=5
Surplus	30
Share of surplus	10	10	10
Final settlement	-60+10=-50	25+10=35	5+10=15

So, -50+35+15 = 0, so everything works out.

Each player has now received a combination of goods and cash that are worth $10 more than what she would have considered a fair share.