Fair Settlement

Question

How can the cost for a rare coin and the cost for an heirloom ring be fairly distributed between three people under a sealed bid?

The rules are:

(1) No person can insist on getting any item.

(2) Each person must be willing to accept cash in lieu of items.

(3) Each person must bid independently without regard to other persons.

Answer 1

SETTLEMENT

ITEMS	PERSON A BID	PERSON B BID	PERSON C BID
Coin (Rare)	$80	$60	$60
Ring (Heirloom)	$70	$60	$75
Secret Bid Value	$150	$120	$135

Since A bid higher on the coin and C bid higher on the ring, they are afforded possession. But the value of their bid must be considered to fairly distribute the items. Thus, A’s cost share for the coin is $150/3 = $50 and C’s cost share for the ring is $45 ($135/3). The income for distribution is $60. A owes $30 which is the difference between the bid and the fair share or $80 - $50. C owes $30 because he bid $75 and his cost share is $35 ($75 - $45). B gets $40 cash from the $60 given by A and C leaving a surplus of $20 which is distributed evenly between the three persons.

Shares	$150/3 = $50	$120/3 = $40	$135/3 = $45
Owes for Item	$30	Receives $40	$30
Surplus	$6.67	$6.67	$6.67
Final Cost	6.67-30 = -$23.33 for the coin	40 + 6.67 = $46.67 cash	6.67-30  = -$23.33 for the ring

 

Finally, A pays $23.33 to B for the coin. C pays $23.33 to B for the ring. B gets an equal amount of cash from A and C. That’s fair, right?