Final answer:
The problem presents a fair division scenario using sealed bids, where four heirs must divide two items of an estate. Player A's original fair share is $450, calculated from the total bids provided. The final allocation ensures each heir's share is balanced, with adjustments made through payments for fairness.
Step-by-step explanation:
The question involves a problem regarding fair division using the method of sealed bids, which falls under game theory, a branch of applied mathematics. The scenario involves four heirs (A, B, C, and D) who need to divide an estate, consisting of a desk and a vanity, fairly among themselves. Each heir submits a bid on both items, and these bids are used to determine the fair division of the assets.
Let's calculate Player A's original fair share of the estate:
- Sum of all bids for the desk: 320 (A) + 280 (B) + 200 (C) + 300 (D) = $1100
- Sum of all bids for the vanity: 220 (A) + 200 (B) + 120 (C) + 160 (D) = $700
- Total value of the estate as per the bids: $1100 (Desk) + $700 (Vanity) = $1800
- Player A's original fair share: $1800 / 4 heirs = $450
In the initial allocation, the sum of Player A's bids (320 + 220) is $540, which is above their fair share of $450. To compensate, Player A would typically pay or receive money to balance it out. In the final allocation, Player A would either get one of the items they bid on and pay or receive money to ensure the overall value they receive is equal to their fair share, or they could receive both items and pay the excess amount to the estate if their combined bid is higher than their share.
Regarding the division method, if one item is favored by all, the bids are used to allow the highest bidder to win and balance payments are made to ensure each heir receives their fair share of the overall estate. In essence, sealed bids provide a way to reveal the subjective values each heir places on each item, thus enabling a fair division based on these valuations.