This problem involves using the "Sealed Bids" method to allocate items (in this case, four vehicles) to three children (Abby, Ben, and Carla). Each child has placed a bid on the four items. The goal is to allocate the items fairly based on their bids and make necessary payments or compensations from or to the estate.

Here’s the breakdown of how the sealed bids method works:

1. Step 1: Determine Fair Share

   • Each person’s "fair share" is the total value of their bids divided by the number of people involved.

   Abby's total bid: $7 + 12 + 3 + 7 = 29$
   Ben's total bid: $9 + 14 + 6 + 3 = 32$
   Carla's total bid: $8 + 10 + 5 + 4 = 27$

   Each person’s fair share is their total bid divided by 3 (since there are three people).

   • Abby's fair share: $\frac{29}{3} = 9.67$ thousand dollars.
   • Ben's fair share: $\frac{32}{3} = 10.67$ thousand dollars.
   • Carla's fair share: $\frac{27}{3} = 9$ thousand dollars.

2. Step 2: Allocation

   • Each item is awarded to the highest bidder:
     • Motorcycle: Ben bids 9 (higher than Abby's 7 and Carla's 8), so Ben gets the motorcycle.
     • Car: Ben bids 14 (higher than Abby's 12 and Carla's 10), so Ben gets the car.
     • Tractor: Carla bids 5 (higher than Abby's 3 and Ben's 6), so Carla gets the tractor.
     • Boat: Abby bids 7 (higher than Ben's 3 and Carla's 4), so Abby gets the boat.

3. Step 3: Payments

   • Compare the value of items each person receives with their fair share.
   • Ben receives both the motorcycle and car, with a total value of $9 + 14 = 23$, but his fair share is only $10.67$. Therefore, Ben needs to pay $23 - 10.67 = 12.33$ thousand dollars to the estate.
   • Carla receives the tractor, valued at 5, but her fair share is $9$. Therefore, Carla is owed $9 - 5 = 4$ thousand dollars.
   • Abby receives the boat, valued at 7, but her fair share is $9.67$. Therefore, Abby is owed $9.67 - 7 = 2.67$ thousand dollars.

4. Step 4: Redistribution of Excess

   • After payments, excess money is redistributed to those who received less than their fair share.

   So for Abby’s allocation:

   • Abby gets the Boat.
   • She is owed money by the estate: $9.67 - 7 = 2.67$ thousand dollars.

Would you like more details on this, or have any further questions?

Related Questions:

1. How does the sealed bid method ensure fairness in inheritance disputes?
2. What happens if two people place identical bids in a sealed bid auction?
3. Could Abby have received any other item based on the bids? Why or why not?
4. How is excess money redistributed after the sealed bids process?
5. If Abby had bid differently on the tractor, how might that have affected the final allocation?

Tip: Always compare each person’s total bids with the number of items when calculating fair shares in sealed bids!