Welcome back to class! In Section 8-3, we are diving into a topic that applies mathematics to a very human problem: conflict resolution. Specifically, we are looking at Fair Division. Whether it is dividing an estate among siblings, splitting assets in a divorce, or even just sharing a cake, math provides logical frameworks to ensure every party feels they received a fair portion of the goods.

This section covers three primary methods for determining fair shares:

1. The Divide-and-Choose Procedure

This is the most intuitive method, often used by children. One person divides the item into two parts, and the other person chooses which part they want. This ensures fairness because the divider is motivated to make the parts as equal as possible, knowing they will likely be left with the smaller piece if they don't.

  • The Lone-Divider Method: This is an extension of divide-and-choose applied to three people. One person cuts the assets into three piles, and the other two choose. If the choosers pick different piles, the divider gets the remaining one. If they pick the same pile, further division is required.

2. The Adjusted Winner Procedure

This method is excellent for two people dividing discrete items (like a house, car, or furniture). Each person is given 100 points to distribute privately among the items based on how much they value them.

  • Step 1: Initially, each item is given to the person who bid the most points on it.
  • Step 2: Calculate the total points each person received. The person with more points is the Leader; the other is the Trailer.
  • Step 3: To make it fair, items (or percentages of items) are transferred from the Leader to the Trailer. We calculate ratios ($ \frac{\text{Leader's Bid}}{\text{Trailer's Bid}} $) to determine which items to move first (starting with the lowest ratio).
  • Step 4: Finally, we divide the Critical Item to equalize the point totals using this equation:

$$ \text{Trailer's Score} + p(\text{Trailer's Bid}) = \text{Leader's Score} - p(\text{Leader's Bid}) $$

Here, $p$ represents the percentage of the critical item transferred to the Trailer.

3. The Knaster Procedure (Method of Sealed Bids)

This method is commonly used for inheritance involving more than two heirs and large items like real estate or cars. It uses a cash pot, often called a "kitty," to balance the value.

  1. Each heir submits a secret dollar bid for the item.
  2. The item goes to the highest bidder.
  3. For a fair division among $n$ people, the fair share is defined as $ \frac{\text{Bid}}{n} $.
  4. The winner pays the difference between their bid and their fair share into the kitty. Non-winners withdraw their fair share (cash) from the kitty.
  5. Any surplus cash remaining in the kitty is divided equally among all parties.

These notes contain detailed examples, such as the division of a family farm and an inheritance between John and Faye. Please review the step-by-step calculations in the attached PDF before taking the quiz.

Next Steps:

  • Review the detailed examples in the attached PDF.
  • Watch the lecture video for a walkthrough of the equations.
  • Complete the Section 8-3 Quiz.