Section 8-3: Fair Division - What is a Fair Share?
Welcome to the review of Section 8-3! In this section, we delve into the fascinating world of fair division. Often, dividing assets, whether in a divorce, inheritance, or business dissolution, can be tricky. This section equips you with mathematically sound methods to ensure fairness and avoid disputes. Let's explore!
Key Concepts:
- Understanding Fair Division: It's all about finding mathematically robust ways to divide assets in various scenarios, aiming for a perceived sense of equity among all parties involved.
Methods of Fair Division:
- Divide-and-Choose Procedure:
This simple yet effective method works best with two parties. One person divides the item(s) into two parts, and the other person gets to choose which part they want. This ensures that the divider tries to make both parts as equal in value as possible, since they don't know which portion they will receive.
- The Lone-Divider Method:
The Lone-Divider Method applies the divide-and-choose procedure to parties of 3 or more. One person (Person 1) divides the assets into three parts, and then Persons 2 and 3 choose between the piles. If they both choose a different pile, then Person 1 gets the remaining pile. If they choose the same pile, then Person 1 chooses which pile they want and Persons 2 and 3 mix the remaining piles and perform the divide-and-choose procedure on the mixed remaining piles.
- Adjusted Winner Procedure:
This method is used when dividing multiple items between two people. Each person assigns points to each item, with the total points adding up to 100. The items are initially awarded to the person who bid the most points for them. Then, an adjustment process ensures both parties end up with an equal proportion of the total points.
Here's a step-by-step breakdown:
- Step 1: Initial Division: Award each item to the highest bidder. The person with the higher score is the point leader, and the other is the trailer.
- Step 2: Tied Items: Tied items go to the leader.
- Step 3: Calculate Ratios: Calculate the ratio of the leader's bid to the trailer's bid for each item the leader possesses. The ratio formula is expressed as follows: $$ rac{\text{leader's bid}}{\text{trailer's bid}}$$.
- Step 4: Transfer Items: Transfer items from the leader to the trailer based on increasing ratios, making sure not to reverse who leads. The item that reverses who leads is the critical item.
- Step 5: Divide the Critical Item: Find the percentage, $p$, of the critical item to give the trailer from the leader using the following formula: Trailer's Score + $p$ x Trailer's Bid = Leader's Score - $p$ x Leader's Bid
Example: Two siblings dividing inheritance: Guitar, Jewelry, Car, Library, Cash. Both siblings assign 100 points to allocate between them. The initial assignment is as follows:
Item My Points Sister's Points Guitar 35 10 Jewelry 10 10 Car 20 40 Library 15 10 Cash 20 30 I get the Guitar and the Library (50 points), and my sister gets the Car and the Cash (70 points). The Jewelry goes to my sister, as it's tied. Thus, my sister is 80% satisfied, while I'm at 50%, so we need to make adjustments. First, we must calculate the ratio of my sister's items relative to my items. These are listed in increasing order.
Ratio for Jewelry: $\frac{10}{10}$ = 1
Ratio for Cash: $\frac{30}{20}$ = 1.5
Ratio for Car: $\frac{40}{20}$ = 2.0
First, Jewelry is transferred to me. My points are now at 60 and my sister's are at 70. The next item (Cash) is the critical item because it will change the point leader. To determine how much to transfer, we can use the formula:
My score + $p$ percent of my cash bid = Sister's score - $p$ percent of her cash bid.
Which gives us the equation: 60 + 20$p$ = 70 - 30$p$, or $p$ = 0.20.
Thus, I get the Guitar, the Library, and 20% of the Cash. My sister gets the car and 80% of the cash. The final scores are:
My total points: 35 + 10 + 15 + 20 x 0.20 = 64.
My sister's total points: 40 + 30 x 0.80 = 64.
Both are 64% satisfied.
- The Knaster Procedure:
This procedure is suitable for dividing items among multiple parties, without necessarily dividing ownership. Each party assigns a monetary value to each item (their bid) without knowing the other parties' bids. The item goes to the highest bidder, who then pays the others their fair share in cash.
Fair division is not just a mathematical exercise; it's about ensuring everyone feels valued and respected. Keep practicing these methods, and you'll become a master of fair division!