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What are the components of Fair Division?
1. There is something of value to be divided, called the booty.
2. There are parties seeking a share of the booty.
3. The players are willing participants and accept the outcome as binding.
4. Players act rationally and conform to their value system.
5. Players have no advantageous information on other players.
6. Players are entitled to a minimal share that is proportional.
What is a fair share?
Let S = the value something being shared. Let N = the number of participants
Comments:
1. Fair shares are relative. What is fair to one is no necessarily fair to another.
This is due to personal likes and dislikes.
2. Participants must accept a fair share even if it is not the portion they like most.
The goal is to get each member a fair share, not to maximize their share.
Fair Division Types
Discrete Division occurs when the booty consists of non- divisible items like a boat, car, house, painting, jewelry, etc.
Continuous Division occurs when the booty consists of items that can be cut up in an infinite number of ways. Continuous division can be applied to partitioning land, money, and food.
Divider–Chooser Method
This method is used anytime there is a continuous fair-division problem involving two participants. This method is also called “You Cut - I Choose”, and as the name suggests, the first participant cuts the metaphorical cake into two portions, and the second player gets to pick which piece they want. This method guarantees that each participant receives at least 50% in value according to their personal value system. Recall, the metaphorical cake can be anything of value which needs to be distributed, such as a house, car, money, etc.
Example 1:
Doug and Carla go to the county fair and win a chocolate-strawberry cake in a raffle. The cake is half chocolate and half strawberry as shown in the image. Doug and Carla decide to share the cake using the Divider-Chooser method.
This problem is really an exercise in negotiations.
1. Doug and Carla could be two people, two countries, or two business entities, and they want a fair share of the cake.
2. The cake represents something of value for which these two entities have expressed interest. The cake could represent land, money, ownership of a corporations, etc.
3. Proceed with this problem not as friends, but as negotiators looking to acquire a
fair share of the booty. The goal is not to maximize your share, and not to harm the other entity.
4. Likes and dislikes of each entity are private and cannot be leveraged against one another.
Suppose that Doug is the divider and Carla is the Chooser.
If Doug likes chocolate and strawberry the same, then how would Doug cut the cake?
There are three scenarios shown above. Which of these cuts would Doug make according to his personal value system? The answer, all are valid, meaning that Doug could potentially make any of the cuts shown above. If fact, there are infinitely many ways in which Doug could cut this cut and secure his fair share.
Let's assume that Doug makes cut #3.
Suppose Carla loves strawberry but hates chocolate. Which slice would Carla choose?
The answer should be obvious, Carla would choose Slice #2 because it has more strawberry.
Quantifying the Results
What percent of the cake’s value does each player feel they received?
Doug = 50%
Carla ≈ 66.67%
Explanation:
Doug likes both chocolate and strawberry the same, therefore the chocolate half of the cake holds 50% 0f the cake value and the strawberry half of the cake also holds 50% of the cake value. Therefore, as long as Doug receives 180˚ of the cake or more he will have received a fair share of the cake.
Carla loves strawberry and hates chocolate. This means that 100% of the cake value is in the strawberry half and 0% of the cake value is in the chocolate half. As long as Carla receives 90˚ of the strawberry cake, she will have received a fair share and be satisfied. Since Carla gets to choose which slice she wants, she chooses slice #2 which contains 120˚ of strawberry. This generates the following proportion and calculates that Carla feels as though she received 66.67% of the cake value.
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