Three partners are dividing a plot of land among themselves using the lone-divider method. After the divider D divides the land into three shares s1, s2 and s3, the choosers C1 and C2 submit their bids for these shares.
Suppose that the chooser’s bids are C1 = {s3}: C2 = {s1, s2}. Describe two different fair divisions of the land.
To describe two different fair divisions of the land using the lone-divider method, we need to consider the bids of the choosers and allocate the shares in a way that satisfies certain fairness criteria.
First Fair Division:
- In this division, we allocate the shares based on the chooser's bids.
- Since C1 has bid for share s3 and C2 has bid for shares s1 and s2, we assign s3 to C1 and split s1 and s2 equally between C2.
- Therefore, the first fair division will be: C1 = {s3}, C2 = {s1/2, s2/2}.
Second Fair Division:
- In this division, we allocate the shares in a way that prohibits envy between the choosers, meaning neither of them would prefer the other's share over their own.
- To achieve this, we can use the "envy-free" method, where each chooser ranks the shares in order of their preference, and we assign shares in such a way that no chooser envies the other.
- Let's assume C1's ranking is s1, s2, s3 (in order of preference) and C2's ranking is s3, s2, s1.
- We assign the shares based on the rankings. C1 gets their top choice, which is s1. C2 gets their second choice, which is s2, and the remaining share s3 goes to C2 as it's their top choice.
- Therefore, the second fair division will be: C1 = {s1}, C2 = {s2, s3}.
These two divisions demonstrate different ways to achieve fairness in allocating the land shares among the three partners using the lone-divider method.