Question

1. There are four objects and four people. Object A is held by person WA, B by XB, C by YC and D by ZD.   Here are their preferences:

WA   XB   YC   ZD

B    C    A    A

A    D   C    C

C    A   B    D

D    B    D   B

Use the Shapley-Scarf method of one-sided matching (where people point at the holder of their favorite object and we look for circles of pointing) to redistribute the objects.  

Answer 1

In Shapley-Scarf method, each person point towards the person possessing his most preferred item.

In this case, for example, WA prefers to have house B and house B is currently held by XB. So, WA point towards XB.

Similarly, XB points towards YC, YC point towards WA and ZD point towards WA. This is shown in table below,

Now, we have to look at the longest possible circle. Here, WA --> XB --> YC --> WA is the longest complete circle.

In the complete circle, switching happens. WA gets B (from XB), XB gets C (from YC) and YC gets A (from WA). They get their first preferences and exit from market, leaving ZD with D.

Thus the final possessions will be,

WA --> B

XB --> C

YC --> A

ZD --> D