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I could use some help with this: Assume that Arthur, Bruce, Clark and Diana are looking for jobs. The jobs available are dolphin-rider, detective, radiologist and cattle roper.

Arthur's preferences: dolphin-rider, detective, radiologist, cattle roper. Bruce's preferences: dolphin-rider, detective, radiologist, cattle roper. Clark's preferences: dolphin-rider, cattle roper, radiologist, detective. Diana's preferences: detective, radiologist, cattle roper, dolphin-rider.

Their preferences aren't the only criteria, though. There is the question of skills:

Arthur's skills: dolphin-rider, radiologist, cattle roper, detective. Bruce's skills detective, cattle roper, dolphin-rider, radiologist. Clark's skills: radiologist, dolphin-rider, cattle roper, detective. Diana's skills: cattle roper, detective, dolphin-rider, radiologist.

It is clear that a matching will be an assignment of each person to a unique job. (a) Give a nice intuitive definition of a "stable matching" in this context. (b) Find a stable matching. Here is what I was thinking: (a). A stable matching in this context would be the four people will be assigned to a job they have the highest skill in and one they will prefer in order to keep them happy and productive. (b). Arthur is a Dolphin-rider, Bruce is a detective, Clark is a radiologist, and Diane is a cattle-roper.

Not sure if that would be a suitable definition, or if my answer for b would work for my definition.

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  • $\begingroup$ You can almost treat the skills as "which job prefers this person the most?" $\endgroup$ – Vladhagen Nov 12 '13 at 19:02
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I would say that a stable matching consists of placing each person in a job such that if they desire to go to another job, the person in that job has higher skill at it than they do. (i.e. the job does not "desire" them.)

A:Dolphin (He desires it most and he is most skilled at it)

B:Detective (Wants Dolphin Rider more, but Arthur is more skilled)

C:Radiologist (Desires more to ride dolphins, but he is not the most skilled at the job. Also would rather be a cattle roper, but Diana is more skilled.)

D:Cattle Roper. (Desires more to be a Radiologist, but Clark is more skilled. Desires more to be a detective, but Bruce is more skilled.)

So it looks like you have a nice write up.

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