5 Better Off Alone In the stable marriage problem, suppose that some men and wom
ID: 2257194 • Letter: 5
Question
5 Better Off Alone In the stable marriage problem, suppose that some men and women have standards and would not just settle for anyone. In other words, in addition to the preference orderings they have, they prefer being alone to being with some of the lower-ranked individuals (in their own preference list). A pairing could ultimately have to be partial, i.e., some individuals would remain single. The notion of stability here should be adjusted a little bit. A pairing is stable if there is no paired individual who prefers being single over being with his/her current partner there is no paired man and single woman (or paired woman and single man) that would both prefer to be with each other over being single or with his/her current partner there is no paired man and paired woman that would both prefer to be with each other over their current partners, and . there is no single man and single woman that would both prefer to be with each other over being single.Explanation / Answer
Each individual is in love with the online mate. The individual puts his preference list after the lasts person he/she is willing to marry.
To illustrate this, let us see the following example: there are 3 men and 3 women. The preference list for men is:
Man
Preference List
M1
W1>W2
M2
W2>W1>W3
M3
W3
Preference list for women:
Woman
Preference List
W1
M1
W2
M3>M2>M1
W3
M2>M3>M1
In this example, M1 is willing to marry W1 & W2. He likes W1 better than W2, but he would be rather single than being paired with W3. On the other hand, M2 does not like to be single, he likes W2 then W1 and then W3. M3 has a high standard he will either marry W3 or remain single.
If we introduce an online mate for each person, for men lets say it is OM1,OM2, OM3 and for women it is OW1,OW2, OW3. Now the preference list for online mates will become
When the online mates are paired with men and women, stable pairing has to be found. Each individual who is not matched remains single.
This pairing concludes that stable pairing happens only in individual’s set up. If an individual is paired with an online mate, he/she would prefer to be with the online mate rather than human or being single. If an individual prefers to be paired with individual he/she prefers to be paired with an individual rather than being single or being with an online mate.
(b) Suppose a man M1 is paired with a woman W1 in stable pairing S and he is unpaired in stable pairing T.
A man M1 is unpaired with woman W1 in T. W1 must be paired with M2, whom she prefers over M1.
If W1 is paired with a man whom she does not prefer over M1, then it would be a rogue couple- a contradiction.
No man or woman can be paired or unpaired in one stable pairing. They have to be paired in all stable pairings or unpaired in all stable pairings.
Man
Preference List
M1
W1>W2
M2
W2>W1>W3
M3
W3
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