The stable matching problem as described in the text assumes

The stable matching problem, as described in the text, assumes that all men and women have a fully ordered list of preferences. In this problem, we will consider a version of the problem in which men and women can be indifferent between certain options. As before, we have a set M of n men and a set W of n women. Assume each man and each woman ranks the members of the opposite gender, but now we allow ties in the ranking. For example (with n = 4), a woman could say that m1 is ranked in first place; second place is a tie between m2 and m3 (she has no preference between them); and m4 is in last place. We will say that w prefers m to m0 if m is ranked higher than m0 on her preference list (they are not tied). With indifferences in the rankings, there could be two natural notions of stability. And for each, we can ask about the existence of stable matchings.

(a) A strong instability in a perfect matching S consists of a man m and a woman w, such that each of m and w prefers the other to their partner in S. Does there always exist a perfect matching with no strong instability? Either give an example of a set of men and women with preference lists for which every perfect matching has a strong instability or give an algorithm that is guaranteed to find a perfect matching with no strong instability.

(b) A weak instability in a perfect matching S consists of a man m and a woman w, such that their partners in S are w 0 and m0 , respectively, and one of the following holds:

• m prefers w to w 0 , and w either prefers m to m0 or is indifferent between these two choices; or

• w prefers m to m0 , and m either prefers w to w 0 or is indifferent between these two choices.

In other words, the pairing between m and w is either preferred by both, or preferred by one while the other is indifferent. Does there always exist a perfect matching with no weak instability? Either give an example of a set of men and women with preference lists for which every perfect matching has a weak instability; or give an algorithm that is guaranteed to find a perfect matching with no weak instability.

Solution

Humans are the most unpredictable creatures on earth because we let emotions rule over pure logic. As long as humans have emotions, you will never be able to predict with absolute certainty how any one person (let alone an entire group of people) will react to the same circumstances.

Likewise, you cannot predict with absolute certainty how the same person will react to the same set of circumstances on a different day. You can formulate probabilities that if A, B, and C occur then X, Y, and Z should happen (cause and effect.) But there are just too many variables within humans involving societal norms and morés, environment, upbringing, chemical imbalances within the body, DNA and genome aberrations, etc. etc.