There is a setof singles, and a matchmaker wants to figure out a way to match people together to help form marriages.
Call the set of men M, and the set of women W. Each man has preferences over the women in W, which we can denote P.
Now first, the matching has to be individually rational: this means that if a person is matched to a person of the opposite sex, then they should prefer being matched to that person to staying single.
Next, suppose we match all the men and women together, but (for example) m1 and w2 prefer each other to the people whom they were matched with. Then m1 and w2 can form a blocking pair and block the matching u, since they would rather be matched with each other. Thus, a stable matching u is one in which no people block it.
Now the thing about our current setup, with M and W, is that there's ALWAYS a stable matching in the market - we can find a matching which no individual or pair will block. In fact, there is a male-optimal and female-optimal matching.
Now, suppose you're in a marriage market, and you currently have an optimal matching u, but some people in your sex are considering extending their preferences to find some unattractive people of the opposite sex preferable to staying single. Do you want this?
Call the set of men M, and the set of women W. Each man has preferences over the women in W, which we can denote P.
Now first, the matching has to be individually rational: this means that if a person is matched to a person of the opposite sex, then they should prefer being matched to that person to staying single.
Next, suppose we match all the men and women together, but (for example) m1 and w2 prefer each other to the people whom they were matched with. Then m1 and w2 can form a blocking pair and block the matching u, since they would rather be matched with each other. Thus, a stable matching u is one in which no people block it.
Now the thing about our current setup, with M and W, is that there's ALWAYS a stable matching in the market - we can find a matching which no individual or pair will block. In fact, there is a male-optimal and female-optimal matching.
Now, suppose you're in a marriage market, and you currently have an optimal matching u, but some people in your sex are considering extending their preferences to find some unattractive people of the opposite sex preferable to staying single. Do you want this?
