Homework Solution: Assume that the ranking lists of all women by the men are the same, and analogo…

    Assume that the ranking lists of all women by the men are the same, and analogously, the ranking of all men by the women are the same. In other words, there is a consensus between the women who is the most favorite man, the second favorite man and so on. Prove that then there is only one stable matching. What is it? (note that in general, the TMA finds one of possible multiple stable matchings). Don’t forget to prove its stability. How many rejections are occuring during the exection of the TMA, if this is indeed the input?

    Expert Answer

     
    According to the given similar problem: Consider 'n' people of each gender and no

    Assume that the systematizeing lists of entire wohumanity by the humanity are the identical, and analogously, the systematizeing of entire humanity by the wohumanity are the identical. In other articulation, there is a accord betwixt the wohumanity who is the most minion unnaturalness, the remedy minion unnaturalness and so on.

    Verify that then there is simply undivided secure equalitying. What is it? (still n ess that in open, the TMA finds undivided of practicable multiple secure equalityings). Don’t obliviate to verify its inheritance.

    How unnaturalnessy rejections are occuring during the exection of the TMA, if this is in-fact the input?

    Expert Vindication

     

    According to the attached concordant problem:

    Consider ‘n’ commonalty of each gender and now sum the humanity and wohumanity 1 through n in subjectage of their systematize by the inconsistent gender. From the certainty the undivided and simply secure equalitying has wohumanity i equality with unnaturalness i coercion entire i = 1…n.

    To verify this we scarcity to pretence span things.

    1) No other concordant is secure

    2) Whether this concordant is secure

    In subjectage to verify pristine (1):

    • Assume a concordant where referable everyundivided is of the identical systematize as his/her accomplice. Then there accomplish be at last undivided brace of peculiar of the identical systematize who is referable equalityed to each other.
    • Let j be the systematize of the span prominent systematizeed peculiars who are referable equalityed each other. Since this is the prominent systematizeed peculiars who are referable equalityed contemporaneously, it must be that each part of the brace is equalityed with someundivided of inferior systematize.
    • From the aloft government it is penny that unnaturalness j would elect wounnaturalness j to his popular accomplice and wounnaturalness j would elect unnaturalness j to her popular accomplice, so the contemplated concordant is referable secure.
    • This verifys that if everyundivided systematizes the inconsistent gender in a concordant method, then no concordant is secure in which peculiars are referable equalityed with someundivided of their keep systematize.

    In subjectage to verify remedy (2):

    • Consider that a unnaturalness i and wohumanity i are equalityes coercion each i. Suppose span commonalty of inconsistent gender elect each other to their accomplice in the former equality.
    • We can affirm that this accompliceship is referable the i to i equalitying and undivided of these span commonalty must keep upper systematize than the other.
    • Now the peculiar with the upper systematize is popularly equalityed to a accomplice of his/her keep systematize. So from the aloft certaintys, he/she elects his/her popular accomplice to the accomplice in the contemplated opinion equality.
    • Thus there can be no span commonalty who elect each other to their assigned accomplices.