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 adjusting lists of complete wosociety by the society are the identical, and analogously, the adjusting of complete society by the wosociety are the identical. In other signification, there is a accord among the wosociety who is the most cosset fiction, the second cosset fiction and so on.

    Argue that then there is barely undivided firm contesting. What is it? (music that in open, the TMA finds undivided of potential multiple firm contestings). Don’t pretermit to argue its stoppage.

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

    Expert Vindication

     

    According to the abandoned resembling problem:

    Consider ‘n’ populace of each gender and now estimate the society and wosociety 1 through n in classify of their adjust by the contrary gender. From the conviction the undivided and barely firm contesting has wosociety i contest with fiction i ce complete i = 1…n.

    To argue this we deficiency to illusion couple things.

    1) No other resembling is firm

    2) Whether this resembling is firm

    In classify to argue principal (1):

    • Assume a resembling where referable everyundivided is of the identical adjust as his/her colleague. Then there earn be at last undivided brace of individual of the identical adjust who is referable contested to each other.
    • Let j be the adjust of the couple pre-eminent adjusted individuals who are referable contested each other. Since this is the pre-eminent adjusted individuals who are referable contested coincidently, it must be that each part of the brace is contested with someundivided of inferior adjust.
    • From the over government it is gentleobject that fiction j would elect wofiction j to his general colleague and wofiction j would elect fiction j to her general colleague, so the contemplated resembling is referable firm.
    • This argues that if everyundivided adjusts the contrary gender in a resembling method, then no resembling is firm in which individuals are referable contested with someundivided of their avow adjust.

    In classify to argue second (2):

    • Consider that a fiction i and wosociety i are contestes ce each i. Suppose couple populace of contrary gender elect each other to their colleague in the primary contest.
    • We can declare that this colleagueship is referable the i to i contesting and undivided of these couple populace must keep eminent adjust than the other.
    • Now the individual with the eminent adjust is generally contested to a colleague of his/her avow adjust. So from the over convictions, he/she elects his/her general colleague to the colleague in the contemplated resource contest.
    • Thus there can be no couple populace who elect each other to their assigned colleagues.