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Can you help with this statement?

Let N be a set of agents in a two-sided matching market. Apply a version of the deferred acceptance alghoritm and suppose it produces the allocation µ in which agent i is matched with itself. Suppose now that we delete agent i from the set of agents and from the preferences of each of the other agents. (a) is µ stable also in the new model? (b) If we apply the same version of deferred acceptance in the new model, will the outcome will be necessarily the same as before (that is the allocation µ)?

I think that answer (a) is because if i is matched with itself is impossible to find a blocking coalition. I think also that the answer (b) is yes. Because if it is unmatched it means that either she prefers to remain alone or no one wants her. Any suggestions?

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  • $\begingroup$ Are you talking about 1-1 matchings? $\endgroup$
    – Michael Greinecker
    May 25 at 16:41
  • $\begingroup$ Yes, 1:1 matching $\endgroup$
    – userF
    May 25 at 16:42
  • $\begingroup$ If you have learned that already, this follows from side optimality and the lone wolf/rural hospital theorem. $\endgroup$
    – Michael Greinecker
    May 25 at 16:46
  • $\begingroup$ Can you give me some references on this? I am not aware of it $\endgroup$
    – userF
    May 25 at 17:08
  • $\begingroup$ I learned this thigs first from the game theory book by Maschler, Solan, and Zamir $\endgroup$
    – Michael Greinecker
    May 25 at 17:10

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