Descending the Stable Matching Lattice: How many Strategic Agents are required to turn Pessimality to Optimality?
Abstract
The set of stable matchings induces a distributive lattice. The supremum of the stable matching lattice is the boyoptimal (girlpessimal) stable matching and the infimum is the girloptimal (boypessimal) stable matching. The classical boyproposal deferredacceptance algorithm returns the supremum of the lattice, that is, the boyoptimal stable matching. In this paper, we study the smallest group of girls, called the {\em minimum winning coalition of girls}, that can act strategically, but independently, to force the boyproposal deferredacceptance algorithm to output the girloptimal stable matching. We characterize the minimum winning coalition in terms of stable matching rotations and show that its cardinality can take on any value between $0$ and $\left\lfloor \frac{n}{2}\right\rfloor$, for instances with $n$ boys and $n$ girls. Our main result is that, for the random matching model, the expected cardinality of the minimum winning coalition is $(\frac{1}{2}+o(1))\log{n}$. This resolves a conjecture of Kupfer \cite{Kup18}.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.15748
 arXiv:
 arXiv:2007.15748
 Bibcode:
 2020arXiv200715748N
 Keywords:

 Mathematics  Combinatorics