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A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one. [3] The stable matching problem has also been called the stable marriage problem, using a metaphor of marriage between men and women, and many sources describe the Gale–Shapley algorithm in terms of marriage proposals. However ...
Algorithms to solve the hospitals/residents problem can be hospital-oriented (as the NRMP was before 1995) [15] or resident-oriented. This problem was solved, with an algorithm, in the same original paper by Gale and Shapley, in which the stable marriage problem was solved. [9]
Gale's 1962 paper with Lloyd Shapley on the stable marriage problem provides the first formal statement and proof of a problem that has far-reaching implications in many matching markets. The resulting Gale–Shapley algorithm is currently being applied in New York and Boston public school systems in assigning students to schools. In 2012 The ...
The Gale–Shapley algorithm can be used to construct two special lattice elements, its top and bottom element. Every finite distributive lattice can be represented as a lattice of stable matchings. The number of elements in the lattice can vary from an average case of e − 1 n ln n {\displaystyle e^{-1}n\ln n} to a worst-case of exponential.
Robert W. Irving [1] extended the Gale–Shapley algorithm as shown below to provide such a weakly stable matching in time, where n is the size of the stable marriage problem. Ties in the men and women's preference lists are broken arbitrarily. Preference lists are reduced as the algorithm proceeds.
Top trading cycle (TTC) is an algorithm for trading indivisible items without using money. It was developed by David Gale and published by Herbert Scarf and Lloyd Shapley . [ 1 ] : 30–31
The algorithm consists of two phases. In Phase 1, participants propose to each other, in a manner similar to that of the Gale-Shapley algorithm for the stable marriage problem. Each participant orders the other members by preference, resulting in a preference list—an ordered set of the other participants.
The game is a potential game (Monderer and Shapley 1996-a,1996-b) The game has generic payoffs and is 2 × N (Berger 2005) Fictitious play does not always converge, however. Shapley (1964) proved that in the game pictured here (a nonzero-sum version of Rock, Paper, Scissors), if the players start by choosing (a, B), the play will cycle ...