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If such a pair exists, the matching is not stable, in the sense that the members of this pair would prefer to leave the system and be matched to each other, possibly leaving other participants unmatched. A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one. [3]
If the stable roommates problem instance has a stable matching, then there is a stable matching contained in any one of the stable tables. Any stable subtable of a stable table, and in particular any stable subtable that specifies a stable matching as in 2, can be obtained by a sequence of rotation eliminations on the stable table.
In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements
Matching cards are removed from the game when paired. Concentration is a round game in which all of the cards are laid face down on a surface and two cards are flipped face up over each turn. The object of the game is to turn over pairs of matching cards. Concentration can be played with any number of players or as a solitaire or patience game ...
Matching games are games that require players to match similar elements. Participants need to find a match for a word, picture, tile or card. For example, students place 30 word cards; composed of 15 pairs, face down in random order. Each person turns over two cards at a time, with the goal of turning over a matching pair, by using their memory.
The winner is the player who has eventually collected a pair of every rank. Jokers can be used to create a pair by asking another player if they have any jokers in their hand. Two jokers form one pair. Jokers can be used as wild cards. Instead of going round in a circle, the turn switches to the last player who said "Go Fish".
A pairing-strategy for Breaker requires a set of element-pairs such that: All pairs are pairwise-disjoint; Every winning-set contains at least one pair. Whenever Maker picks an element of a pair, Breaker picks the other element of the same pair. At the end, Breaker has an element in each pair; by condition 2, he has an element in each winning-set.
A major reason that the blossom algorithm is important is that it gave the first proof that a maximum-size matching could be found using a polynomial amount of computation time. Another reason is that it led to a linear programming polyhedral description of the matching polytope, yielding an algorithm for min-weight matching. [4]