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10.4 Stable Matchings using visual studio .net toprint qr code on asp.net web,windows application Recommended GS1 barcodes for mobile apps The stable matching pro .net framework qr codes blem was introduced as a model of how to assign students to colleges. Since its introduction, it has been the object of intensive study by both computer scientists and economists.

In computer science it used as vehicle for illustrating basic ideas in the analysis of algorithms. In economics it is used as a stylized model of labor markets. It has a direct real-world counterpart in the procedure for matching medical students to residencies in the United States.

The simplest version of the problem involves a set M of men and a set W of women. Each m M has a strict preference ordering over the elements of W and each w W has a strict preference ordering over the men. As before the preference ordering of agent i will be denoted i and x i y will mean that agent i ranks x above y.

A matching is an assignment of men to women such that each man is assigned to at most one woman and vice versa. We can accommodate the possibility of an agent choosing to remain single as well. This is done by including for each man (woman) a dummy woman (man) in the set W (M) that corresponds to being single (or matched with oneself).

With this construction we can always assume that . M. = . W . . As in the house alloc Visual Studio .NET QR Code 2d barcode ation problem a group of agents can subvert a prescribed matching by opting out.

In a manner analogous to the house allocation problem, we can de ne a blocking set. A matching is called unstable if there are two men m, m and two women w, w such that. (i) m is matched to w, (ii) m is matched to w , and (iii) w m w and m w m The pair (m, w ) is cal qr barcode for .NET led a blocking pair. A matching that has no blocking pairs is called stable.

. mechanism design without money Example 10.8 The preference orderings for the men and women are shown in the table below m1 m2 m3 w1 w2 w3 w2 w1 w3 w1 w3 w2 w1 w2 w3 m1 m3 m2 m3 m1 m2 m1 m3 m2 Consider the matching { .net vs 2010 QR Code 2d barcode (m1 , w1 ), (m2 , w2 ), (m3 , w3 )}. This is an unstable matching since (m1 , w2 ) is a blocking pair.

The matching {(m1 , w1 ), (m3 , w2 ), (m2 , w3 )}, however, is stable. Given the preferences of the men and women, is it always possible to nd a stable matching Remarkably, yes, using what is now called the deferred acceptance algorithm. We describe the male-proposal version of the algorithm.

De nition 10.9 (Deferred Acceptance Algorithm, male-proposals) First, each man proposes to his top-ranked choice. Next, each woman who has received at least two proposals keeps (tentatively) her top-ranked proposal and rejects the rest.

Then, each man who has been rejected proposes to his top-ranked choice among the women who have not rejected him. Again each woman who has at least two proposals (including ones from previous rounds) keeps her top-ranked proposal and rejects the rest. The process repeats until no man has a woman to propose to or each woman has at most one proposal.

At this point the algorithm terminates and each man is assigned to a woman who has not rejected his proposal. Notice that no man is assigned to more than one woman. Since each woman is allowed to keep only one proposal at any stage, no woman is assigned to more than one man.

Therefore the algorithm terminates in a matching. We illustrate how the (male-proposal) algorithm operates using Example 10.8 above.

In the rst round, m1 proposes to w2 , m2 to w1 , and m3 to w1 . At the end of this round w1 is the only woman to have received two proposals. One from m3 and the other from m2 .

Since she ranks m3 above m2 , she keeps m3 and rejects m2 . Since m3 is the only man to have been rejected, he is the only one to propose again in the second round. This time he proposes to w3 .

Now each woman has only one proposal and the algorithm terminates with the matching {(m1 , w2 ), (m2 , w3 ), (m3 , w2 )}. It is easy to verify that the matching is stable and that it is different from the one presented earlier. Theorem 10.

10 The male propose algorithm terminates in a stable matching.. proof Suppose not. Then there exists a blocking pair (m1 , w1 ) with m1 matched to w2 , say, and w1 matched to m2 . Since (m1 , w1 ) is blocking and w1 m1 w2 , in the proposal algorithm, m1 would have proposed to w1 before w2 .

Since m1 was not matched with w1 by the algorithm, it must be because w1 received a proposal from a man that she ranked higher than m1 . Since the algorithm matches her to m2 it follows that m2 w1 m1 . This contradicts the fact that (m1 , w1 ) is a blocking pair.

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