Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 18 To recap Tuesday We set up the basic model of two sided one to one matching Two finite populations call them Men and Women who want to match to mates from the other group or stay alone Strict preferences over the other group or being alone We defined a matching as a pairing up of some men and women We said a matching was stable if nobody is paired up who would rather be alone no pair m w would both rather be with each other than with their assigned mate note how helpful strict preferences are in simplifying these conditions We proved the result from Gale and Shapley who proved the existence of a stable matching for any set of preferences by providing an algorithm that finds one step through the deferred acceptance algorithm We showed that not only does the deferred acceptance algorithm end on a stable matching but when the men are proposing it ends on the stable matching that every man in the population weakly prefers to every other stable matching the men optimal stable matching And we showed that if all the men unanimously weakly prefer a stable matching to another stable matching 0 then all the women weakly prefer 0 to So all the players on one side of the market have aligned preferences over stable matchings and roughly they all agree on which matching is best but their preferences are opposed to the preferences of the players on the other side of the market if there is more than one stable matching We wrapped up by introducing the notion of a lattice and claiming that the set of stable matchings formed a lattice but we didn t have time to prove it so that s where we ll pick up today 1 Lattice Theorem Tuesday we got nearly all the way through proving the Lattice Theorem For a general set with a partial order we can define the meet of two points as their least upper bound that is the point if it exists which is lower than every point which is higher than both x and y And we can define the join as the greatest lower bound the point if it exists which is higher than every point which is lower than both x and y A lattice is any set with a partial order which is closed under meet and join that is a partially ordered set is a lattice if for any two points in their set the subset of points that are greater than both has a minimal point and the subset of points which are less than both has a maximal point What we re trying to show is that with the partial order provided by the preferences of the men that is one matching is weakly greater than another if every man weakly prefers it the set of stable matchings of a given marriage market turns out to be a lattice Formally take some marriage market and let and 0 be two stable matchings Define a new mapping M W M W such that m is whichever of m and 0 m is preferred by m w is whichever of w and 0 w is least preferred by w So we can think of as the pairwise max of and 0 from the mens perspective and the pairwise min from the womens perspective Then it turns out that is not just an arbitrary map but it s a matching and it turns out to be a stable matching We could do the same thing with a new mapping which matches men to the worse of m and 0 m and women to the better of w and 0 w and this would also be a stable matching This means that for any two stable matchings and 0 sup 0 and inf 0 are also stable matchings so the set of stable matchings is closed under sup and inf and is therefore a lattice which gives us a bunch of nice mathematical structure 2 Last class we managed to slog through the proof that is a matching that is that w m if and only if m w We still need to show it s a stable matching That is we need to show that if and 0 are stable matchings then is individually rational and unblocked Individual rationality follows from individual rationality of and 0 Whoever you match to under you must have matched to under either or 0 so if those are IR so is So now suppose there was a pair m w that blocked If m prefers w to m then by definition he prefers w to both m and 0 m And if w prefers to m to w then she prefers to m to at least one of w and 0 w If she prefers m to w and he prefers her to m then they would have blocked similarly with 0 in the other case So is a stable matching It s not hard to show that it s the minimal matching that is men preferred to both and 0 Every man just does exactly as well under as under his favorite of and 0 so be any other matching stable or not which is preferred by all men to and 0 has to be bigger than So is the least upper bound of and 0 since it s a stable matching the set of stable matchings is closed under the supremum operator And we can do the same with the infimum And so we learn that the set of stable matchings is a lattice It actually has some additional properties as a lattice technically it s a complete distributive lattice Roth and Sotomayor give an example taken from Knuth where the lattice theorem was proved first I think of a marriage market four men four women and a set of preferences which has 10 stable matchings and shows how they are arranged according to the partial order 1 2 3 4 5 6 7 8 9 10 While you could come up with an example where there are more than two matchings in an indifference class the rest of this structure has to hold in a lattice for any indifference class with more than one point there must be a single point in the next group up and in the next group down otherwise there would not be a well defined sup or inf 3 Another cool result Roth and Sotomayor give this as a corollary of a different result but it follows pretty directly from the lattice result we just proved Take two matching and 0 and let M and M 0 be the set of men who match to some woman don t end up alone under each and W and W 0 likewise Let be the meet of and 0 as defined above let M and W be the men and women who end up matched up Now if m matches under it s because he prefers his mate to being alone so if …