DCS/CSCI 2350: Social & Economic Networks Matching Markets - PDF document

4/19/16 ¡ DCS/CSCI 2350: Social & Economic Networks Matching Markets Readings: Ch. 10 of EK & Wikipedia (stable marriage) Mohammad T . Irfan 1 ¡

4/19/16 ¡ Alvin Roth Nobel Prize 2012 Stable marriage problem u http://en.wikipedia.org/wiki/ Stable_marriage_problem Lloyd Shapley Nobel Prize 2012 2 ¡

4/19/16 ¡ Applications u Medical residency matching u Kidney exchange program Applications 3 ¡

4/19/16 ¡ Stable marriage problem u Given n men and n women, where each man ranked all women and each woman ranked all men, find a stable matching. u Stable matching u Following cannot happen: X and Y are not matched to each other, but they prefer each other over their matched partners. u Is there a matching such that u Everyone is married (perfect matching) u The matching is stable u Yes, Gale-Shapley algorithm (1962) Gale-Shapley algorithm u Men-proposing version (men-optimal) u As long as there is a “free” (unmatched) man u A free man X proposes to his top-ranked woman Y who hasn’t yet rejected him u That woman Y keeps her most preferred suitor and rejects the other (if any), who becomes a free man u Demo u http://mathsite.math.berkeley.edu/smp/smp.html u Exhibit Walkthrough 4 ¡

4/19/16 ¡ Matching market Starter model: Buyers mark goods acceptable or not Bipartite matching problem Each link: The room is “acceptable” by the student 5 ¡

4/19/16 ¡ Perfect matching u Choice of edges in the bipartite graph so that each node is the endpoint of exactly one of the chosen edges. Dark edges are the chosen edges—also known as the assignment Constricted set u Delete edge (Room3, Vikram) from the previous example u A set of nodes S is constricted if N(S) its neighbor set N(S) has less S number of nodes u |N(S)| < |S| u Constricted set è Perfect matching is impossible u Reverse is also true! 6 ¡

4/19/16 ¡ Matching theorem Konig (1931), Hall (1935) u Gives a characterization of perfect matching u A bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching if and only if it contains a constricted set. Review u Perfect matching A bipartite graph One perfect matching Another perfect matching u Matching Theorem/Hall’s Theorem u There exists a perfect matching if and only if there’s no constricted set 7 ¡

4/19/16 ¡ Review u Stable marriage problem u Given the rank lists of men and women, find a stable perfect matching u “Accept or not” model of dorm room allocation u Find a perfect matching u Model with valuations for the rooms u Find a perfect matching that maximizes the sum of the valuations u Social welfare = sum of the valuations Example of model with valuations u Many different perfect matchings: 70 Alice Room 1 70, 20, 10 80 20 Bob 10 Room 2 80, 20, 0 10 40 Cindy Room 3 50, 40, 10 Social welfare = 130 Social welfare = 100 How to find a perfect matching that maximizes the social welfare? Optimal assignment 8 ¡

4/19/16 ¡ More general matching markets Valuations and optimal assignment Model u n sellers, each is selling a house u p i = price of seller i’s house u n buyers u v ij = buyer j’s valuation of seller i’s house (or house i) u (v ij – p i ) is buyer j’s payoff if he buys house i u Assumption: buyers are not stupid u They are only interested in buying houses that maximize their payoff u That maximum payoff must also be non-negative u Preferred seller graph u Bipartite graph between buyers and sellers where every edge encodes a buyer’s maximum payoff (>= 0) 9 ¡

4/19/16 ¡ What we want u A perfect matching in the preferred seller graph u Market clearing prices (MCP): The set of prices at which we get a perfect matching u It would be awesome if the perfect matching is also an optimal assignment u Maximizes social welfare (i.e., sum of the buyers’ valuations in that assignment) Next u Show: Any MCP gives an optimal assignment u Does an MCP always exist? u Constructive proof (by an algorithm) u Single-item auction as a matching market 10 ¡