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

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DCS/CSCI 2350: Social & Economic Networks

Matching Markets

Readings: Ch. 10 of EK & Wikipedia (stable marriage)

Mohammad T . Irfan

4/19/16 ¡ 2 ¡ Alvin Roth Nobel Prize 2012

Stable marriage problem

u http://en.wikipedia.org/wiki/

Stable_marriage_problem Lloyd Shapley Nobel Prize 2012

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Applications

u Medical residency matching u Kidney exchange program

Applications

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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

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Matching market

Starter model: Buyers mark goods acceptable or not

Bipartite matching problem

Each link: The room is “acceptable” by the student

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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

its neighbor set N(S) has less number of nodes

u |N(S)| < |S|

u Constricted set è Perfect

matching is impossible

u Reverse is also true!

S N(S)

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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.

u Perfect matching u Matching Theorem/Hall’s Theorem

u There exists a perfect matching if and only if

there’s no constricted set

Review

A bipartite graph One perfect matching Another perfect matching

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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:

Social welfare = 130

Alice 70, 20, 10 Bob 80, 20, 0 Cindy 50, 40, 10 Room 1 Room 2 Room 3 10 80 40

Social welfare = 100

70 20 10

How to find a perfect matching that maximizes the social welfare? Optimal assignment

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More general matching markets

Valuations and optimal assignment

Model

u n sellers, each is selling a house

u pi = price of seller i’s house

u n buyers

u vij = buyer j’s valuation of seller i’s house (or house i) u (vij – pi) 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)

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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