Recitation 2: Game Theory and Intro to Matching
EconW4260: Market Design
Prepared by Janet Lu and Danyan Zha
February 7, 2016
Last recitation, we considered efficiency concepts assuming that we know everyones preferences. However, if we do not know everyo

Recitation 1: Efficiency
EconW4260: Market Design
Prepared by Janet Lu and Danyan Zha
January 24, 2016
As market designers, we want to design allocation mechanisms that achieve efficiency, fairness, and possibly revenue goals and non-manipulability. Today

1
MATCHING THEORY
Yeon-Koo Che
Columbia University
2
Two-Sided Matching
Theory on how individuals on one side (say men) match with
individuals on the other side (say women), given
Individuals preferences for potential partners on the other side.
Lack o

1
SPONSORED SEARCH
AUCTIONS
Yeon-Koo Che
2
3
4
5
Sponsored Search Auctions
Sponsored search is a huge auction market
Google revenue in 2012: $50.18 billion 2013: $59.83 billion.
Hal Varian, Google Chief Economist Most people dont realize that all
that

1
AUCTION THEORY
Yeon-Koo Che
2
Auctions in Practice
Search Engine
Ebay
Christies & Sothebys
Spectrum (FCC)
Oil & gas lease
Used-car
Foreclosure homes
Livestock
Treasury Bills
Fish
Flower
IPO
3
Auctions Around the World
Dutch Flower Auction:

1
MATCHING WITH
TRANSFERS
Yeon-Koo Che
2
From Matching to Markets
So far - allocating items to people (or people to people)
without money.
Today: efficient allocation with prices.
Define efficiency.
Auction designs to achieve efficient allocation.
As

1
SCHOOL CHOICE
Yeon-Koo Che
2
Background
Neighborhood schools: Traditionally, students are
assigned to public schools according to where they
live.
Limited and unequal freedom of choices
Starting with Minnesota in 1987, several school
districts adopted

1
MATCHING WITH
OBJECTS
Yeon-Koo Che
2
House allocation problems
In some matching markets, only one side of the
market has preferences (or we care mostly about the
preferences of one side).
Examples
students picking housing on campus
students choosing

1
KIDNEY EXCHANGE
Yeon-Koo Che
2
Kidney Exchange
Transplants are standard treatment for patients with failed
kidneys.
Shortage of kidneys
Over 100,476 patients in the waitlist (as of Feb14, 2016).
3000 patients added each month
Median waiting time (f

APPLICATION: NRMP
Yeon-Koo Che
NRMP redesign
We will discuss redesign of NRMP algorithm in 1990s as
a case study.
We will see how basic matching theory can be used for
economic design.
But we will also see why the basic theory is not enough
due to additi

1
Introduction to Market Design
Yeon-Koo Che
Columbia University
January, 2016
What we we shall do in this course
Designing mechanisms (or institutions) to allocate
scarce resources and creating marketplaces / platforms
using theory.
Start with an alloc

EconW4260
Due: February 17, 2016
Problem Set #2
1. (10 points) Consider a marriage problem. Prove that the men-optimal
stable matching is women-pessimalnamely, each woman matches with the
worst achievable man in the men-optimal stable matching.
2. (10 poi

EconW4260
Due: May 2nd, 2016
Problem Set #4 Solutions
1. (a) The lowest market clearing prices are $3, $1 and $0. The lowest
price for the bottom position is 0. The price p2 for the middle
position must be set to make bidder 3 indifferent. Bidder 3 makes

EconW4260
Due: February 17, 2016
Sketch of Answers to PS#2
1. (10 points) Consider a marriage problem. Prove that the men-optimal
stable matching is women-pessimalnamely, each woman matches with the
worst achievable man in the men-optimal stable matching.

Econ W4260
Total: 70 points
Model Answers for Problem Set #1
1. (20 points) The Pareto efficient allocations are any allocations cfw_(xA , yB ), (xB , yB )
such that xA = 1, xB = 0, yA [0, 2], and yB = 2 yA or such that xA = 0,
xB = 1, yA [0, 1), and yB =

EconW4260
Due: February 1, 2016
Problem Set #1
In many realistic circumstances, some goods are indivisible. Suppose there are two
goods, X and Y . Good X is indivisible in that a consumer can consume either one
or zero units. Good Y is money that is compl

EconW4260
Due: April 11, 2016
Problem Set #3
1. There are three agents A, B, C and two items X, Y in an assignment
problem. The values of the agents are given in the following table.
A
B
C
X
Y
6.3 4.2
3.1 4.8
5.4 2.3
(a) Find all competitive equilibrium p

EconW4260
Due: April 11, 2016
Problem Set #3
1. (20pts) There are three agents A, B, C and two items X, Y in an assignment problem. The agents values are given in the following table.
X
Y
A 6.3 4.2
B 3.1 4.8
C 5.4 2.3
(a) Find all competitive equilibrium

EconW4260
Due: May 2nd, 2016
Problem Set #4
1. (40pts) A search engine has three positions to sell. The positions receive 300, 200 and 100 clicks per day, respectively. There are three
advertisers, with perclick values $10, $7 and $2. If bidder i obtains