Matching and Market Design:
Introduction, Bipartite Matching
NICOLE IMMORLICA, MICROSOFT RESEARCH NE
NICOLE IMMORLICA, MSR NE RESEARCHER
Background. Computer science, economics, applied research.
Ph.D. 2005 (in computer science).
Thesis: Computing with Strategic Agents.
Postdocs 2005-2008. Studied applications of:
theory to advertising markets at MSR,
combinatorics to economics at CWI.
Professor 2008-2012 (of computer science).
Taught courses and advised students in CS-econ.
Researcher 2012-present. Study CS-econ issues
from the very theoretical to the very applied.
NICOLE IMMORLICA, MSR NE RESEARCHER
Research. What are realistic utopias for selfish societies?
Mechanism Design: Can we systematically allocate
scare resources to people who need it the most?
Or sell the resources and generate high revenue?
Social Networks: What behavioral patterns can be
supported by various structures? What information
can be learned? How does behavior impact structure?
Market Design: What outcomes will be observed
in matching markets, and how can we facilitate
the matching process in practical settings?
COURSE LOGISTICS
Course. CS 286r: Topics at the Interface of Computer Science
and Economics – Matching and Market Design.
Website. www.Immorlica.com/marketDesign/Harvard
Lectures. Fridays, 9am-noon, MD 221.
Includes a 20 minute coffee break halfway through,
and you’re all invited to join me for lunch afterwards.
Teachers. Nicole Immorlica, Ran Shorrer, Brendan Lucier,
Scott Kominers, and more!
Workload. Readings/participation (around 20 pages/week),
2 problem sets (around 6 hrs each), course project (3 stages).
Outline
1. Introduction: markets in practice and theory,
discussion of market design
2. Bipartite Matching: elementary definitions,
max cardinality & max weight matchings
Part 1:
Introduction.
Markets
agents
objects
Markets are a medium of exchange.
Markets
kidneys
high
ad
slots
schools
workers
agents
objects
students
firms
patients
advertisers
Sponsored
Kidney
School
LaborExchange
Markets
Choice
Search
Examples
Traditional markets: shopping malls, eBay, ad
auctions, FCC spectrum auctions
School choice: allocation of students to schools
via centralized city-run program
Labor markets: NRMP, cadet-branch matching
Kidney exchange: matching of kidney donors to
compatible recipients
Market Design
Al Roth
Market Design
Market design involves a responsibility for
detail, a need to deal with all of a market’s
complications, not just its principle features.
Designers therefore cannot work only with
the simple conceptual models used for
theoretical insights into the general working
of markets. Instead, market design calls for
an engineering approach.
– from The Economist as an Engineer, by Al Roth
Market Design
…this paper makes the case that experimental
and computational economics are natural
complements to game theory in the work of
design. The paper also argues that some of the
challenges facing both markets involve dealing
with related kinds of complementarities, and
that this suggests an agenda for future
theoretical research.
– from The Economist as an Engineer, by Al Roth
Market Design
Develop simple theory,
…to deal with complexity in practice.
Computer Science
&
Economics
On Spherical Cows …
Computer Science: computability
P versus NP
P: set of problems that can be solved efficiently
(i.e., in polynomial time).
NP: set of problems whose solutions can be checked efficiently.
Computer Science: computability
3
6
8
1
2
9
4
3
2
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9
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9
Computer Science: computability
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Computer Science: computability
Traveling Salesman: find smallest route that visits each capitol.
Computer Science: computability
Student social
network:
Study groups of size two? Size three?
Economics: rationality
Many systems are composed of many
independent self-interested agents
These agents are
• rational, i.e. they act in their own self-interest
• and reason strategically, i.e. they take into
account the actions of others
Let’s play a game
Experiment: The median game.
1. Guess an integer in [1, …, 100].
2. Write your number on a piece of paper.
P R I Z E : The people whose numbers are closest
to 2/3 of the median win.
The Median Game
25
45
Jose
0
Julian
50
Bruce
Marcos
69
Nicole
Calculating
the winner:
winner!
1. Sort the numbers: 0, 25, 45, 50, 69
2. Pick the middle one (the median): 45
3. Compute 2/3 of the median: 30
Are you a winner?
Questions
Given computability & rationality assumptions:
• How will selfish agents behave?
• What properties emerge as a result of selfish
behavior?
• Is it possible to formulate the rules of the
system to encourage socially-optimal
behavior?
Market Design
Develop simple theory,
…to deal with complexity in practice.
Market Design Complexities
Thickness: need to attract a sufficient proportion
of potential market participants to come
together ready to transact with one another.
– from What have we learned from market design?, by Al Roth
Market Design Complexities
Congestion: must provide enough time or fast enough
transactions so that market participants can consider
enough alternatives to arrive at satisfactory ones.
– from What have we learned from market design?, by Al Roth
Market Design Complexities
Simplicity: must make it easy to participate in
market as opposed to transacting outside of the
marketplace or engaging in strategic behavior
that reduces overall welfare.
– from What have we learned from market design?, by Al Roth
Market Design Complexities
Others:
• Asset to be traded
• Nature of contracts
• Medium of exchange
• Measure of performance
• Need for design
• Market culture
• Fairness and repugnance
– from 1/22/14 post on The Leisure of the Theory Class, by Ricky Vohra
Example: traditional markets
Practice:
FCC spectrum auctions, eBay, ad auctions etc.:
sellers offer goods and services, buyers
purchase via posted prices or auctions
Example: traditional markets
Issues:
Strategic behavior, complex agent preferences,
price discovery, packages and deals
Example: traditional markets
Theory:
Existence of market-clearing prices,
approximately optimal simple mechanisms,
techniques to aid price discovery
Example: school choice
Practice:
Boston, New York City, etc:
students submit preferences about different
schools; matched based on “priorities” (e.g.,
test scores, geography, sibling matches)
Example: school choice
Issues:
NYC too slow to clear; Boston strategically
complicated, result in unstable matches, many
complaints in school boards
Example: school choice
Theory:
theorists proposed alternate mechanisms
including the Gale-Shapley algorithm for stable
marriage, schools adopt these
Example: entry-level labor markets
Practice:
National Residency Matching Program (NRMP):
physicians look for residency programs at
hospitals in the United States
Example: entry-level labor markets
Issues:
1950
1990
decentralized,
centralized
unraveling,
clearinghouse,
exploding offers 95% voluntary
inefficiencies
participation
dropping participation
sparks redesign to
accommodate couples,
system still in use
Example: entry-level labor markets
Theory:
NRMP central clearinghouse algorithm
corresponds to Gale-Shapley algorithm
Example: kidney exchange
Practice:
In 2005:
• 75,000 patients waiting for transplants
• 16,370 transplants performed (9,800 from
deceased donors, 6,570 from living donors)
• 4,200 patients died while waiting
Example: kidney exchange
Issues:
Source and allocation of kidneys:
• cadaver kidneys: centralized matching
mechanism based on priority queue
• living donors: patient must identify donor,
needs to be compatible
• other: angel donors, black market sales
Example: kidney exchange
Theory:
living donor exchanges:
patient 1
donor 1
patient 2
donor 2
Example: kidney exchange
Theory:
living donor exchanges:
adopted mechanism uses top-trading cycles,
theory of maximum matching, results in
improved welfare (many more transplants)
Part 2:
Bipartite Matching.
Matching
Boys
Girls
Questions.
1) What’s the most # of agents we can accommodate?
2) How can we find this allocation?
Matching
Bipartite Graph:
Boys
Girls
left vertices
right vertices
edges
Matching
right vertices
Bipartite Graph:
0
1
left vertices
edges
Matching
matching = a set of edges that share no vertices.
Matching
maximal
How to find a maximum matching?
Idea: add edges until we can’t anymore.
Matching
maximal
maximum
How to find a maximum matching?
Idea: add edges until we can’t anymore.
Not maximum, but close!
Matching
maximal
maximum
Defn. A soln. S to a maximization problem is an α-approximation
if its value is at least an α fraction of the optimal value.
Thm. Maximal matching (½)-approximates maximum matching.
Matching
How to find a maximum matching?
Idea: add edges until we can’t anymore,
allowing people to push each other out.
Matching
augmenting path = path between exposed vertices
Theorem. Matching is maximum iff no augmenting paths.
Matching
matching = a set of edges that share no vertices.
vertex cover = a set of vertices such that each edge
is incident to at least one vertex in the set.
Matching
Theorem. Maximum matching equals minimum vertex cover.
Matching
Theorem. Maximum matching equals
minimum vertex cover (bipartite graphs).
Proof. Matching = M, cover = C.
1. |M| ≤ |C|.
Matching
Theorem. Maximum matching equals
minimum vertex cover (bipartite graphs).
Proof. Matching = M, cover = C.
2. |max M| ≥ |min C| (constructive).
Matching
Theorem. Maximum matching equals
minimum vertex cover (bipartite graphs).
Proof. Matching = M, cover = C.
2. |max M| ≥ |min C| (constructive).
augmenting path = path between exposed vertices
Matching
Theorem. Maximum matching equals
minimum vertex cover (bipartite graphs).
Proof. Matching = M, cover = C.
2. |max M| ≥ |min C| (constructive).
Key.
vertices reachable from left-side exposed vertices.
proposed vertex cover.
Matching
Theorem. Maximum matching equals minimum vertex cover.
Question. When can we match everyone?
Perfect Matching
Question. When can we match everyone?
Defn. A matching is perfect if every vertex is matched.
Hall’s Marriage Theorem
Boys
Girls
Defn. A matching is perfect if every vertex is matched.
Thm. A perfect matching exists if and only if every set of girls
likes at least as large a set of boys.
Hall’s Marriage Theorem
Boys
Boys in Cover
Girls
Girls not in Cover
Condition. Every set of girls likes at least as large a set of boys.
Prf. |Cover| = |Boys in Cover| + |Girls in Cover|
≥ |Girls not in Cover| + |Girls in Cover|
= |Girls|
and Girls is a cover, so |min Cover| ≤ |Girls|.
Matching in Random Graphs
Agents
Items
Theorem. If each agent likes at least 2log n items, then with good
probability there is a way to assign everyone an item they like.
Matching in Random Graphs
Theorem. If each agent likes at least 2ln(n) items, then with good
probability there is a way to assign everyone an item they like.
Intuition. Deferred randomness.
1) Pr[ unique choice in market of size k ] = (1 – 1/k)k-1 ≥ 1/3
2) Constant fraction of market clears in each step.
3) Entire market clears in about log n steps.
Matching in Random Graphs
Theorem. If each agent likes k < ½ ln(n) items, then with good
probability someone is unassigned.
Intuition. Some item is liked by nobody.
E[ # unliked items]
for k < ½ ln(n).
= n ∙ Pr[ item is unliked ]
= n ∙ (1 – 1/n)nk
= n ∙ e-k
> n ∙ n-½
>1
Weighted Matching
Agents
$4
Items
$6
$8
Questions.
1) What’s the most value we can create?
2) How can we find this allocation?
Weighted Matching
Agents
$4
Items
Agents
$6
$8
WLOG, assume complete bipartite graph.
Look for max-weight matching .
Items
Primal-Dual Approach
L
3
3
“budgets” y(.)
“value” or
“weight” w(.)
2
3
1
2
3
3
2
3
1
2
1
R
0
0
“prices” y(.)
0
0
“dual” variable y maps vertices to numbers such that
for every edge e = (u,v), w(e) ≤ y(u) + y(v) (y non-negative).
Primal-Dual Approach
L
“weight” w(.)
3
3
2
3
2
“budgets” y(.)
R
0
1
0
“prices” y(.)
3
3
2
3
1
2
1
0
0
w(e) ≤ y(u) + y(v) implies ∑e in M w(e) ≤ ∑e in LUR y(v)
Primal-Dual Approach
L
“weight” w(.)
3
3
2
3
2
“budgets” y(.)
R
0
1
0
“prices” y(.)
3
3
2
3
1
2
1
0
0
Feasible Dual: for every M, y(.), ∑e in M w(e) ≤ ∑e in LUR y(v)
Certificate of Optimality: find M, y(.) s.t. this holds with equality
Hungarian Algorithm
Algorithm maintains invariants
1) Feasibility of dual: w(e) ≤ y(u) + y(v)
2) Tightness: if e=(u,v) is in M, then w(e) = y(u) + y(v)
Algorithm:
Initialize y(v) = max weight for v in L; y(v) for v in R = 0; M = {.}.
Repeat:
1) Augment matching: if there’s an augmenting path in
subgraph of tight edges, use it to augment matching M.
2) Dual adjustment: if M is not perfect, adjust dual variable y(.)
to make more edges tight.
Until M is maximum or duals reach zero.
Augmentation Step
new matching edges
L
4
5
3
R
6
5
4
3
5
3
2
1
matching
edge
2
0
Find augmenting paths in subgraph of tight edges.
Dual Adjustment Step
L
4
3 5
3
R
6
5
4
3
5
3
2
1
2
0
matching edges
tight edges
Update dual variables to make more tight edges:
1) Orient matching edges right-to-left, tight edges left-to-right.
2) Find set Z of vertices reachable from exposed vertices of L.
3) Decrease dual of v in L ∩ Z; increase dual of v in R ∩ Z until
an edge goes tight.
Dual Adjustment Step
L
R
1
2
1
2
3
1
0
2
3
3
2
2
1
2
1
0
0
matching edges
tight edges
Dual Adjustment Step
L
R
0 1
1 2
2
3
1 2
1
0 1
2
3
2 3
2
2
1
2
1
0
0
matching edges
tight edges
Dual Adjustment Step
L
R
0
1
2
3
2
1
2
1
3
2
2
2
2
1
0
1
0
matching edges
tight edges
Correctness
Algorithm maintains invariants
1) Tight: w(e) = y(u) + y(v), for e in M
matching edges can’t cross in or
out of red set.
2) Dual feasibility: w(e) ≤ y(u) + y(v)
edges that cross from inside red
set to outside it cannot be tight.
matching edges
tight edges
reachable from
exposed vertices
L
R
-Δ
+Δ
Optimality
Uncovered vertices all have zero duals
at end of algorithm:
1) Once a vertex is covered, it
remains so throughout algorithm.
2) So uncovered left-vertices have
duals that decrease at same rate
and reach zero simultaneously
3) And uncovered right-vertices have
zero duals initially, never change.
Thus, by tightness invariant, weight of
matching equals dual value and so it
must be optimal.
L
R
-Δ
+Δ
Example
L
R
3
0
2
3
1
3
0
2
3
3
0
2
2
1
3
1
0
matching edges
tight edges
Example
L
R
3
0
2
3
1
3
0
2
3
3
0
2
2
1
3
1
0
matching edges
tight edges
Example
L
R
3
0
2
3
1
3
0
2
3
3
0
2
2
1
3
1
0
matching edges
tight edges
Example
L
R
0
2
2 3
3
1
3
0
2
3
3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
0
2
2 3
3
1
3
0
2
3
3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
0
2
2 3
3
1
3
0
2
3
3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
2
1 2 3
0 1
3
1
2 3
0
2
3
3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
2
1 2 3
0 1
3
1
2 3
0
2
3
3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
2
0 1 2 3
0 1 2
3
1
1 2 3
0 1
2
3
2 3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
2
0 1 2 3
0 1 2
3
1
1 2 3
0 1
2
3
2 3
0
2
2
1
2 3
1
0
matching edges
tight edges
Example
L
R
2
-1 0 1 2 3
0 1 2 3
3
1
0 1 2 3
0 1 2
2
3
1 2 3
0 1
2
2
1
1 2 3
1
0
matching edges
tight edges
Example
L
R
2
-1 0 1 2 3
0 1 2 3
3
1
0 1 2 3
0 1 2
2
3
1 2 3
0 1
2
2
1
1 2 3
1
0
matching edges
tight edges
Example
L
R
2
-1 0 1 2 3
0 1 2 3
3
1
0 1 2 3
0 1 2
2
3
1 2 3
0 1
2
2
1
1 2 3
1
0
matching edges
tight edges