Men Cheating in the Gale-Shapley
Stable Matching Algorithm
Chien-Chung Huang
Dartmouth College
Motivations & Results

Cheating Strategies in the Stable
Marriage problem


Gale-Shapley algorithm
 Deterministic/Randomized strategies
 Strengthening of Dubins-Freedman theorem
Random Stable Matching
 Group strategies ensuring that every cheating
man has a probability which majorizes the
original one
Here Comes the Story…
Adam, Bob, Carl, David
Geeta, Heiki, Irina, Fran
Adam
Fran
Carl, David, Bob, Adam
Irina, Fran, Heiki, Geeta
Bob
Geeta
Geeta, Fran, Heiki, Irina
Carl
Carl, Bob, David, Adam
Heiki
Irina, Heiki, Geeta, Fran
David
Adam, Carl, David, Bob
Irina
Search for a Matching
X
Geeta
Adam
David
Heiki
Geeta prefers Carl to Adam!
Blocking Pair
X
Bob
Irina
Carl
Fran
Carl likes Geeta better than Fran!
Stable Matching
Adam
Heiki
David
Irina
Unfortunately,
Irina loves David better!
Stable
Matching:
Bob
and Irina
are not aa matching
blocking pair
without blocking pairs
Bob
Fran
Bob likes Irina better than Fran!
Carl
Geeta
Goal
Adam
Bob
Carl
David
Fran
The stable marriage problem
(Gale and Shapley, American
Mathematical Monthly, 1962)
Geeta
Heiki
Irina
Deciding a Stable Matching

Gale-Shapley Stable Matching
algorithm


Men Propose, women accept/reject
Random Stable Matching
Gale-Shapley Algorithm
Geeta, Heiki, Irina, Fran
Fran
Adam
Carl > Adam
Irina, Fran, Heiki, Geeta
This is a stable matching
Bob
Geeta
Geeta, Fran, Heiki, Irina
Carl
Heiki
Irina, Heiki, Geeta, Fran
David
David > Bob
Irina
Cheating in the Gale-Shapley
Stable Matching

Women-Cheating Strategies (I)

Gale and Sotomayor
(American Mathematical
Monthly~1985)

Geeta
Strategy: Every woman declares men ranking
lower than her best possible partner
unacceptable
:
Carl
David
Best possible
partner
X
Bob
X
Adam
Cheating in the Gale-Shapley
Stable Matching (cont’d)

Women-Cheating Strategies (II)


Geeta
Teo, Sethuraman, and Tan (IPCO 1999)
For a sole cheating woman, they give her optimal
strategies, both when truncation is allowed and
when it is not.
:
Carl
David
Best possible
partner
X
Bob
X
Adam
Cheating in the Gale-Shapley
Stable Matching (cont’d)

Can men cheat?

Bad news 1: For men, individually,
being truthful is a dominant strategy
Cheating in the Gale-Shapley
Stable Matching (cont’d)

Can men cheat together?
Unfortunately…bad news 2

Dubins-Freedman Theorem

(1981, Roth 1982)
--A subset of men cannot falsify their lists
so that everyone of them gets a better
partner than in the Gale-Shapley stable
matching
Can we get around it??
Our Results (Gale-Shapley Algorithm)

The Coalition Strategy


An impossibility result on the randomized
coalition strategy


a nonempty subset of liars get better partners
and no one gets hurt.
Some liars never profit
Randomized cheating strategy ensuring
that the expected rank of the partner of
every liar improves

Liars must be willing to take risks
Our Results (Random Stable Matching)


Variant Scenario: suppose stable matching
is chosen at random
A modified coalition strategy

Ensures that the probability distribution over
partners majorizes the original one
Coalition Strategy (Characterization)
Cabal (core): a set of men
who exchange their partners
Coalition
Carl
C = (K, A(k))
Bob
Accomplices: other
fellow men falsify
their lists to help
them
Adam
Coalition Strategy (cont’d)

Envy graph
Adam
A directed cycle is a
potential coalition
Carl
Bob
David
Coalition Strategy (cont’d)


Coalition strategy is the only strategy in
which liars help one another without hurting
themselves
It is impossible that some men cheat to help
one another by hurting truthful people
However, that does not mean that you will never
get hurt by being truthful

By Dubins-Freedman theorem, some
accomplices still don’t have the motivation
to lie
Men’s Classification


Cabalists: men who belong to the
cabal of one coalition
Hopeless men: men who do not
belong to any cabal of the coalitions
These men cannot benefit from the coalition
strategy
Randomized Coalition Strategy



Motivation: some people (accomplices)
do not profit from cheating
League: Each man in the league has a
set of pure strategies.
A successful randomized strategy
should guarantee every liar:


Positive Expectation Gain
Elimination of Risk
Organizing a League


A league can only be realized by a
mixture of coalitions
Find a union of coalitions ci=(ki,A(ki))
so that the league:
Coalition
2=K
2,1A(k
2)1)
CoalitionCC
, A(k
1=K
K
A(k12)
Adam
L = Ui Ki = Ui A(ki) A(kK)
1 2
Bob
Carl
David
Unfortunately…


Every coalition must involve at least
one hopeless man
Hence, it is impossible to organize a
league
Remark

Dubins-Freedman Theorem is more
robust than we imagined


Bad news 3: Even a randomized
coalition strategy cannot circumvent it
The motivation issue still remains:
some men just don’t have a reason to
help
In pursuit of motivation


Suppose liars are willing to take some risk
Let us relax the second requirement of a
randomized strategy


Victim strategy: Some victim (man) has to
sacrifice himself to help others
A randomized strategy is possible in this
case
Positive Expectation Gain
Elimination of Risk
Random Stable Matching

Origin: a question raised by Roth &
Vate (Economic Theory, 1991)
Observation


When men use the coalition
strategy, all original stable
matchings remain stable.
The coalition strategy creates
many new stable matchings
New Men-optimal Matching
(by the coalition strategy)
Men-optimal Matching
Women-optimal Matching
A Variant of Coalition Strategy


Make sure that in all the new
stable matchings, all men in
the coalition are getting
partners as good as the
original stable matching.
For the cheater(s), the new
probability distribution
majorizes the old one
New Men-optimal Matching
(by the coalition strategy)
Men-optimal Matching
Women-optimal Matching
Remark

In the random stable
matching, it is possible that
all cheating men improve
(probabilistically).
New Men-optimal Matching
(by coalition strategy)
Men-optimal Matching
Women-optimal Matching
Conclusion



Cheating Strategies for men in the GaleShapley stable matching algorithm
(deterministic and randomized)
Strengthening of Dubins-Freedman
theorem
Strategies for Random Stable Matching
Voila, C’est tout


Thanks for your attention
Questions?