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

Fran

Geeta

Irina

Heiki

Geeta, Heiki, Irina, Fran

Irina, Fran, Heiki, Geeta

Geeta, Fran, Heiki, Irina

Irina, Heiki, Geeta, Fran

Adam, Bob, Carl, David

Carl, David, Bob, Adam

Carl, Bob, David, Adam

Adam, Carl, David, Bob

Search for a Matching

Adam Geeta

Bob Irina Carl Fran

David Heiki

Carl likes Geeta better than Fran!

Geeta prefers Carl to Adam!

X

X

Blocking Pair

Stable Matching

Adam Heiki

Bob Fran GeetaCarl

IrinaDavid

Bob likes Irina better than Fran!

Unfortunately, Irina loves David better!

Stable Matching: a matching without blocking

pairs

Bob and Irina are not a blocking pair

Goal

Adam

Bob

Carl

David

Fran

Geeta

Irina

Heiki

The stable marriage problem (Gale and Shapley, American Mathematical Monthly, 1962)

Deciding a Stable Matching

Gale-Shapley Stable Matching algorithm Men Propose, women accept/reject

Random Stable Matching

Gale-Shapley Algorithm

Adam

Heiki

Bob

Fran

Geeta

Carl

IrinaDavid

Geeta, Heiki, Irina, Fran

Irina, Fran, Heiki, Geeta

Geeta, Fran, Heiki, Irina

Irina, Heiki, Geeta, Fran

Carl > Adam

David > Bob

This is a stable matching

Cheating in the Gale-Shapley Stable Matching

Women-Cheating Strategies (I) Gale and Sotomayor (American Mathematical

Monthly~1985)

Strategy: Every woman declares men ranking lower than her best possible partner unacceptable

AdamBobCarl DavidGeeta:

Best possible partner

X X

Cheating in the Gale-Shapley Stable Matching (cont’d)

Women-Cheating Strategies (II) 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.

AdamBobCarl DavidGeeta: X X

Best possible partner

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 a nonempty subset of liars get better partners and no one gets hurt.

An impossibility result on the randomized coalition strategy 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

Cabal (core): a set of men who exchange their partners

Coalition Strategy (Characterization)

C = (K, A(k))Coalition

Accomplices: other fellow men falsify their lists to help them

AdamBobCarl

Coalition Strategy (cont’d) Envy graph

Adam

Bob

Carl

David

A directed cycle is a potential coalition

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

By Dubins-Freedman theorem, some accomplices still don’t have the motivation to lie

However, that does not mean that you will never get hurt by being truthful

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:

L = Ui Ki = Ui A(ki)

Adam Bob Carl David

Coalition C1=K1,

A(k1)K1 A(k1)K2A(k2)

Coalition C2=K2,

A(k2)

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 Men-optimal Matching

Women-optimal Matching

New Men-optimal Matching (by the coalition strategy)

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

Men-optimal Matching

Women-optimal Matching

New Men-optimal Matching (by the coalition strategy)

Remark

In the random stable matching, it is possible that all cheating men improve (probabilistically).

Men-optimal Matching

Women-optimal Matching

New Men-optimal Matching (by coalition strategy)

Conclusion

Cheating Strategies for men in the Gale-Shapley 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?