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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
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