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… AdamBob CarlDavidFran 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 Geeta Carl Irina David 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 AdamBob CarlDavidFran 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 Irina David 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 AdamBob CarlDavid Geeta : Best possible partner XX

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. AdamBob CarlDavid Geeta : XX 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

Some Preliminaries  Adam’s preference list Adam Fran Geeta Irina Heiki : “Left”“Right”M 0 -partner

Observation (I)  Shifting women from the left to the right of men’s lists will not make any man worse off Adam Fran Geeta Irina Heiki :

Observation (II)  Permuting the left and the right portions of the preference lists will not change the outcome Adam Fran Geeta Irina Heiki :

Observation (III)  What about shifting women from the right to the left of a list? Adam Fran Geeta Irina Heiki :

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 AdamBob Carl

Coalition Strategy (cont’d)  Step1: Organize the “cabal” (core) of a coalition  K = (m 1,m 2,…m k ) such that m i prefers m i-1 ’s partner to his own  The goal of the cabal is that under the Gale-Shapley algorithm, m i can be matched to m i-1 ’s partner

Coalition Strategy (cont’d)  Step2: Enlist the help of “accomplices”  Man m is an accomplice of the cabal K if there exists a man m i in K whose preferred partner in the cabal prefers m over m i  The accomplices should avoid proposing to the women preferred by the men in the cabal By shifting that woman from the left to the right of his list

Coalition Strategy (cont’d)  Envy graph AdamBob CarlDavid 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 c i =(k i,A(k i )) so that the league: L = U i K i = U i A(k i ) AdamBob CarlDavid Coalition C 1 =K 1, A(k 1 ) K1K1 A(k 1 ) K2K2 A(k 2 ) Coalition C 2 =K 2, A(k 2 )

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?