Stable Matching - Heinz Nixdorf Institut · 2012-11-14 · Stable MatchingExistence, Computation,...

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Stable Matching Existence, Computation, Convergence Correlated Preferences Stable Matching Algorithmic Game Theory Alexander Skopalik Algorithmic Game Theory Stable Matching

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Transcript of Stable Matching - Heinz Nixdorf Institut · 2012-11-14 · Stable MatchingExistence, Computation,...

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Matching

    Algorithmic Game Theory

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Marriage

    Set of Women Y

    ( , , )

    ( , , )

    ( , , )

    Set of Men X

    ( , , )

    ( , , )

    ( , , )

    Every person has a preference list (left/right is most/least preferred).

    No polygamy - at most one match per person!

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Marriage

    Set of Women Y

    ( , , )

    ( , , )

    ( , , )

    Set of Men X

    ( , , )

    ( , , )

    ( , , )

    Every person has a preference list (left/right is most/least preferred).

    No polygamy - at most one match per person!

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Marriage

    Set of Women Y

    ( , , )

    ( , , )

    ( , , )

    Set of Men X

    ( , , )

    ( , , )

    ( , , )

    Every person has a preference list (left/right is most/least preferred).

    No polygamy - at most one match per person!

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Marriage

    Set of Women Y

    ( , , )

    ( , , )

    ( , , )

    Set of Men X

    ( , , )

    ( , , )

    ( , , )

    Every person has a preference list (left/right is most/least preferred).

    No polygamy - at most one match per person!

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Matching

    I Set X of m men, set Y of n women

    I Each x ∈ X has a preference order �x over all matches y ∈ Y.

    I Each y ∈ Y has a preference order �y over all matches x ∈ X .

    I For each person being unmatched is the least preferred state, i.e., eachperson wants to be matched rather than unmatched.

    I A matching S ⊆ X ×Y is a set of pairs {x , y}, where each person appearsin at most one pair.

    I For a matching M we denote by M(x) ∈ Y the match of man x ∈ X in M.We denote M(x) = ∗ if x is unmatched in M. Similar definition of M(y).

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Stable Matching

    When is a matching stable? What is a hazard to stability?

    I In a matching M, a pair {x , y} is blockingpair if and only if x and y prefer each otherto y ′ = M(x) and x ′ = M(y), respectively.

    I M is a stable matching if and only if itadmits no blocking pair.

    x ′ y

    x y′

    Applications

    Residents/Hospitals College Admission Job Market etc.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Matching Game

    Let us formulate the model as a matching game. One side is active, picksstrategies and receives payoffs, whereas the passive side only reacts to thestrategies. W.l.o.g. we assume the active side are the men:

    I Player set is X , every player has strategy space YI If x ∈ X picks strategy y , he proposes to woman y .I Woman y picks from all proposals the most preferred one.

    I Payoff px(y) for man x playing strategy y :

    Let �x= (y1, . . . , yn) be his preference list.

    px(y) =

    {k m matched to wn−k , k ∈ {0, . . . , n − 1}−1 m unmatched

    Consider a state of the game and assume x deviates to y ′. This is a protitabledeviation if and only if y ′ is more preferred by x and accepts his proposal, i.e.,{x , y ′} is a blocking pair. Hence, stable matchings are exactly the pure Nashequilibria of the matching game.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Existence and Computation

    Algorithm 1: Gale-Shapley Algorithm with Man-Proposal

    Initialize �′x=�x for all x ∈ Xwhile there is an unmatched man x ∈ X with �′x 6= ∅ do

    Every man x ∈ X proposes to topmost woman in �′xEvery woman y ∈ Y keeps most preferred proposal and rejects all othersEvery woman matches to man of kept proposalIf his proposal is rejected, man x removes topmost entry from �′x

    Theorem (Gale, Shapley 1962)

    A stable matching always exists and can be computed in time O(nm).

    Proof:Consider Algorithm 1. Obviously, it can be implemented to run in time O(nm).It computes a matching M, as each man proposes to at most one woman at atime and each woman keeps at most one proposal.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Convergence

    It is straightforward to verify that over the run of the algorithm

    I for a man, the preference of proposed women is strictly decreasing, and

    I for a woman, the preference of matched partners is strictly increasing.

    Assume for contradiction M has a blocking pair {x , y} with y �x M(x) andx �y M(y). x must have proposed to y and got rejected, so y must keep aproposal of some better man x ′ �y x . Hence, her match in M can only bebetter than x ′. Thus, M(y) �y x ′ �y x , a contradiction.

    With a reformulation of this idea we show convergence in the matching game.

    TheoremFor every matching game and every initial matching M0, there is a sequence of2nm best-response improvement steps to a stable matching.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Convergence

    Proof:The sequence has two phases. In Phase 1, only matched men are allowed toplay best responses. Denote by X the set of matched men in M. The followingfunction keeps decreasing over phase 1:

    Φ(M) =∑x∈X

    (n − px(M(x))). (rank of x ’s partner in �x)

    Suppose x ∈ X plays a best response.I x remains matched, improves rank of partner by at least 1.

    I Some x ′ ∈ X can get unmatched, less players in X , Φ drops by at least 1.

    Thus, Φ drops by at least 1 in every iteration. As 1 ≤ Φ(M) ≤ nm, phase 1terminates after at most nm iterations.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Convergence

    In Phase 2, only unmatched men are allowed to play best responses. Denote byY the set of matched women in M. The following function keeps increasingover phase 2:

    Ψ(M) =∑y∈Y

    py (M(y)).

    Suppose an unmatched man x plays a best response.

    I x gets matched to y ∈ Y , py increases by at least 1.I x gets matched to y 6∈ Y , y enters Y .

    Thus, Ψ grows by at least 1 in every iteration. As 1 ≤ Ψ(M) ≤ nm, phase 2terminates after at most nm iterations.

    To show that the final matching is stable, observe that throughout phase 2 nomatched man can improve. When unmatched x gets matched to y , this onlyincreases her payoff. Assuming that there was no blocking pair with any of thematched men before, there is no blocking pair after x and y are matched,because x played a best response and y ’s payoff is even higher now.Finally, there are no blocking pairs with unmatched men as phase 2 is over.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Potential Game

    Our previous proof involves two potential functions. So are matching gamespotential games? The following result shows a negative answer.

    Proposition

    Best-response dynamics in the matching game can cycle.

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    ( , , )

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Random Dynamics

    Matching games belong to the class of weakly acyclic games, in which fromevery state we always have at least one improvement sequence to a pure NE. Incontrast, in potential games from every state all improvement sequences reacha pure NE.

    An interesting consequence of weak acyclicity is that if we pick improvementmoves at random, we execute a random walk over the states of the game. It isguaranteed to reach an absorbing state (i.e., a pure NE) in the limit withprobability 1, because ultimately – by chance – for some state we will correctlyexecute the one improvement sequence that leads us to the pure NE.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Random Dynamics

    How long are such random improvement sequences? How long does it take ifwe choose in each step an improvement move uniformly at random? What if,in addition, we restrict to best-response improvement moves instead ofarbitrary better-response moves?

    Theorem (Ackermann, Goldberg, Mirrokni, Röglin, Vöcking 2011)

    There is a matching game with n men and n women and an initial matchingM0 such that, with probability 1− 2−Ω(n), random dynamics starting from M0need 2Ω(n) steps to reach a stable matching.

    This result holds for both random better- and best-response dynamics.

    Devastatingly, with high probability the convergence time will be exponential...

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Correlated Preferences

    An intuitive case of matching is when both players receive the same payofffrom a match. Then each match has a positive edge-weight, and this weight isgiven to both players if they match along this edge. This is referred to ascorrelated or weighted matching.

    In a correlated matching game, we have px(y) = py (x) = pxy > 0 for all x ∈ Xand y ∈ Y, and px(∗) = py (∗) = 0.

    Definition (Ordinal Potential Game)

    We call a strategic game Γ = (N , (Σi )i∈N , (ci )i∈N ) ordinal potential game ifthere exists a function Φ: Σ→ R such that for every i ∈ N , for everyS−i ∈ Σ−i , and every Si , S ′i ∈ Σi :

    ci (Si , S−i ) > ci (S′i , S−i ) ⇒ Φ(Si ,S−i ) > Φ(S ′i ,S−i ) .

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Ordinal Potential Games

    The improvement in Φ does not necessarily mirror the exact amount by whichplayer i reduces his cost (also it might behave arbitrarily if i increases his cost).It only decreases strictly whenever player i strictly decreases his cost. Obviously,the existence of an ordinal potential Φ suffices to guarantee existence of a pureNE and convergence of every improvement sequence in a finite game.

    TheoremEvery correlated matching game is an ordinal potential game.

    Proof:For a matching M, define the following function

    Φ(M) = (px1,y1 , . . . , pxk ,yk ),

    where (xi , yi ) ∈ M are all matched pairs sorted in non-increasing order ofpayoffs, i.e., for i ≤ j it holds pxi ,yi ≥ pxj ,yj .

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Correlated Matching is Potential Game

    We let Φ(M) > Φ(M ′) if the vector of match payoffs in M is lexicographicallylarger than in M ′. Intuitively, M has more higher payoff edges than M ′.

    Assume in state M, man x executes an improvement move and creates pair{x , y}. Thus, pxy > px,M(x) and pxy > pM(y),y . All pairs that get removed havepayoff < pxy . Hence, lexicographically the sorted vector of pair payoffsincreases – we add a higher payoff pair and delete pairs with strictly smallerpayoff. This proves that correlated matching is an ordinal potential game.

    Above we showed that even in general matching games from every initialmatching there exists a short sequence of best-response steps to a stablematching. Our last theorem shows that for correlated matching randomdynamics converge in expected polynomial time.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Random Dynamics in Correlated Matching

    TheoremFor every correlated matching game and initial matching M0, randombetter-response dynamics converge to a stable matching in expectedpolynomial time.

    Proof:Suppose we resolve a blocking pair {x , y} with maximum payoff pxy .

    I There cannot exist a new blocking pair with payoff > pxy . Such a blockingpair is not altered by {x , y} and must have been present before. However,pxy was the one with largest benefit.

    Hence, we have the invariant that if we resolve a blocking pair {x , y} ofmaximum payoff x and y never become part of a blocking pair again.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

  • Stable Matching Existence, Computation, Convergence Correlated Preferences

    Random Dynamics in Correlated Matching

    If we choose improvement moves uniformly at random, then with probability atleast 1/nm we pick a move that corresponds to a blocking pair with maximumpayoff. If this pair is resolved, the players remain matched to each other untilthe end because of the above argument. Such a move occurs every O(nm)rounds in expectation, and after O(min{n,m}) such moves, we have reached astable matching.

    Note that the last argument works similarly if we take random best-responsemoves, as a blocking pair of maximum payoff obviously represents a bestresponse for the man.

    Alexander Skopalik Algorithmic Game Theory

    Stable Matching

    Stable MatchingExistence, Computation, ConvergenceCorrelated Preferences