Stable Matching Existence, Computation, Convergence Correlated Preferences. Stable Matching. Algorithmic Game Theory.

Transcription

1 Existence, Computation, Convergence Correlated Preferences

2 Existence, Computation, Convergence Correlated Preferences Stable Marriage Set of Women Y Set of Men X

3 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).

4 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!

5 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).

6 Existence, Computation, Convergence Correlated Preferences Set X of m men, set Y of n women Each x X has a preference order x over all matches y Y. Each y Y has a preference order y over all matches x X. For each person being unmatched is the least preferred state, i.e., each person wants to be matched rather than unmatched. A matching S X Y is a set of pairs {x, y}, where each person appears in at most one pair. 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).

7 Existence, Computation, Convergence Correlated Preferences When is a matching stable? What is a hazard to stability? In a matching M, a pair {x, y} is blocking pair if and only if x and y prefer each other to y = M(x) and x = M(y), respectively. x y M is a stable matching if and only if it admits no blocking pair. x y Applications Residents/Hospitals College Admission Job Market etc.

8 Existence, Computation, Convergence Correlated Preferences Matching Game Let us formulate the model as a matching game. One side is active, picks strategies and receives payoffs, whereas the passive side only reacts to the strategies. W.l.o.g. we assume the active side are the men: Player set is X, every player has strategy space Y If x X picks strategy y, he proposes to woman y. Woman y picks from all proposals the most preferred one. Payoff p x(y) for man x playing strategy y: Let x= (y 1,..., y n) be his preference list. { k m matched to w n k, k {0,..., n 1} p x(y) = 1 m unmatched Consider a state of the game and assume x deviates to y. This is a protitable deviation 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 Nash equilibria of the matching game.

9 Existence, Computation, Convergence Correlated Preferences Existence and Computation Algorithm 1: Gale-Shapley Algorithm with Man-Proposal Initialize x= x for all x X while there is an unmatched man x X with x do Every man x X proposes to topmost woman in x Every woman y Y keeps most preferred proposal and rejects all others Every woman matches to man of kept proposal If 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 a time and each woman keeps at most one proposal.

10 Existence, Computation, Convergence Correlated Preferences Convergence It is straightforward to verify that over the run of the algorithm for a man, the preference of proposed women is strictly decreasing, and 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) and x y M(y). x must have proposed to y and got rejected, so y must keep a proposal of some better man x y x. Hence, her match in M can only be better than x. Thus, M(y) y x y x, a contradiction. With a reformulation of this idea we show convergence in the matching game. Theorem For every matching game and every initial matching M 0, there is a sequence of 2nm best-response improvement steps to a stable matching.

11 Existence, Computation, Convergence Correlated Preferences Convergence Proof: The sequence has two phases. In Phase 1, only matched men are allowed to play best responses. Denote by X the set of matched men in M. The following function keeps decreasing over phase 1: Φ(M) = x X(n p x(m(x))). (rank of x s partner in x) Suppose x X plays a best response. x remains matched, improves rank of partner by at least 1. 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 1 terminates after at most nm iterations.

12 Existence, Computation, Convergence Correlated Preferences Convergence In Phase 2, only unmatched men are allowed to play best responses. Denote by Y the set of matched women in M. The following function keeps increasing over phase 2: Ψ(M) = p y (M(y)). y Y Suppose an unmatched man x plays a best response. x gets matched to y Y, p y increases by at least 1. x gets matched to y Y, y enters Y. Thus, Ψ grows by at least 1 in every iteration. As 1 Ψ(M) nm, phase 2 terminates after at most nm iterations. To show that the final matching is stable, observe that throughout phase 2 no matched man can improve. When unmatched x gets matched to y, this only increases her payoff. Assuming that there was no blocking pair with any of the matched 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.

13 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

14 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

15 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

16 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

17 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

18 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

19 Existence, Computation, Convergence Correlated Preferences Potential Game Our previous proof involves two potential functions. So are matching games potential games? The following result shows a negative answer. Proposition Best-response dynamics in the matching game can cycle.

20 Existence, Computation, Convergence Correlated Preferences Random Dynamics Matching games belong to the class of weakly acyclic games, in which from every state we always have at least one improvement sequence to a pure NE. In contrast, in potential games from every state all improvement sequences reach a pure NE. An interesting consequence of weak acyclicity is that if we pick improvement moves at random, we execute a random walk over the states of the game. It is guaranteed to reach an absorbing state (i.e., a pure NE) in the limit with probability 1, because ultimately by chance for some state we will correctly execute the one improvement sequence that leads us to the pure NE.

21 Existence, Computation, Convergence Correlated Preferences Random Dynamics How long are such random improvement sequences? How long does it take if we choose in each step an improvement move uniformly at random? What if, in addition, we restrict to best-response improvement moves instead of arbitrary better-response moves? Theorem (Ackermann, Goldberg, Mirrokni, Röglin, Vöcking 2011) There is a matching game with n men and n women and an initial matching M 0 such that, with probability 1 2 Ω(n), random dynamics starting from M 0 need 2 Ω(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...

22 Existence, Computation, Convergence Correlated Preferences Correlated Preferences An intuitive case of matching is when both players receive the same payoff from a match. Then each match has a positive edge-weight, and this weight is given to both players if they match along this edge. This is referred to as correlated or weighted matching. In a correlated matching game, we have p x(y) = p y (x) = p xy > 0 for all x X and y Y, and p x( ) = p y ( ) = 0. Definition (Ordinal Potential Game) We call a strategic game Γ = (N, (Σ i ) i N, (c i ) i N ) ordinal potential game if there exists a function Φ: Σ R such that for every i N, for every S i Σ i, and every S i, S i Σ i : c i (S i, S i ) > c i (S i, S i ) Φ(S i, S i ) > Φ(S i, S i ).

23 Existence, Computation, Convergence Correlated Preferences Ordinal Potential Games The improvement in Φ does not necessarily mirror the exact amount by which player 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 pure NE and convergence of every improvement sequence in a finite game. Theorem Every correlated matching game is an ordinal potential game. Proof: For a matching M, define the following function Φ(M) = (p x1,y 1,..., p xk,y k ), where (x i, y i ) M are all matched pairs sorted in non-increasing order of payoffs, i.e., for i j it holds p xi,y i p xj,y j.

24 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 lexicographically larger 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, p xy > p x,m(x) and p xy > p M(y),y. All pairs that get removed have payoff < p xy. Hence, lexicographically the sorted vector of pair payoffs increases we add a higher payoff pair and delete pairs with strictly smaller payoff. This proves that correlated matching is an ordinal potential game. Above we showed that even in general matching games from every initial matching there exists a short sequence of best-response steps to a stable matching. Our last theorem shows that for correlated matching random dynamics converge in expected polynomial time.

25 Existence, Computation, Convergence Correlated Preferences Random Dynamics in Correlated Matching Theorem For every correlated matching game and initial matching M 0, random better-response dynamics converge to a stable matching in expected polynomial time. Proof: Suppose we resolve a blocking pair {x, y} with maximum payoff p xy. There cannot exist a new blocking pair with payoff > p xy. Such a blocking pair is not altered by {x, y} and must have been present before. However, p xy was the one with largest benefit. Hence, we have the invariant that if we resolve a blocking pair {x, y} of maximum payoff x and y never become part of a blocking pair again.

26 Existence, Computation, Convergence Correlated Preferences Random Dynamics in Correlated Matching If we choose improvement moves uniformly at random, then with probability at least 1/nm we pick a move that corresponds to a blocking pair with maximum payoff. If this pair is resolved, the players remain matched to each other until the 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 a stable matching. Note that the last argument works similarly if we take random best-response moves, as a blocking pair of maximum payoff obviously represents a best response for the man.