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Mixed Matching Markets or Union rates and free contracts Winfried Hochstättler TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA
Summary • Stable Matching • Men Propose – Women Dispose • Assignment Game • Firms Propose – Worker Negotiate • Unifying Models • And Algorithms
men women • A pair is blocking, if ~ and • A marriage is stable, if it has no blocking pair Stable Marriages (Gale, Shapley 1962) Preference lists (by weights) • Man i likes woman j with weight aij • Woman j likes man i with weight bij • A perfect matching is called a marriage • If i and j are matched they receive a payoff of uij = aij resp. vij = bij
Men Propose – Women Dispose (1962) Every man proposes to his favourite woman that has not already turned him down. Each woman with at least one proposal, engages to her favourite proposer and turns other proposers down. When all women are engaged, then the matching is stable.
Why is the matching stable? • Assume • Since Man2 prefers Woman1 to his fiancee, he has proposed to her and she has turned him down. • When Woman1 turned Man2 down, she preferred her present proposer to him. • Woman with a proposal can only improve during the algorithm, a contradiction.
Men propose – Women Dispose • Yields a „Man-optimal“ solution • each man gets his favourite among all woman he is matched to in some stable matching • Can be implemented to run in . • Input Data are two Matrices A and B
4 3 4 Assignment Game (Shapley and Shubik 1972) Firms • We have n firms and n workers. • A contract between a firm and a worker yields an added value • The input data is a square matrix encoding all possible added values. • Objective: find a perfect matching and an allocation of the added values. Workers • A perfect matching together with an allocation of the edge weights is stable, if there is no pair such that
Lineare Programming Duality A matching and an allocation is stable if and only if This is the dual program of maximum weighted bipartit matching. A stable solution can be found by linear programming resp. by the Hungarian method.
Firms Propose – Worker Negotiate firm worker 3 1 4
Firms Propose – Worker Negotiate firm worker 3 4
Firms Propose – Worker Negotiate • Is a Primal-Dual Algorithm where the subroutine for MaxCardinality Matching is non-standard • Instead of making a partial injective map (a matching) a total injective map (a perfect matching) we try to turn a total map into a total injective map. • Yields a „Firm-Optimal“ solution (dual variables) • Can be implemented to run in • Input Data is an -matrix C.
Towards a Unifying Model • Roth and Sotomayor (1991) • Wrote a book on two-sided matching markets; pointed out structural similarities between the stable solutions of stable matching and assignment games; asked for a unifying model. • Eriksson and Karlander (2000) • Presented a model and a pseudopolynomial time (auction-)algorithm to compute stable outcomes for integer data. • Sotomayor (2000) • „non-constructive“ proof of the existence of stable outcomes in the general case. • Hochstättler, Jin and Nickel (2006) • derived two algorithms from the above.
The Eriksson-Karlander-Model • Firms and workers are either • flexible (wages are individually negotiated) • or rigid (wages according to a fixed rate) • The graph now has flexible edges (both contracters flexible)and rigid edges (at least one rigid contractor) • Input Data: Two Matrices , and flags • for the players. Flexible contracts have side payments. • Distribution of the added value in a flexible contract: • In a rigid contract:
Stable Outcomes • An outcome is called feasible, if and • sum up to the weight of • An edge is called a blocking pair in if • is a rigid edge and as well as or • a flexible edge and In both cases: i and j improve when they cooperate. • There always exists an outcome without blocking pairs (stable outcome).
A New Model (Nickel, Schiess, WH, 2008) • Edges are • flexible (wages are individually negotiated) • or rigid (wages according to a fixed rate) • The graph now for each pair of players has as well a flexible edge as a rigid edge. • Input Data: Three Matrices and . • Distribution of the added value in a flexible contract: • In a rigid contract:
Stable Outcomes An outcome is called feasible, if and sum up to the weight of An edge is called a blocking pair in if the rigid edge of has as well as or or the flexible edge satisfies In both cases: i and j improve when they cooperate. There always exists an outcome without blocking pairs (stable outcome). Proven algorithmically.
Special Cases • and : • Assignment Game • : • Stable Matching • Eriksson and Karlander • Set and if an edge is flexible, if an edge is rigid.
The Algorithm • During the algorithm we maintain a (partial) map of proposals • And a preliminary payoff • Such that defining if resp. if the payoff has no blocking pair. • We then maximize We use augmenting path methods and a dual update procedure for similar to the Hungarian method.
The Augmenting Path procedure • Augmentation digraph : • favorite blocking partners: edges maximizing resp. • The map maps each firm to a favourite blocking partner (backward edges) • Augmentation: • Workers with a best rigid proposal turn all rigid proposals down except for the best one. • Workers with a best flexible proposal turn all rigid proposals down. • Find a dipath from a worker with several proposals to • - a jobless worker, a rigid edge, an insolvent firm or • - a worker with a rigid proposal • If no such path exists: • - perform Hungarian payoff update
Analysis • Invariants of the algorithm: • Each firm always makes one proposal. • Payoffs of firms are computed from and • is non-increasing. • is non-decreasing. • Complexity: • is augmented. • A rigid edge is dismissed. • A firm becomes insolvent.