Pretty Good Structure, 2009, Paris

Mixed Matching Markets Mixed Matching Markets

Union rates and free contractsUnion rates and free contracts

Winfried Hochstättler

or

Pretty Good Structure, 2009, Paris

SummarySummary

Stable MatchingMen Propose – Women Dispose

Assignment GameFirms Propose – Worker Negotiate

Unifying ModelsAnd Algorithms

Pretty Good Structure, 2009, Paris

Stable Marriages (Gale, Shapley 1962)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 women

A pair is blocking, if ~ and

A marriage is stable, if it has no blocking pair

Pretty Good Structure, 2009, Paris

Men Propose – Women Dispose (1962)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.

Pretty Good Structure, 2009, Paris

Why is the matching 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.

Pretty Good Structure, 2009, Paris

Men propose – Women DisposeMen 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

O(n2)(n £ n)

Pretty Good Structure, 2009, Paris

Assignment Game (Shapley and Shubik Assignment Game (Shapley and Shubik 1972)1972)

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.

Firms Workers

4

4

3

A perfect matching together with an allocation of the edge weights is stable, if there is no pair such that

Pretty Good Structure, 2009, Paris

Lineare Programming DualityLineare Programming Duality

A matching and an allocation is stable if and only if

8i 8j : ui + vj ¸ ®i j

minP n

i=1 ui +P n

j =1 vj

This is the dual program of maximum weighted bipartit matching.

maxP n

i=1

P nj =1 ®i j xi j

subject toP n

i=1 xi j = 1 8jP nj =1 xi j = 1 8i

A stable solution can be found by linear programming resp. by the Hungarian method.

Pretty Good Structure, 2009, Paris

Firms Propose – Worker NegotiateFirms Propose – Worker Negotiate

firm worker

3 2 7 2

5 3 5 2

0 5 6 2

2 3 5 1

3 2 7 2

5 3 5 2

0 5 6 2

2 3 5 1

1

3 2 6 2

5 3 4 2

0 5 5 2

2 3 4 1

3 -1 3 2

5 0 1 2

0 2 2 2

2 0 1 1

3 4

Pretty Good Structure, 2009, Paris

Firms Propose – Worker NegotiateFirms Propose – Worker Negotiate

firm worker 33 4

3 -1 3 2

5 0 1 2

0 2 2 2

1 0 1 1

Pretty Good Structure, 2009, Paris

Firms Propose – Worker NegotiateFirms 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.

O(n3)(n £ n)

Pretty Good Structure, 2009, Paris

Towards a Unifying ModelTowards 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.O(n4)

Pretty Good Structure, 2009, Paris

Firms and workers are eitherflexible (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)

The Eriksson-Karlander-ModelThe Eriksson-Karlander-Model

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:

Pretty Good Structure, 2009, Paris

Stable OutcomesStable 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).

Pretty Good Structure, 2009, Paris

Edges areflexible (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.

A New Model (Nickel, Schiess, WH, A New Model (Nickel, Schiess, WH, 2008)2008)

Input Data: Three Matrices and .

Distribution of the added value in a flexible contract:

In a rigid contract:

ESCAPE 2007, Hangzhou

Stable OutcomesStable 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.

Pretty Good Structure, 2009, Paris

and :Assignment Game

:Stable Matching

Eriksson and KarlanderSet and if an edge is flexible,

if an edge is rigid.

Special CasesSpecial Cases

Pretty Good Structure, 2009, Paris

The Algorithm 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.

Pretty Good Structure, 2009, Paris

The Augmenting Path procedureThe 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

Pretty Good Structure, 2009, Paris

AnalysisAnalysis

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.

Pretty Good Structure, 2009, Paris

Thank you for your attention.Thank you for your attention.