Mixed Matching Markets
or
Union rates and free contracts
Winfried Hochstättler
Pretty Good Structure, 2009, Paris
Summary
Stable Matching
Men Propose – Women Dispose
Assignment Game
Firms Propose – Worker Negotiate
Unifying Models
And Algorithms
Pretty Good Structure, 2009, Paris
Stable Marriages (Gale, Shapley 1962)
men
women
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
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)
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?
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 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 O(n2. )
Input Data are two (n £ n) Matrices A and B
Pretty Good Structure, 2009, Paris
Assignment Game (Shapley and Shubik 1972)
Firms
Workers
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.
4
3
A perfect matching together with an
allocation
of the edge weights is
stable, if there is no pair
such that
4
Pretty Good Structure, 2009, Paris
Lineare Programming Duality
A matching and an allocation is stable if and only if
Pn
Pn
min i = 1 ui + j = 1 vj
8i 8j : ui + vj ¸ ®i j
This is the dual program of maximum weighted bipartit
matching.
Pn Pn
max
® x
P in= 1 j = 1 i j i j
subject t o P i = 1 x i j = 1
8j
n
8i
j = 1 xi j = 1
A stable solution can be found by linear programming
resp. by the Hungarian method.
Pretty Good Structure, 2009, Paris
Firms Propose – Worker Negotiate
firm
worker
3
1
4
3
2
-1
7
3
6
2
5
3
0
5
1
4
2
0
5
2
6
2
5
2
2
3
0
5
1
4
1
Pretty Good Structure, 2009, Paris
Firms Propose – Worker Negotiate
firm
worker
3
3
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 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)
3
O(n
)
Can be implemented to run in
Input Data is an (n £ n) -matrix C.
Pretty Good Structure, 2009, Paris
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 O(n 4 ) algorithms from the above.
Pretty Good Structure, 2009, Paris
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:
Pretty Good Structure, 2009, Paris
Stable Outcomes
An outcome
is called feasible, if
sum up to the weight of
and
An edge
if
is called a blocking pair in
is a rigid edge and
a flexible edge and
as well as
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
or
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:
Pretty Good Structure, 2009, Paris
Stable Outcomes
An outcome
is called feasible, if
sum up to the weight of
and
An edge
if
is called a blocking pair in
the rigid edge of
has
or the flexible edge satisfies
as well as
In both cases: i and j improve when they cooperate.
There always exists an outcome without blocking pairs
(stable outcome). Proven algorithmically.
ESCAPE 2007, Hangzhou
or
Special Cases
and
:
Assignment Game
:
Stable Matching
Eriksson and Karlander
Set
and
if an edge is rigid.
Pretty Good Structure, 2009, Paris
if an edge is flexible,
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 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
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.
Pretty Good Structure, 2009, Paris
Thank you for your attention.
Pretty Good Structure, 2009, Paris