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The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples

Marzieh Barkhordar Elahe Ghasempour

Sharif University of Technology

December 24,2017

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 1 / 25

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 2 / 25

Introduction

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 3 / 25

Introduction

Stable Marriage Problem

Sometimes, you have to take what you can get. Thats the message of the stable marriage problem, whose mathematical solution pairs potential partners in such a way that none will divorce.

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 4 / 25

Definitions

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 5 / 25

Definitions

Notations

A matching µ : A → 2X : function between a set of contracts X and a set of agents A µ(a) : set of contracts matched to agent a xA ⊆ A : two or more agents are involved in contract x Xa {x ∈ X : a ∈ xA} : set of contracts that agent a could be matched to >a : strict preferences of agent a ∈ A Cha : 2Xa → 2Xa : returns the agent a’s choice set Example 1 If an agent a has the preferences {x1, x2} >a {x3} >a ∅ over contracts then Cha{x1, x2, x3} = {x1, x2} and Cha({x1, x3}) = {x3}.

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 6 / 25

Definitions

Stable Matching Problem

A matching µ is individually rational if for any a ∈ A that Cha(µ(a)) = µ(a). A matching µ as stable if for every unused contract at least one agent a ∈ xA does not want to sign it: x ∈ Cha(µ(a) ∪ {x})

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 7 / 25

Definitions

Preferences

Four restrictions on agent preferences : The law of aggregate demand : |Cha(U ∪ V )| ≥ |Cha(U)| for all U, V ⊆ X Substitutable preferences : if x, x′ ∈ X, B ⊆ X and x ∈ Cha(B) implies x ∈ Cha(B ∪ x′) Responsive preferences : Suppose agent a would choose at most ka contracts. If the contracts can be listed x1 <a x2 <a . . . <a xn such that if i < j and |B| < ka then B ∪ {xi} <a B ∪ {xj} Ordinal preferences : Agents only ever want at most one contract, this is described formally as |Cha(S)| ≤ 1∀S ∈ Qa

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 8 / 25

Definitions

Matching Market

A matching market is bipartite if the set of agents A can be partitioned into a set of doctors D and hospitals H such that no two doctors or two hospital share contracts. Resident Matching Problem with Couples (RMPC) : The market is partitioned into single doctors and couples in the set D and the positions in the set H Each single doctor and position at a hospital will be an agent. Each couple will be an agent that signs at most one contract with

• ne or two hospital positions

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 9 / 25

Definitions

Cont’d

Single application : one contract with one hospital where only one member is employed. Joint application : one contract with two hospitals where each member is employed by a different hospital All agents, including the couples, have ordinal preferences over contracts

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 10 / 25

Definitions

Example

Example 2 Consider one couple c (comprising of cf and cm ), doctor d and two hospitals h1 and h2. Contracts : x1 = ((cm, h1), (cf, h2)), x2 = ((cm, h1)), x3 = ((d, h1)) Hospital h1 prefers cm over d and the couple c prefers x1 over x2. Clearly h1 is indifferent between x1 and x2 Therefore we break ties in the hospitals preferences over contracts in favor of the couples Preference list of the hospital h1 : x1 >h1 x2 >h1 x3 .

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 11 / 25

Formulating Stable Matching as an Integer Program

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 12 / 25

Formulating Stable Matching as an Integer Program

Definitions

zx ∈ {0, 1} : shows whether contract x is used. ya,B ∈ {0, 1} : whether the set B of contracts is allocated to agent a

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 13 / 25

Formulating Stable Matching as an Integer Program

Stable Matching Linear Program

Set constraint : Each agent is allocated exactly one set of contracts

• B∈Qa

ya,B = 1 ∀a ∈ A Matching constraint : If a contract x ∈ X is used then all agents a ∈ xA must sign it. zx =

• B∈Qa:x∈B

ya,B ∀x ∈ X, ∀a ∈ xA Non-negativity constraint : zx, ya,B ≥ 0 Stability constraint : If a contract x ∈ X is unused then at least

• ne agent a ∈ xA does not want to sign it.

zx +

• a∈xA
• B∈Qa:x∈Cha(B∪x)

ya,B ≥ 1 ∀x ∈ X

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 14 / 25

Formulating Stable Matching as an Integer Program

Objectives

Maximum employment objective : Maximize the number of contracts signed max

• x∈X

zx Stable marriage objective : Maximize a linear function over the contracts max

• x∈X

rxzx Doctor-optimal objective : Maximize the utility of the doctors max

• d∈D
• B∈Qd

ud(B)yd,B Arbitrary objective : Maximize a linear function over sets of contracts allocated to agents max

• a∈A
• B∈Qa

fa(B)ya,B

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 15 / 25

An Approximation Rural Hospital Theorem With Couples

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 16 / 25

An Approximation Rural Hospital Theorem With Couples

Upper bound

Rural Hospital Theorem Consider a bipartite market with single doctors and hospitals that have

• rdinal preferences i.e. a RMPC with |C| = 0. In every stable matching

the same doctors and hospitals will be matched. That is: nµ(a) = nµ′(a) For any two stable matchings µ and µ′.

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 17 / 25

An Approximation Rural Hospital Theorem With Couples

Cont’d

nµ(a) =

• x∈µa

(|xA| − 1) : employment level of agent a in matching µ m(µ, µ′) =

• c∈C

I(µ(c) = µ(c′)) : number of couples that change position between stable matching µ and µ′ Approximate Rural Hospital Theorem with Couples For any two stable matchings µ and µ′ in a Resident Matching Problem with Couples (RMPC) the following inequality holds:

• a∈A

|nµ(a) − nµ′(a)| ≤ 2m(µ, µ′)

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 18 / 25

An Approximation Rural Hospital Theorem With Couples

Example

Example The couple c comprises of two doctors cm and cf The three doctors are d1 , d2 and d3

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 19 / 25

An Approximation Rural Hospital Theorem With Couples

Upper Bound

The area of the dot is pro- portional to the number of observations. The red line is the upper bound produced by our theorem (y = 2x), the black line the best fit of the line (y = 1.6x)

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 20 / 25

With Are Solutions to the Linear Program Integer?

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 21 / 25

With Are Solutions to the Linear Program Integer?

Doctor-optimal and hospital-optimal rounding

Definition Consider some z that is a feasible solution to the Stable Matching Linear Program. The matching µz is the doctor-optimal rounding of the fractional solution z. We define µz(d) = Chd({x : zx > 0}) ∀d ∈ D and µz(h) = {x ∈ Xh : x ∈ µz(xD)} ∀h ∈ H. Theorem If the market is bipartite and agents preferences satisfy substitutes and the law of aggregate demand then µz and µz are stable matchings. Corollary The SMLP with the doctor-optimal (or hospital-optimal) objective has a unique integer optimum z when the market is bipartite and agent preferences satisfy substitutes and the law of aggregate demand.

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 22 / 25

Stable Matching Linear Program With Couples

Outline

1

Introduction

2

Definitions

3

Formulating Stable Matching as an Integer Program

4

An Approximation Rural Hospital Theorem With Couples

5

With Are Solutions to the Linear Program Integer?

6

Stable Matching Linear Program With Couples

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 23 / 25

Stable Matching Linear Program With Couples

Methods

The three methods we use to find a stable matching are : A linear program with the doctor-optimal objective A integer program with the doctor-optimal objective Sorted deferred acceptance (SoDA) a deferred acceptance algorithm that accommodates couples.

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 24 / 25

Stable Matching Linear Program With Couples

Thank you.

Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 25 / 25