The Stable Matching Linear Program and an Approximate Rural Hospital - PowerPoint PPT Presentation

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 Introduction 1 Definitions 2 Formulating Stable Matching as an Integer Program 3 An Approximation Rural Hospital Theorem With Couples 4 With Are Solutions to the Linear Program Integer? 5 Stable Matching Linear Program With Couples 6 Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 2 / 25

Introduction Outline Introduction 1 Definitions 2 Formulating Stable Matching as an Integer Program 3 An Approximation Rural Hospital Theorem With Couples 4 With Are Solutions to the Linear Program Integer? 5 Stable Matching Linear Program With Couples 6 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 Introduction 1 Definitions 2 Formulating Stable Matching as an Integer Program 3 An Approximation Rural Hospital Theorem With Couples 4 With Are Solutions to the Linear Program Integer? 5 Stable Matching Linear Program With Couples 6 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 → 2 X : function between a set of contracts X and a set of agents A µ ( a ) : set of contracts matched to agent a x A ⊆ A : two or more agents are involved in contract x X a � { x ∈ X : a ∈ x A } : set of contracts that agent a could be matched to > a : strict preferences of agent a ∈ A Ch a : 2 X a → 2 X a : returns the agent a ’s choice set Example 1 If an agent a has the preferences { x 1 , x 2 } > a { x 3 } > a ∅ over contracts then Ch a { x 1 , x 2 , x 3 } = { x 1 , x 2 } and Ch a ( { x 1 , x 3 } ) = { x 3 } . 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 Ch a ( µ ( a )) = µ ( a ) . A matching µ as stable if for every unused contract at least one agent a ∈ x A does not want to sign it: x �∈ Ch a ( 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 : | Ch a ( U ∪ V ) | ≥ | Ch a ( U ) | for all U, V ⊆ X Substitutable preferences : if x, x ′ ∈ X, B ⊆ X and x �∈ Ch a ( B ) implies x �∈ Ch a ( B ∪ x ′ ) Responsive preferences : Suppose agent a would choose at most k a contracts. If the contracts can be listed x 1 < a x 2 < a . . . < a x n such that if i < j and | B | < k a then B ∪ { x i } < a B ∪ { x j } Ordinal preferences : Agents only ever want at most one contract, this is described formally as | Ch a ( S ) | ≤ 1 ∀ S ∈ Q a 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 one 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 c f and c m ), doctor d and two hospitals h 1 and h 2 . Contracts : x 1 = (( c m , h 1 ) , ( c f , h 2 )) , x 2 = (( c m , h 1 )) , x 3 = (( d, h 1 )) Hospital h 1 prefers c m over d and the couple c prefers x 1 over x 2 . Clearly h 1 is indifferent between x 1 and x 2 Therefore we break ties in the hospitals preferences over contracts in favor of the couples Preference list of the hospital h 1 : x 1 > h 1 x 2 > h 1 x 3 . 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 Introduction 1 Definitions 2 Formulating Stable Matching as an Integer Program 3 An Approximation Rural Hospital Theorem With Couples 4 With Are Solutions to the Linear Program Integer? 5 Stable Matching Linear Program With Couples 6 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 z x ∈ { 0 , 1 } : shows whether contract x is used. y a,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 � y a,B = 1 ∀ a ∈ A B ∈ Q a Matching constraint : If a contract x ∈ X is used then all agents a ∈ x A must sign it. � z x = ∀ x ∈ X, ∀ a ∈ x A y a,B B ∈ Q a : x ∈ B Non-negativity constraint : z x , y a,B ≥ 0 Stability constraint : If a contract x ∈ X is unused then at least one agent a ∈ x A does not want to sign it. � � z x + y a,B ≥ 1 ∀ x ∈ X a ∈ x A B ∈ Q a : x �∈ Ch a ( B ∪ 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 z x x ∈ X Stable marriage objective : Maximize a linear function over the contracts � max r x z x x ∈ X Doctor-optimal objective : Maximize the utility of the doctors � � u d ( B ) y d,B max d ∈ D B ∈ Q d Arbitrary objective : Maximize a linear function over sets of contracts allocated to agents � � max f a ( B ) y a,B a ∈ A B ∈ Q a 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 Introduction 1 Definitions 2 Formulating Stable Matching as an Integer Program 3 An Approximation Rural Hospital Theorem With Couples 4 With Are Solutions to the Linear Program Integer? 5 Stable Matching Linear Program With Couples 6 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 ordinal preferences i.e. a RMPC with | C | = 0 . In every stable matching the same doctors and hospitals will be matched. That is: n µ ( a ) = n µ ′ ( 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 � n µ ( a ) = ( | x A | − 1) : employment level of agent a in matching µ x ∈ µ a � m ( µ, µ ′ ) = I ( µ ( c ) � = µ ( c ′ )) : number of couples that change c ∈ C 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: � | n µ ( a ) − n µ ′ ( a ) | ≤ 2 m ( µ, µ ′ ) a ∈ A Marzieh Barkhordar, Elahe Ghasempour The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples December 24,2017 18 / 25