Stable Matching, Friendship, and Altruism Elliot Anshelevich - PowerPoint PPT Presentation

Stable Matching, Friendship, and Altruism Elliot Anshelevich Rensselaer Polytechnic Institute (RPI), Troy, New York Joint work with: Onkar Bhardwaj (RPI), Sanmay Das (WashU), Martin Hoefer (MPI), Yonatan Naamad (Princeton)

Stable Matching Also known as “stable marriage" Classic game theory and algorithmic problem Applications: residents and hospitals, students and schools, kidney matching, ...

Motivation Stable matching with cardinal utilities Students told to choose project partners for class Stable partner assignment: No two students should want to leave their present partner and be partners with each other (at least one of them should be unwilling)

Motivation Stable matching with cardinal utilities Students told to choose project partners for class Stable partner assignment: No two students should want to leave their present partner and be partners with each other (at least one of them should be unwilling) Eve Eve Eve Adam Adam Adam 100 100 100 STABLE Unstable 90 90 90 90 90 90 Assignment Assignment 55 Romeo 55 Romeo 55 Romeo Juliet Juliet Juliet

Stable Matching with Cardinal Utilities Model Undirected graph, weights on the edges (denoted r uv ) Nodes told to choose their partners u , v partners then both get reward r uv No partner then 0 reward Stability No “blocking pair" ( x , y ) a blocking pair if x prefers y over its current partner and vice versa. For now, higher preference = more reward

I. First Goal of this Talk Understand some basic properties of this "nice" stable matching model Does a stable matching exist? What is the quality of stable matchings? Can we improve their quality? Eve Eve Eve Adam Adam Adam 100 100 100 STABLE Unstable 90 90 90 90 90 90 Assignment Assignment 55 Romeo 55 Romeo 55 Romeo Juliet Juliet Juliet

I. First Goal of this Talk Understand some basic properties of this "nice" stable matching model Does a stable matching exist? Yes: Greedy matchings are stable. What is the quality of stable matchings? Can we improve their quality? Eve Eve Eve Adam Adam Adam 100 100 100 STABLE Unstable 90 90 90 90 90 90 Assignment Assignment 55 Romeo 55 Romeo 55 Romeo Juliet Juliet Juliet

Quality of Stable Matchings Eve Eve Eve Adam Adam Adam 100 100 100 STABLE Unstable 90 90 90 90 90 90 Assignment Assignment 55 Romeo 55 Romeo 55 Romeo Juliet Juliet Juliet Quality of a matching Value of matching: v ( M ) = � ( uv ) ∈ M r uv v ( M OPT ) Price of Anarchy: PoA = v ( M worst , stable ) v ( M OPT ) Price of Stability: PoS = v ( M best , stable )

I. First Goal of this Talk Understand some basic properties of this "nice" stable matching model Does a stable matching exist? Yes: Greedy matchings are stable. What is the quality of stable matchings? Can we improve their quality? Bounds on PoA, PoS v ( M ) = � ( uv ) ∈ M r uv v ( M OPT ) Price of Anarchy = v ( M worst , stable ) v ( M OPT ) Price of Stability = v ( M best , stable )

I. First Goal of this Talk Understand some basic properties of this "nice" stable matching model Does a stable matching exist? Yes: Greedy matchings are stable. In fact, stable matchings are exactly the greedy matchings. What is the quality of stable matchings? Can we improve their quality? Bounds on PoA, PoS v ( M ) = � ( uv ) ∈ M r uv v ( M OPT ) Price of Anarchy = v ( M worst , stable ) v ( M OPT ) Price of Stability = v ( M best , stable )

I. First Goal of this Talk Understand some basic properties of this "nice" stable matching model Does a stable matching exist? Yes: Greedy matchings are stable. In fact, stable matchings are exactly the greedy matchings. What is the quality of stable matchings? Can we improve their quality? Bounds on PoA, PoS v ( M ) = � ( uv ) ∈ M r uv v ( M OPT ) PoA, PoS ≤ 2 Price of Anarchy = v ( M worst , stable ) Tight Bounds v ( M OPT ) Price of Stability = v ( M best , stable )

Quality of Stable Matchings Bounds on PoA, PoS v ( M ) = � ( uv ) ∈ M r uv v ( M OPT ) PoA, PoS ≤ 2 Price of Anarchy = v ( M worst , stable ) Tight Bounds v ( M OPT ) Price of Stability = v ( M best , stable ) v v u u 11 11 Only Stable Optimum Matching 10 10 10 10 Matching (but unstable) Value = 11 Value = 20 w w z z

II. Main topic of this talk: Friendship and Altruism What if nodes do not care about only their own reward? They care about well-being of their friends (to some extent) Does it improve the quality of stable matchings?

II. Main topic of this talk: Friendship and Altruism What if nodes do not care about only their own reward? They care about well-being of their friends (to some extent) Does it improve the quality of stable matchings? Example Suppose the utility (or happiness) of nodes also counts the reward of their neighbors. v v u u 11 11 (u,v) no more Optimum Matching 10 10 10 10 (now stable!) a blocking pair w w z z

Utility Definition More formally, Utility: U ( u ) = R ( u ) + � v � = u α d ( u , v ) · R ( v ) A node cares α k about well-being of nodes k-hops away 1 ≥ α 1 ≥ α 2 ≥ · · · ≥ α diam ( G ) ≥ 0 ⇒ More distance means less care

Utility Definition More formally, Utility: U ( u ) = R ( u ) + � v � = u α d ( u , v ) · R ( v ) A node cares α k about well-being of nodes k-hops away 1 ≥ α 1 ≥ α 2 ≥ · · · ≥ α diam ( G ) ≥ 0 ⇒ More distance means less care Example utility calculation 1 Suppose α d ( u , v ) = d ( u , v ) , then: R ( w ) R ( x ) R ( y ) R ( z ) U ( u ) = R ( u ) + R ( v ) + + + + u v w x y z 2 3 4 5 3 3 5 5 5 4 1 2 3 = 1 + 1 + + + + 2 3 4 5

Stable Matching with Friendship or Altruism v ( M OPT ) v ( M OPT ) Recall, PoS = v ( M best , stable ) and PoA = v ( M worst , stable ) Stable Matching still exists Price of Anarchy still at most 2

Stable Matching with Friendship or Altruism v ( M OPT ) v ( M OPT ) Recall, PoS = v ( M best , stable ) and PoA = v ( M worst , stable ) Stable Matching still exists Price of Anarchy still at most 2 Theorem 2 + 2 α 1 With Friendship, PoS ≤ 1 + 2 α 1 + α 2 ( . . . a tight bound)

Bounds with Friendship Theorem 2 + 2 α 1 With Friendship, PoS ≤ 1 + 2 α 1 + α 2 v u Remarks: – Better than the bound of 2 without friendship biswivel – Only α 1 and α 2 matter w z – PoA stays the same 2 α 1 = α 2 = 1 / 2 ⇒ PoS ≤ 1 . 2 1.8 1.6 A little friendship makes a PoS 1.4 large difference for PoS. 1.2 1 0 0.2 0.4 0.6 0.8 1 alpha 1

Proof Sketch of PoS Bound (1/3) Algorithm Start with matching M = optimum matching Select best blocking pair, say ( u , v ) 1 – maximum r uv among all blocking pairs Make u , v partners (dropping their current partners) 2 Repeat 3

Proof Sketch of PoS Bound (1/3) Algorithm Start with matching M = optimum matching Select best blocking pair, say ( u , v ) 1 – maximum r uv among all blocking pairs Make u , v partners (dropping their current partners) 2 Repeat 3 Without Friendship or Altruism Converges to stable matching in linear time Output: a stable matching within factor of 2 of optimal

Proof Sketch of PoS Bound (1/3) Algorithm Start with matching M = optimum matching Select best relaxed blocking pair, say ( u , v ) 1 – maximum r uv among all relaxed blocking pairs Make u , v partners (dropping their current partners) 2 Repeat 3

Proof Sketch of PoS Bound (1/3) Algorithm Start with matching M = optimum matching Select best relaxed blocking pair, say ( u , v ) 1 – maximum r uv among all relaxed blocking pairs Make u , v partners (dropping their current partners) 2 Repeat 3 Will show Algorithm terminates after O ( m 2 ) iterations Output: a stable matching of high quality

Proof Sketch of PoS Bound (2/3) Types of blocking pairs v v unmatched u u v u swivel biswivel u v w z z swivel

Proof Sketch of PoS Bound (2/3) Types of blocking pairs v v unmatched u u v u swivel biswivel u v w z z swivel Relaxed blocking pairs v u unmatched v u v u relaxed biswivel (slightly weaker conditions) swivel u v w z (same conditions) swivel z (same conditions) ignores ( v , z ) and ( u , w ) edges

Proof Sketch of PoS Bound (1/3) Algorithm Start with matching M = optimum matching Select best relaxed blocking pair, say ( u , v ) 1 v u – maximum r uv among all relaxed blocking pairs Make u , v partners (dropping their current 2 biswivel partners) w z Repeat 3 Convergence Algorithm terminates after O ( m 2 ) iterations Output: a stable matching of high quality

Proof Sketch of PoS Bound (3/3) Trace trajectories of edges under algorithm execution w v u z x y Figure: Trajectory ( uv ) → ( vw ) → ( wx ) → ( xz ) Quality of matching can decrease only when relaxed biswivel Cannot decrease indefinitely because the trajectory of an edge can have at most one relaxed-biswivel – Conditions for relaxed biswivel bounds the decrease

Proof Sketch of PoS Bound Algorithm Start with matching M = optimum matching Select best relaxed blocking pair, say ( u , v ) 1 u v – maximum r uv among all relaxed blocking pairs Make u , v partners (dropping their current 2 biswivel partners) w z Repeat 3 Convergence Algorithm terminates after O ( m 2 ) iterations 2 + 2 α 1 Output: a stable matching within factor 1 + 2 α 1 + α 2 of optimum