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CMU 15-896 Matching 1: Kidney exchange Teacher: Ariel Procaccia 2 Donor 2 Exchange Patient 2 Kidney 15896 Spring 2016: Lecture 13 Patient 1 Donor 1 Incentives A decade ago kidney exchanges were carried out by individual hospitals

  1. CMU 15-896 Matching 1: Kidney exchange Teacher: Ariel Procaccia

  2. 2 Donor 2 Exchange Patient 2 Kidney 15896 Spring 2016: Lecture 13 Patient 1 Donor 1

  3. Incentives • A decade ago kidney exchanges were carried out by individual hospitals • Today there are nationally organized exchanges; participating hospitals have little other interaction • It was observed that hospitals match easy-to- match pairs internally, and enroll only hard-to- match pairs into larger exchanges • Goal: incentivize hospitals to enroll all their pairs 15896 Spring 2016: Lecture 13 3

  4. The strategic model • Undirected graph (only pairwise matches!) Vertices = donor-patient pairs o Edges = compatibility o Each player controls subset of vertices o • Mechanism receives a graph and returns a matching • Utility of player = # its matched vertices • Target: # matched vertices • Strategy: subset of revealed vertices But edges are public knowledge o • Mechanism is strategyproof (SP) if it is a dominant strategy to reveal all vertices 15896 Spring 2016: Lecture 13 4

  5. OPT is manipulable 15896 Spring 2016: Lecture 13 5

  6. OPT is manipulable 15896 Spring 2016: Lecture 13 6

  7. Approximating SW • Theorem [Ashlagi et al. 2010]: No deterministic SP mechanism can give a approximation • Proof: We just proved it! • Theorem [Kroer and Kurokawa 2013]: No � randomized SP mechanism can give a � approximation • Proof: Homework 2 15896 Spring 2016: Lecture 13 7

  8. SP mechanism: Take 1 • Assume two players • The M ATCH {{1},{2}} mechanism: Consider matchings that maximize the o number of “internal edges” Among these return a matching with max o cardinality 15896 Spring 2016: Lecture 13 8

  9. Another example 15896 Spring 2016: Lecture 13 9

  10. Guarantees • M ATCH {{1},{2}} gives a 2-approximation Cannot add more edges to matching o For each edge in optimal matching, one of o the two vertices is in mechanism’s matching • Theorem (special case): M ATCH {{1},{2}} is strategyproof for two players 15896 Spring 2016: Lecture 13 10

  11. Proof of theorem � � � � matching when player 1 is • � honest, = matching when player �′ 1 hides vertices �′ � consists of paths and even- • � length cycles, each consisting of alternating edges �′ � ∩ What’s wrong with the �′ illustration on the right? � ∩ �′ 15896 Spring 2016: Lecture 13 11

  12. Proof of theorem • Consider a path in , denote its edges in by and its edges in by • For �� � � � � � � �� � , suppose � • �� �� �� �� � • It holds that �� �� � is max cardinality • �� �� � � � • � �� �� �� �� � 15896 Spring 2016: Lecture 13 12

  13. Proof of theorem � � � � � • Suppose �� �� � • �� �� Every subpath within � is of o even length We can pair the edges of �� o � , except maybe the first and �� and the last • � �� �� � � � � �� �� 15896 Spring 2016: Lecture 13 13

  14. The case of 3 players 15896 Spring 2016: Lecture 13 14

  15. SP mechanism: Take 2 • Let � be a bipartition of the � players • The M ATCH  mechanism: Consider matchings that maximize the o number of “internal edges” and do not have any edges between different players on the same side of the partition Among these return a matching with max o cardinality (need tie breaking) 15896 Spring 2016: Lecture 13 15

  16. Eureka? • Theorem [Ashlagi et al. 2010]: M ATCH  is strategyproof for any number of players and any partition • Recall: for , M ATCH {{1},{2}} guarantees a 2-approx 15896 Spring 2016: Lecture 13 16

  17. Eureka? Poll 1: approximation guarantees given by M ATCH  for and ? 1. 2. 3. More than 4. 15896 Spring 2016: Lecture 13 17

  18. The mechanism • The M IX - AND -M ATCH mechanism: Mix: choose a random partition  o Match: Execute M ATCH  o • Theorem [Ashlagi et al. 2010]: M IX - AND - M ATCH is strategyproof and guarantees a 2-approximation • We only prove the approximation ratio 15896 Spring 2016: Lecture 13 18

  19. Proof of theorem ∗ optimal matching • � such that • Create a matching is max cardinality on each � , and � � ∗ ∗ �� �� �� �� � ��� � ��� � ∗∗ � max cardinality on each � � o For each path � in � ∗ Δ� ∗∗ , add � ∩ � ∗∗ to �′ if o � ∗∗ has more internal edges than � ∗ , otherwise add � ∩ � ∗ to �′ For every internal edge �′ gains relative to � ∗ , it o loses at most one edge overall ∎ 15896 Spring 2016: Lecture 13 19

  20. Proof of theorem � be the output of • Fix and let M ATCH  • The mechanism returns max cardinality across subject to being max cardinality internally, therefore � � � � �� �� �� �� � �∈� � ,�∈� � � �∈� � ,�∈� � 15896 Spring 2016: Lecture 13 20

  21. Proof of theorem � 1 � � � � � � 2 � � � � �� � � �� � � �∈� � ,�∈� � � 1 � � � 2 � � � � �� � � �� � � �∈� � ,�∈� � � � 1 � � � � �� 2 � � � � �� � � �∈� � ,�∈� � � � 1 ∗ � 1 � ∗ � � � �� 2 � � �� � � � �� 2 � � �� � ��� � ��� � 1 ∗ � 1 � 1 2 � ∗ ∎ ∗ 2 � � �� 2 � � �� � ��� 15896 Spring 2016: Lecture 13 21

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