CSC2556 Lecture 6 Kidney Exchange Cake-Cutting [Some - PowerPoint PPT Presentation

CSC2556 Lecture 6 Kidney Exchange Cake-Cutting [Some illustrations due to: Ariel Procaccia] CSC2556 - Nisarg Shah 1

Announcements • Project proposal ➢ Due: Mar 06 by 11:59PM ➢ I’ll soon put up a few sample project ideas. ➢ If you have trouble finding a project idea, meet me. • Structure ➢ Problem space introduction ➢ High-level research question ➢ Prior work ➢ Detailed goals • Length: Ideally 1 page (2 pages max) CSC2556 - Nisarg Shah 2

Kidney Exchange CSC2556 - Nisarg Shah 3

Kidney Exchange Donor 1 Donor 2 Patient 1 Patient 2 CSC2556 - Nisarg Shah 4

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 CSC2556 - Nisarg Shah 5

The strategic model • Undirected graph, only pairwise matches ➢ Vertex = donor-patient pair ➢ Edge = compatibility • Each agent controls a subset of vertices ➢ Possible strategy: hide some vertices (match internally), and only reveal others ➢ Utility of agent = # its matched vertices (self-matched + matched by mechanism) CSC2556 - Nisarg Shah 6

The strategic model • Mechanism: ➢ Input: revealed vertices by agents (edges are public) ➢ Output: matching • Target: # matched vertices • Strategyproof (SP): If no agent benefits from hiding vertices irrespective of what other agents do. CSC2556 - Nisarg Shah 7

OPT is manipulable CSC2556 - Nisarg Shah 8

OPT is manipulable CSC2556 - Nisarg Shah 9

Approximating SW • Theorem [Ashlagi et al. 2010]: No deterministic SP mechanism can give a 2 − 𝜗 approximation • Proof: ➢ No perfect matching exists. ➢ Any algorithm must either leave a blue node or a gray node unmatched. CSC2556 - Nisarg Shah 10

Approximating SW • Theorem [Ashlagi et al. 2010]: No deterministic SP mechanism can give a 2 − 𝜗 approximation • Proof: ➢ Suppose it leaves a blue node unmatched o If the blue agent hides two nodes as follows, the mechanism is forced to return a matching of size 1 when a matching of size 2 exists. CSC2556 - Nisarg Shah 11

Approximating SW • Theorem [Ashlagi et al. 2010]: No deterministic SP mechanism can give a 2 − 𝜗 approximation • Proof: ➢ Suppose it leaves a gray node unmatched o If the gray agent hides two nodes as follows, the mechanism is forced to return a matching of size 1 when a matching of size 2 exists. CSC2556 - Nisarg Shah 12

Approximating SW • Theorem [Kroer and Kurokawa 2013]: No randomized 6 5 − 𝜗 approximation. SP mechanism can give a • Proof: Homework! CSC2556 - Nisarg Shah 13

SP mechanism: Take 1 • Assume two agents • M ATCH {{1},{2}} mechanism: ➢ Consider matchings that maximize the number of “internal edges” for each agent. ➢ Among these return, a matching with max overall cardinality. CSC2556 - Nisarg Shah 14

Another example CSC2556 - Nisarg Shah 15

Guarantees • M ATCH {{1},{2}} gives a 2-approximation ➢ Cannot add more edges to matching ➢ For each edge in optimal matching, one of the two vertices is in mechanism ’ s matching • Theorem (special case): M ATCH {{1},{2}} is strategyproof for two agents. CSC2556 - Nisarg Shah 16

Proof 𝑊 𝑊 1 2 • 𝑁 = matching when player 1 is 𝑁 𝑁′ honest, 𝑁′ = matching when 𝑁′ player 1 hides vertices 𝑁 • 𝑁Δ𝑁′ consists of paths and even- 𝑁 length cycles, each consisting of alternating 𝑁, 𝑁′ edges 𝑁′ 𝑁 ∩ What ’ s wrong with the 𝑁′ illustration on the right? 𝑁 ∩ 𝑁′ CSC2556 - Nisarg Shah 17

Proof • Consider a path in 𝑁Δ𝑁′ , denote its edges in 𝑁 by 𝑄 and its edges in 𝑁′ by 𝑄′ • Consider sets 𝑄 11 , 𝑄 22 , 𝑄 12 containing edges of 𝑄 among 𝑊 1 , among 𝑊 2 , and between 𝑊 1 - 𝑊 2 ➢ Same for 𝑄′ 11 , 𝑄′ 22 , 𝑄′ 12 ′ • Note that 𝑄 11 ≥ 𝑄 11 ➢ Property of the algorithm CSC2556 - Nisarg Shah 18

Proof ′ • Case 1: 𝑄 11 = 𝑄 11 ′ • Agent 2 ’ s vertices don ’ t change, so 𝑄 22 = 𝑄 22 ′ • 𝑁 is max cardinality ⇒ 𝑄 12 ≥ 𝑄 12 • 𝑉 1 𝑄 = 2 𝑄 11 + 𝑄 12 ′ ′ = 𝑉 1 (𝑄 ′ ) ≥ 2 𝑄 + 𝑄 11 12 CSC2556 - Nisarg Shah 19

Proof 𝑊 𝑊 1 2 ′ • Case 2: 𝑄 11 > 𝑄 11 ′ • 𝑄 12 ≥ 𝑄 − 2 12 ➢ Every sub-path within 𝑊 2 is of even length ′ , ➢ Pair up edges of 𝑄 12 and 𝑄 12 except maybe the first and the last • 𝑉 1 𝑄 = 2 𝑄 11 + 𝑄 12 ′ ′ ≥ 2 𝑄 + 1 + 𝑄 − 2 11 12 = 𝑉 1 𝑄 ′ ∎ CSC2556 - Nisarg Shah 20

The case of 3 players CSC2556 - Nisarg Shah 21

SP Mechanism: Take 2 • Let Π = Π 1 , Π 2 be a bipartition of the players • M ATCH  mechanism: ➢ Consider matchings that maximize the 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 cardinality (need tie breaking) CSC2556 - Nisarg Shah 22

Eureka? • Theorem [Ashlagi et al. 2010]: M ATCH  is strategyproof for any number of agents and any partition Π . • Recall: For 𝑜 = 2 , M ATCH {{1},{2}} is a 2-approximation • Question: 𝑜 = 3 , M ATCH {{1},{2,3}} approximation? 1. 2 2. 3 3. 4 4. More than 4 CSC2556 - Nisarg Shah 23

The Mechanism • The M IX - AND -M ATCH mechanism: ➢ Mix: choose a random partition  ➢ Match: Execute M ATCH  • Theorem [Ashlagi et al. 2010]: M IX - AND -M ATCH is strategyproof and a 2-approximation. • We only prove the approximation ratio. CSC2556 - Nisarg Shah 24

Proof • 𝑁 ∗ = optimal matching • Claim: I can create a matching 𝑁 ′ such that ➢ 𝑁′ is max cardinality on each 𝑊 𝑗 , and ′ + ∗ + ∗ | 1 1 ′ ➢ σ 𝑗 𝑁 𝑗𝑗 2 σ 𝑗≠𝑘 𝑁 𝑗𝑘 ≥ σ 𝑗 𝑁 𝑗𝑗 2 σ 𝑗≠𝑘 |𝑁 𝑗𝑘 ➢ 𝑁 ∗∗ = max cardinality on each 𝑊 𝑗 ➢ For each path 𝑄 in 𝑁 ∗ Δ𝑁 ∗∗ , add 𝑄 ∩ 𝑁 ∗∗ to 𝑁′ if 𝑁 ∗∗ has more internal edges than 𝑁 ∗ , otherwise add 𝑄 ∩ 𝑁 ∗ to 𝑁′ ➢ For every internal edge 𝑁′ gains relative to 𝑁 ∗ , it loses at most one edge overall ∎ CSC2556 - Nisarg Shah 25

Proof • Fix Π and let 𝑁 Π be the output of M ATCH  • The mechanism returns max cardinality across Π subject to being max cardinality internally, therefore Π + Π ≥ ෍ ′ + ′ ෍ 𝑁 𝑗𝑗 ෍ 𝑁 𝑗𝑘 𝑁 𝑗𝑗 ෍ 𝑁 𝑗𝑘 𝑗 𝑗∈Π 1 ,𝑘∈Π 2 𝑗 𝑗∈Π 1 ,𝑘∈Π 2 CSC2556 - Nisarg Shah 26

Proof = 1 Π + 𝔽 𝑁 Π Π 2 𝑜 ෍ ෍ 𝑁 𝑗𝑗 ෍ 𝑁 𝑗𝑘 Π 𝑗 𝑗∈Π 1 ,𝑘∈Π 2 ≥ 1 ′ + ′ 2 𝑜 ෍ ෍ 𝑁 𝑗𝑗 ෍ 𝑁 𝑗𝑘 Π 𝑗 𝑗∈Π 1 ,𝑘∈Π 2 ′ + 1 ′ = ෍ 𝑁 𝑗𝑗 2 𝑜 ෍ ෍ 𝑁 𝑗𝑘 𝑗 Π 𝑗∈Π 1 ,𝑘∈Π 2 ′ + 1 ∗ + 1 ′ ∗ = ෍ 𝑁 𝑗𝑗 2 ෍ 𝑁 𝑗𝑘 ≥ ෍ 𝑁 𝑗𝑗 2 ෍ 𝑁 𝑗𝑘 𝑗 𝑗≠𝑘 𝑗 𝑗≠𝑘 ≥ 1 ∗ + 1 = 1 ∗ 2 𝑁 ∗ 2 ෍ 𝑁 𝑗𝑗 2 ෍ 𝑁 𝑗𝑘 ∎ 𝑗 𝑗≠𝑘 CSC2556 - Nisarg Shah 27

Cake-Cutting CSC2556 - Nisarg Shah 28

Cake-Cutting • A heterogeneous, divisible good ➢ Heterogeneous: it may be valued differently by different individuals ➢ Divisible: we can share/divide it between individuals • Represented as [0,1] ➢ Almost without loss of generality • Set of players 𝑂 = {1, … , 𝑜} • Piece of cake 𝑌 ⊆ [0,1] ➢ A finite union of disjoint intervals CSC2556 - Nisarg Shah 29

Agent Valuations • Each player 𝑗 has a valuation 𝑊 𝑗 that is very much like a probability distribution over [0,1] 𝛽 β • Additive: For 𝑌 ∩ 𝑍 = ∅ , 𝑊 𝑗 𝑌 + 𝑊 𝑗 𝑍 = 𝑊 𝑗 𝑌 ∪ 𝑍 β 𝛽 + 𝛾 • Normalized: 𝑊 0,1 = 1 𝑗 𝛽 • Divisible: ∀𝜇 ∈ [0,1] and 𝑌 , ∃𝑍 ⊆ 𝑌 s.t. 𝑊 𝑗 𝑍 = 𝜇𝑊 𝑗 (𝑌) 𝜇𝛽 CSC2556 - Nisarg Shah 30

Fairness Goals • An allocation is a disjoint partition 𝐵 = (𝐵 1 , … , 𝐵 𝑜 ) of the cake • We desire the following fairness properties from our allocation 𝐵 : • Proportionality (Prop): 𝑗 𝐵 𝑗 ≥ 1 ∀𝑗 ∈ 𝑂: 𝑊 𝑜 • Envy-Freeness (EF): ∀𝑗, 𝑘 ∈ 𝑂: 𝑊 𝑗 𝐵 𝑗 ≥ 𝑊 𝑗 (𝐵 𝑘 ) CSC2556 - Nisarg Shah 31

Fairness Goals • Prop: ∀𝑗 ∈ 𝑂: 𝑊 Τ 𝑗 𝐵 𝑗 ≥ 1 𝑜 • EF: ∀𝑗, 𝑘 ∈ 𝑂: 𝑊 𝑗 𝐵 𝑗 ≥ 𝑊 𝑗 𝐵 𝑘 • Question: What is the relation between proportionality and EF? Prop ⇒ EF 1. EF ⇒ Prop 2. Equivalent 3. Incomparable 4. CSC2556 - Nisarg Shah 32

C UT - AND -C HOOSE • Algorithm for 𝑜 = 2 players • Player 1 divides the cake into two pieces 𝑌, 𝑍 s.t. Τ 𝑊 1 𝑌 = 𝑊 1 𝑍 = 1 2 • Player 2 chooses the piece she prefers. • This is EF and therefore proportional. ➢ Why? CSC2556 - Nisarg Shah 33