cs234r
Markets for Networks and Crowds
BRENDAN LUCIER, MICROSOFT RESEARCH NE
NICOLE IMMORLICA, MICROSOFT RESEARCH NE
Lecture 3:
Itai Ashlagi, Yash Kanoria, and Jacob Leshno.
Unbalanced Random Matching Markets: The
Stark Effect of Competition, Journal of Political
Economy, forthcoming.
Unbalanced Markets:
Setting:
𝑛 men, 𝑛 + 𝑘 women, random complete preferences
• Let 𝑅𝑀 𝜇 = ave. rank of matched men in 𝜇.
• Let 𝑅𝑊 𝜇 = ave. rank of matched women in 𝜇.
𝑚1 : 𝑤1 ≻ 𝑤2 ≻ 𝑤3 ≻ 𝑤4
𝑚2 : 𝑤1 ≻ 𝑤3 ≻ 𝑤4 ≻ 𝑤2
𝑤1 : 𝑚3 ≻ 𝑚2 ≻ 𝑚1
𝑤2 : 𝑚1 ≻ 𝑚2 ≻ 𝑚3
𝑚3 : 𝑤2 ≻ 𝑤4 ≻ 𝑤3 ≻ 𝑤1
𝑤3 : 𝑚1 ≻ 𝑚3 ≻ 𝑚2
𝑤4 : 𝑚1 ≻ 𝑚2 ≻ 𝑚3
𝑘 = 1,
𝑅𝑀 𝜇 = 5 3 ,
𝑅𝑊 𝜇 = 2.
Unbalanced Markets:
Let 𝜇𝑀 (𝜇𝑊 ) be the man (woman)-opt. matching.
Prior work [Pittel’89], [Knuth-Motwani-Pittel’90]:
Proposing side does well in balanced (𝑘 = 0) markets.
• Pr 𝑅𝑀 𝜇𝑀 > 32 ln 𝑛 → 0 as 𝑛 grows
• 𝐸 𝑅𝑀 𝜇𝑀 ≤ ln 𝑛
• 𝐸 𝑅𝑊 𝜇𝑀 ≤ 𝑛 ln 𝑛
Unbalanced Markets:
Let 𝜇𝑀 (𝜇𝑊 ) be the man (woman)-opt. matching.
This paper:
Short side does well in all stable matchings (𝑘 > 0).
• 𝑅𝑀 𝜇 ≤ 3 ln 𝑛𝑘 , 𝑅𝑊 𝜇 ≥ 3 ln 𝑛𝑛 𝑘 for any 𝜇
And close to optimal.
• 𝑅𝑀 𝜇𝑊 ≤ 1 + 𝑜 1 𝑅𝑀 𝜇𝑀 ,
• 𝑅𝑊 𝜇𝑊 ≥ 1 − 𝑜 1 𝑅𝑊 𝜇𝑀
Simulation w/n=40
(the authors vary the # of men; this talk varies # of women)
Simulation w/n=40
(the authors vary the # of men; this talk varies # of women)
Core Idea:
Rejection chains:
Start from man-optimal 𝜇𝑀 .
Iteratively initiate rejection chains to reach 𝜇𝑊 .
Count total # proposals in process.
Analysis:
Let 𝑆 = 𝑤 ∶ 𝜇 𝑤 best stable mate .
When 𝑤 ∈ 𝑆 receives a proposal, algorithm terminates.
Improvement Cycle
𝑆
𝑆
Analysis:
Let 𝑆 = 𝑤 ∶ 𝜇 𝑤 best stable mate .
When 𝑤 ∈ 𝑆 receives a proposal, algorithm terminates.
𝑆
Terminal Phase
𝑆
𝑆
Analysis:
Let 𝑆 = 𝑤 ∶ 𝜇 𝑤 best stable mate .
When a 𝑤 ∈ 𝑆 receives a proposal, can’t be stable.
𝑆
Terminal Phase
𝑆
𝑆
𝑆
𝑆
Analysis:
Consider starting rejection chain from woman 𝑤
who received most proposals in 𝜇𝑀 .
• Phase likely to be terminal since single woman
accepts all proposals, but 𝑤 only accepts
those she prefers to the ones she’s received.
• Pr 𝑚 picks 𝑤 ≈ Pr[𝑚 picks single] = 1 𝑛
• Expected length of chain ≈ 𝑛.
𝑆 large implies future rejection chains are short.
INFORMAL Discussion:
Simulations indicate results hold with correlation.
INFORMAL Discussion:
But advantage diminishes as correlation increases.
INFORMAL Discussion:
Similarly, paper shows impact of
• Many-to-one
• Short lists
Question: What makes a side short?