Matching Markets with Ordinal Preferences
TIFR, May 2013
Matching Markets
𝜋1
𝜋2
𝜋3
• N agents, N items, N complete preferences.
• Outcome:
Agent-Item Matching
Outline of Talk
• Mechanisms
– Random Serial Dictatorship (RSD)
– Rank Maximal Matching (RMM)
• Welfare
– Ordinal Welfare Factor
– Rank Approximation
• Truthfulness
– Dealing with randomness.
Random Serial Dictatorship
• Agents arrive in a random permutation and
pick their best unallocated item.
𝜋 = (2,1,3)
…
𝜋 = (3,2,1)
Choice 1
Choice 2
Choice 3
Rank Maximal Matching
• Maximize #(top choice), then Maximize #(top 2),...
• Polytime computable.
Irving, 2003
Irving, Kavitha, Melhorn, Michail, Paluch, 2004.
Social Welfare
• Pareto Optimality.
No other outcome makes everyone happier.
• RMM leads to a Pareto Optimal outcome.
• RSD leads to ex-post Pareto Optimal outcome.
Social Welfare
• Cardinal Welfare
Each pair associated with cardinal number.
Social welfare = Sum of utilities.
• What to do when no numbers are known?
Ordinal Welfare Factor (OWF)
• Outcome 𝑀 is 𝛼-efficient, if for any 𝑀′,
#agents with 𝑀 ≥ 𝑀′ ≥ 𝛼𝑁
• Problem: Everyone has same ordering.
(1, 1)
(2, 2)
M = (3, 3)
…
(N,N)
(1, N)
(2, 1)
M’ = (3, 2)
…
(N,N-1)
𝛼 < 1/𝑁
Ordinal Welfare Factor (OWF)
• Randomization.
A distribution 𝑴 is 𝛼-efficient, if for any other
distribution 𝑴′,
𝐄𝐱𝐩 𝑀←𝑴,𝑀′←𝑴′ [#agents with (𝑀 ≥ 𝑀′ )] ≥ 𝛼𝑁
• Mechanism has OWF 𝛼 if it returns an 𝛼efficient distribution.
Symmetric “Bad” Example
• Every agent has same preference order.
• 𝑴 is uniform over all matchings.
• Fix matching 𝑀′ = { 1,1 , 2,2 … , 𝑁, 𝑁 },
∀𝑖,
• 𝑴 is
𝐏𝐫 𝑀←𝑴
1
1
+
2
2𝑛
𝑖
𝑀 ≥𝑖 𝑀 ≥
𝑁
-efficient.
′
Performance of Mechanisms
• Theorem. RSD has OWF ≥ 1/2
Bhalgat, C, Khanna 2011.
• RMM is deterministic.
Many agents can be made better off at the
expense of one agent.
Strengths and Weaknesses
• Comparative Measure.
• Notion of “approximation”.
Quantify mechanisms.
• Not good for deterministic mechanisms.
• No notion of “how much better off”.
Rank Approximation
• Let 𝑀𝑖∗ maximize #(agents getting top i)
𝑛𝑖 ≔ 𝑀𝑖∗
• 𝑀 is 𝛼-rank approximate if
#(agents getting top 𝑖 in 𝑀) ≥ 𝛼𝑛𝑖 .
• Mechanism has 𝛼-rank approximation if it
returns an 𝛼-rank approximate matching.
Connection to Cardinal Welfare
• Homogenous agents:
Each agent has same cardinal profile
𝑢1 > 𝑢2 > ⋯ > 𝑢𝑁
• 𝑀 is 𝛼-rank approximate implies
𝛼-approximation for homogenous agents.
Performance of Mechanisms
• Theorem. RMM has ½-rank approximation.
1
- Maximal/Maximum ≥
2
- Optimal.
• RSD is not 𝛼-approximate for any constant 𝛼.
≈ 𝑁
Choice 1
Strengths and Weaknesses
• Deterministic mechanisms can have good rank
approximation.
• Cardinal welfare for homogenous agents.
• Could improve many while hurting only a few.
• No good rank appx known in non-matching
setting.
Truthfulness
• If an agent lies, he gets a worse item.
If an agent lies, he doesn’t get a better item.
• Issues with randomized mechanisms.
What are worse and better distributions?
• Hierarchy of truthfulness.
Randomization vs Truthfulness
Universally Truthful.
Distribution over deterministic mechanisms
Strongly Truthful. (Gibbard, 77)
Lying gives a stochastically dominated allocation.
Lex Truthful. (?)
Lying gives a lexicographically dominated allocation.
Weakly Truthful. (Bogomolnaia-Moulin, 01)
Lying can’t give stochastically dominating allocation.
Lex Truthful Implementation
• A deterministic algorithm A can be 𝜖-lextruthful implemented if there is a randomized
mechanism M such that
– M is Lex Truthful.
– With probability > (1-𝜖), outcome of M is same as
that of A
Theorem. Any pseudomonotone algorithm A
is 𝜖-lex-implementable, for any 𝜖 > 0.
C, Swamy 2013
Pseudomonotonicity
A 𝜋𝑖 , 𝜋−𝑖 = 𝑀
𝜋𝑖
M(i)
M’(i)
M’(i) is below M(i) in 𝜋𝑖
A 𝜋′𝑖 , 𝜋−𝑖 = 𝑀′
𝜋𝑖
𝜋′𝑖
b
M’(i)
M(i)
b
or there’s b above M’(i) in 𝜋𝑖
which has been demoted.
Performance of Mechanisms
• RSD is Universally Truthful.
Under certain conditions, it is the only
strongly truthful mechanism. (Larsson, 94)
• RMM satisfies pseudomonotonicity.
Therefore, it can be 𝜖-LT implemeneted.
Summary
• Welfare definitions unclear in ordinal settings.
Saw two notions.
Generalizes to Social choice settings.
• Truthfulness of randomized mechanisms also
tricky. Hierarchy of truthfulness.
• Can results be extended to general settings?