1. Matching Markets with Ordinal Preferences
25 minutes
TIFR, May 2013

2. Matching Markets
𝜋 1
𝜋 2
𝜋 3
Problem Statement. Fish out references.
N agents, N items, N complete preferences.
Outcome: Agent-Item Matching

3. Outline of Talk Mechanisms Welfare Truthfulness
Random Serial Dictatorship (RSD)
Rank Maximal Matching (RMM)
Welfare
Ordinal Welfare Factor
Rank Approximation
Truthfulness
Dealing with randomness.

4. Random Serial Dictatorship
Agents arrive in a random permutation and pick their best unallocated item.
𝜋= (2,1,3)
Choice 1
Choice 2
Choice 3
Definition.
Strongly truthful.
Uniqueness. …
𝜋= (3,2,1)

5. Rank Maximal Matching
Maximize #(top choice), then Maximize #(top 2),...
Polytime computable.
RSD and PS have bad rank approximation
Good Rank approximation for RMM
Irving, 2003 Irving, Kavitha, Melhorn, Michail, Paluch, 2004.

6. 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.
How would we define social welfare? (Break into two slides?)
Pareto Optimality
Cardinality. Why is it not desirable? 1. Bad numbers. 2. Arrow
Two notions:
Condorcet? Ordinal Welfare Factor?
Rank Approximation – connection to cardinal welfare

7. Social Welfare
Cardinal Welfare Each pair associated with cardinal number.
Social welfare = Sum of utilities.
What to do when no numbers are known?
How would we define social welfare? (Break into two slides?)
Pareto Optimality
Cardinality. Why is it not desirable? 1. Bad numbers. 2. Arrow
Two notions:
Condorcet? Ordinal Welfare Factor?
Rank Approximation – connection to cardinal welfare

8. Ordinal Welfare Factor (OWF)
Outcome 𝑀 is 𝛼-efficient, if for any 𝑀′,
Problem: Everyone has same ordering.
#agents with 𝑀≥ 𝑀 ′ ≥𝛼𝑁
(1, 1)
(2, 2)
(3, 3) …
(N,N)
(1, N)
(2, 1)
(3, 2) …
(N,N-1)
M =
M’ =
𝛼<1/𝑁

9. Ordinal Welfare Factor (OWF)
Randomization. A distribution 𝑴 is 𝛼-efficient, if for any other distribution 𝑴′,
Mechanism has OWF 𝛼 if it returns an 𝛼-efficient distribution.
𝐄𝐱 𝐩 𝑀←𝑴, 𝑀 ′ ← 𝑴 ′ [#agents with (𝑀≥ 𝑀 ′ )]≥𝛼𝑁

10. Symmetric “Bad” Example
Every agent has same preference order.
𝑴 is uniform over all matchings.
Fix matching 𝑀 ′ ={ 1,1 , 2,2 …, 𝑁,𝑁 },
𝑴 is 𝑛 -efficient.
∀𝑖, 𝐏 𝐫 𝑀←𝑴 𝑀 ≥ 𝑖 𝑀 ′ ≥ 𝑖 𝑁

11. Performance of Mechanisms
Theorem. RSD has OWF ≥ 1/2
RMM is deterministic. Many agents can be made better off at the expense of one agent.
Bhalgat, C, Khanna 2011.

12. Strengths and Weaknesses
Comparative Measure.
Notion of “approximation”. Quantify mechanisms.
Not good for deterministic mechanisms.
No notion of “how much better off”.

13. Rank Approximation
Let 𝑀 𝑖 ∗ maximize #(agents getting top i) 𝑛 𝑖 ≔ 𝑀 𝑖 ∗
𝑀 is 𝛼-rank approximate if #(agents getting top 𝑖 in 𝑀) ≥𝛼 𝑛 𝑖 .
Mechanism has 𝛼-rank approximation if it returns an 𝛼-rank approximate matching.
At the end of this slide state that there’s no relation between the two.

14. Connection to Cardinal Welfare
Homogenous agents: Each agent has same cardinal profile 𝑢 1 > 𝑢 2 >⋯> 𝑢 𝑁
𝑀 is 𝛼-rank approximate implies 𝛼-approximation for homogenous agents.

15. Performance of Mechanisms
Theorem. RMM has ½-rank approximation. - Maximal/Maximum ≥ Optimal.
RSD is not 𝛼-approximate for any constant 𝛼.
≈ 𝑁
Choice 1

16. 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.

17. 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.
Definition of truthfulness (small slide)

18. 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.
Various notions under randomization.
Stochastic dominance
Weak SD
Lex.
Weakly Truthful. (Bogomolnaia-Moulin, 01)
Lying can’t give stochastically dominating allocation.

19. Lex Truthful Implementation
A deterministic algorithm A can be 𝜖-lex-truthful 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

20. Pseudomonotonicity A 𝜋 𝑖 , 𝜋 −𝑖 =𝑀 A 𝜋′ 𝑖 , 𝜋 −𝑖 =𝑀′ 𝜋 𝑖 𝜋 𝑖 𝜋′ 𝑖
M(i) M’(i)
b M’(i) M(i)
M’(i) is below M(i) in 𝜋 𝑖
𝜋′ 𝑖 b
ADD a proof slide?
RMM satisfies pseudomonotonicity; proof of the theorem.
or there’s b above M’(i) in 𝜋 𝑖
which has been demoted.

21. 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.

22. 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?

23. Recall RSD and OWF Agents arrive in a random order 𝜎.
At time 𝑡, agent 𝜎[𝑡] chooses his best item among the unallocated items.
Induces a distribution on matchings.
Fix any matching 𝑀’.
𝑀′(𝑎): item allocated to agent a in M’.
Theorem: Expected number of agents a getting item M’(a) or better is ≥𝑁/2.
Bhalgat, C, Khanna 2011.

24. OWF of RSD 𝑀′(𝑎): item allocated to agent a in M’.
Call 𝑎 dead at time 𝑡 if 𝑀′(𝑎) allocated by 𝑡 and 𝑎 hasn’t arrived by time t.
𝐷 𝑡 : expected dead agents at time t.
𝐴𝐿 𝐺 𝑡 : expected number of “happy” agents at 𝑡.
𝐴𝐿 𝐺 𝑡+1 −𝐴𝐿 𝐺 𝑡 ≥1 − 𝐷 𝑡 𝑁−𝑡
Claim: 𝐷 𝑡 ≤ 𝑡+2 𝑁−𝑡 𝑁+1 ≈ 𝑡 𝑁−𝑡 𝑁

25. Proof of Claim Claim: 𝐷 𝑡 ≤ 𝑡+2 𝑁−𝑡 𝑁+1 ≈ 𝑡 𝑁−𝑡 𝑁
𝐸 𝑎 :𝑎 hasn’t arrived by time t.
𝐷 𝑡 = 𝒂 𝐏𝐫 𝐸 𝑎 ∧ 𝐹 𝑎
𝐹 𝑎 :𝑀′(𝑎) allocated by time t.
𝐏𝐫 𝐸 𝑎 = 𝑁−𝑡 𝑁
a 𝐏𝐫 𝐹 𝑎 = 𝑖 𝐏𝐫[ Item 𝑖 allocated by time 𝑡]=𝑡
Independent, then done. ≈ Negatively correlated.