Matching Markets with Ordinal Preferences

TIFR, May 2013

Matching Markets

• N agents, N items, N complete preferences.

• Outcome: Agent-Item Matching

𝜋 1

𝜋 2

𝜋 3

Outline of Talk

• Mechanisms – Random Serial Dictatorship (RSD)– Rank Maximal Matching (RMM)

• Welfare – Ordinal Welfare Factor– Rank Approximation

• Truthfulness– Dealing with randomness.

• Agents arrive in a random permutation and pick their best unallocated item.

Random Serial Dictatorship

Choice 1Choice 2Choice 3

(2,1,3)

…

(3,2,1)

• Maximize #(top choice), then Maximize #(top 2),...

• Polytime computable.

Rank Maximal Matching

Irving, 2003Irving, 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 WelfareEach pair associated with cardinal number.

Social welfare = Sum of utilities.

• What to do when no numbers are known?

• Outcome is -efficient, if for any ,

• Problem: Everyone has same ordering.

#agents with

Ordinal Welfare Factor (OWF)

(1, 1)(2, 2)(3, 3)…(N,N)

M =

(1, N)(2, 1)(3, 2)…(N,N-1)

M’ = 𝛼<1/𝑁

Ordinal Welfare Factor (OWF)

• Randomization. A distribution is -efficient, if for any other distribution ,

• Mechanism has OWF if it returns an -efficient distribution.

agents with

Symmetric “Bad” Example

• Every agent has same preference order.• is uniform over all matchings.• Fix matching ,

• is -efficient.

∀ 𝑖 ,𝐏𝐫 {𝑀←𝑴 } [𝑀 ≥𝑖𝑀′ ]≥ 𝑖

𝑁

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.

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 maximize #(agents getting top i)

• is -rank approximate if #(agents getting top in ) .

• Mechanism has -rank approximation if it returns an -rank approximate matching.

Connection to Cardinal Welfare

• Homogenous agents: Each agent has same cardinal profile

• is -rank approximate implies -approximation for homogenous agents.

Performance of Mechanisms

• Theorem. RMM has ½-rank approximation.- Maximal/Maximum

- Optimal.

• RSD is not -approximate for any constant .

Choice 1≈√𝑁

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.

Weakly Truthful. (Bogomolnaia-Moulin, 01)

Lying can’t give stochastically dominating allocation.

Lex Truthful. (?)

Lying gives a lexicographically dominated allocation.

Lex Truthful Implementation

• A deterministic algorithm A can be -lex-truthful implemented if there is a randomized mechanism M such that

– M is Lex Truthful.– With probability > (1-), outcome of M is same as

that of A

Theorem. Any pseudomonotone algorithm Ais -lex-implementable, for any

C, Swamy 2013

Pseudomonotonicity

A A

M(i) M’(i)𝜋 𝑖 b M’(i) M(i)𝜋 𝑖

b 𝜋 ′ 𝑖M’(i) is below M(i) in

or there’s b above M’(i) in 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 -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?