NUS - Top Trading Cycles in Prioritized Matching: An Irrelevance … · 2018. 7. 25. · Top...

Minimizing Justified Envy in School Choice:The Design of New Orleans’ One App

A. Abdulkadiroglu, Y-K. Che, P. Pathak, A. Roth, and O. Tercieux

Top Trading Cycles in Prioritized Matching: AnIrrelevance of Priorities in Large Markets

Y-K. Che and O. Tercieux

Workshop on Matching, Search and Market Design

July 26, 2018

(Workshop on Matching, Search and Market Design)Minimizing Justified Envy in School Choice: The Design of New Orleans’ One App A. Abdulkadiroglu, Y-K. Che, P. Pathak, A. Roth, and O. Tercieux Top Trading Cycles in Prioritized Matching: An Irrelevance of Priorities in Large Markets Y-K. Che and O. TercieuxJuly 26, 2018 1 / 47

Priority based resource allocation

Matching of agents to indivisible objects based on their (expressed)preferences and priorities.

Examples: school choice, housing allocation, organ exchange;

Two (conflicting) goals:

1 Pareto-efficiency: maximizing agents’ welfare

2 Stability: respecting agents’ priorities/eliminating justified envy.

Our questions

In practice mainly two mechanisms are recommended by economists:(e.g., Boston)

Deferred Acceptance (Gale-Shapley)(eventually adopted by several school districts, e.g., Boston)

Top Trading Cycle (Shapley-Scarf, Abdulkadiroglu-Sonmez)(eventually adopted by several school districts, e.g., New Orleans)

Why are these two mechanisms (DA and TTC) so prominent?

Our questions

In practice mainly two mechanisms are recommended by economists:(e.g., Boston)

Deferred Acceptance (Gale-Shapley)(eventually adopted by several school districts, e.g., Boston)

Top Trading Cycle (Shapley-Scarf, Abdulkadiroglu-Sonmez)(eventually adopted by several school districts, e.g., New Orleans)

Why are these two mechanisms (DA and TTC) so prominent?

Our questions

Fix one of the criterion, fulfill as much as possible the other.

If the policy maker prefers the elimination of justified envy. DA is the“natural” choice (maximizes efficiency)

On the other hand, if the policy maker prefers efficiency, alternativeabound: TTC, SD,...

I Many Pareto-efficient mechanisms, how do they compare in terms ofjustified envy?

Raises an even more basic question. TTC has been recommended(and adopted). TTC ”uses” priorities but in what sense does itimprove on mechanisms ignoring priorities? (Like RSD)

Our questions

Fix one of the criterion, fulfill as much as possible the other.

If the policy maker prefers the elimination of justified envy. DA is the“natural” choice (maximizes efficiency)

On the other hand, if the policy maker prefers efficiency, alternativeabound: TTC, SD,...

I Many Pareto-efficient mechanisms, how do they compare in terms ofjustified envy?

Raises an even more basic question. TTC has been recommended(and adopted). TTC ”uses” priorities but in what sense does itimprove on mechanisms ignoring priorities? (Like RSD)

Preview

In one-to-one environments:I there is no Pareto efficient and strategy proof mechanism that has less

justified envy than TTC.

In the many-to-one environment:I TTC (and existing variants) are all envy-dominated by alternative

Pareto efficient and strategy proof mechanisms.I In expectation, TTC has less justified envy than RSD

In Large MarketsI Strong equivalence result between TTC and RSD in large economies.I Large nb. objects: TTC is not significantly different from RSD in terms

of justified envy

A priority based allocation problem consists of:

1. a set of individuals I = {i1, ..., in},2. a set of objects O = {o1, ..., om},3. a vector copies for each object q = (qo1 , ..., qom),

4. a list of strict individuals preferences P = (Pi1 , ...,Pin), and

5. a list of strict object priorities �= (�o1 , ...,�om).

µ(i) ∈ O ∪ {i} is individual i ’s match

Pareto efficiency, justified envy (blocking), matching mechanism,strategy-proofness

Top Trading Cycle (TTC) mechanism

Step 1:

Each individual points to his most preferred object

Each object points to the individual with the highest priority

There exists at least one cycle and no cycles intersect.

Individuals in a cycle get the object they point.

Remove assigned individuals; adjust number of copies for objects involvedin cycles.

Step n:

Each individual points to his most preferred objectamong remaining ones

Each object points to the individual with the highest priorityamong remaining ones

There exists at least one cycle and no cycles intersect.

Individuals in a cycle get the object they point.

Remove assigned individuals / adjust number of copies for objects involvedin cycles.

Comparing Mechanisms

Definition

A mechanism ϕ1 has less justified envy than ϕ2 at priority profile �, iffor any preference profile P and individual-object pair (i , o), if pair (i , o)blocks ϕ1(P,�), then pair (i , o) blocks ϕ2(P,�).

A mechanism ϕ1 has less justified envy than ϕ2 if it has less justifiedenvy than ϕ2 at each priority profile �

Definition

A mechanism ϕ1 has strictly less justified envy than ϕ2 if ϕ1 has lessenvy than ϕ2, but ϕ2 does not have less envy than ϕ1.

Definition

Minimal Envy

Since the concept of less justified envy defines a preorder, our lastdefinition describes the minimal element of that order.

Definition

Given a class of mechanisms C, ϕ is justified envy minimal in C if thereis no other mechanism ψ in C that has strictly less envy than ϕ.

Main Result

Theorem

Suppose each object comes in single copy. Let ϕ be a Pareto efficient andstrategy-proof mechanism. If ϕ has less justified envy than TTC at �,then ϕ(·,�) = TTC (·,�).

Corollary

Suppose each object comes in single copy. TTC is justified envyminimal in the class of Pareto-efficient and strategy-proof mechanisms.

Discussion

One cannot further reduce justified envy in TTC without sacrificingefficiency or strategy-proofness.

This result does not imply that TTC is the only justified-envy minimalmechanism in the class of Pareto efficient and strategy-proofmechanisms.

However, consider a general class of SP+PE mechanisms:

I fo :�o→�o be an arbitrary function that transforms its priority intoanother (possibly same or distinct) priority. Let f = (fo)o .

I f = (fo)oI Let ϕ(·,�) = TTC (·, f (�))

Theorem

Suppose fo(�o) 6=�o for some object o. Then, the mechanismϕ(·,�) = TTC (·, f (�)) is not justified-envy minimal.

Discussion

One cannot further reduce justified envy in TTC without sacrificingefficiency or strategy-proofness.

This result does not imply that TTC is the only justified-envy minimalmechanism in the class of Pareto efficient and strategy-proofmechanisms.

However, consider a general class of SP+PE mechanisms:

I fo :�o→�o be an arbitrary function that transforms its priority intoanother (possibly same or distinct) priority. Let f = (fo)o .

I f = (fo)oI Let ϕ(·,�) = TTC (·, f (�))

Theorem

Suppose fo(�o) 6=�o for some object o. Then, the mechanismϕ(·,�) = TTC (·, f (�)) is not justified-envy minimal.

Many-to-one

The three TTC variations

TTC-Counters, TTC-Clinch and Trade and Equitable TTC are notgenerally comparable in terms of justified envy in general.

Furthermore

Theorem

Suppose there is an object with more than one copy. Then,for each of the three mechanisms, there exists a justified-envy minimal,Pareto efficient, and strategy-proof mechanism that has strictly lessjustified envy than that mechanism.

The three TTC variations

TTC-Counters, TTC-Clinch and Trade and Equitable TTC are notgenerally comparable in terms of justified envy in general.

Furthermore

Theorem

Suppose there is an object with more than one copy. Then,for each of the three mechanisms, there exists a justified-envy minimal,Pareto efficient, and strategy-proof mechanism that has strictly lessjustified envy than that mechanism.

Back to the more basic question

Can we compare TTC and RSD?

In general, TTC does not have less justified envy than RSD at someprofile of priorities.

How can we capture our intuition that TTC has less envy than RSD?On average/in expectation?

Example with n = 2: Relevance of Priorities

I = {1, 2} and O = {o1, o2}. Suppose both prefer o1 over o2. (If not,the same between TTC and RSD.)

I Under TTC, o1 is assigned to its top-priority agent. ⇒ No justifiedenvy.

I But under RSD, the assignment is random, so there is 1/2 chance of“justified envy.”

NB: Well-known equivalence result (originally byAbdulkadiroglu-Sonmez, but more appropriately Pathak-Sethuraman(2011)) does not apply to the joint distribution of assignment.

Random Priorities – TTC vs Random Serial Dictatorship

Draw priorities randomly, run TTC and RSD

TTC and RSD have the same distribution of ranks for individuals

However, the likelihood with which justified envy arises is differentunder two mechanisms:

For mechanisms M = {TTC ,RSD} and any pair (i , o), NM(i , o) isthe number of individuals that i justifiably envies at o.

Theorem

Given a individual-object pair (i , o), NRSD(i , o) first-order stochasticallydominates NTTC (i , o).

In addition, if, under TTC, individual i prefers object o to his assignmentwith strictly positive probability, then NRSD(i , o) strictly stochasticallydominates NTTC (i , o).

Corollary

The expected number of individuals with justified envy, the expectednumber of blocking pairs, and the expected number of individuals eachindividual justifiably envies are all smaller under TTC than under RSD.

TTC has ”probabilistically” strictly less justified envy than RSD.

Theorem

Corollary

Theorem

Corollary

Theorem

Corollary

Intuition for the finite market

TTC induces the same probability as RSD for any agent i to envyanother j over some o. (Why? the equivalence result)

Whether that envy is justified depends on whether o assigned via ashort cycle (j → o → j) or a long cycle (j → o → ...→ j):

1 Short cycle ⇒ No [since if i had higher priority, then o would point to ibefore it does to j .]

2 Long cycle ⇒ With prob 1/2 (just like under RSD) [ since in that casej ’s assignment to o “has nothing to do with” her priority.]

Since short cycle occurs with positive probability, TTC inducesprobabilistically less justified envy than RSD.

Comparing Mechanisms in New Orleans

Comparing Mechanisms in Boston

Large Markets

Large markets

What can we expect in environments with large number of objects(very much like NYC)?

I random priorities and random preferences

I n individuals and objects (one-to-one) where n→∞

Example with large n: Asymptotic Irrelevance of Priorities

Expected (normalized) Rank Achieved by Objects in TIC, RSD

� g C.

U') U')

-e- RSD

2 10 100 1000 10000

General Result: Asymptotic Irrelevance of Priorities

Let Rk be the normalized rank [i.e., rank/n] enjoyed by k (either an agentor an object) under TTC.

Theorem

{(Roj )nj=1, (Rik )nk=1} converges in distribution to {(U[0, 1])n, (Rik )nk=1} as

n→∞, where U[0, 1] is uniform [0, 1].

Corollary

In the limit, TTC induces the same distribution of the ranks enjoyed by allagents and objects, and thus the same incidence of justified envy, as RSD.

General Result: Asymptotic Irrelevance of Priorities

Let Rk be the normalized rank [i.e., rank/n] enjoyed by k (either an agentor an object) under TTC.

Theorem

{(Roj )nj=1, (Rik )nk=1} converges in distribution to {(U[0, 1])n, (Rik )nk=1} as

n→∞, where U[0, 1] is uniform [0, 1].

Corollary

In the limit, TTC induces the same distribution of the ranks enjoyed by allagents and objects, and thus the same incidence of justified envy, as RSD.

Intuition for the Asymptotic Irrelevance

Key Observation 1: Let R∗o be the priority rank of the agent that opoints to when it is assigned under TTC. Then, the priority rank oenjoys under TTC ∼ U{R∗o + 1, ..., n} conditional on the object beingassigned via a long cycle.

Key Observation 2:

(a) The proportion of objects assigned via long cyclesp−→ 1.

(b) The proportion of objects o for which R∗o < log(n)

p−→ 1.

In words, virtually all objects are assigned via long cycles and point tovery high priority individuals (in relative ranks) when assigned.

⇒ In a sufficiently large market, the objects’ normalized ranks are eachdistributed according to U[0, 1], independently of the ranks enjoyedby the agents.

Key Observation 2:

p−→ 1.

Key Observation 2:

p−→ 1.

Key Observation 2:

p−→ 1.

Key Observation 2:

p−→ 1.

To obtain (a), we characterize the probabilistic structure of TTC

(i) the number of agents and objects follow a simple Markov chain: nottrivial due to the conditioning issue.

(ii) the number of rounds required for TTC is sublinear in n.(iii) the expected number of objects assigned via short cycles per round is

bounded (by 2).

⇒ Combining (ii) and (iii) give (a)

To obtain (b), we imagine a new mechanism TTC∗—same as TTCexcept that objects are assigned agents that objects point to. Paretoefficiency from objects’ perspective leads to (b) [see Che and Tercieux(TE, 2018)].

Key Observation 2:

p−→ 1.

To obtain (a), we characterize the probabilistic structure of TTC

(i) the number of agents and objects follow a simple Markov chain: nottrivial due to the conditioning issue.

(ii) the number of rounds required for TTC is sublinear in n.(iii) the expected number of objects assigned via short cycles per round is

bounded (by 2).

⇒ Combining (ii) and (iii) give (a)

To obtain (b), we imagine a new mechanism TTC∗—same as TTCexcept that objects are assigned agents that objects point to. Paretoefficiency from objects’ perspective leads to (b) [see Che and Tercieux(TE, 2018)].

Robustness

Correlated Preferences:I Weak correlation: ui (o) = uo + ξio , where uo and ξio have bounded

support. Then, our result is robust.

I Extreme correlation: All have identical preferences. TTC admits fewerjustified envy than does RSD.

Ultimately clarifies when priorities are relevant under TTC.I One-to-one setting: TTC is using priorities in an “optimal” way.

I Many-to-one setting: TTC is using priorities in a meaningful way butnot optimal.

I In Large Markets: We should not hope for much gains from TTCcompared to RSD.

Thank you!

NUS - Top Trading Cycles in Prioritized Matching: An Irrelevance … · 2018. 7. 25. · Top...

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