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Efficiency and Stability in Large Matching Markets Yeon-Koo Che (Columbia) and Olivier Tercieux (PSE) September 18, 2014 Toronto Workshop Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 1 / 47 Introduction An

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Efficiency and Stability in Large Matching Markets

Yeon-Koo Che (Columbia) and Olivier Tercieux (PSE) September 18, 2014 Toronto Workshop

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 1 / 47

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Introduction

An important class of resource allocation problems involves “matching without transfers” assignment of students to public school allocation of social housing assignment of teachers to schools assignment of organs to patients in need In practice, those markets are often organized in a centralized way.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 2 / 47

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Objectives

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 3 / 47

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Objectives

1 Pareto-efficiency: satisfying the preferences of the agents. Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 3 / 47

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Objectives

1 Pareto-efficiency: satisfying the preferences of the agents.

Attained by Random Serial Dictatorship (RSD), Top Trading Cycles (TTC), etc.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 3 / 47

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Objectives

1 Pareto-efficiency: satisfying the preferences of the agents.

Attained by Random Serial Dictatorship (RSD), Top Trading Cycles (TTC), etc.

2 Stability: respecting agents’ priorities (aka “no justified envy”, or

“fairness”).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 3 / 47

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Objectives

1 Pareto-efficiency: satisfying the preferences of the agents.

Attained by Random Serial Dictatorship (RSD), Top Trading Cycles (TTC), etc.

2 Stability: respecting agents’ priorities (aka “no justified envy”, or

“fairness”).

Attained by Gale and Shapley’s Deferred Acceptance Algorithm (DA).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 3 / 47

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Conflicts

Impossibility: No algorithm achieves both Pareto-efficient and No Justified Envy (Roth, 82).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 4 / 47

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Conflicts

Impossibility: No algorithm achieves both Pareto-efficient and No Justified Envy (Roth, 82). = ⇒ Prominent mechanisms achieve one objective at the “minimal” sacrifice of the other.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 4 / 47

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Conflicts

Impossibility: No algorithm achieves both Pareto-efficient and No Justified Envy (Roth, 82). = ⇒ Prominent mechanisms achieve one objective at the “minimal” sacrifice of the other. DA is stable and efficient among stable mechanisms (Gale and Shapley, 62) (Boston, Hong Kong, New York, Paris...)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 4 / 47

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Conflicts

Impossibility: No algorithm achieves both Pareto-efficient and No Justified Envy (Roth, 82). = ⇒ Prominent mechanisms achieve one objective at the “minimal” sacrifice of the other. DA is stable and efficient among stable mechanisms (Gale and Shapley, 62) (Boston, Hong Kong, New York, Paris...) Top Trading Cycle is efficient and envy minimal (Abdulkadiroglu, Che, Tercieux, 13) (San Francisco, New Orleans,...)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 4 / 47

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Research Questions

How do alternative PE mechanisms differ in utilitarian efficiency and payoff distribution?

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 5 / 47

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Research Questions

How do alternative PE mechanisms differ in utilitarian efficiency and payoff distribution? (Examples of PE mechanisms: Serial Dictatorship / random Serial Dictatorship, Hylland and Zeckhauser, Top-trading Cycles, YRMH-IGYT, Abdulkadiroglu and Sonmez TTC, Hierarchical Exchange)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 5 / 47

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Research Questions

How do alternative PE mechanisms differ in utilitarian efficiency and payoff distribution? (Examples of PE mechanisms: Serial Dictatorship / random Serial Dictatorship, Hylland and Zeckhauser, Top-trading Cycles, YRMH-IGYT, Abdulkadiroglu and Sonmez TTC, Hierarchical Exchange) What is the optimal way to resolve the tradeoff of the two goals?

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 5 / 47

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Research Questions

How do alternative PE mechanisms differ in utilitarian efficiency and payoff distribution? (Examples of PE mechanisms: Serial Dictatorship / random Serial Dictatorship, Hylland and Zeckhauser, Top-trading Cycles, YRMH-IGYT, Abdulkadiroglu and Sonmez TTC, Hierarchical Exchange) What is the optimal way to resolve the tradeoff of the two goals? Attaining one at the minimal sacrifice of the other may not be the best if the sacrifice is significant and/or if one can approximately achieve both.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 5 / 47

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Large Market with Random Preferences

To make progress, we add some structure to the environment: Large markets:

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 6 / 47

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Large Market with Random Preferences

To make progress, we add some structure to the environment: Large markets: Realistic in the applications mentioned.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 6 / 47

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Large Market with Random Preferences

To make progress, we add some structure to the environment: Large markets: Realistic in the applications mentioned. In New York, 100,000 students apply each year to 500 schools; In medical matching, 20,000 doctors and 3,000-4,000 programs

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 6 / 47

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Large Market with Random Preferences

To make progress, we add some structure to the environment: Large markets: Realistic in the applications mentioned. In New York, 100,000 students apply each year to 500 schools; In medical matching, 20,000 doctors and 3,000-4,000 programs Random preference structure: individuals draw preferences at random with some correlation (to be specified).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 6 / 47

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Setting

Finite set of individuals I and finite set of objects O to be matched

For simplicity, |I| = |O| = n

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 7 / 47

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Setting: preferences

Each i ∈ I receives utility from object o ∈ O Ui(o) = U(uo, ξio) where uo is the common value component The uo are in [0, 1] Let X n(·) be its distribution and X(·) its limit

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 8 / 47

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Distribution of common values (finite example)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 9 / 47

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Setting: preferences

Each i ∈ I receives utility from object o ∈ O Ui(o) = U(uo, ξio) ξio is the idiosyncratic shock on i’s preferences for object o The {ξio}i,o is a collection of iid random variable Distribution takes values in [0, ¯ ξ] ⊂ R U(·, ·) takes values in R+, is strictly increasing and continuous All objects are acceptable (utility of the outside option is normalized to 0)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 10 / 47

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Setting: objects’ preferences (agents’ priorities)

First part: arbitrary.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 11 / 47

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Setting: objects’ preferences (agents’ priorities)

First part: arbitrary. Second part: Each o ∈ O receives utility from individual i ∈ I : Vo(i) = V (ηio)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 11 / 47

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Setting: objects’ preferences (agents’ priorities)

First part: arbitrary. Second part: Each o ∈ O receives utility from individual i ∈ I : Vo(i) = V (ηio) Purely idiosyncratic preferences.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 11 / 47

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Setting: matching

A matching µ is a bijective mapping µ : I ∪ O → I ∪ O such that µ(I) ⊂ O and

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 12 / 47

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Setting: matching

A matching µ is a bijective mapping µ : I ∪ O → I ∪ O such that µ(I) ⊂ O and

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 12 / 47

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Setting: matching

A matching µ is a bijective mapping µ : I ∪ O → I ∪ O such that µ(I) ⊂ O and µ(i) = o if and only if µ(o) = i

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 12 / 47

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Setting: matching

A matching µ is a bijective mapping µ : I ∪ O → I ∪ O such that µ(I) ⊂ O and µ(i) = o if and only if µ(o) = i

A matching is Pareto-efficient if no individual i can be made strictly better-off without hurting another individual.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 12 / 47

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Setting: matching

A matching µ is a bijective mapping µ : I ∪ O → I ∪ O such that µ(I) ⊂ O and µ(i) = o if and only if µ(o) = i

A matching is Pareto-efficient if no individual i can be made strictly better-off without hurting another individual. A matching µ is stable if there is no pair (i, o) where i would prefer o to his match µ(i) and o would assign higher priority to i rather than to his match µ(o)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 12 / 47

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Setting: matching

A matching µ is a bijective mapping µ : I ∪ O → I ∪ O such that µ(I) ⊂ O and µ(i) = o if and only if µ(o) = i

A matching is Pareto-efficient if no individual i can be made strictly better-off without hurting another individual. A matching µ is stable if there is no pair (i, o) where i would prefer o to his match µ(i) and o would assign higher priority to i rather than to his match µ(o)

A matching mechanism ˜ µ maps “states” into matchings, where a state refers to the profile of preferences together with the profile of priorities.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 12 / 47

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PE mechanism: Serial Dictatorship (SD) mechanism

A serial dictatorship mechanism SDf specifies an ordering f : {1, 2, 3, ..., n} → I, where f (i) is the ith “dictator” f (1) chooses his favorite object f (2) chooses his favorite object among the remaining ones and so on....

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 13 / 47

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PE Mechanism: Top Trading Cycle (TTC) mechanism

Assume objects have preferences / priorities Step 1: Each individual points to his most preferred object Each object points to its most preferred individual

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 14 / 47

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PE Mechanism: Top Trading Cycle (TTC) mechanism

Assume objects have preferences / priorities Step 1: Each individual points to his most preferred object Each object points to its most preferred individual There exists at least one cycle and no cycles intersect.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 14 / 47

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PE Mechanism: Top Trading Cycle (TTC) mechanism

Assume objects have preferences / priorities Step 1: Each individual points to his most preferred object Each object points to its most preferred individual There exists at least one cycle and no cycles intersect. Remove cycles. Individuals in a cycle get the object they point to.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 14 / 47

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PE Mechanism: Top Trading Cycle (TTC) mechanism

Assume objects have preferences / priorities Step 1: Each individual points to his most preferred object Each object points to its most preferred individual There exists at least one cycle and no cycles intersect. Remove cycles. Individuals in a cycle get the object they point to. Step t = 2, ...: Repeat the same procedure with the remaining economy.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 14 / 47

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Utilitarian efficiency of PE mechanisms

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 15 / 47

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Utilitarian Efficiency of Pareto Efficiency

Let U∗ := 1

0 U(u, ¯

ξ)dX(u) be the utilitarian upper bound;

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 16 / 47

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Utilitarian Efficiency of Pareto Efficiency

Let U∗ := 1

0 U(u, ¯

ξ)dX(u) be the utilitarian upper bound; In our example: U∗ = ∑K

k=1 xkU(uk, ¯

ξ).

Theorem

Let µ be a Pareto-efficient matching mechanism. 1 n ∑

i∈I

Ui(µ(i))

p

− → U∗,

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 16 / 47

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SLIDE 41

Utilitarian Efficiency of Pareto Efficiency

Let U∗ := 1

0 U(u, ¯

ξ)dX(u) be the utilitarian upper bound; In our example: U∗ = ∑K

k=1 xkU(uk, ¯

ξ).

Theorem

Let µ be a Pareto-efficient matching mechanism. 1 n ∑

i∈I

Ui(µ(i))

p

− → U∗, i.e., for any δ > 0, Pr

  • 1

n ∑

i∈I

Ui(µ(i)) − U∗

  • < δ
  • → 1 as n → ∞.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 16 / 47

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Implication in terms of distribution of payoffs:

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 17 / 47

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Implication in terms of distribution of payoffs:

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 18 / 47

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Implication in terms of distribution of payoffs:

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 18 / 47

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Sketch of proof

Intuition given for the case where X(·) is degenerate (i.e. we only have the idiosyncratic component) A PE mechanism ˜ µ can be implemented by a serial dictatorship mechanism with a particular serial order ˜ f For arbitrarily small ε, δ > 0, define the random set: ¯ I := {i ∈ I

  • Ui( ˜

µ(i)) ≤ U(u0, ¯ ξ) − ε and ˜ f (i) ≤ (1 − δ)|O|}. We show via applying a random graph theory result that | ¯ I| n

p

− → 0.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 19 / 47

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Sketch of proof

Intuition given for the case where X(·) is degenerate (i.e. we only have the idiosyncratic component) A PE mechanism ˜ µ can be implemented by a serial dictatorship mechanism with a particular serial order ˜ f For arbitrarily small ε, δ > 0, define the random set: ¯ I := {i ∈ I

  • Ui( ˜

µ(i)) ≤ U(u0, ¯ ξ) − ε and ˜ f (i) ≤ (1 − δ)|O|}. We show via applying a random graph theory result that | ¯ I| n

p

− → 0. (We show ¯ I to be a shorter side of an independent set of an associated random graph, which vanishes.)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 19 / 47

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Detour: Random bipartite graph

A random bipartite graph G(V1, V2, p) : V1 is the set of vertices on one side V2 is the set of vertices on the other side and The set of edges is random: An edge (i, j) ∈ V1 × V2 is added with probability p.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 20 / 47

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Size of an independent set

Given a (deterministic) bipartite graph G(V1, V2, E), W1 × W2 ⊆ V1 × V2 is an independent set if (i, j) ∈ W1 × W2 = ⇒ (i, j) / ∈ E.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 21 / 47

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Size of an independent set

Given a (deterministic) bipartite graph G(V1, V2, E), W1 × W2 ⊆ V1 × V2 is an independent set if (i, j) ∈ W1 × W2 = ⇒ (i, j) / ∈ E.

Theorem (Extension of Bollobas and Erd¨

  • s (1975))

Let W1 × W2 be an independent set in a random bipartite graph G(V1, V2, p) where 0 < p < 1 Pr {min {|W1| , |W2|} < κ ln n} → 1 as n → ∞. (where κ is a strictly positive constant)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 21 / 47

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Sketch of proof

Now that we have ¯ I := {i ∈ I

  • Ui( ˜

µ(i)) ≤ ¯ ξ − ε and ˜ f (i) ≤ (1 − δ)|O|} let us define ¯ O := {o ∈ O

  • ˜

f ( ˜ µ(o)) ≥ (1 − δ)|O|}. Build an associated random bipartite graph Random variables {ξio} induce a random graph on I × O where (i, o) is an edge iff ξio > ¯ ξ − ε

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 22 / 47

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Sketch of proof

Now that we have ¯ I := {i ∈ I

  • Ui( ˜

µ(i)) ≤ ¯ ξ − ε and ˜ f (i) ≤ (1 − δ)|O1|} let us define ¯ O := {o ∈ O

  • ˜

f ( ˜ µ(o)) ≥ (1 − δ)|O|},

  • bjects assigned to agents with “bad” serial orders.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 23 / 47

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Sketch of proof

  • Claim. ¯

I × ¯ O is an independent set in the associated random graph.

  • Proof. Otherwise, if (i, o) ∈ ¯

I × ¯ O is an edge then

  • 1. (i, o) ∈ ¯

I × ¯ O = ⇒ Ui(o) > Ui( ˜ µ(i))

  • 2. o ∈ ¯

O = ⇒ when i gets to choose, o is still available = ⇒ i picks ˜ µ(i) while a better object o is available.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 24 / 47

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Sketch of proof

  • Claim. ¯

I × ¯ O is an independent set in the associated random graph.

  • Proof. Otherwise, if (i, o) ∈ ¯

I × ¯ O is an edge then

  • 1. (i, o) ∈ ¯

I × ¯ O = ⇒ Ui(o) > Ui( ˜ µ(i))

  • 2. o ∈ ¯

O = ⇒ when i gets to choose, o is still available = ⇒ i picks ˜ µ(i) while a better object o is available. Contradiction.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 24 / 47

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Stability versus efficiency

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 25 / 47

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Asymptotic Efficiency and Stability

Matching mechanism ˜ µ is asymptotically efficient if for any ˜ µ′ which Pareto-dominates ˜ µ and any ǫ > 0 |Iǫ( ˜ µ′| ˜ µ)| |I|

p

− → 0, where Iǫ( ˜ µ′| ˜ µ) := {i ∈ I|Ui( ˜ µ′(i)) − Ui( ˜ µ(i)) > ǫ}.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 26 / 47

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Asymptotic Efficiency and Stability

Matching mechanism ˜ µ is asymptotically efficient if for any ˜ µ′ which Pareto-dominates ˜ µ and any ǫ > 0 |Iǫ( ˜ µ′| ˜ µ)| |I|

p

− → 0, where Iǫ( ˜ µ′| ˜ µ) := {i ∈ I|Ui( ˜ µ′(i)) − Ui( ˜ µ(i)) > ǫ}. Matching mechanism ˜ µ is asymptotically stable if, for any ǫ > 0 |Jǫ| |I × O|

p

− → 0, where Jǫ := {(i, o) ∈ I × O|Ui(o) − Ui( ˜ µ(i)) > ǫ and Vo(i) − Vo( ˜ µ(o)) > ǫ}.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 26 / 47

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Asymptotic Instability of TTC

If X(·) is degenerate (i.e., only one tier of objects), TTC is asymptotically stable:

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 27 / 47

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Asymptotic Instability of TTC

If X(·) is degenerate (i.e., only one tier of objects), TTC is asymptotically stable: Our first result implies that all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) But if we add tiers on objects/correlation in individuals’ preferences, TTC is not asymptotically stable (even with this weaker notion).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 27 / 47

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Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 28 / 47

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Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 29 / 47

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Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 30 / 47

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SLIDE 62

Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 31 / 47

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Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 32 / 47

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SLIDE 64

Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 33 / 47

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SLIDE 65

Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 34 / 47

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SLIDE 66

Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 35 / 47

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SLIDE 67

Asymptotic Instability of TTC

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 36 / 47

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SLIDE 68

Asymptotic Inefficiency of DA

With only one tier of objects (X(·) degenerate), all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) (Wilson (72), Knuth (76), Pittel (89, 92), Compte-Jehiel (07)...)

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 37 / 47

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SLIDE 69

Asymptotic Inefficiency of DA

With only one tier of objects (X(·) degenerate), all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) (Wilson (72), Knuth (76), Pittel (89, 92), Compte-Jehiel (07)...) ⇒Hence, asymptotically, DA is efficient.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 37 / 47

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SLIDE 70

Asymptotic Inefficiency of DA

With only one tier of objects (X(·) degenerate), all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) (Wilson (72), Knuth (76), Pittel (89, 92), Compte-Jehiel (07)...) ⇒Hence, asymptotically, DA is efficient. But if we add tiers on the side of objects / correlation in individuals’ preferences, DA is not asymptotically efficient.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 37 / 47

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SLIDE 71

Asymptotic Inefficiency of DA

With only one tier of objects (X(·) degenerate), all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) (Wilson (72), Knuth (76), Pittel (89, 92), Compte-Jehiel (07)...) ⇒Hence, asymptotically, DA is efficient. But if we add tiers on the side of objects / correlation in individuals’ preferences, DA is not asymptotically efficient.

  • Assume there are two tiers and tier 1 objects are uniformly better than

tier 2 objects.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 37 / 47

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SLIDE 72

Asymptotic Inefficiency of DA

With only one tier of objects (X(·) degenerate), all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) (Wilson (72), Knuth (76), Pittel (89, 92), Compte-Jehiel (07)...) ⇒Hence, asymptotically, DA is efficient. But if we add tiers on the side of objects / correlation in individuals’ preferences, DA is not asymptotically efficient.

  • Assume there are two tiers and tier 1 objects are uniformly better than

tier 2 objects.

  • Inefficiency can be seen more clearly with the McVitie-Wilson version
  • f DA: Serialize the agents, and each agent applies to an object “one

at a time.”

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 37 / 47

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SLIDE 73

Asymptotic Inefficiency of DA

With only one tier of objects (X(·) degenerate), all individuals get a payoff arbitrarily close to the upper bound U(u0

1, ¯

ξ) (Wilson (72), Knuth (76), Pittel (89, 92), Compte-Jehiel (07)...) ⇒Hence, asymptotically, DA is efficient. But if we add tiers on the side of objects / correlation in individuals’ preferences, DA is not asymptotically efficient.

  • Assume there are two tiers and tier 1 objects are uniformly better than

tier 2 objects.

  • Inefficiency can be seen more clearly with the McVitie-Wilson version
  • f DA: Serialize the agents, and each agent applies to an object “one

at a time.”

  • Apply Ashlagi, Kanoria, and Leshno (2013).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 37 / 47

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Asymptotically inefficiency of DA

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 38 / 47

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SLIDE 75

Asymptotically inefficiency of DA

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 39 / 47

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SLIDE 76

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

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SLIDE 77

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively. Consider a (bit more general) model with finite tiers on the objects (the payoffs can overlap across tiers).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

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SLIDE 78

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively. Consider a (bit more general) model with finite tiers on the objects (the payoffs can overlap across tiers). The algorithm is parametrized by an integer β(n). Consider the market composed of individuals I and objects O.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

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SLIDE 79

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively. Consider a (bit more general) model with finite tiers on the objects (the payoffs can overlap across tiers). The algorithm is parametrized by an integer β(n). Consider the market composed of individuals I and objects O.

Start running the McVitie-Wilson version of Gale-Shapley’s algorithm.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

slide-80
SLIDE 80

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively. Consider a (bit more general) model with finite tiers on the objects (the payoffs can overlap across tiers). The algorithm is parametrized by an integer β(n). Consider the market composed of individuals I and objects O.

Start running the McVitie-Wilson version of Gale-Shapley’s algorithm. Keep track of the number of offers made by each individual.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

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SLIDE 81

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively. Consider a (bit more general) model with finite tiers on the objects (the payoffs can overlap across tiers). The algorithm is parametrized by an integer β(n). Consider the market composed of individuals I and objects O.

Start running the McVitie-Wilson version of Gale-Shapley’s algorithm. Keep track of the number of offers made by each individual. When there is an individual who has made more than β(n) offers, finalize the matching, i.e., any object gets matched with the individual he tentatively holds if any.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

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SLIDE 82

Achieving Both: DA with Circuit Breaker

We modify DA to prevent the agents from competing excessively. Consider a (bit more general) model with finite tiers on the objects (the payoffs can overlap across tiers). The algorithm is parametrized by an integer β(n). Consider the market composed of individuals I and objects O.

Start running the McVitie-Wilson version of Gale-Shapley’s algorithm. Keep track of the number of offers made by each individual. When there is an individual who has made more than β(n) offers, finalize the matching, i.e., any object gets matched with the individual he tentatively holds if any.

Iterate the process until we exhaust the market.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 40 / 47

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SLIDE 83

Result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 41 / 47

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SLIDE 84

Result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

Theorem

The mechanism is “asymptotically incentive compatible”: Truthtelling is an ǫ-Bayes Nash equilibrium.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 41 / 47

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SLIDE 85

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

1 whp, the β(n) most preferred objects of all individuals are in O1. We

condition on this event

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 42 / 47

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SLIDE 86

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

1 whp, the β(n) most preferred objects of all individuals are in O1. We

condition on this event

2 whp, all objects in O1 are assigned without the circuit breaker being

triggered (i.e., no agent makes more than log(n)2 offers) (by the classical results, due to Pittel and others).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 42 / 47

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SLIDE 87

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

1 whp, the β(n) most preferred objects of all individuals are in O1. We

condition on this event

2 whp, all objects in O1 are assigned without the circuit breaker being

triggered (i.e., no agent makes more than log(n)2 offers) (by the classical results, due to Pittel and others). And we know the circuit breaker will be triggered right after all objects in O1 are matched, so no object outside O1 is matched by the first stage.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 42 / 47

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SLIDE 88

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

1 whp, the β(n) most preferred objects of all individuals are in O1. We

condition on this event

2 whp, all objects in O1 are assigned without the circuit breaker being

triggered (i.e., no agent makes more than log(n)2 offers) (by the classical results, due to Pittel and others). And we know the circuit breaker will be triggered right after all objects in O1 are matched, so no object outside O1 is matched by the first stage.

3 whp, individuals matched in this step get high idiosyncratic payoffs Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 42 / 47

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SLIDE 89

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable.

1 whp, the β(n) most preferred objects of all individuals are in O1. We

condition on this event

2 whp, all objects in O1 are assigned without the circuit breaker being

triggered (i.e., no agent makes more than log(n)2 offers) (by the classical results, due to Pittel and others). And we know the circuit breaker will be triggered right after all objects in O1 are matched, so no object outside O1 is matched by the first stage.

3 whp, individuals matched in this step get high idiosyncratic payoffs 4 whp, almost all objects in O1 get high idiosyncratic payoffs (by the

classical results).

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 42 / 47

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SLIDE 90

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable. We iterate the reasoning for other tiers.

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 43 / 47

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SLIDE 91

Intuition for the result

Theorem

Let ˜ µ be the matching mechanism obtained by this procedure for

  • log(n)2 ≤ β(n) = o(n). ˜

µ is asymptotically efficient and asymptotically stable. Hence,

1 whp, almost all objects get high payoffs =

⇒ asymptotically stable

2 whp, all individuals are assigned objects that yield high idiosyncratic

payoffs = ⇒ asymptotically efficient

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 44 / 47

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SLIDE 92

Simulations

We simulate a situation where common values uniformly distributed from [0, 1] the idiosyncratic payoff ξio uniformly distributed from [0, 1]

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 45 / 47

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SLIDE 93

Conclusion

While there is an (asymptotically) efficient and stable matching mechanism: two of the prominent mechanisms fail to find this matching Alternative mechanism which limits competition seem to perform better In practice, students can only report a small number of objects in their list of preferences. This also limits the total number of offers that agents can make, and this may an unexpected good effect on the performance of the mechanism

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 46 / 47

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SLIDE 94

Conclusion: Data ongoing

Abdulkadiroglu, Pathak and Roth (2006) have studied NYC data for the entrance in high school Under DA: out of 80,000 students 5,000 can be made better-off by letting them exchange their assignments Under TTC: out of 80,000, 55,000 are part of a blocking pair This suggests that DA and TTC are indeed not close to be efficient or stable in the field. We are currently running our alternative algorithm on NYC data...

Yeon-Koo Che and Olivier Tercieux Efficiency and Stability in the Large Toronto 47 / 47