Pareto Improving Segmentation of Multi-product Markets Nima - - PowerPoint PPT Presentation

Pareto Improving Segmentation

• f Multi-product Markets

Nima Haghpanah joint with Ron Siegel May 4, 2019

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Market Segmentation

Firms can segment the market based on available data

◮ age, sex, location, browsing history, ...

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Market Segmentation

Firms can segment the market based on available data

◮ age, sex, location, browsing history, ...

Effects of segmentation on consumers?

◮ Segmentation can harm consumers ◮ Can segmentation benefit consumers?

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Does a Pareto Improving Segmentation Exist?

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Does a Pareto Improving Segmentation Exist?

Menu

Product Bundle1 Bundle2 . . . $5 $10 $8 . . .

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Does a Pareto Improving Segmentation Exist?

Menu

Product Bundle1 Bundle2 . . . $5 $10 $8 . . . consumer c

Un-segmented market: CS(c)

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Does a Pareto Improving Segmentation Exist?

consumer c

Un-segmented market: CS(c) Menu1 Menu2 Menu3 Menu4

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Does a Pareto Improving Segmentation Exist?

consumer c

Un-segmented market: CS(c) Menu1 Menu2 Menu3 Menu4 Segmented market: CS(c, )

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Does a Pareto Improving Segmentation Exist?

consumer c

Un-segmented market: CS(c) Menu1 Menu2 Menu3 Menu4 Segmented market: CS(c, ) ∃? segmentation s.t. CS(c, ) ≥ CS(c) for all c, & > for some c

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A Single Product Example (with Unit Demands)

1 − q 1 q 2

“Market”: Valuation v:

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A Single Product Example (with Unit Demands)

A Pareto Improving (PI) segmentation exists if market q is inefficient 1 − q 1 q 2

“Market”: Valuation v:

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A Single Product Example (with Unit Demands)

A Pareto Improving (PI) segmentation exists if market q is inefficient

◮ q ∈ (0.5, 1)

1 − q 1 q 2

“Market”: Valuation v:

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A Single Product Example (with Unit Demands)

A Pareto Improving (PI) segmentation exists if market q is inefficient

◮ q ∈ (0.5, 1)

1 − q 1 q 2

“Market”: Valuation v: All markets:

q

0.5 1

Optimal price: 1 2

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A Single Product Example (with Unit Demands)

A Pareto Improving (PI) segmentation exists if market q is inefficient

◮ q ∈ (0.5, 1)

1 − q 1 q 2

“Market”: Valuation v: All markets:

q

0.5 1

Optimal price: 1 2

Surplus q

• f v = 2

0.5 1

1

q

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A Single Product Example (with Unit Demands)

A Pareto Improving (PI) segmentation exists if market q is inefficient

◮ q ∈ (0.5, 1): Segment to q′ ≤ 0.5 and q′′ > q

1 − q 1 q 2

“Market”: Valuation v: All markets:

q

0.5 1

Optimal price: 1 2

Surplus q

• f v = 2

0.5 1

1

q q′′ q′

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A Single Product Example (with Unit Demands)

A Pareto Improving (PI) segmentation exists if market q is inefficient

◮ q ∈ (0.5, 1): Segment to q′ ≤ 0.5 and q′′ > q ◮ Holds with any number of valuations

1 − q 1 q 2

“Market”: Valuation v: All markets:

q

0.5 1

Optimal price: 1 2

Surplus q

• f v = 2

0.5 1

1

q

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Screening Example: Qualities L and H

1 − q 1 q 2

“Market”: vH: vL: 0.75 1

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75 1 − q 1 q 2

“Market”: vH: vL: 0.75 1

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75 1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75 1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

Surplus q

• f high

type

0.25 0.75 1

1 0.25

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75 1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

Surplus q

• f high

type

0.25 0.75 1

1 0.25

q =

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75 1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

Surplus q

• f high

type

0.25 0.75 1

1 0.25

q q′′ q′

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75 1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

Surplus q

• f high

type

0.25 0.75 1

1 0.25

q q′′ q′

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75

◮ PI segmentation exists for all but one inefficient markets

1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

Surplus q

• f high

type

0.25 0.75 1

1 0.25

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Screening Example: Qualities L and H

PI segmentation ∄ for inefficient market q = 0.75

◮ PI segmentation exists for all but one inefficient markets

Two types, any # of qualities (characterize optimal mechanisms)

◮ PI segmentation exists for all but finitely many inefficient markets

1 − q 1 q 2

“Market”: vH: vL: 0.75 1

q

0.25 0.75 1

p(H) = 1 1.75 2 p(L) = 0.75

Surplus q

• f high

type

1

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Model

◮ Types: t ∈ T (finite) ◮ Alternatives: a ∈ A (finite)

◮ e.g., qualities, quantities, configurations ◮ e.g., bundles: {1}, {2}, {1, 2}

◮ Valuations: v(t, a) ◮ Costs: c(a)

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Model

◮ Types: t ∈ T (finite) ◮ Alternatives: a ∈ A (finite)

◮ e.g., qualities, quantities, configurations ◮ e.g., bundles: {1}, {2}, {1, 2}

◮ Valuations: v(t, a) ◮ Costs: c(a)

Mechanism: a : T → ∆(A), p : T → R

◮ IC: E[v(t, a(t))] − p(t) ≥ E[v(t, a(t′))] − p(t′) ◮ IR: E[v(t, a(t))] − p(t) ≥ 0

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Markets and Segmentations

Market f ∈ ∆(T)

◮ Mechanism optimal if maximizes Ef [p(t) − c(t)] ◮ CS(t, f ): surplus of type t in “the” optimal mechanism for market f

◮ (fix arbitrary selection rule when multiple optimal mechanisms) 7 / 15

Markets and Segmentations

Market f ∈ ∆(T)

◮ Mechanism optimal if maximizes Ef [p(t) − c(t)] ◮ CS(t, f ): surplus of type t in “the” optimal mechanism for market f

◮ (fix arbitrary selection rule when multiple optimal mechanisms)

Segmentation µ of f : µ ∈ ∆(∆(T)) s.t. Eµ[f ′(t)] = f (t), ∀t

◮ e.g., two types, Eµ[q′] = q

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Markets and Segmentations

Market f ∈ ∆(T)

◮ Mechanism optimal if maximizes Ef [p(t) − c(t)] ◮ CS(t, f ): surplus of type t in “the” optimal mechanism for market f

◮ (fix arbitrary selection rule when multiple optimal mechanisms)

Segmentation µ of f : µ ∈ ∆(∆(T)) s.t. Eµ[f ′(t)] = f (t), ∀t

◮ e.g., two types, Eµ[q′] = q

Segmentation µ of f is Pareto improving if

• 1. ∀f ′ ∈ Supp(µ): ∀t ∈ Supp(f ′), CS(t, f ′) ≥ CS(t, f )
• 2. ∃f ′ ∈ Supp(µ): ∃t ∈ Supp(f ′),

>

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Markets and Segmentations

Market f ∈ ∆(T)

◮ Mechanism optimal if maximizes Ef [p(t) − c(t)] ◮ CS(t, f ): surplus of type t in “the” optimal mechanism for market f

◮ (fix arbitrary selection rule when multiple optimal mechanisms)

Segmentation µ of f : µ ∈ ∆(∆(T)) s.t. Eµ[f ′(t)] = f (t), ∀t

◮ e.g., two types, Eµ[q′] = q

Segmentation µ of f is Pareto improving if

• 1. ∀f ′ ∈ Supp(µ): ∀t ∈ Supp(f ′), CS(t, f ′) ≥ CS(t, f )
• 2. ∃f ′ ∈ Supp(µ): ∃t ∈ Supp(f ′),

> (f ′ ≻CS f if 1 and 2)

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Main Result: PI Segmentations Exist?

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Main Result: PI Segmentations Exist?

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Main Result: PI Segmentations Exist?

Recall: If ∃ PI segmentation of f ⇒ f is inefficient

◮ (∃t, a(t) /

∈ arg maxa′ v(t, a′) − c(a′))

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Main Result: PI Segmentations Exist?

Recall: If ∃ PI segmentation of f ⇒ f is inefficient

◮ (∃t, a(t) /

∈ arg maxa′ v(t, a′) − c(a′))

Theorem

For almost all inefficient markets, PI segmentations exist.

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Main Result: PI Segmentations Exist?

Recall: If ∃ PI segmentation of f ⇒ f is inefficient

◮ (∃t, a(t) /

∈ arg maxa′ v(t, a′) − c(a′))

Theorem

For almost all inefficient markets, PI segmentations exist. {f : f inefficient & ∄ PI segmentation} ⊆ a finite union of hyperplanes

◮ H(a) = {f | t a(t) · f (t) = 0} (a(t) = 0 for some t)

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Main Result: PI Segmentations Exist?

Recall: If ∃ PI segmentation of f ⇒ f is inefficient

◮ (∃t, a(t) /

∈ arg maxa′ v(t, a′) − c(a′))

Theorem

For almost all inefficient markets, PI segmentations exist. {f : f inefficient & ∄ PI segmentation} ⊆ a finite union of hyperplanes

◮ H(a) = {f | t a(t) · f (t) = 0} (a(t) = 0 for some t)

f (1) f (2) 1 1

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Main Result: PI Segmentations Exist?

Recall: If ∃ PI segmentation of f ⇒ f is inefficient

◮ (∃t, a(t) /

∈ arg maxa′ v(t, a′) − c(a′))

Theorem

For almost all inefficient markets, PI segmentations exist. {f : f inefficient & ∄ PI segmentation} ⊆ a finite union of hyperplanes

◮ H(a) = {f | t a(t) · f (t) = 0} (a(t) = 0 for some t)

f (1) f (2) 1 1

f (1) − 3f (2) = 0 (a = (1, −3))

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Main Result: PI Segmentations Exist?

Recall: If ∃ PI segmentation of f ⇒ f is inefficient

◮ (∃t, a(t) /

∈ arg maxa′ v(t, a′) − c(a′))

Theorem

For almost all inefficient markets, PI segmentations exist. {f : f inefficient & ∄ PI segmentation} ⊆ a finite union of hyperplanes

◮ H(a) = {f | t a(t) · f (t) = 0} (a(t) = 0 for some t)

f (1) f (2) 1 1

f (1) − 3f (2) = 0 (a = (1, −3)) Surplus

q

• f high

type 0.25 0.75 1

1 0.25

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Proof Outline

Given inefficient market f

• 1. ∃f1 s.t. f1 ≻CS f

◮ ∀t ∈ Supp(f1) : CS(t, f ′) ≥ CS(t, f ), (∃t, >)

Define f2 s.t. f = ǫf1 + (1 − ǫ)f2

• 2. Small ǫ, generic f : OptMech(f ) = OptMech(f2)

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f .

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f . Assume: 1. c = 0 2. ∃t ∀a, v(t, a) < v(t, a) 3. ∃¯ a ∀t, v(t, a) < v(t, ¯ a)

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f . Assume: 1. c = 0 2. ∃t ∀a, v(t, a) < v(t, a) 3. ∃¯ a ∀t, v(t, a) < v(t, ¯ a) v(·, ¯ a) v(·, a)

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f . Assume: 1. c = 0 2. ∃t ∀a, v(t, a) < v(t, a) 3. ∃¯ a ∀t, v(t, a) < v(t, ¯ a) v(·, ¯ a) v(·, a) t

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f . Assume: 1. c = 0 2. ∃t ∀a, v(t, a) < v(t, a) 3. ∃¯ a ∀t, v(t, a) < v(t, ¯ a)

1 Some type t is assigned ¯

a

2 Type t pays strictly more than v(t, ¯

a)

3 f1(t) = 1 − δ, f1(t) = δ for small enough δ ◮ optimal to sell ¯

a at price v(t, ¯ a)

v(·, ¯ a) v(·, a) t

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f . Assume: 1. c = 0 2. ∃t ∀a, v(t, a) < v(t, a) 3. ∃¯ a ∀t, v(t, a) < v(t, ¯ a)

1 Some type t is assigned ¯

a

2 Type t pays strictly more than v(t, ¯

a)

3 f1(t) = 1 − δ, f1(t) = δ for small enough δ ◮ optimal to sell ¯

a at price v(t, ¯ a)

v(·, ¯ a) v(·, a) t t

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Step 1: Every Inefficient Market Is Dominated

Lemma

If f is inefficient, then there exists an efficient f1 s.t. f1 ≻CS f . Assume: 1. c = 0 2. ∃t ∀a, v(t, a) < v(t, a) 3. ∃¯ a ∀t, v(t, a) < v(t, ¯ a)

1 Some type t is assigned ¯

a

2 Type t pays strictly more than v(t, ¯

a)

3 f1(t) = 1 − δ, f1(t) = δ for small enough δ ◮ optimal to sell ¯

a at price v(t, ¯ a)

Intuition: allocation of t distorted to extract rents from t v(·, ¯ a) v(·, a) t t

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Step 1, General Proof Idea

market f : v(·, a1) v(·, a2)

1 3 2

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Step 1, General Proof Idea

market f : v(·, a1) v(·, a2)

1 3 2 a1 a2 a1

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Step 1, General Proof Idea

p(a1) p(a2) market f : v(·, a1) v(·, a2)

1 3 2 a1 a2 a1

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Step 1, General Proof Idea

p(a1) p(a2) market f : v(·, a1) v(·, a2)

1 3 2 a1 a2 a1

v(·, a1) v(·, a2)

1 3 2 a1 a2 a2

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Step 1, General Proof Idea

p(a1) p(a2) market f : v(·, a1) v(·, a2)

1 3 2 a1 a2 a1

p(a1) p(a2) v(·, a1) v(·, a2)

1 3 2 a1 a2 a2

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Step 1, General Proof Idea

p(a1) p(a2) market f : v(·, a1) v(·, a2)

1 3 2 a1 a2 a1

p(a1) p(a2) v(·, a1) v(·, a2)

1 3 2 a1 a2 a2

In addition to types 2 and 3

◮ include all other types in the “chain of information rents”

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Step 1, General Proof Idea

p(a1) p(a2) market f : v(·, a1) v(·, a2)

1 3 2 a1 a2 a1

p(a1) p(a2) ∃? market f1 v(·, a1) v(·, a2)

1 3 2 a1 a2 a2

In addition to types 2 and 3

◮ include all other types in the “chain of information rents”

Final step: ensure mechanism is optimal for some market f1

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Recall Proof Outline

Given inefficient market f

• 1. ∃f1 s.t. f1 ≻CS f

◮ ∀t ∈ Supp(f1) : CS(t, f ′) ≥ CS(t, f ), (∃t, >)

Define f2 s.t. f = ǫf1 + (1 − ǫ)f2

• 2. Small ǫ, generic f : OptMech(f ) = OptMech(f2)

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2)

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2) The set of all IC & IR mechanisms (a, p) is a polytope

◮ The set of all payment rules is a projection (thus a polytope itself)

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2) The set of all IC & IR mechanisms (a, p) is a polytope

◮ The set of all payment rules is a projection (thus a polytope itself)

p(1) p(2)

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2) The set of all IC & IR mechanisms (a, p) is a polytope

◮ The set of all payment rules is a projection (thus a polytope itself)

p(1) p(2) f

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2) The set of all IC & IR mechanisms (a, p) is a polytope

◮ The set of all payment rules is a projection (thus a polytope itself)

p(1) p(2) f f2

13 / 15

Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2) The set of all IC & IR mechanisms (a, p) is a polytope

◮ The set of all payment rules is a projection (thus a polytope itself)

p(1) p(2) f

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Step 2: Small Perturbation Preserves Optimal Mechanisms

f = (1 − ǫ)f1 + ǫf2

Lemma

Fix any f1. For generic f , there exists ǫ small enough such that OptMech(f ) = OptMech(f2) The set of all IC & IR mechanisms (a, p) is a polytope

◮ The set of all payment rules is a projection (thus a polytope itself)

p(1) p(2) f f2

13 / 15

Recall Proof Outline

Given inefficient market f

• 1. ∃f1 s.t. f1 ≻CS f

◮ ∀t ∈ Supp(f1) : CS(t, f ′) ≥ CS(t, f ), (∃t, >)

Define f2 s.t. f = ǫf1 + (1 − ǫ)f2

• 2. Small ǫ, generic f : OptMech(f ) = OptMech(f2)

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Conclusions

There is scope for intervention:

◮ PI segmentations generically exist ◮ (do not exist for some markets) ◮ Key idea: Every inefficient market is dominated

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Conclusions

There is scope for intervention:

◮ PI segmentations generically exist ◮ (do not exist for some markets) ◮ Key idea: Every inefficient market is dominated

Thanks!

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