When Is Pure Bundling Optimal? Nima Haghpanah (Penn State) Joint - - PowerPoint PPT Presentation

When Is Pure Bundling Optimal?

Nima Haghpanah (Penn State)

Joint work with Jason Hartline (Northwestern)

October 26, 2018

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Multi-product Monopolist’s Optimal Selling Strategy?

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Multi-product Monopolist’s Optimal Selling Strategy?

◮ Sell Separately: Offer each product for a price

Sell Separately

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Multi-product Monopolist’s Optimal Selling Strategy?

◮ Sell Separately: Offer each product for a price ◮ Pure Bundling: Offer only the grand bundle of all products

Sell Separately Pure Bundling

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Multi-product Monopolist’s Optimal Selling Strategy?

◮ Sell Separately: Offer each product for a price ◮ Pure Bundling: Offer only the grand bundle of all products ◮ Mixed Bundling: Offer a menu of bundles and prices

Sell Separately Pure Bundling

Mixed Bundling

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Multi-product Monopolist’s Optimal Selling Strategy?

◮ Sell Separately: Offer each product for a price ◮ Pure Bundling: Offer only the grand bundle of all products ◮ Mixed Bundling: Offer a menu of bundles and prices

Sell Separately Pure Bundling

Mixed Bundling This paper: When is Pure Bundling Optimal?

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The Model

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The Model

Single seller, products 1 to k, single buyer

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The Model

Single seller, products 1 to k, single buyer

◮ Bundle b ⊆ {1, . . . , k} ◮ vb value for bundle b ◮ Type v = (vb)b⊆{1,...,k}

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The Model

Single seller, products 1 to k, single buyer

◮ Bundle b ⊆ {1, . . . , k} ◮ vb value for bundle b ◮ Type v = (vb)b⊆{1,...,k}∼ µ

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The Model

Single seller, products 1 to k, single buyer

◮ Bundle b ⊆ {1, . . . , k} ◮ vb value for bundle b ◮ Type v = (vb)b⊆{1,...,k}∼ µ

Mechanism:

◮ menu of (price, bundle)

Bundle Price $4 b $5 b′ . . . . . .

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The Model

Single seller, products 1 to k, single buyer

◮ Bundle b ⊆ {1, . . . , k} ◮ vb value for bundle b ◮ Type v = (vb)b⊆{1,...,k}∼ µ

Mechanism:

◮ menu of (price, bundle)

Bundle Price $4 b $5 b′ $4.5 lottery over b, b′ . . . . . .

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The Model

Single seller, products 1 to k, single buyer

◮ Bundle b ⊆ {1, . . . , k} ◮ vb value for bundle b ◮ Type v = (vb)b⊆{1,...,k}∼ µ

Mechanism:

◮ menu of (price, bundle)

Pure Bundling Mechanism: Bundle Price $4 b $5 b′ $4.5 lottery over b, b′ . . . . . . Bundle Price $4 {1, . . . , k}

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Examples: Additive Values v{1,2} = v{1} + v{2}

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5. v{2} v{1} 0.8 0.2 0.2 0.8

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

v{2} v{1} 0.8 0.2 0.2 0.8

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

v{2} v{1} 0.8 0.2 0.2 0.8 Stigler ’63, Adams & Yellen ’76: Bundle if values negatively correlated

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

Example 2: v{1}, v{2} i.i.d U[0, 1] v{2} v{1} Stigler ’63, Adams & Yellen ’76: Bundle if values negatively correlated

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

Example 2: v{1}, v{2} i.i.d U[0, 1] Sell Separately: 0.5 0.5 {1} {2} {1, 2} v{2} v{1} Stigler ’63, Adams & Yellen ’76: Bundle if values negatively correlated

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

Example 2: v{1}, v{2} i.i.d U[0, 1] Pure Bundling: 0.81 0.81 {1, 2} v{2} v{1} Stigler ’63, Adams & Yellen ’76: Bundle if values negatively correlated

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

Example 2: v{1}, v{2} i.i.d U[0, 1]

◮ Optimal:

Bundle Price 0.86 {1, 2} 0.66 {1} 0.66 {2}

Mixed Bundling: 0.66 0.66 {1} {2} {1, 2} v{2} v{1} Stigler ’63, Adams & Yellen ’76: Bundle if values negatively correlated

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Examples: Additive Values v{1,2} = v{1} + v{2}

Example 1: (v{1}, v{2}) = (.8, .2) probability 0.5, (.2, .8) probability 0.5.

◮ Pure bundling optimal

Example 2: v{1}, v{2} i.i.d U[0, 1]

◮ Optimal:

Bundle Price 0.86 {1, 2} 0.66 {1} 0.66 {2}

Mixed Bundling: 0.66 0.66 {1} {2} {1, 2} v{2} v{1} Stigler ’63, Adams & Yellen ’76: Bundle if values negatively correlated McAfee et al. ’89: Pure bundling generically not optimal

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Special Case: Two Identical Products

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

v1 vgb

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

Mechanism: v1 vgb $5 Two Units

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

Mechanism: v1 vgb $5 Two Units $3 One Unit

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

Mechanism: Pure Bundling (PB) Mechanism: v1 vgb $5 Two Units $3 One Unit $4 Two Units

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

Mechanism: Pure Bundling (PB) Mechanism: v1 vgb $5 Two Units $3 One Unit $4 Two Units Main Result:

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

Mechanism: Pure Bundling (PB) Mechanism: v1 vgb $5 Two Units $3 One Unit $4 Two Units Main Result:

◮ PB optimal if v1/vgb “stochastically nondecreasing” in vgb ◮ PB not optimal if v1/vgb “stochastically decreasing” in vgb

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Special Case: Two Identical Products

Values (v1, vgb) ∼ µ

◮ (vgb need not = 2v1)

Mechanism: Pure Bundling (PB) Mechanism: v1 vgb $5 Two Units $3 One Unit $4 Two Units Main Result:

◮ PB optimal if v1/vgb “stochastically nondecreasing” in vgb

◮ High vgb implies high “relative utility”

◮ PB not optimal if v1/vgb “stochastically decreasing” in vgb

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Intuition

PB optimal if high vgb implies high “relative utility” v1/vgb

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Intuition

PB optimal if high vgb implies high “relative utility” v1/vgb v1 vgb v1 vgb

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Intuition

PB optimal if high vgb implies high “relative utility” v1/vgb p v1 vgb v1 vgb

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Intuition

PB optimal if high vgb implies high “relative utility” v1/vgb p v1 vgb v1 vgb p p − δ v1 vgb

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Intuition

PB optimal if high vgb implies high “relative utility” v1/vgb p v1 vgb v1 vgb p p − δ p v1 vgb v1 vgb p p − δ

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A Special Case: Types “on a Path”

v1 vgb

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A Special Case: Types “on a Path”

v1 vgb V1(vgb)

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb v1 vgb V1(vgb) vgb

r(vgb)

v1

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb; e.g., r(vgb) = ˆ r v1 vgb ˆ r

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb v1 vgb V1

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. v1 vgb V1

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. v1 vgb r′ v′

gb

r vgb

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. Two types:

1 r′ ≥ r: PB optimal (∀µ)

v1 vgb r′ v′

gb

r vgb

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. Two types:

1 r′ ≥ r: PB optimal (∀µ) 2 r′ < r: PB not optimal (∃µ)

v1 vgb r′ v′

gb

r vgb

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. Two types:

1 r′ ≥ r: PB optimal (∀µ) 2 r′ < r: PB not optimal (∃µ) ◮ vgb = Pr[v ′]v ′

gb

v1 vgb r′ v′

gb

r vgb

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. v1 vgb

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A Special Case: Types “on a Path”

Ratio (relative utility) r(vgb) := v1/vgb

Proposition

Given “Path” V1, PB is optimal ∀µ iff r monotone nondecreasing. Stokey’79, Acquisti and Varian’05:

◮ PB optimal if r constant

v1 vgb ˆ r

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb v1 vgb vgb r v1

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb

◮ (r, vgb) ∼ ˆ

µ instead of (v1, vgb) ∼ µ v1 vgb vgb r vgbr

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb

◮ (r, vgb) ∼ ˆ

µ instead of (v1, vgb) ∼ µ

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. v1 vgb

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb

◮ (r, vgb) ∼ ˆ

µ instead of (v1, vgb) ∼ µ

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ≥ ˆ

r | vgb) nondecreasing in vgb (stochastic dominance) v1 vgb ˆ r vgb

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb

◮ (r, vgb) ∼ ˆ

µ instead of (v1, vgb) ∼ µ

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ≥ ˆ

r | vgb) nondecreasing in vgb (stochastic dominance) v1 vgb ˆ r vgb→

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb

◮ (r, vgb) ∼ ˆ

µ instead of (v1, vgb) ∼ µ

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ≥ ˆ

r | vgb) nondecreasing in vgb (stochastic dominance) Curve: v1 vgb

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Main Theorem (Two Identical Products)

Ratio (relative utility) r := v1/vgb

◮ (r, vgb) ∼ ˆ

µ instead of (v1, vgb) ∼ µ

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ≥ ˆ

r | vgb) nondecreasing in vgb (stochastic dominance) Curve: v1 vgb r′ v′

gb

r vgb

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb.

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb.

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ∈ ˆ

R | vgb) nondecreasing in vgb for all “upper sets” ˆ R.

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ∈ ˆ

R | vgb) nondecreasing in vgb for all “upper sets” ˆ R. rb rb′ 1 1

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ∈ ˆ

R | vgb) nondecreasing in vgb for all “upper sets” ˆ R. rb rb′ 1 1 ˆ R

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Main Theorem: Any Number of Products

Products 1 to k, (vb)b⊆{1,...,k}

◮ ∀b, define ratio rb = vb/vgb ∈ [0, 1]. Let r = (rb)b⊆{1,...,k}.

Theorem

PB is

◮ optimal

if r stochastically nondecreasing in vgb.

◮ not optimal if r stochastically

decreasing in vgb. r stochastically nondecreasing in vgb:

◮ Pr(r ∈ ˆ

R | vgb) nondecreasing in vgb for all “upper sets” ˆ R. rb rb′ 1 1 ˆ R

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Example 1: Complementarities

Two products, values v{1}, v{2}, v{1,2}

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Example 1: Complementarities

Two products, values v{1}, v{2}, v{1,2} described by x, y1, y2 vb = x · (y111∈b + y212∈b + (1 − y1 − y2)11,2∈b)

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Example 1: Complementarities

Two products, values v{1}, v{2}, v{1,2} described by x, y1, y2 vb = x · (y111∈b + y212∈b + (1 − y1 − y2)11,2∈b)

◮ x: intensity ◮ y1: values for product 1 only (y2 for product 2) ◮ y1 + y2 > 1 ⇒ substitutes: v{1} + v{2} > v{1,2}.

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Example 1: Complementarities

Two products, values v{1}, v{2}, v{1,2} described by x, y1, y2 vb = x · (y111∈b + y212∈b + (1 − y1 − y2)11,2∈b)

◮ x: intensity ◮ y1: values for product 1 only (y2 for product 2) ◮ y1 + y2 > 1 ⇒ substitutes: v{1} + v{2} > v{1,2}. (y1 + y2 > 1 ⇒

complements; y1 + y2 = 1 ⇒ additive)

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Example 1: Complementarities

Two products, values v{1}, v{2}, v{1,2} described by x, y1, y2 vb = x · (y111∈b + y212∈b + (1 − y1 − y2)11,2∈b)

◮ x: intensity ◮ y1: values for product 1 only (y2 for product 2) ◮ y1 + y2 > 1 ⇒ substitutes: v{1} + v{2} > v{1,2}. (y1 + y2 > 1 ⇒

complements; y1 + y2 = 1 ⇒ additive)

Corollary

PB is

◮ optimal

if (y1, y2) stochastically nondecreasing in x.

◮ not optimal if (y1, y2) stochastically

decreasing in x.

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Example 1: Complementarities

Two products, values v{1}, v{2}, v{1,2} described by x, y1, y2 vb = x · (y111∈b + y212∈b + (1 − y1 − y2)11,2∈b)

◮ x: intensity ◮ y1: values for product 1 only (y2 for product 2) ◮ y1 + y2 > 1 ⇒ substitutes: v{1} + v{2} > v{1,2}. (y1 + y2 > 1 ⇒

complements; y1 + y2 = 1 ⇒ additive)

Corollary

PB is

◮ optimal

if (y1, y2) stochastically nondecreasing in x.

◮ not optimal if (y1, y2) stochastically

decreasing in x. PB optimal if high value consumers consider products more substitutable

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Recall Additive Example

v{2} v{1} 0.8 0.2 0.2 0.8

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Recall Additive Example

Additivity & perfect negative correlation ⇒ vgb constant ⇒ r trivially stochastically nondecreasing in vgb ⇒ PB optimal v{2} v{1} 0.8 0.2 0.2 0.8

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Recall Additive Example

Additivity & perfect negative correlation ⇒ vgb constant ⇒ r trivially stochastically nondecreasing in vgb ⇒ PB optimal v{2} v{1} 0.8 0.2 0.2 0.8 Folklore: Bundle if v{1}, v{2} negatively correlated

◮ v1, v2: disutility from getting smaller bundle (compared to {1, 2}) ◮ Reinterpretation: Bundle if disutilities negatively correlated

Our result: Bundle if v1/vgb and vgb positively correlated

◮ 1 − v1/vgb: relative disutility from getting smaller bundle ◮ Bundle if relative disutility and vgb negatively correlated

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Example 2: Cobb Douglas Utilities

◮ k divisible products 1, . . . , k

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Example 2: Cobb Douglas Utilities

◮ k divisible products 1, . . . , k ◮ Bundle: b = (b1, . . . , bk), bi ∈ [0, 1]

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Example 2: Cobb Douglas Utilities

◮ k divisible products 1, . . . , k ◮ Bundle: b = (b1, . . . , bk), bi ∈ [0, 1] ◮ A type specified by x, y1, . . . , yk

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Example 2: Cobb Douglas Utilities

◮ k divisible products 1, . . . , k ◮ Bundle: b = (b1, . . . , bk), bi ∈ [0, 1] ◮ A type specified by x, y1, . . . , yk

v(b) = x

• i

byi

i

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Example 2: Cobb Douglas Utilities

◮ k divisible products 1, . . . , k ◮ Bundle: b = (b1, . . . , bk), bi ∈ [0, 1] ◮ A type specified by x, y1, . . . , yk

v(b) = x

• i

byi

i

Corollary

PB is

◮ optimal

if (y1, . . . , yk) stochastically nondecreasing in x.

◮ not optimal if (y1, . . . , yk) stochastically

decreasing in x.

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Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss

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Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)]

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Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss = v − 1−F(v)

f (v)

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)]

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Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss = v − 1−F(v)

f (v)

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)] max

mechanism (x,p) Ev[x(v) · φ(v)]

s.t. 0 ≤ x(v) ≤ 1, ∀v, incentive compatibility

13 / 19

Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss = v − 1−F(v)

f (v)

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)] max

mechanism (x,p) Ev[x(v) · φ(v)]

s.t. 0 ≤ x(v) ≤ 1, ∀v, incentive compatibility v φ(v)

13 / 19

Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss = v − 1−F(v)

f (v)

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)] max

mechanism (x,p) Ev[x(v) · φ(v)]

s.t. 0 ≤ x(v) ≤ 1, ∀v, incentive compatibility v φ(v)

13 / 19

Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss = v − 1−F(v)

f (v)

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)] max

mechanism (x,p) Ev[x(v) · φ(v)]

s.t. 0 ≤ x(v) ≤ 1, ∀v, incentive compatibility v φ(v) x∗(v) p 1

13 / 19

Envelope Analysis and Virtual Values

v Single dimension: “virtual value” φ(v) = v - revenue loss = v − 1−F(v)

f (v)

Lemma (Myerson’81)

Revenue of any IC mechanism is Ev[x(v) · φ(v)]

Theorem (Myerson’81; Riley and Zeckhauser’83)

Posting a price for the item is the optimal mechanism max

mechanism (x,p) Ev[x(v) · φ(v)]

s.t. 0 ≤ x(v) ≤ 1, ∀v, incentive compatibility v φ(v) x∗(v) p 1

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Envelope Analysis and Curves

Lemma

Revenue of any IC mechanism is Ev[x1(v) · φ1(v) + xgb(v) · φgb(v)] V1 v1 vgb

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Envelope Analysis and Curves

Lemma

Revenue of any IC mechanism is Ev[x1(v) · φ1(v) + xgb(v) · φgb(v)]

◮ φgb(v) = vgb − 1−Fgb(vgb) fgb(vgb) ◮ φ1(v) = V1(vgb) − V ′ 1(vgb) 1−Fgb(vgb) fgb(vgb)

where Fgb, fgb are c.d.f and p.d.f of vgb V1 v1 vgb

14 / 19

Envelope Analysis and Curves

Lemma

Revenue of any IC mechanism is Ev[x1(v) · φ1(v) + xgb(v) · φgb(v)]

◮ φgb(v) = vgb − 1−Fgb(vgb) fgb(vgb) ◮ φ1(v) = V1(vgb) − V ′ 1(vgb) 1−Fgb(vgb) fgb(vgb)

where Fgb, fgb are c.d.f and p.d.f of vgb Property:

◮ If r(vgb) nondecreasing then r(vgb)φgb(vgb) ≥ φ1(vgb)

V1 v1 vgb

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Envelope Analysis and Curves

Lemma

Revenue of any IC mechanism is Ev[x1(v) · φ1(v) + xgb(v) · φgb(v)]

◮ φgb(v) = vgb − 1−Fgb(vgb) fgb(vgb) ◮ φ1(v) = V1(vgb) − V ′ 1(vgb) 1−Fgb(vgb) fgb(vgb)

where Fgb, fgb are c.d.f and p.d.f of vgb Property:

◮ If r(vgb) nondecreasing then r(vgb)φgb(vgb) ≥ φ1(vgb)

V1 v1 vgb φ1 vgb rφgb φgb v∗

gb

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Envelope Analysis and Curves

Lemma

Revenue of any IC mechanism is Ev[x1(v) · φ1(v) + xgb(v) · φgb(v)]

◮ φgb(v) = vgb − 1−Fgb(vgb) fgb(vgb) ◮ φ1(v) = V1(vgb) − V ′ 1(vgb) 1−Fgb(vgb) fgb(vgb)

where Fgb, fgb are c.d.f and p.d.f of vgb Property:

◮ If r(vgb) nondecreasing then r(vgb)φgb(vgb) ≥ φ1(vgb) ◮ If further φgb is increasing then x∗ is optimal

V1 v1 vgb v∗

gb

x∗

gb = 1

x∗

1 = 0

x∗

gb = 0

x∗

1 = 0

φ1 vgb rφgb φgb v∗

gb

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Beyond Regularity

If ratio r increasing, then only “downward” IC constraints bind V1 v1 vgb v v′ v′′

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Beyond Regularity

If ratio r increasing, then only “downward” IC constraints bind Generalized virtual value: ˆ φ(v) = v −

• v′: IC from v′ to v binds

λ(ν′)(v′ − v), V1 v1 vgb v v′ v′′ ˆ φ(v)

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Beyond Regularity

If ratio r increasing, then only “downward” IC constraints bind Generalized virtual value: ˆ φ(v) = v −

• v′: IC from v′ to v binds

λ(ν′)(v′ − v), Thus r ˆ φgb ≥ ˆ φ1, and x∗

1 = 0.

V1 v1 vgb v v′ v′′ ˆ φ(v)

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Beyond Paths: Orthogonalization

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Beyond Paths: Orthogonalization

Two paths V1, ˆ V1 (both with monotone ratio), same marginal Fgb V1 v1 vgb ˆ V1 v1 vgb

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Beyond Paths: Orthogonalization

Two paths V1, ˆ V1 (both with monotone ratio), same marginal Fgb

◮ Let p∗ = maxp p(1 − Fgb(p))

V1 v1 vgb ˆ V1 v1 vgb

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Beyond Paths: Orthogonalization

Two paths V1, ˆ V1 (both with monotone ratio), same marginal Fgb

◮ Let p∗ = maxp p(1 − Fgb(p)) ◮ PB with price p∗ is opt for each instance

V1 v1 vgb ˆ V1 v1 vgb

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Beyond Paths: Orthogonalization

Two paths V1, ˆ V1 (both with monotone ratio), same marginal Fgb

◮ Let p∗ = maxp p(1 − Fgb(p)) ◮ PB with price p∗ is opt for each instance ◮ Consider their mixture:

v1 vgb = α× +(1 − α)× V1 v1 vgb ˆ V1 v1 vgb

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Beyond Paths: Orthogonalization

Two paths V1, ˆ V1 (both with monotone ratio), same marginal Fgb

◮ Let p∗ = maxp p(1 − Fgb(p)) ◮ PB with price p∗ is opt for each instance ◮ Consider their mixture:

◮ Profit ≤ profit if seller “knows” the curve ◮ So optimal to PB with price p∗

v1 vgb = α× +(1 − α)× V1 v1 vgb ˆ V1 v1 vgb

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Beyond Paths: Orthogonalization

Two paths V1, ˆ V1 (both with monotone ratio), same marginal Fgb

◮ Let p∗ = maxp p(1 − Fgb(p)) ◮ PB with price p∗ is opt for each instance ◮ Consider their mixture:

◮ Profit ≤ profit if seller “knows” the curve ◮ So optimal to PB with price p∗

v1 vgb = α× +(1 − α)× V1 v1 vgb ˆ V1 v1 vgb Question: When can a distribution be decomposed?

1 to ratio-monotone curves 2 with same marginal Fgb 16 / 19

When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) r vgb ˆ r 1

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When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) ⇔ “contour lines” nondecreasing r vgb ˆ r 1

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When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) ⇔ “contour lines” nondecreasing q = 1 q = 3/4 q = 2/4 q = 1/4 r vgb ˆ r 1

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When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) ⇔ “contour lines” nondecreasing Decompose distribution µ into {µ | q}q∈[0,1] q = 1 q = 3/4 q = 2/4 q = 1/4 r vgb ˆ r 1

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When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) ⇔ “contour lines” nondecreasing Decompose distribution µ into {µ | q}q∈[0,1]

1 Support of each µ | q ratio-monotone

q = 1 q = 3/4 q = 2/4 q = 1/4 r vgb ˆ r 1

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When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) ⇔ “contour lines” nondecreasing Decompose distribution µ into {µ | q}q∈[0,1]

1 Support of each µ | q ratio-monotone 2 q independent from vgb

q = 1 q = 3/4 q = 2/4 q = 1/4 r vgb ˆ r 1

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When Can a Distribution be Decomposed?

r stochastically nondecreasing in vgb (Pr(r ≥ ˆ r | vH) ↑ in vgb) ⇔ “contour lines” nondecreasing Decompose distribution µ into {µ | q}q∈[0,1]

1 Support of each µ | q ratio-monotone 2 q independent from vgb

Strassen ’65, Kamae et al. ’77: generalization to higher dimensions q = 1 q = 3/4 q = 2/4 q = 1/4 r vgb ˆ r 1

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Related Work

Technically:

◮ Orthogonalization: Eso, Szentes ’07; Pavan et al. ’14 ◮ Wilson ’93, Armstrong ’96: fixed paths ◮ Carroll ’16: virtual values, fixed paths

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Related Work

Technically:

◮ Orthogonalization: Eso, Szentes ’07; Pavan et al. ’14 ◮ Wilson ’93, Armstrong ’96: fixed paths ◮ Carroll ’16: virtual values, fixed paths

Bundling: Mostly additive values

◮ Fang and Norman ’06: Pure bundling vs. selling separately ◮ Daskalakis et al. ’17: PB optimal if values i.i.d [c, c + 1] for large c

◮ Pavlov ’11, Menicucci et al. ’15: Other i.i.d distributions

◮ McAfee and McMillan ’88, Manelli and Vincent ’06: optimality of

deterministic mechanisms

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Main Result

PB optimal if high vgb implies high “relative utility” v1/vgb p v1 vgb v1 vgb p p − δ p v1 vgb v1 vgb p p − δ

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Main Result

PB optimal if high vgb implies high “relative utility” v1/vgb p v1 vgb v1 vgb p p − δ p v1 vgb v1 vgb p p − δ Thanks!

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