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

Pareto Improving Segmentation of Multi-product Markets Nima Haghpanah joint with Ron Siegel May 4, 2019 1 / 15

Market Segmentation Firms can segment the market based on available data ◮ age, sex, location, browsing history, ... 2 / 15

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? 2 / 15

Does a Pareto Improving Segmentation Exist? 3 / 15

Does a Pareto Improving Segmentation Exist? Menu Product $5 Bundle 1 $10 Bundle 2 $8 . . . . . . 3 / 15

Does a Pareto Improving Segmentation Exist? Un-segmented market: CS ( c ) consumer c Menu Product $5 Bundle 1 $10 Bundle 2 $8 . . . . . . 3 / 15

Does a Pareto Improving Segmentation Exist? Un-segmented market: CS ( c ) consumer c Menu 1 Menu 2 Menu 3 Menu 4 3 / 15

Does a Pareto Improving Segmentation Exist? Un-segmented market: CS ( c ) Segmented market: CS ( c , ) consumer c Menu 1 Menu 2 Menu 3 Menu 4 3 / 15

Does a Pareto Improving Segmentation Exist? Un-segmented market: CS ( c ) Segmented market: CS ( c , ) ∃ ? segmentation s.t. CS ( c , ) ≥ CS ( c ) for all c , & > for some c consumer c Menu 1 Menu 2 Menu 3 Menu 4 3 / 15

A Single Product Example (with Unit Demands) 1 − q q “Market”: Valuation v : 1 2 4 / 15

A Single Product Example (with Unit Demands) A Pareto Improving (PI) segmentation exists if market q is inefficient 1 − q q “Market”: Valuation v : 1 2 4 / 15

A Single Product Example (with Unit Demands) A Pareto Improving (PI) segmentation exists if market q is inefficient ◮ q ∈ (0 . 5 , 1) 1 − q q “Market”: Valuation v : 1 2 4 / 15

A Single Product Example (with Unit Demands) A Pareto Improving (PI) segmentation exists if market q is inefficient ◮ q ∈ (0 . 5 , 1) Optimal price: 1 2 1 − q q “Market”: q All markets: 0 0 . 5 1 Valuation v : 1 2 4 / 15

A Single Product Example (with Unit Demands) A Pareto Improving (PI) segmentation exists if market q is inefficient ◮ q ∈ (0 . 5 , 1) Optimal price: 1 2 1 − q q “Market”: q All markets: 0 0 . 5 1 Valuation v : 1 2 Surplus 1 of v = 2 q q 0 0 . 5 1 4 / 15

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 Optimal price: 1 2 1 − q q “Market”: q All markets: 0 0 . 5 1 Valuation v : 1 2 Surplus 1 of v = 2 q q ′ q q ′′ 0 0 . 5 1 4 / 15

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 Optimal price: 1 2 1 − q q “Market”: q All markets: 0 0 . 5 1 Valuation v : 1 2 Surplus 1 of v = 2 q q 0 0 . 5 1 4 / 15

Screening Example: Qualities L and H 1 − q q “Market”: v H : 1 2 v L : 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 1 − q q “Market”: v H : 1 2 v L : 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : Surplus 0 . 75 1 1 of high type 0 . 25 q 0 0 . 25 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : Surplus 0 . 75 1 1 of high type 0 . 25 q q = 0 0 . 25 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : Surplus 0 . 75 1 1 of high type 0 . 25 q q ′ q q ′′ 0 0 . 25 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : Surplus 0 . 75 1 1 of high type 0 . 25 q q ′ q q ′′ 0 0 . 25 0 . 75 1 5 / 15

Screening Example: Qualities L and H PI segmentation ∄ for inefficient market q = 0 . 75 ◮ PI segmentation exists for all but one inefficient markets p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : Surplus 0 . 75 1 1 of high type 0 . 25 q 0 0 . 25 0 . 75 1 5 / 15

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 p ( H ) = 1 1 . 75 2 1 − q q “Market”: p ( L ) = 0 . 75 q v H : 0 0 . 25 0 . 75 1 1 2 v L : Surplus 0 . 75 1 of high type q 1 5 / 15

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 ) 6 / 15

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 6 / 15

Markets and Segmentations Market f ∈ ∆( T ) ◮ Mechanism optimal if maximizes E f [ 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 E f [ 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. E µ [ f ′ ( t )] = f ( t ) , ∀ t ◮ e.g., two types, E µ [ q ′ ] = q 7 / 15

Markets and Segmentations Market f ∈ ∆( T ) ◮ Mechanism optimal if maximizes E f [ 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. E µ [ f ′ ( t )] = f ( t ) , ∀ t ◮ e.g., two types, E µ [ 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 ′ ) , > 7 / 15

Markets and Segmentations Market f ∈ ∆( T ) ◮ Mechanism optimal if maximizes E f [ 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. E µ [ f ′ ( t )] = f ( t ) , ∀ t ◮ e.g., two types, E µ [ 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) 7 / 15

Main Result: PI Segmentations Exist? 8 / 15

Main Result: PI Segmentations Exist? 8 / 15

Main Result: PI Segmentations Exist? Recall: If ∃ PI segmentation of f ⇒ f is inefficient ◮ ( ∃ t , a ( t ) / ∈ arg max a ′ v ( t , a ′ ) − c ( a ′ )) 8 / 15

Main Result: PI Segmentations Exist? Recall: If ∃ PI segmentation of f ⇒ f is inefficient ◮ ( ∃ t , a ( t ) / ∈ arg max a ′ v ( t , a ′ ) − c ( a ′ )) Theorem For almost all inefficient markets, PI segmentations exist. 8 / 15

Main Result: PI Segmentations Exist? Recall: If ∃ PI segmentation of f ⇒ f is inefficient ◮ ( ∃ t , a ( t ) / ∈ arg max a ′ 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 ) 8 / 15

Main Result: PI Segmentations Exist? Recall: If ∃ PI segmentation of f ⇒ f is inefficient ◮ ( ∃ t , a ( t ) / ∈ arg max a ′ 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 (2) 1 1 f (1) 8 / 15