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

A Special Case: Types “on a Path” Ratio (relative utility) r ( v gb ) := v 1 / v gb Proposition Given “Path” V 1 , PB is optimal ∀ µ iff r monotone nondecreasing. v 1 r Two types: 1 r ′ ≥ r : PB optimal ( ∀ µ ) r ′ 2 r ′ < r : PB not optimal ( ∃ µ ) ◮ v gb = Pr [ v ′ ] v ′ v gb gb v ′ v gb gb 7 / 19

A Special Case: Types “on a Path” Ratio (relative utility) r ( v gb ) := v 1 / v gb Proposition Given “Path” V 1 , PB is optimal ∀ µ iff r monotone nondecreasing. v 1 v gb 7 / 19

A Special Case: Types “on a Path” Ratio (relative utility) r ( v gb ) := v 1 / v gb Proposition Given “Path” V 1 , PB is optimal ∀ µ iff r monotone nondecreasing. v 1 ˆ r Stokey’79, Acquisti and Varian’05: ◮ PB optimal if r constant v gb 7 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb v 1 v 1 r v gb v gb 8 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb ◮ ( r , v gb ) ∼ ˆ µ instead of ( v 1 , v gb ) ∼ µ v 1 v gb r r v gb v gb 8 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb ◮ ( r , v gb ) ∼ ˆ µ instead of ( v 1 , v gb ) ∼ µ Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . v 1 v gb 8 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb ◮ ( r , v gb ) ∼ ˆ µ instead of ( v 1 , v gb ) ∼ µ Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . v 1 r stochastically nondecreasing in v gb : ◮ Pr ( r ≥ ˆ r | v gb ) nondecreasing in v gb (stochastic dominance) r ˆ v gb v gb 8 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb ◮ ( r , v gb ) ∼ ˆ µ instead of ( v 1 , v gb ) ∼ µ Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . v 1 r stochastically nondecreasing in v gb : ◮ Pr ( r ≥ ˆ r | v gb ) nondecreasing in v gb (stochastic dominance) r ˆ v gb v gb → 8 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb ◮ ( r , v gb ) ∼ ˆ µ instead of ( v 1 , v gb ) ∼ µ Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . v 1 r stochastically nondecreasing in v gb : ◮ Pr ( r ≥ ˆ r | v gb ) nondecreasing in v gb (stochastic dominance) Curve: v gb 8 / 19

Main Theorem (Two Identical Products) Ratio (relative utility) r := v 1 / v gb ◮ ( r , v gb ) ∼ ˆ µ instead of ( v 1 , v gb ) ∼ µ Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . v 1 r stochastically nondecreasing in v gb : r ′ ◮ Pr ( r ≥ ˆ r | v gb ) nondecreasing in v gb (stochastic dominance) r Curve: v gb v ′ v gb gb 8 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . 9 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . 9 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . 9 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . r stochastically nondecreasing in v gb : ◮ Pr ( r ∈ ˆ R | v gb ) nondecreasing in v gb for all “upper sets” ˆ R . 9 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . r b 1 r stochastically nondecreasing in v gb : ◮ Pr ( r ∈ ˆ R | v gb ) nondecreasing in v gb for all “upper sets” ˆ R . r b ′ 1 9 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . r b 1 r stochastically nondecreasing in v gb : ˆ R ◮ Pr ( r ∈ ˆ R | v gb ) nondecreasing in v gb for all “upper sets” ˆ R . r b ′ 1 9 / 19

Main Theorem: Any Number of Products Products 1 to k , ( v b ) b ⊆{ 1 ,..., k } ◮ ∀ b , define ratio r b = v b / v gb ∈ [0 , 1]. Let r = ( r b ) b ⊆{ 1 ,..., k } . Theorem PB is ◮ optimal if r stochastically nondecreasing in v gb . ◮ not optimal if r stochastically decreasing in v gb . r b 1 r stochastically nondecreasing in v gb : ˆ R ◮ Pr ( r ∈ ˆ R | v gb ) nondecreasing in v gb for all “upper sets” ˆ R . r b ′ 1 9 / 19

Example 1: Complementarities Two products, values v { 1 } , v { 2 } , v { 1 , 2 } 10 / 19

Example 1: Complementarities Two products, values v { 1 } , v { 2 } , v { 1 , 2 } described by x , y 1 , y 2 v b = x · ( y 1 1 1 ∈ b + y 2 1 2 ∈ b + (1 − y 1 − y 2 ) 1 1 , 2 ∈ b ) 10 / 19

Example 1: Complementarities Two products, values v { 1 } , v { 2 } , v { 1 , 2 } described by x , y 1 , y 2 v b = x · ( y 1 1 1 ∈ b + y 2 1 2 ∈ b + (1 − y 1 − y 2 ) 1 1 , 2 ∈ b ) ◮ x : intensity ◮ y 1 : values for product 1 only ( y 2 for product 2) ◮ y 1 + y 2 > 1 ⇒ substitutes: v { 1 } + v { 2 } > v { 1 , 2 } . 10 / 19

Example 1: Complementarities Two products, values v { 1 } , v { 2 } , v { 1 , 2 } described by x , y 1 , y 2 v b = x · ( y 1 1 1 ∈ b + y 2 1 2 ∈ b + (1 − y 1 − y 2 ) 1 1 , 2 ∈ b ) ◮ x : intensity ◮ y 1 : values for product 1 only ( y 2 for product 2) ◮ y 1 + y 2 > 1 ⇒ substitutes: v { 1 } + v { 2 } > v { 1 , 2 } . ( y 1 + y 2 > 1 ⇒ complements; y 1 + y 2 = 1 ⇒ additive) 10 / 19

Example 1: Complementarities Two products, values v { 1 } , v { 2 } , v { 1 , 2 } described by x , y 1 , y 2 v b = x · ( y 1 1 1 ∈ b + y 2 1 2 ∈ b + (1 − y 1 − y 2 ) 1 1 , 2 ∈ b ) ◮ x : intensity ◮ y 1 : values for product 1 only ( y 2 for product 2) ◮ y 1 + y 2 > 1 ⇒ substitutes: v { 1 } + v { 2 } > v { 1 , 2 } . ( y 1 + y 2 > 1 ⇒ complements; y 1 + y 2 = 1 ⇒ additive) Corollary PB is ◮ optimal if ( y 1 , y 2 ) stochastically nondecreasing in x. ◮ not optimal if ( y 1 , y 2 ) stochastically decreasing in x. 10 / 19

Example 1: Complementarities Two products, values v { 1 } , v { 2 } , v { 1 , 2 } described by x , y 1 , y 2 v b = x · ( y 1 1 1 ∈ b + y 2 1 2 ∈ b + (1 − y 1 − y 2 ) 1 1 , 2 ∈ b ) ◮ x : intensity ◮ y 1 : values for product 1 only ( y 2 for product 2) ◮ y 1 + y 2 > 1 ⇒ substitutes: v { 1 } + v { 2 } > v { 1 , 2 } . ( y 1 + y 2 > 1 ⇒ complements; y 1 + y 2 = 1 ⇒ additive) Corollary PB is ◮ optimal if ( y 1 , y 2 ) stochastically nondecreasing in x. ◮ not optimal if ( y 1 , y 2 ) stochastically decreasing in x. PB optimal if high value consumers consider products more substitutable 10 / 19

Recall Additive Example v { 2 } 0 . 8 0 . 2 v { 1 } 0 . 2 0 . 8 11 / 19

Recall Additive Example Additivity & perfect negative correlation ⇒ v gb constant ⇒ r trivially stochastically nondecreasing in v gb ⇒ PB optimal v { 2 } 0 . 8 0 . 2 v { 1 } 0 . 2 0 . 8 11 / 19

Recall Additive Example Additivity & perfect negative correlation ⇒ v gb constant ⇒ r trivially stochastically nondecreasing in v gb ⇒ PB optimal v { 2 } 0 . 8 0 . 2 v { 1 } 0 . 2 0 . 8 Folklore: Bundle if v { 1 } , v { 2 } negatively correlated ◮ v 1 , v 2 : disutility from getting smaller bundle (compared to { 1 , 2 } ) ◮ Reinterpretation: Bundle if disutilities negatively correlated Our result: Bundle if v 1 / v gb and v gb positively correlated ◮ 1 − v 1 / v gb : relative disutility from getting smaller bundle ◮ Bundle if relative disutility and v gb negatively correlated 11 / 19

Example 2: Cobb Douglas Utilities ◮ k divisible products 1 , . . . , k 12 / 19

Example 2: Cobb Douglas Utilities ◮ k divisible products 1 , . . . , k ◮ Bundle: b = ( b 1 , . . . , b k ) , b i ∈ [0 , 1] 12 / 19

Example 2: Cobb Douglas Utilities ◮ k divisible products 1 , . . . , k ◮ Bundle: b = ( b 1 , . . . , b k ) , b i ∈ [0 , 1] ◮ A type specified by x , y 1 , . . . , y k 12 / 19

Example 2: Cobb Douglas Utilities ◮ k divisible products 1 , . . . , k ◮ Bundle: b = ( b 1 , . . . , b k ) , b i ∈ [0 , 1] ◮ A type specified by x , y 1 , . . . , y k � b y i v ( b ) = x i i 12 / 19

Example 2: Cobb Douglas Utilities ◮ k divisible products 1 , . . . , k ◮ Bundle: b = ( b 1 , . . . , b k ) , b i ∈ [0 , 1] ◮ A type specified by x , y 1 , . . . , y k � b y i v ( b ) = x i i Corollary PB is ◮ optimal if ( y 1 , . . . , y k ) stochastically nondecreasing in x. ◮ not optimal if ( y 1 , . . . , y k ) stochastically decreasing in x. 12 / 19

Envelope Analysis and Virtual Values Single dimension: v “virtual value” φ ( v ) = v - revenue loss 13 / 19

Envelope Analysis and Virtual Values Single dimension: v “virtual value” φ ( v ) = v - revenue loss Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] 13 / 19

Envelope Analysis and Virtual Values Single dimension: v = v − 1 − F ( v ) “virtual value” φ ( v ) = v - revenue loss f ( v ) Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] 13 / 19

Envelope Analysis and Virtual Values Single dimension: v = v − 1 − F ( v ) “virtual value” φ ( v ) = v - revenue loss f ( v ) Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] mechanism ( x , p ) E v [ x ( v ) · φ ( v )] max s.t. 0 ≤ x ( v ) ≤ 1 , ∀ v , incentive compatibility 13 / 19

Envelope Analysis and Virtual Values Single dimension: v = v − 1 − F ( v ) “virtual value” φ ( v ) = v - revenue loss f ( v ) Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] mechanism ( x , p ) E v [ x ( v ) · φ ( v )] max φ ( v ) s.t. 0 ≤ x ( v ) ≤ 1 , ∀ v , v incentive compatibility 13 / 19

Envelope Analysis and Virtual Values Single dimension: v = v − 1 − F ( v ) “virtual value” φ ( v ) = v - revenue loss f ( v ) Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] mechanism ( x , p ) E v [ x ( v ) · φ ( v )] max φ ( v ) s.t. 0 ≤ x ( v ) ≤ 1 , ∀ v , v incentive compatibility 13 / 19

Envelope Analysis and Virtual Values Single dimension: v = v − 1 − F ( v ) “virtual value” φ ( v ) = v - revenue loss f ( v ) Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] x ∗ ( v ) mechanism ( x , p ) E v [ x ( v ) · φ ( v )] max 1 φ ( v ) s.t. 0 ≤ x ( v ) ≤ 1 , ∀ v , v incentive compatibility p 13 / 19

Envelope Analysis and Virtual Values Single dimension: v = v − 1 − F ( v ) “virtual value” φ ( v ) = v - revenue loss f ( v ) Lemma (Myerson’81) Revenue of any IC mechanism is E v [ x ( v ) · φ ( v )] Theorem (Myerson’81; Riley and Zeckhauser’83) Posting a price for the item is the optimal mechanism x ∗ ( v ) mechanism ( x , p ) E v [ x ( v ) · φ ( v )] max 1 φ ( v ) s.t. 0 ≤ x ( v ) ≤ 1 , ∀ v , v incentive compatibility p 13 / 19

Envelope Analysis and Curves Lemma Revenue of any IC mechanism is E v [ x 1 ( v ) · φ 1 ( v ) + x gb ( v ) · φ gb ( v )] v 1 V 1 v gb 14 / 19

Envelope Analysis and Curves Lemma Revenue of any IC mechanism is E v [ x 1 ( v ) · φ 1 ( v ) + x gb ( v ) · φ gb ( v )] ◮ φ gb ( v ) = v gb − 1 − F gb ( v gb ) f gb ( v gb ) 1 ( v gb ) 1 − F gb ( v gb ) ◮ φ 1 ( v ) = V 1 ( v gb ) − V ′ f gb ( v gb ) where F gb , f gb are c.d.f and p.d.f of v gb v 1 V 1 v gb 14 / 19

Envelope Analysis and Curves Lemma Revenue of any IC mechanism is E v [ x 1 ( v ) · φ 1 ( v ) + x gb ( v ) · φ gb ( v )] ◮ φ gb ( v ) = v gb − 1 − F gb ( v gb ) f gb ( v gb ) 1 ( v gb ) 1 − F gb ( v gb ) ◮ φ 1 ( v ) = V 1 ( v gb ) − V ′ f gb ( v gb ) where F gb , f gb are c.d.f and p.d.f of v gb Property: ◮ If r ( v gb ) nondecreasing then r ( v gb ) φ gb ( v gb ) ≥ φ 1 ( v gb ) v 1 V 1 v gb 14 / 19

Envelope Analysis and Curves Lemma Revenue of any IC mechanism is E v [ x 1 ( v ) · φ 1 ( v ) + x gb ( v ) · φ gb ( v )] ◮ φ gb ( v ) = v gb − 1 − F gb ( v gb ) f gb ( v gb ) 1 ( v gb ) 1 − F gb ( v gb ) ◮ φ 1 ( v ) = V 1 ( v gb ) − V ′ f gb ( v gb ) where F gb , f gb are c.d.f and p.d.f of v gb Property: ◮ If r ( v gb ) nondecreasing then r ( v gb ) φ gb ( v gb ) ≥ φ 1 ( v gb ) v 1 V 1 φ gb r φ gb v ∗ φ 1 gb v gb v gb 14 / 19

Envelope Analysis and Curves Lemma Revenue of any IC mechanism is E v [ x 1 ( v ) · φ 1 ( v ) + x gb ( v ) · φ gb ( v )] ◮ φ gb ( v ) = v gb − 1 − F gb ( v gb ) f gb ( v gb ) 1 ( v gb ) 1 − F gb ( v gb ) ◮ φ 1 ( v ) = V 1 ( v gb ) − V ′ f gb ( v gb ) where F gb , f gb are c.d.f and p.d.f of v gb Property: ◮ If r ( v gb ) nondecreasing then r ( v gb ) φ gb ( v gb ) ≥ φ 1 ( v gb ) ◮ If further φ gb is increasing then x ∗ is optimal v 1 V 1 φ gb r φ gb x ∗ gb = 0 x ∗ gb = 1 v ∗ φ 1 gb x ∗ 1 = 0 x ∗ 1 = 0 v gb v ∗ v gb gb 14 / 19

Beyond Regularity If ratio r increasing, then only “downward” IC constraints bind v 1 V 1 v ′′ v ′ v 0 v gb 15 / 19

Beyond Regularity If ratio r increasing, then only “downward” IC constraints bind Generalized virtual value: λ ( ν ′ )( v ′ − v ) , ˆ � φ ( v ) = v − v ′ : IC from v ′ to v binds v 1 V 1 v ′′ v ′ v ˆ φ ( v ) 0 v gb 15 / 19

Beyond Regularity If ratio r increasing, then only “downward” IC constraints bind Generalized virtual value: λ ( ν ′ )( v ′ − v ) , ˆ � φ ( v ) = v − v ′ : IC from v ′ to v binds Thus r ˆ φ gb ≥ ˆ φ 1 , and x ∗ 1 = 0. v 1 V 1 v ′′ v ′ v ˆ φ ( v ) 0 v gb 15 / 19

Beyond Paths: Orthogonalization 16 / 19

Beyond Paths: Orthogonalization Two paths V 1 , ˆ V 1 (both with monotone ratio), same marginal F gb v 1 v 1 V 1 ˆ V 1 v gb v gb 16 / 19

Beyond Paths: Orthogonalization Two paths V 1 , ˆ V 1 (both with monotone ratio), same marginal F gb ◮ Let p ∗ = max p p (1 − F gb ( p )) v 1 v 1 V 1 ˆ V 1 v gb v gb 16 / 19

Beyond Paths: Orthogonalization Two paths V 1 , ˆ V 1 (both with monotone ratio), same marginal F gb ◮ Let p ∗ = max p p (1 − F gb ( p )) ◮ PB with price p ∗ is opt for each instance v 1 v 1 V 1 ˆ V 1 v gb v gb 16 / 19

Beyond Paths: Orthogonalization Two paths V 1 , ˆ V 1 (both with monotone ratio), same marginal F gb ◮ Let p ∗ = max p p (1 − F gb ( p )) ◮ PB with price p ∗ is opt for each instance ◮ Consider their mixture: v 1 v 1 v 1 V 1 = α × +(1 − α ) × ˆ V 1 v gb v gb v gb 16 / 19

Beyond Paths: Orthogonalization Two paths V 1 , ˆ V 1 (both with monotone ratio), same marginal F gb ◮ Let p ∗ = max p p (1 − F gb ( 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 ∗ v 1 v 1 v 1 V 1 = α × +(1 − α ) × ˆ V 1 v gb v gb v gb 16 / 19

Beyond Paths: Orthogonalization Two paths V 1 , ˆ V 1 (both with monotone ratio), same marginal F gb ◮ Let p ∗ = max p p (1 − F gb ( 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 ∗ v 1 v 1 v 1 V 1 = α × +(1 − α ) × ˆ V 1 v gb v gb v gb Question: When can a distribution be decomposed? 1 to ratio-monotone curves 2 with same marginal F gb 16 / 19

When Can a Distribution be Decomposed? r stochastically nondecreasing in v gb ( Pr ( r ≥ ˆ r | v H ) ↑ in v gb ) r 1 ˆ r 0 v gb 17 / 19

When Can a Distribution be Decomposed? r stochastically nondecreasing in v gb ( Pr ( r ≥ ˆ r | v H ) ↑ in v gb ) ⇔ “contour lines” nondecreasing r 1 ˆ r 0 v gb 17 / 19