Ironing, Sweeping, and Multivariate Majorization Optimal Mechanisms for Mass-Produced Goods (with Jacob Goeree (UNSW) and Ningyi Sun (UNSW)) Nick Bedard (WLU) Virtual MD Seminar, Oct. 26 Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 1 / 35
Introduction We consider monopoly that sells an excludable, non-rivalrous good - For profit public goods - Many mass-produced goods fit this framework - E.g. newspapers, songs, movies, books, iPhones, television Monopolist chooses single quality level to be enjoyed by all consumers Monopolist can restrict access to the good Buyers’ valuations are interdependent: private and common values - Could go either way: higher type for i may raise/lower j ’s value Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 2 / 35
Main Difficulty The problem is naturally irregular - i.e. incentive constraints cannot be substituted out Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 3 / 35
Main Difficulty The problem is naturally irregular - i.e. incentive constraints cannot be substituted out Mussa and Rosen’s (1978) and Myerson’s (1981) “ironing” - Constructive approach but only for unidimensional case Rochet and Chon´ e’s (1998) “sweeping” - Works for multidimensional case but not constructive Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 3 / 35
Main Difficulty We develop a constructive multidimensional approach to ironing - Extend majorization theory to higher dimensions - Based on Kuhn-Tucker theory - Implement ironing via simple quadratic minimization problems Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 4 / 35
Main Difficulty Seller’s problem: � 2 q ( x ) 2 � q ( x ) MR ( x ) − 1 max E q ( x ) is non-decreasing Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 5 / 35
Main Difficulty Seller’s problem: � 2 q ( x ) 2 � q ( x ) MR ( x ) − 1 max E q ( x ) is non-decreasing q ( x ) = max { 0 , MR ( x ) } not IC if MR ( x ) is decreasing in some directions Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 5 / 35
Main Difficulty Seller’s problem: � 2 q ( x ) 2 � q ( x ) MR ( x ) − 1 max E q ( x ) is non-decreasing q ( x ) = max { 0 , MR ( x ) } not IC if MR ( x ) is decreasing in some directions Ironing: construct non-decreasing MR ( x ) such that � 2 q ( x ) 2 � q ( x ) MR ( x ) − 1 q ( x ) = arg max ˜ E q ( x ) solves the original problem Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 5 / 35
Main Difficulty Seller’s problem: � 2 q ( x ) 2 � q ( x ) MR ( x ) − 1 max E q ( x ) is non-decreasing q ( x ) = max { 0 , MR ( x ) } not IC if MR ( x ) is decreasing in some directions E.g. MR 1 MR 2 MR � �� � � �� � � �� � � 0 � � 0 � � 0 � 9 0 1 2 10 2 1 10 1 + 9 10 11 = 10 20 12 2 11 2 0 1 2 2 12 4 Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 5 / 35
Main Difficulty x Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 6 / 35
Main Difficulty x Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 6 / 35
Main Difficulty 1 1 0 . 5 0 . 5 1 1 x 2 x 2 0 0 . 5 0 0 . 5 0 . 5 0 . 5 x 1 x 1 1 1 Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 7 / 35
Model Buyers i = { 1 , . . . , n } with types x i - Types drawn independently according to distribution F i ( x i ) - Highest type ¯ x i Seller chooses - quality: q ( x ) - access rights: { η 1 ( x ) , . . . , η n ( x ) } - transfers: { t 1 ( x ) , . . . , t n ( x ) } Buyer i ’s utility: u i ( x ) = v i ( x ) q ( x ) η i ( x ) − t i ( x ) Assume v i ( x i , x − i ) increasing in x i for all x − i ∈ X − i Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 8 / 35
Seller’s Problem �� � ( q, η , t ) E max t i ( x ) − C ( q ( x )) i ∈ N Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 9 / 35
Seller’s Problem �� � ( q, η , t ) E max t i ( x ) − C ( q ( x )) i ∈ N subject to (ex post) incentive compatibility: for all i ∈ N , x ∈ X u i ( x ) ∈ arg max u i (ˆ x i , x − i ) ˆ x i ∈ X i and individual rationality: for all i ∈ N , x ∈ X u i ( x ) ≥ 0 Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 9 / 35
Seller’s Problem Proposition Mechanism ( q, η , t ) is incentive compatible if and only if, for all i ∈ N , x ∈ X , q ( x ) η i ( x ) non-decreasing in x i and � t i ( x ) = v i ( x ) q ( x ) η i ( x ) − ∆ i v i ( s i , x − i ) q ( s i , x − i ) η i ( s i , x − i ) s i < x i ◦ ∆ i v i ( s i , x − i ) = v i ( s + i , x − i ) − v i ( s i , x − i ) and s + i is one type higher than s i Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 10 / 35
Seller’s Problem Seller’s problem can be written as � �� � � max q ( x ) MR i ( x ) η i ( x ) − C q ( x ) E ( q, η ): X → R ≥ 0 × [0 , 1] n i ∈ N q ( x ) η i ( x ) non-decreasing for all i where MR i ( x ) = v i ( x ) − ∆ i v i ( x ) 1 − F i ( x i ) f i ( x i ) Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 11 / 35
Seller’s Problem: Public Access and Quadratic Costs Seller’s problem can be written as � 2 q ( x ) 2 � � MR i ( x ) − 1 max E q ( x ) q : X → R ≥ 0 i ∈ N q ( x ) non-decreasing for all i where MR i ( x ) = v i ( x ) − ∆ i v i ( x ) 1 − F i ( x i ) f i ( x i ) Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 12 / 35
Seller’s Problem: Public Access and Quadratic Costs Seller’s problem can be written as � 2 q ( x ) 2 � � MR i ( x ) − 1 max q ( x ) E q : X → R ≥ 0 i ∈ N ∆ q ( x ) ≥ 0 for all i where MR i ( x ) = v i ( x ) − ∆ i v i ( x ) 1 − F i ( x i ) f i ( x i ) Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 12 / 35
Seller’s Problem: Public Access Saddle-point problem � � 2 q ( x ) 2 � � � � MR i ( x ) − 1 min max q ( x ) + λ i ( x )∆ i q ( x ) E λ : X → R ≥ 0 q : X → R ≥ 0 i ∈ N i ∈ N x ∈ X Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 13 / 35
Seller’s Problem: Public Access Saddle-point problem � � 2 q ( x ) 2 � � � � MR i ( x ) − 1 min max E q ( x ) − ∆ i λ i ( x ) q ( x ) λ : X → R ≥ 0 q : X → R ≥ 0 i ∈ N i ∈ N x ∈ X x i , x − i ) = 0 and ∆ i v i ( s i , x − i ) = v i ( s i , x − i ) − v i ( s − where λ i (¯ i , x − i ) and s − i is one type higher than s i . Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 13 / 35
Seller’s Problem: Public Access Saddle-point problem � 2 q ( x ) 2 � � � � MR i ( x ) − ∆ i λ i ( x ) − 1 min max q ( x ) E f i ( x ) q : X → R ≥ 0 λ : X → R ≥ 0 i ∈ N Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 13 / 35
Seller’s Problem: Public Access Saddle-point problem � 2 q ( x ) 2 � � � � MR i ( x ) − ∆ i λ i ( x ) − 1 min max E q ( x ) f i ( x ) λ : X → R ≥ 0 q : X → R ≥ 0 � �� � i ∈ N this “majorizes” MR i in i direction Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 13 / 35
Univariate Majorization For g : X → R , h : X → R , g majorizes h in coordinate i , denoted g ≻ i h , if Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 14 / 35
Univariate Majorization For g : X → R , h : X → R , g majorizes h in coordinate i , denoted g ≻ i h , if (i) for any x i ∈ X we have E [ g ( s, x − i ) | s ≤ x i ] ≤ E [ h ( s, x − i ) | s ≤ x i ] Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 14 / 35
Univariate Majorization For g : X → R , h : X → R , g majorizes h in coordinate i , denoted g ≻ i h , if (i) for any x i ∈ X we have E [ g ( s, x − i ) | s ≤ x i ] ≤ E [ h ( s, x − i ) | s ≤ x i ] and (ii) E [ g ( x )] = E [ h ( x )] Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 14 / 35
Univariate Majorization g ( x ) h ( x ) x x 0 1 0 1 Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 15 / 35
Univariate Majorization g ( x ) h ( x ) x x 0 1 0 1 E ( h ( y ) | y ≤ x ) E ( g ( y ) | y ≤ x ) x 0 1 Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 15 / 35
Univariate Majorization (i) MR i ( x ) − ∆ i λ i ( x ) has lower lower-sums than MR i ( x ) : f i ( x i ) Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 16 / 35
Univariate Majorization (i) MR i ( x ) − ∆ i λ i ( x ) has lower lower-sums than MR i ( x ) : for all x i ∈ X i f i ( x i ) � � � MR i ( s, x − i ) − ∆ i λ i ( x ) � s ≤ x i E f i ( s i ) Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 16 / 35
Univariate Majorization (i) MR i ( x ) − ∆ i λ i ( x ) has lower lower-sums than MR i ( x ) : for all x i ∈ X i f i ( x i ) � � � MR i ( s, x − i ) − ∆ i λ i ( x ) � s ≤ x i E f i ( s i ) � � − � = E MR ( s, x − i ) | s ≤ x i s ≤ x i ∆ i λ i ( s, x − i ) Nick Bedard (WLU) Optimal Mechanisms for Mass-Produced Goods 16 / 35
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