The Interval Structure of Optimal Disclosure Yingni Guo, Eran Shmaya - PowerPoint PPT Presentation

The Interval Structure of Optimal Disclosure Yingni Guo, Eran Shmaya MSU Feb 02 2018

An example An online platform promotes a product to a customer. Customer’s payoff from buying depends on an unknown s uniform on [0 , 1]. Customer’s payoff from buying is u ( s ) = s − 3 / 4. Platform’s payoff is 1. Not buying gives both a payoff of 0. density 1 s 3 0 1 4 1 / 33

An example An online platform promotes a product to a customer. Customer’s payoff from buying depends on an unknown s uniform on [0 , 1]. Customer’s payoff from buying is u ( s ) = s − 3 / 4. Platform’s payoff is 1. Not buying gives both a payoff of 0. density 1 s 1 3 0 1 2 4 1 / 33

An example Customer reads some product reviews/report and acquires private information about s . We refer to the private information as Customer’s type. Given s , Customer’s type is H with prob. s and L with prob. 1 − s . density 1 L H s 0 1 2 / 33

An example H type’s belief (density) is f ( s | H ) = 2 s . L type’s is f ( s | L ) = 2(1 − s ). f ( s | H ) f ( s | L ) 2 2 s s 0 1 0 1 3 / 33

An example H type’s belief (density) is f ( s | H ) = 2 s . L type’s is f ( s | L ) = 2(1 − s ). f ( s | H ) f ( s | L ) 2 2 s s 0 . 42 1 0 1 3 / 33

An example H type’s belief (density) is f ( s | H ) = 2 s . L type’s is f ( s | L ) = 2(1 − s ). f ( s | H ) f ( s | L ) 2 2 s s 0 . 42 1 0 . 62 1 3 / 33

An example: optimal disclosure Only H type buys π H π L 3 π L s 0 1 4 Both types buy 4 / 33

An example: optimal disclosure Only H type buys π H π L 3 π L s 0 1 4 Both types buy Nested intervals U-shaped cutoff mechanism 4 / 33

An example: optimal disclosure Only H type buys π H π L 3 π L s 0 1 4 Both types buy type 1 Nested intervals U-shaped cutoff mechanism z ( s ) 0 s 0 1 4 / 33

An example: optimal disclosure Only H type buys π H π L 3 π L s 0 1 4 Both types buy type 1 Nested intervals U-shaped cutoff mechanism z ( s ) 0 s 0 1 A lobbyist sways a legislator’s position on an issue. A media outlet promotes a candidate or an agenda. 4 / 33

Related literature Information design (mostly single receiver): Rayo and Segal (2010), Kamenica and Gentzkow (2011) Aumann and Maschler (1995), Ely (2017) Kolotilin (2016), Kolotilin, Li, Mylovanov, and Zapechelnyuk (2016) Gentzkow and Kamenica (2016), Dworczak and Martini (2016) Information design with multiple receivers: Lehrer, Rosenberg and Shmaya (2010, 2013), Bergemann and Morris (2016a, 2016b), Mathevet, Perego and Taneva (2016), Taneva (2016) Schnakenberg (2015), Alonso and Cˆ amara (2016), Chan et al. (2016), Guo and Bardhi (2016), Arieli and Babichenko (2016) Cheap talk with privately informed receiver: Seidmann (1990), Watson (1996), Olszewski (2004), Chen (2009), Lai (2014) 5 / 33

Roadmap 1 Environment and main results 2 A simple algorithm 3 Sketch of proof

Environment One Sender and one Receiver. s ∈ S ⊂ R : set of states. Receiver’s utility from accepting is u : S → R (0 if Receiver rejects). u is monotone increasing. t ∈ T ⊂ R : set of types. Lowest type is t . f : distribution over S × T . Assumption: f ( s , t ) satisfies increasing monotone likelihood ratio, i.e., f ( s , t ) f ( s , t ′ ) (weakly) increases in s for every t ′ < t . 6 / 33

A (disclosure) mechanism is a triple: ( X , κ, r ) recommendation set of Markov kernel function signals from S to X r : X × T → { 1 , 0 } When the state is s , the mechanism randomizes a signal x according to κ ( s , · ); recommends that type t accept if and only if r ( x , t ) = 1. 7 / 33

– Fully reveal the state: X = S and κ ( s , · ) = δ s . – Reveal whether s is above or below π : X = { above , below } ; for s � π , κ ( s , · ) = δ above ; for s < π , κ ( s , · ) = δ below . – For B ⊂ S , reveal whether the state is in B or not. – Randomize: X = { above , below , null } ; for s � π , κ ( s , · ) = 1 / 2 δ above + 1 / 2 δ null ; for s < π , κ ( s , · ) = 1 / 2 δ below + 1 / 2 δ null . 8 / 33

A strategy for type t is σ : X → { 0 , 1 } . 9 / 33

A strategy for type t is σ : X → { 0 , 1 } . σ ∗ t = r ( · , t ) is the strategy that follows the recommendation for type t . 9 / 33

A strategy for type t is σ : X → { 0 , 1 } . σ ∗ t = r ( · , t ) is the strategy that follows the recommendation for type t . A mechanism is publicly incentive-compatible if for every t �� � � σ ∗ t ∈ arg max f ( s , t ) u ( s ) σ ( x ) κ ( s , d x ) d s . 9 / 33

A strategy for type t is σ : X → { 0 , 1 } . σ ∗ t = r ( · , t ) is the strategy that follows the recommendation for type t . A mechanism is publicly incentive-compatible if for every t �� � � σ ∗ t ∈ arg max f ( s , t ) u ( s ) σ ( x ) κ ( s , d x ) d s . If type t obeys the recommendation, his acceptance probability at s , is � ρ ( s , t ) = r ( x , t ) κ ( s , d x ) . 9 / 33

Sender’s problem is �� � �� Maximize f ( s , t ) r ( x , t ) κ ( s , d x ) d s d t � �� � = ρ ( s , t ) among all publicly IC mechanisms. 10 / 33

Structural theorem A mechanism is a cutoff mechanism if X = T ∪ {∞} , r ( x , t ) = 1 ↔ t � x . 11 / 33

Structural theorem A mechanism is a cutoff mechanism if X = T ∪ {∞} , r ( x , t ) = 1 ↔ t � x . Every publicly IC mechanism is essentially a cutoff one. 11 / 33

Structural theorem A mechanism is a cutoff mechanism if X = T ∪ {∞} , r ( x , t ) = 1 ↔ t � x . Every publicly IC mechanism is essentially a cutoff one. T ∪ ∞ T ∪ ∞ ∞ ∞ H H L L s s 0 1 0 1 11 / 33

Structural theorem A mechanism recommends t to accept on an interval if ρ ( s , t ) 1 s π π 12 / 33

Structural theorem Theorem 1: The optimal publicly IC mechanism is a cutoff mechanism that recommends that each type accept on an interval. 13 / 33

Private incentive compatibility Receiver doesn’t observe x . He reports t ′ and observes r ( x , t ′ ). 14 / 33

Private incentive compatibility Receiver doesn’t observe x . He reports t ′ and observes r ( x , t ′ ). σ ( r ( x , t ′ )) for some type t ′ ∈ T and some ¯ σ = ¯ σ : { 0 , 1 } → { 0 , 1 } . 14 / 33

Private incentive compatibility Receiver doesn’t observe x . He reports t ′ and observes r ( x , t ′ ). A mechanism is privately incentive-compatible if for every t � �� � σ ∗ t ∈ arg max f ( s , t ) u ( s ) σ ( x ) κ ( s , d x ) d s , σ ( r ( x , t ′ )) for some type t ′ ∈ T and some ¯ over σ = ¯ σ : { 0 , 1 } → { 0 , 1 } . 14 / 33

Privately Publicly IC IC ⋆ 15 / 33

Equivalence theorem Privately Publicly IC IC ⋆ Theorem 2: No privately IC mechanism gives Sender a higher payoff than the optimal publicly IC mechanism. 15 / 33

Roadmap 1 Environment and main results 2 A simple algorithm 3 Sketch of proof

Binary-type case: a simple algorithm T = { H , L } , S = [0 , 1]. 16 / 33

Binary-type case: a simple algorithm T = { H , L } , S = [0 , 1]. u is strictly increasing and u ( ζ ) = 0 for some ζ ∈ (0 , 1). 16 / 33

Binary-type case: a simple algorithm T = { H , L } , S = [0 , 1]. u is strictly increasing and u ( ζ ) = 0 for some ζ ∈ (0 , 1). π H π L ζ π L π H s 0 1 16 / 33

Binary-type case: a simple algorithm T = { H , L } , S = [0 , 1]. u is strictly increasing and u ( ζ ) = 0 for some ζ ∈ (0 , 1). π H π L ζ π L π H s 0 1 Sender has one IC constraint for each type: � π L � π H Maximize f ( s , L ) d s + f ( s , H ) d s π H ,π L ,π L ,π H π L π H � π L subject to f ( s , L ) u ( s ) d s � 0 , π L � π L � π H f ( s , H ) u ( s ) d s + f ( s , H ) u ( s ) d s � 0 . π H π L 16 / 33

Binary-type case: pooling vs separating Proposition: Pooling is optimal if and only if f (1 , L ) − f ( π ∗ L , L ) < 1 − u ( π ∗ f (1 , H ) L , H ) L ) u (1) , f ( π ∗ � 1 where π ∗ L is such that L f ( s , L ) u ( s ) d s = 0. In this case the mechanism π ∗ recommends to both types to accept on [ π ∗ L , 1]. 17 / 33

Binary-type case: pooling vs separating Proposition: Pooling is optimal if and only if f (1 , L ) − f ( π ∗ L , L ) < 1 − u ( π ∗ f (1 , H ) L , H ) L ) u (1) , f ( π ∗ � 1 where π ∗ L is such that L f ( s , L ) u ( s ) d s = 0. In this case the mechanism π ∗ recommends to both types to accept on [ π ∗ L , 1]. π ∗ ζ s 0 1 L 17 / 33

Binary-type case: a comparison with cavU approach Revisit the Platform-Customer example: u ( s ) = s − 3 f ( s , H ) = s , f ( s , L ) = 1 − s , 4 . The problem amounts to an infinite-dimensional LP problem: � 1 Maximize ( g L ( s ) + sg H ( s )) d s g L ( s ) ≥ 0 , g H ( s ) ≥ 0 0 subject to g L ( s ) + g H ( s ) � 1 , ∀ s , � 1 � � s − 3 (1 − s ) g L ( s ) d s � 0 , 4 0 � 1 � � s − 3 s g H ( s ) d s � 0 . 4 0 18 / 33

Continuous-type example: a screening perspective T = [0 , 1], S = [ − 1 , 1]. u ( s ) = − η < 0 for every s < 0 and u ( s ) � 0 for s � 0. f : T → R + such that f ( s , t ) = f ( t ) for every s < 0. 19 / 33