Single-Crossing Di ff erences on Distributions Navin Kartik SangMok Lee Daniel Rappoport September 2017 SCD on Distributions Kartik, Lee, Rappoport
Introduction (1) Single Crossing Di ff erences is central to MCS 8 a, a 0 2 A : v ( a, θ ) � v ( a 0 , θ ) is single crossing in θ in type 8 A 0 ✓ A ( ) choices are monotonic | {z } strong set order Agent may be faced with lotteries over A • directly or indirectly (e.g., in a game) • e.g., Crawford and Sobel ’82: what if S does not know R ’s prefs? For vNM agent, Single Crossing Expectational Di ff erences 8 P, Q 2 ∆ A : E P [ v ( a, θ )] � E Q [ v ( a, θ )] is SC in θ Not assured by SCD over A SCD on Distributions Kartik, Lee, Rappoport
Introduction (2) Our results: 1 Characterize v ( a, θ ) that have SCED A Takeaway SCED ( v ( a, θ ) ⇠ u ( a ) + f ( θ ) w ( a ) , with f monotonic ) | {z } often 2 Establish SCED ( ) MCS on ∆ A 3 Applications In achieving (1): Characterize sets of functions whose linear combinations are SC A characterization of MLRP (known, but apparently not well) SCD on Distributions Kartik, Lee, Rappoport
Literature More related (elaborate later) : Kushnir and Liu 2017 Quah and Strulovici ECMA 2012, Choi and Smith JET 2016 Karlin 1968 book Milgrom and Shannon ECMA 1994 Less related: Milgrom RAND 1981 Athey QJE 2002 SCD on Distributions Kartik, Lee, Rappoport
Main Results SCD on Distributions Kartik, Lee, Rappoport
Setting A is some space (outcomes/allocations) • talk as if A finite; avoiding technical details • ∆ A is set of all prob. measures ( Θ , ) is a partially-ordered space (types) • is reflexive, transitive, antisymmetric • contains upper and lower bounds for all pairs • some results are trivial when | Θ | 2 v : A ⇥ Θ ! R (payo ff fn) R Expected Utility: V ( P, θ ) ⌘ A v ( a, θ )d P Expectational Di ff erence: D P,Q ( θ ) ⌘ V ( P, θ ) � V ( Q, θ ) SCD on Distributions Kartik, Lee, Rappoport
Single Crossing Definition f : Θ → R is 1 single crossing from below if ( ∀ θ l < θ h ) f ( θ l ) ≥ ( > )0 = ⇒ f ( θ h ) ≥ ( > )0 . 2 single crossing from above if ( ∀ θ l < θ h ) f ( θ l ) ≤ ( < )0 = ⇒ f ( θ h ) ≤ ( < )0 . 3 single crossing if it is SC from below or from above. E.g., f ( · ) > 0 is SC from below and above. SCD on Distributions Kartik, Lee, Rappoport
SC Expectational Di ff erences Definition Let X be arbitrary. f : X ⇥ Θ ! R has SC Di ff erences (SCD) if 8 x, x 0 2 X : f ( x, θ ) � f ( x 0 , θ ) is single crossing in θ . Not quite the usual definition; X need not be ordered Definition v has SC Expectational Di ff erences (SCED) if V : ∆ A ⇥ Θ ! R has SCD. D P,Q ( θ ) is SC for all lotteries P, Q SCED is an ordinal property of prefs over ∆ A When | A | = 2 , equiv. to v having SCD SCD on Distributions Kartik, Lee, Rappoport
6 SCD ) SCED = a = 2 2 1.5 a = 1 E @ v H ., q LD H 1 ê 2 L@ a = 2 D + H 1 ê 2 L@ a = 0 D 1 0.5 a = 0 0 - 1 0 1 q SCD on Distributions Kartik, Lee, Rappoport
Main Result Theorem v has SCED if and only if v ( a, θ ) = g 1 ( a ) f 1 ( θ ) + g 2 ( a ) f 2 ( θ ) + c ( θ ) , (1) with f 1 , f 2 each SC and ratio ordered. If f 1 , f 2 > 0 , then RO ( ) f 1 /f 2 monotonic; and SC trivial Then interpret as: two prefs s.t. each θ ’s pref is a convex combination, with weight shifting monotonically in θ But f 1 , f 2 need not be positive (nor single-signed) D P,Q ( θ ) = α 1 f 1 ( θ ) + α 2 f 2 ( θ ) for some α 2 R 2 (1) = ) Is such D P,Q single crossing? SCD on Distributions Kartik, Lee, Rappoport
Ratio Ordering Definition Let f 1 , f 2 : Θ ! R each be SC. 1 f 1 ratio dominates f 2 if (i) ( 8 θ l θ h ) f 1 ( θ l ) f 2 ( θ h ) f 1 ( θ h ) f 2 ( θ l ) , (ii) omitted nuances. details 2 f 1 and f 2 are ratio ordered if f 1 ratio dominates f 2 or vice-versa. If both are (str. + ) densities, simply likelihood ratio ordering Defn does not assume either f i has constant sign • ( 8 f ) f and � f are ratio ordered SCD on Distributions Kartik, Lee, Rappoport
Geometric Interpretation � ) to f ( θ h ) f 1 RD f 2 = ) ( 8 θ l < θ h ) f ( θ l ) rotates clockwise ( 180 ( f ( θ 0 ) , 0) ⇥ ( f ( θ 00 ) , 0) = k f ( θ 0 ) kk f ( θ 00 ) k sin( r ) e 3 = � f 1 ( θ 0 ) f 2 ( θ 00 ) � f 1 ( θ 00 ) f 2 ( θ 0 ) � e 3 � ) Ratio ordering = ) f ( θ ) rotates monotonically ( 180 ( = modulo nuances point (ii) SCD on Distributions Kartik, Lee, Rappoport
Linear Combinations Lemma Lemma Let f 1 , f 2 : Θ ! R each be SC. α 1 f 1 ( θ ) + α 2 f 2 ( θ ) is SC 8 α 2 R 2 f 1 , f 2 are ratio ordered. ( ) A characterization of LR ordering (for str. + densities) Strict Coe ff s of opp signs are key f 1 and f 2 need not be SC in the same direction (e.g., f 1 = � f 2 ) SCD on Distributions Kartik, Lee, Rappoport
Linear Combinations Lemma Lemma Let f 1 , f 2 : Θ ! R each be SC. α 1 f 1 ( θ ) + α 2 f 2 ( θ ) is SC 8 α 2 R 2 f 1 , f 2 are ratio ordered. ( ) Intuition: ( ( = ) SCD on Distributions Kartik, Lee, Rappoport
Linear Combinations Lemma Lemma Let f 1 , f 2 : Θ ! R each be SC. α 1 f 1 ( θ ) + α 2 f 2 ( θ ) is SC 8 α 2 R 2 f 1 , f 2 are ratio ordered. ( ) Intuition: ( = ) ) SCD on Distributions Kartik, Lee, Rappoport
Linear Combinations of Multiple Functions Necess. direction of Thm requires aggregating many SC functions Proposition Consider { f i } n i =1 , where each f i : Θ ! R is SC. P i α i f ( x i , θ ) is SC 8 α 2 R n if and only if 9 i, j s.t. 1 Ratio Ordering: f i and f j are ratio ordered; 2 Spanning: ( 8 k ) f k ( · ) = λ k f i ( · ) + γ k f j ( · ) . intuition SCD on Distributions Kartik, Lee, Rappoport
Main Result: SCED Characterization Theorem v has SCED if and only if v ( a, θ ) = g 1 ( a ) f 1 ( θ ) + g 2 ( a ) f 2 ( θ ) + c ( θ ) , with f 1 , f 2 each SC and ratio ordered. Su ffi ciency follows from Linear Combinations Lemma: ⇥R R ⇤ ⇥R R ⇤ D P,Q ( θ ) = g 1 d P � g 1 d Q f 1 ( θ ) + g 2 d P � g 2 d Q f 2 ( θ ) SCD on Distributions Kartik, Lee, Rappoport
Main Result: SCED Characterization Theorem v has SCED if and only if v ( a, θ ) = g 1 ( a ) f 1 ( θ ) + g 2 ( a ) f 2 ( θ ) + c ( θ ) , with f 1 , f 2 each SC and ratio ordered. Idea underlying necessity: Consider A = { a 0 , . . . , a n } and v ( a 0 , · ) = 0 . SCED = ) ( 8 a ) v ( a, θ ) is SC ( * δ a and δ a 0 ) 8 λ 2 R n , P i λ i v ( a i , θ ) / P i ( p ( a i ) � q ( a i )) v ( a i , θ ) , where p, q are PMFs SCED = ) every such linear combination is SC Linear Combinations Prop = ) 9 i, j : ( 8 a ) v ( a, · ) = g 1 ( a ) v ( a i , · ) + g 2 ( a ) v ( a j , · ) , with RO (and SC) SCD on Distributions Kartik, Lee, Rappoport
Main Result: SCED Characterization Theorem v has SCED if and only if v ( a, θ ) = g 1 ( a ) f 1 ( θ ) + g 2 ( a ) f 2 ( θ ) + c ( θ ) , with f 1 , f 2 each SC and ratio ordered. While SCED is restrictive, it is satisfied in some familiar cases screening/mech design : v (( q, t ) , θ ) = g 1 ( q ) f ( θ ) � g 2 ( t ) , f monotonic • unless g 1 is constant, f ( · ) must be monotonic voting/communication : v ( a, θ ) = � ( a � θ ) 2 = � a 2 + 2 a θ � θ 2 • for v ( a, θ ) = � | a � θ | d with d > 0 , only d = 2 satisfies SCED signaling : v (( w, e ) , θ ) = w � e/ θ (usually e, θ > 0 ) in all these cases, one f i ( · ) = 1 SCD on Distributions Kartik, Lee, Rappoport
Main Result: SCED Characterization Theorem v has SCED if and only if v ( a, θ ) = g 1 ( a ) f 1 ( θ ) + g 2 ( a ) f 2 ( θ ) + c ( θ ) , with f 1 , f 2 each SC and ratio ordered. Theorem Assume some agreement: ( 9 P, Q ) ( 8 θ ) V ( P, θ ) > V ( Q, θ ) . v has SCED if and only if prefs have a representation ˜ v ( a, θ ) = g 1 ( a ) f 1 ( θ ) + g 2 ( a ) , with f 1 monotonic. SCD on Distributions Kartik, Lee, Rappoport
An MCS Characterization Let f : X ⇥ Θ ! R with ( X, ⌫ ) an ordered set and ( Θ , ) a directed set Assume X is minimal wrt f: ( 8 x 6 = x 0 )( 9 θ ) f ( x, θ ) 6 = f ( x 0 , θ ) Definition f has Monotone Comparative Statics on ( X, ⌫ ) if ( 8 S ✓ X, θ θ 0 ) arg max x 2 S f ( x, θ 0 ) ⌫ SSO arg max x 2 S f ( x, θ ) . Y ⌫ SSO Z if ( 8 y 2 Y, z 2 Z ) ( y _ z 2 Y, y ^ z 2 Z ) Cf. MS ’94: X need not be lattice; monotonicity only in θ but 8 S ✓ X (not only all sublattices) SCD on Distributions Kartik, Lee, Rappoport
An MCS Characterization Let f : X ⇥ Θ ! R with ( X, ⌫ ) an ordered set and ( Θ , ) a directed set Assume X is minimal wrt f: ( 8 x 6 = x 0 )( 9 θ ) f ( x, θ ) 6 = f ( x 0 , θ ) Definition f has Monotone Comparative Statics on ( X, ⌫ ) if ( 8 S ✓ X, θ θ 0 ) arg max x 2 S f ( x, θ 0 ) ⌫ SSO arg max x 2 S f ( x, θ ) . Define a reflexive relation ⌫ SCD on X : x � SCD x 0 if f ( x, θ ) � f ( x 0 , θ ) is SC from only below If f has SCD, ⌫ SCD is an order Proposition f has MCS on ( X, ⌫ ) ( ) f has SCD and ⌫ refines ⌫ SCD . SCD on Distributions Kartik, Lee, Rappoport
SCED and MCS Apply MCS result to our setting; recall D P,Q ( θ ) ⌘ V ( P, θ ) � V ( Q, θ ) Definition P � SCED Q if D P,Q ( · ) is SC from only below; P ⇠ SCED Q if D P,Q ( · ) = 0 . Let e ∆ A be the quotient space defined by ⇠ SCED Corollary V has MCS on ( e ∆ A, ⌫ ) ( ) v has SCED and ⌫ refines ⌫ SCED . A strict version of SCED yields a monotone selection result SSCED SCD on Distributions Kartik, Lee, Rappoport
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