Selling Experiments Dirk Bergemann Alessandro Bonatti Alex Smolin

Selling Experiments
Dirk Bergemann1
Alessandro Bonatti2
1 Yale
University
2 MIT
Sloan
BFI Chicago
July 10th, 2015
Alex Smolin1
Introduction
A decision maker under uncertainty:
has partial (private) information;
seeks to acquire additional information (“data buyer”).
Data seller can augment data buyer’s information.
How much information to provide at what price?
How to provide di¤erent information to di¤erent data buyers?
Interpretation: selling access to a database.
Leading Example: Selling Information about Consumers
…rms can tailor online advertising levels to individual users
targeting requires information about characteristis of
individual users.
data seller, data broker has information on individual users’
attributes.
attributes di¤er in their “sign” and explanatory power.
data broker may o¤er to reveal certain attributes.
…rms may have “…rst-party” information on users
) heterogeneous valuation for a given attribute.
Setting
data seller o¤ers “information product” = experiment
(Blackwell)
reveals information about the decision maker’s payo¤-relevant
state
value of experiment depends on decision maker’s private
information.
decision maker’s beliefs are his type
data seller only has (common) prior over types
Analysis
optimal versioning of an information product
only information product itself is contractible
by contrast, action of decision maker or realized state are not
contractible
data seller has database already installed
seller has zero marginal cost of data provision.
Related Literature
Selling Information
Admati and P‡eiderer (1986, 1990), Es½o and Szentes (2007),
“Selling Cookies” AEJ Micro (2015).
Bundling Information and Products
Johnson and Myatt (2006), Bergemann and Pesendorfer (2007),
“Targeting in Advertising Markets” RAND (2011).
Persuasion
Rayo and Segal (2010), Kamenica and Gentzkow (2011),
Lin (2013), Kolotilin et al. (2014)
Screening
Mussa and Rosen (1978), Myerson (1981), Jullien (2000).
Results
A menu of experiments is o¤ered.
The menu only contains “simple” experiments.
Menu itself is much coarser than possible types.
Systematic distortions in information provided.
Screening facilitated by “directional information.”
Linearity (in probabilities) limits the use of versioning.
Model
Single decision-maker (buyer of information).
Finite states
!2 :
Finite actions
a 2 A:
For most of the talk j j = jAj = 2.
Richer speci…cation of buyer’s private information.
Private Information
Common prior probability over states
2
.
Decision maker privately observes an informative signal.
Decision maker forms an initial belief
Initial beliefs are private information.
2
given the signal.
Alternatives
Distribution of initial beliefs F ( ) (from seller’s point of view).
Database and Experiments
An experiment (information structure) I consists of
I = fS; g
:
! 4S:
Seller o¤ers a menu of experiments M = fI; tg, with
I = fIg
t : I ! R+ :
Database query or “machine” interpretation:
contracting before the state ! is realized;
costless provision of information (data is already stored);
initial and incremental signals are independent conditionally
on the state
Initial and Incremental Information
interim probability
i
= Pr (! = ! i )
likelihood function under experiment I:
ij
= Pr (sj j ! i )
perfect information:
ij
=
1; if i = j
0; if i 6= j
extremal information:
ij
= 1 for some i = j
Buyer’s Payo¤
ex-post utility of decision maker
u (a; !)
without loss of generality: matching action and state:
normalization to f0; 1g :
u (a; !) = 1[a=!] :
Value of Experiments
Buyer’s payo¤ under partial information
u ( ) , max E [u (a; !)] :
a2A
Value of experiment (net value of augmented information)
V (I; ) , EI; [max Es; [u (a; !)]]
a2A
u( ):
Value of Experiments
with symmetry and matching the state
value of experiment I for buyer :
X
max f i ij g
V (I; ) =
j
i
max f i g .
i
Geometry of Value of Experiments
three states ! 1 ; ! 2 ; ! 3
interim beliefs = ( 1 ; 2 )
perfect information experiment
Geometry of Value of Experiments II
three states ! 1 ; ! 2 ; ! 3
interim beliefs = ( 1 ; 2 )
imperfect but extremal experiment
!1
!2
!3
s1
s2
s3
2=3 1=6 1=6
0
1
0
1
0
0
Seller’s Problem
Seller can o¤er a menu of experiments.
Items intended to match initial and incremental information.
Direct mechanism by the Revelation Principle
M = fI ( ) ; t ( )g:
Seller’s objective function, IC and IR constraints:
Z
max
t ( ) dF ( ) ;
fI( );t( )g
s.t.
V (I ( ) ; )
t( )
V I
V (I ( ) ; )
t( )
0 8 :
0
;
t
0
8 ; 0;
Optimality of Coarse Signals
Continuum of information structures I( ).
Each one a potentially complicated map: states ! signals.
Optimally coarse signals (and, later, optimally coarse menu).
Merge signals in I( ) leading to the same action for type .
V (I( ); ) stays constant but V (I( ); 0 ) decreases 8
0
6= .
Lemma (Coarse Signals)
In an optimal menu, every experiment consists of jAj signals.
Binary Model
binary action, binary state
! = !L
! = !H
a = aL a = aH
1
0
0
1
wlog restrict attention to experiments
sL
I=
with the convention
!L
!H
+
sH
1
1
1.
Value of Experiment with Binary Model
Let
= Pr (! = ! H ) :
Value of experiment ( ; )
V ( ; ; )=[
+
(1
)
maxf ; 1
g]+ :
Informative Signals
Perfectly informative experiment:
I=
!L
!H
=
= 1:
sL sH
1
0
0
1
Directionally informative experiment, e.g.,
sL
0
I =
!L
!H
= 1;
< 1:
sH
1
0
1
Directional information valuable for some types, but not for others.
Useful to decision maker who believes that ! H is very likely,
but not useful to agent who believes that ! L is very likely,
because he would still choose aL under both signals.
Value of a Perfectly Informative Experiment
Value of experiment ( ; ) = (1; 1) for type .
( ; ) = (1; 1)
Highest-value buyer is
= 1=2.
Value of a Directionally Informative Experiment
Value of experiment ( ; ) = (1; 2=2) for type .
( ; ) = (1; 1=2)
Distance j
experiment
1=2j is not su¢ cient to describe the value of
Preferences over Experiments
Value of experiment ( ; ) for type
V ( ; ; )=(
) +
maxf ; 1
g:
= baseline informativeness (from payo¤ normalization).
= relative informativeness.
Two “goods” that cannot be produced independently.
Feasible Set of Experiments
Indi¤erence Curves for Given Type
value of experiment is
V ( ; ; )=(
higher types
) +
max f ; 1
have stronger preference for relative
g
Value of Baseline Information
incremental change in the baseline information
while keeping the relative informativeness
constant
V( + ;
+ ; )
V ( ; ; )= ;
8
uniform increase in value of experiment for all types
Set of Optimal Experiments
maximal baseline informativeness for any given relative
informativeness
we can reduce the choice of experiment to a one-dimensional
problem:
Value of an Experiment
Value of experiment ( ; ) for type
V ( ; ; ) = [(
Holding
) +
g]+ :
maxf ; 1
constant, can increase value uniformly 8 .
Lemma (Partially Revealing Signals)
In an optimal menu, every experiment has either
= 1 or
At least one signal perfectly reveals one state.
One-dimensional allocation (relative informativeness)
q( ) , ( )
( ) 2 [ 1; 1] :
= 1.
Buyer’s Utility
Value of experiment q 2 [ 1; 1] for type
V (q; ) = [ q
2 [0; 1]:
maxfq; 0g + minf ; 1
g]+
Experiments q 2 f 1; 1g entirely uninformative.
Single crossing in ( ; q): as in classic nonlinear pricing.
Experiment q = 0 is the most valuable for all types .
Type
= 1=2 has the highest value for any experiment q.
Types disagree on ranking of imperfect experiments.
Optimal Experiments
directionally informative
if q > 0 : useful for agent who thinks ! H is very likely
sL
I=
!L
!H
1
q
0
sH
q
1
if q < 0 : useful for agent who thinks ! L is very likely
I0 =
!L
!H
sL
sH
1
0
q 1+q
TOptimal Experiments with Two Types
Two types:
2 f0:2; 0:7g with equal probability.
Sell full information at reservation value to type
= 0:7.
Net Value of Experiment q ( )
Two-Type Example
IC does not rule out serving type
= 0:2.
Net Value of Experiment q ( )
Two-Type Example
Optimal allocation: q (0:2) =
1=5, and q (0:7) = 0.
Posterior beliefs p (0:2) 2 f1=21; 1g, and p (0:7) 2 f0; 1g.
Net Value of Experiment q ( )
Two-Types: Summary
The optimal menu contains either one or two experiments.
One experiment: one or both types may purchase.
Two experiments:
requires asymmetry in types on opposite sides of 1=2;
no distortion at the top ( closer to 1=2);
no rent at the bottom;
corner solution – no rent at the top.
Geometry of the Problem
Highest type is interior ( = 1=2).
Optimal menu on each side of 1=2: single experiment q = 0.
Easy cases: truncated support or symmetric distribution.
More generally, worry about IC “on both sides.”
Price discrimination within and across two groups.
Continuum of Types
Recall the value of experiment q 2 [ 1; 1] for type
V (q; ) = [ q
maxfq; 0g + minf ; 1
2 [0; 1]:
g]+ .
Single-crossing suggests q increasing in .
Types
= 0 and
Consider type
= 1 receive zero rents.
= 1=2, derive additional condition.
Incentive Compatibility
Rent of type
= 1=2
U (1=2) = U (0) +
Z
0
1=2
V (q; )d = U (1)
Z
1
1=2
V (q; )d :
Incentive Compatibility
Rent of type
= 1=2
U (1=2) =
Z
1=2
(q ( ) + 1)d =
0
Z
1
(q ( )
1=2
Lemma (Implementable Allocations)
The allocation q ( ) is implementable if and only if
q ( ) 2 [ 1; 1] is non-decreasing,
Z 1
and
q ( ) d = 0:
0
Note: a di¤erent kind of constraint.
1)d :
Seller’s Problem
max
q( )
Z
0
1
1
F( )
f( )
q( )
max fq ( ) ; 0g f ( ) d ;
s.t. q ( ) 2 [ 1; 1] non-decreasing,
Z 1
q ( ) d = 0:
0
Piecewise linear (concave) problem with integral constraint.
Absent the integral constraint, corner solutions:
q 2 f 1; 0; 1g, i.e., all-or-nothing information, ‡at price.
E.g., truncated support or symmetric distribution.
Optimal Menu
Jump to Properties
Proposition (Optimal Menu)
An optimal menu consists of at most two experiments.
The …rst experiment is fully informative.
The second experiment (contains a signal that) perfectly
reveals one state.
Seller’s Problem
max
q( )
Z
0
1
1
F( )
f( )
q( )
max fq ( ) ; 0g f ( ) d ;
s.t. q ( ) 2 [ 1; 1] non-decreasing,
Z 1
q ( ) d = 0:
0
Seller’s Problem
max
q( )
Z
1
[( f ( ) + F ( )) q ( )
0
max fq ( ) ; 0g f ( )] d ;
s.t. q ( ) 2 [ 1; 1] non-decreasing,
Z 1
q ( ) d = 0:
0
Consider “virtual values” for each experiment q separately:
(
f( )+F ( )
for q < 0;
( ; q) ,
(
1)f ( ) + F ( ) for q > 0:
= marginal value of going from q( ) =
Problem is not separable: virtual value
1 to 0 to 1:
depends on q:
General Case
Let denote the multiplier on the integral constraint
(shadow cost of providing higher “quantity”).
Let
( ; q) denote the ironed virtual value for experiment q.
Proposition (Optimal Allocation Rule)
Allocation q ( ) is optimal if and only if there exists
0 s.t.
Z q
q ( ) 2 arg max
( ; x)
dx for all ;
q2[ 1;1]
0
has the “pooling property,” and satis…es the integral constraint.
Myerson (1981), Toikka (2011), Luenberger (1969).
Example 1: Uniform Distribution
Virtual Values:
( ; 1) in blue;
( ; 1) in red.
Example 1: Uniform Distribution
Virtual Values:
( ; 1) in blue;
( ; 1) in red.
Example 1: Uniform Distribution
Optimal Menu: Single Experiment
Optimality of Flat Pricing
Proposition (Flat Pricing)
The optimal menu contains only the fully informative experiment
when any of the following conditions hold:
1
the density f ( ) = 0 for all
> 1=2 or
2
the density f ( ) is symmetric around
3
F ( ) + f ( ) and F ( ) + (
< 1=2;
= 1=2.
1)f ( ) are strictly increasing.
A second experiment is o¤ered only if ironing is required.
Non-monotone Density
Probability Density Function: informed types are frequent
Example 2: Combination of Beta Distributions
Probability Density Function
Example 2: Beta Distributions
Virtual Values:
( ; 1) in blue;
( ; 1) in red.
Example 2: Beta Distributions
Ironed Virtual Values
Example 2: Beta Distributions
Ironed Virtual Values
Example 2: Beta Distributions
One More Example
Optimal Menu: Two Experiments (q 2 f :21; 0g)
Properties of the Optimal Menu
Optimal mechanism involves
2 bunching intervals.
Ideally, would sell q = 0 at two di¤erent prices (for
7 1=2).
Symmetric distribution or truncated support ! ‡at pricing.
Second-best menu may contain q = 0 only. . .
. . . or distort the “least pro…table side.”
No further versioning is optimal.
Least informed types 6= most valuable to the seller.
Type
= 1=2 need not get e¢ cient q = 0:
Conclusions: Selling Information
Selling incremental information to privately informed buyers.
Costless acquisition & transmission, free degrading.
“Uninterested seller” – packaging problem.
Bayesian problem for buyers:
Linear in probabilities: limited use of versioning.
Screening across groups through directional information.
More general approach to nonlinear pricing “within” and “across.”
Implications for Observables
How to damage an information good
Should not observe arbitrarily damaged goods.
Directional information: only type-I or II errors.
Should not observe multiple distortions of the same kind.
Directional distortions disclosure of speci…c attributes
(correlated with high- or low- value consumers).
Enriching the Model
More than two states and actions.
Cost of providing information.
Heterogeneous tastes for actions.
Other sources of buyer heterogeneity.
Investment problem with continuous states.
Comparison with richer contracts spaces.
Role of orthogonal vs. correlated information.
Alternatives
Investment
Many States
Many states: ! 1 ; :::; ! N 2
Many actions: a1 ; :::; aN 2 A
Experiments
ij
with
j ij
s1
= 1:
s2
!1
!2
..
.
11
12
13
21
22
23
!N
N1
N2
N3
Perfectly informative experiment:
Extreme points:
s3
ii
ii
= 1 for some i.
= 1 for every i.
Many States
Structure of incentive problem remains piecewise linear.
Proposition (Optimal Experiments)
1
Every optimal experiment contains at most N signals.
2
Every optimal experiment is an extreme point.
3
Any optimal menu contains the perfectly informative
experiment.
Three States, Three Actions, Continuous Types 1
uniform distribution towards the vertices
Three States, Three Actions, Continuous Types 1
Flat pricing under the uniform distribution
Three States, Three Actions, Continuous Types 1
beta distribution on the ! 1 ray
richer menu if distribution di¤ers across rays
Partial and Complete Information
Uninformed types receive complete information:
I0
!1
!2
!3
s1
1
0
0
s2
0
1
0
s3
0
0
1
Partially informed types t1 receive partial information
I1
!1
!2
!3
s1
1
:35
:35
s2
0
:65
0
s3
0
0
:65
Remain partially confused after signal s1 .
Three States, Three Actions, Continuous Types 1
Prior and Posteriors
Three States, Three Actions, Continuous Types 2
Types are uniformly distributed on the orthogonal lines.
Richness of the allocation (vs. types).
Partial information is provided.
Three States, Three Actions, Continuous Types 2
Partial and Complete Information
Uninformed types receive complete information:
!
I0 = 1
!2
!3
s1
1
0
0
s2
0
1
0
s3
0
0
1
Partially informed types ti , here t1 receive partial information
!
I1 = 1
!2
!3
s1
1=2
0
0
s2
1=4
1
0
s3
1=4
0
1
Remain partially confused after signals s2 ; s3 :
Sources of Buyer Heterogeneity
Alternative sources of heterogeneity:
Heterogeneous tastes.
Heterogeneous “stakes of the game.”
Heterogeneous “relevance.”
Di¤erential signal-processing abilities.
Buyer’s private information = beliefs.
Belief heterogeneity: role of mean vs. variance?
Back
The (not so) Natural Model
Back
Buyer faces prediction problem with linear-quadratic payo¤s.
Normally distributed state and signals.
Buyers di¤er in interim beliefs (mean and variance).
) Willingness to pay proportional to incremental precision,
independent of interim mean belief.
No role for “directional” information, or for versioning.
Investment / Product Quality Model
Back
Continuous state ! 2 R; binary action a 2 f0; 1g.
Normalize u(a; !) = a !.
Buyer’s interim pdf is g (! j ), with E [! j ] = :
Assume g (! j ) ordered by rotation around ! = 0.
Restrict attention to binary partitions ! 2 f[ 1; q] ; [q; 1]g.
Resulting V (q; ) has single crossing in (q; ).
Investment / Product Quality Model
A menu is a collection of experiments and prices fq( ); t( )g.
Highest-value experiment for any
is q = 0.
“Robust” features of menu with binary state:
Di¤erent ranking of imperfectly informative experiments.
Integral constraint.
Coarse (2-item) optimal menu.
Compared to contractible-action model: information distortions
(cf. Es½o and Szentes (2007), Li and Shi (2015)).
Example 3: Power Distribution
Probability Density Function
Example 3: Power Distribution
Ironed Virtual Values
Example 3: Power Distribution
Back
Optimal Menu: Two Experiments (q 2 f0; 1=5g)