Public Good Overprovision by a Manipulative
Provider
Gorkem Celik1 Dongsoo Shin2 Roland Strausz3
May 2020
Motivation Model Manipulation Results Welfare Appendix Motivation
What do we do in this paper?
A public good provision mechanism:
Provider (principal) produces a public good consumed by
users (agents) withprivate values.
Users reveal their private info. to Provider’s mechanism. Informationmanipulation opportunities for provider. Examine the e¤ects of manipulation on optimal mechanism. Depending on the parametrization, we observe a combination of
1 Bunching: di¤erent valuations are treated same way. 2 Overprovision: sum of marginal values < marginal cost. 3 Positive rentfor low-valuationusers.
Example: Aggregate litigation procedures
Multiple victims (users) suing the same defendant. Litigation activity = Public Good for multiple victims. Initiated and pursued by a class attorney (provider).
Victims are privately informed about their own claims. Attorney designs an incentive scheme, conditioning her litigation activity and fees on the aggregated information. Manipulation possibility: Attorney contacts the victims separately.
Motivation Model Manipulation Results Welfare Appendix Motivation
Other Examples
Other public goods overprovided at an annoying level Lobbying by an industry trade association.
Managerial services by headquarters of a …rm for its di¤erent divisions.
Related Literature on Manipulation
Concept of "manipulation" is similar to
"Informational Opportunism" in Dequiedt and Martimort (2015, AER)
"Non-credible Auctions" in Akbarpour and Li (2020, Ecm) Manipulation opportunities are more limited here - observability of public good level.
We also have a companion paper:
"Organizational Structures and Manipulable Aggregate Information"
Motivation Model Manipulation Results Welfare Appendix Model
Model
Provider producing public good q at convex costc(q).
Users 1 and 2 have per unit valuesθ1 andθ2. θ1 andθ2 independent random variables.
θk is high(θh)or low(θl) with prob. ϕand(1 ϕ).
∆θ =θh θl >0.
Provider designs asymmetricmechanism: Users send messages (their types).
Messages determine public goodq and paymentsp1 andp2. Users canopt outafter learningqandpk.
Payo¤s:
States of Collective Value
Depending on the user types, one of the three situations arises: High-value State (H): Both users have high valuations.
Public good level isqH. Each user paysphH.
Low-value State (L): Both users have low valuations. Public good level isqL. Each user paysplL.
Intermediate-value State (M): Users have di¤erent valuations. Public good level isqM. High-value user paysphM and
Low-value user paysplM.
A collection of theseq and p’s is anoutcome.
Motivation Model Manipulation Results Welfare Appendix Model
First-Best Levels of Public Good: Known Values
First Best
Choose q( )such that marginal cost equals sum of the marginal values (Samuelson Condition),
H:c0(qH) =2θh
M :c0(qM) =θh+θl
L:c0(qL) =2θl
Benchmark: Non-manipulative Provider
Provider maximizes her expected payo¤ s.t.
low-value user’s ex-post participation constraints
(PClL): θlqL plL 0 and (PClM):θlqM plM 0
and high-value user’sincentive compatibility constraint (ICh)
expected payo¤ to truthful report
| {z }
ϕ(θhqH phH)+(1 ϕ)(θhqM phM)
exp. payo¤ to low type report
| {z }
ϕ(θhqM plM)+(1 ϕ)(θhqL plL)
Low-value user’s ICl and high-value user’s participation
Motivation Model Manipulation Results Welfare Appendix Model
Benchmark: Non-manipulative Provider
Second Best Payments for Non-manipulative Provider
Proposition (1)
Low-value user’s payment determined by participation constraints:
plLn =θlqLn and pnlM =θlqnM
High-value user’s payment determined by incentive compatibility:
ϕpnhH+ (1 ϕ)phMn = ϕ(θhqHn ∆θqMn ) + (1 ϕ) (θhqMn ∆θqLn)
Benchmark: Non-manipulative Provider
Second-Best Output for Non-manipulative Provider
Proposition (1)
The provider distorts the public good levels downwards (except at the top) to reduce the information rent:
c0(qHn) = 2θh [Same as in the …rst best]
c0(qMn ) = θh+θl
ϕ 1 ϕ∆θ
c0(qLn) = 2θl 2
Motivation Model Manipulation Results Welfare Appendix Manipulation
Information Manipulation
Suppose both users reported low values.
So far, we assumed that the provider truthfully transmits these reports to the users.
Possible Manipulation: Low valueL ! Intermediate valueM
Tell each user that the other user reported a high value. Provider’s payo¤ changes
from 2θlqLn c(qLn) to 2θlqMn c(qnM)
Probability ϕ close enough to 1/2, this is a pro…table deviation.
Motivation Model Manipulation Results Welfare Appendix Manipulation
Provider’s Incentive Compatibility
Users can anticipate manipulation after seeing the contract. Provider should persuade them that she will not manipulate. How?
Provider’s Incentive Compatibilitycondition (PIC):
payo¤ when users report low values payo¤ after manipulation 2plL c(qL) 2plM c(qM)
Provider’s Incentive Compatibility
Provider maximizes her expected payo¤ subject to the users’
participation andincentive compatibility constraints, as well as the newProvider’s Incentive Compatibility:
PIC :2plL c(qL) 2plM c(qM)
We examine optimal manipulation-proof outcome. Let’s start with what remains the same as in second best:
No distortion at the top: for high-value state, e¢ cient public good production qH =qH.
Incentive compatibility determines the payments of the high-value userphH and phM.
Motivation Model Manipulation Results Welfare Appendix Results
Optimal Manipulation-Proof Outcome: 1) Bunching
Proposition (2)
Suppose ϕ>1/2. The provider poolsthe low and intermediate
value states by choosing the same public good levels:
c0(qMm) =c0(qLm) =2θl 2
ϕ2 1 ϕ2∆θ.
payments are set as in the second-best:
information rent for high-value user, no rent for low-value user.
PIC trivially satis…ed: No need to manipulate if the contracts do not separate low and intermediate valuations.
Motivation Model Manipulation Results Welfare Appendix Results
Optimal Manipulation-Proof Outcome
What about ϕ<1/2? Then qLn <qL <qMn .
Motivation Model Manipulation Results Welfare Appendix Results
Manipulation-Proof Outcome: 2) Higher Provision
Proposition (4)
Suppose ϕ<1/2.The provider increases both qL and qM above
their second-best levels, such that the PIC holds as equality. payments are set as in the second-best:
information rent for high-value user, zero rent for low-value user.
Public good production is larger than in the second-best for both low-value and intermediate-value states:
Overprovision of Public Good
Motivation Model Manipulation Results Welfare Appendix Results
3) Overprovision and Positive Rent to Low-value User
Provider never setsqMm larger than qH.
Proposition (4) is false if 2θlqLm c(qLm)<2θlqH c(qH).
If so, reduce payment plMm from the low-value user. Proposition (3)
Suppose ϕ<1/2and the above inequality holds. The provider
increases qM to qH (overprovision):
c0(qmL) = 2θl 2
ϕ(1 ϕ)
1 ϕ(1 ϕ)∆θ,
qmM = qHm =qH.
Motivation Model Manipulation Results Welfare Appendix Welfare
Welfare E¤ects: Winners and Losers of Manipulation
Providergets a lower payo¤: "manipulation proofness" as an additional constraint on top of "participation" and "incentive compatibility."
What about theusers?
High-value user’s rent is increasing in low-value user’sq. When ϕ>1/2, indeterminate: qL "butqM #.
When ϕ<1/2, high-value user is making more rent since
qL "andqM ".
When ϕ<1/2, even low-value usercan make a positive rent!
Conclusion
Public good provision mechanism.
Users have private values for the public good.
Provider canmanipulate the aggregated information. Optimal manipulation-proof mechanism for the Provider. Depending on the parametrization, we observe a combination of
1 Bunching: di¤erent valuations are treated same way. 2 Overprovision: sum of marginal values < marginal cost. 3 Positive rentfor low-valuationusers.
Motivation Model Manipulation Results Welfare Appendix Appendix
Low-Value User’s Incentive Compatibility Constraint
expected payo¤ to truth
| {z }
ϕ(θlqM plM)+(1 ϕ)(θlqL plL)
exp. payo¤ to high type report
| {z }
ϕmaxfθlqH phH,0g+(1 ϕ)maxfθlqM phM,0g
Why do we need max operators on RHS of (ICl)?
Low-value user can misreport his type as high, and then opt out afterwards.
Optimal Manipulation-Proof Outcome (Formally)
Provider maximizes her expected payo¤
ϕ2[2phH c(qH)] +2ϕ(1 ϕ) [phM +plM c(qM)]
+ (1 ϕ)2[2plL c(qL)]
by choosing an economic outcome
quantitiesqH,qM,qL and paymentsphH,phM,plM,plLs.t.
constraints of
incentive compatibility ICh andICl
participation PChH,PChM,PClM,PClL
manipulation-proofness PIC
Motivation Model Manipulation Results Welfare Appendix Appendix
Provider’s Maximization (Formally)
Provider maximizes her expected payo¤
ϕ2[2phH c(qH)] +2ϕ(1 ϕ) [phM +plM c(qM)]
+ (1 ϕ)2[2plL c(qL)]
by choosing outcome qH,qM,qL andphH,phM,plM,plL
subject to constraints
(ICh): ϕ(θhqH phH) + (1 ϕ) (θhqM phM)
ϕ(θhqM plM) + (1 ϕ) (θhqL plL)
(PClL):θlqL plL 0
(PClM):θlqM plM 0
Optimal Outcome when high/low values equally likely
For completeness, we give the optimal outcome for ϕ=1/2.
Proposition
Suppose ϕ=1/2. The provider
increases qL above its second-best level qLm >qLn
indeterminacy for the level of qMm 2 [qLm,minfq˜mL,qHg]
payment from low-value user is set to satisfy
Motivation Model Manipulation Results Welfare Appendix Appendix
Optimal Outcome when high/low values equally likely
Downward Manipulation
Another manipulation opportunity for the provider: Suppose both users reported high values.
Provider can tell each user that the other user reported a low value.
Provider’s payo¤ changes
from 2θhqHn c(qHn) to 2θhqMn c(qnM)
