Optimal Disclosure on Crowdfunding Platforms...Optimal Disclosure on Crowdfunding Platforms Giorgi Mekerishvili Economics Department Pennsylvania State University Email: [email protected]

Optimal Disclosure on Crowdfunding Platforms

Giorgi MekerishviliEconomics Department

Pennsylvania State UniversityEmail gzm140psuedu

November 3 2017

Abstract

This paper studies the trade-off between innovation and investor protection oncrowdfunding platforms Informing the investors about the potential risks of a giveninvestment opportunity protects them from failure but comes at the cost of dissuadinginnovation Disclosure requirements dictate the type of information that the innovatorhas to provide to the potential investors A regulator cares about investor protection(investor welfare) and a platform cares about profits The main result is that a reg-ulator would choose disclosure requirements that are not fully informative I provideconditions under which the platformrsquos optimal choice of the disclosure requirementwould coincide with the regulatorrsquos choice Whenever these conditions do not holdonline reputation systems (eg innovator ratings) can substitute for disclosure require-ments ie decentralization of disclosure requirements create a need for better onlinereputation systems Observing that regulation involves a great deal of experimenta-tion I study the optimal dynamic regulatory experiment The results indicate that weneed to be careful in distinguishing fraud from project failure

Contents

1 Introduction 311 When is regulation unnecessary 412 What is the value of reputation systems to SEC 513 Regulatory Experimentation 5

2 Literature Review 6

3 The Model 7

4 Regulated Disclosure Requirements 941 Full Disclosure 1042 No Disclosure 1043 Optimal Disclosure 1144 Discussion 12

5 Unregulated Disclosure Requirements 1351 Regulation vs Deregulation 14

6 Reputation Systems 15

7 Experimentation 17

8 Extensions and Additional Results 2081 Privately Informed Innovator 2082 Not Purely Profit Motivated Platform 22

821 αI = 1 23822 αE = 1 23

83 Innovator Moral Hazard 2384 2 Period Version of the Dynamic Model 2485 Dynamic Model with the Platform 24

9 Conclusion 26

A Appendix 28

B Appendix 29

C Appendix 33

D Appendix 37

1 Introduction

Crowdfunding1 is a form of financing startups that is believed to complement traditionalventure capital investing by motivating further innovation As the Wall Street Journal putit

rdquoCrowdfunding has the potential to revolutionize the financing of small businesstransforming millions of users of social media such as Facebook into overnightventure capitalists and giving life to valuable business ideas that might otherwisego unfunded (WSJ 2013)rdquo

Crowdfunding platforms match innovators with investors who are typically less experiencedthan those involved in venture capital funding (Rainer 2015) For this reason investorprotection is an important aspect of this new market The JOBS act passed by the USHouse of Representatives on March 8 2012 sets out rules for the crowdfunding platformsand ensures investor protection Although its details are still controversial one of the mainobjectives of the JOBS act is to ensure adequate disclosure of entrepreneur information tothe potential investors

Imposing stringent disclosure requirements on entrepreneurs (innovators) may directlybenefit investors by helping them screen innovators and make optimal investment choicesHowever it comes at the cost of dissuading innovation The motivating question for thispaper is how should disclosure requirements be designed to both motivate innovation andoffer sufficient information to investors

The main result of this paper is that the optimal disclosure requirement (policy) may notbe fully informative To see why consider the decision of an investor who obtains perfectinformation about an innovatorrsquos project quality Because the investor cannot commit toinvest in low quality projects investment takes place if and only if the project is of highquality The innovator who initially is unsure about the quality of his project expects thenthat his project is funded if and only if it is of high quality If he attributes sufficiently lowprobability to this contingency he has no incentives to innovate at an early stage or enterthe crowdfunding platform Full disclosure disincentivizes innovation

This issue can be addressed by using a milder disclosure requirement For example theplatform may commit to occasionally hide evidence from the investor that the project is oflow quality and thereby increase the chance of being successfully funded While concealinginformation from the investor makes it more difficult for her to screen entrepreneurs italleviates her commitment problem This insight leads to the main result of this paper

Optimal disclosure need not be full disclosure Partial disclosure compensates forthe investorrsquos lack of commitment and improves social welfare (including makingthe investor strictly better off)

1 Securities and Exchange Commission (SEC) defines crowdfunding as follows

rdquoCrowdfunding generally refers to a financing method in which money is raised through so-liciting relatively small individual investments or contributions from a large number of people(SEC 2017)rdquo

Formally I study an interaction between an innovator and a representative investor on aplatform The platform is committed to disclosure requirements and chooses a fee structure(platform entry fees and payments from the investor if a project is successfully funded) tomaximize expected profits The innovator decides whether to incur a fixed cost of innovationpay the fees and enter the platform The potential quality of the innovation (project) isunknown to the innovator when making the decision If he innovates and enters the platforma signal is revealed to the investor according to the disclosure requirement The investorobserves the signal realization and decides whether to fund the project The investor onlyvalues high quality projects The innovator enjoys a weakly higher payoff in the contingencywhen his project gets funded and is realized to be of high quality compared to the contingencywhen the project is funded and is of low quality

Motivated by the current regulatory practice in this market I consider two cases for whoholds the authority to set disclosure requirements on a platform

bull Regulated Disclosure Requirements The Securities and Exchange Commission (SEC)chooses disclosure requirement and maximizes investorrsquos welfare net of platform fees

bull Unregulated Disclosure Requirements The platform chooses disclosure requirement

Under regulation SEC optimal disclosure requirement depends on the probability thatthe innovatorrsquos project is of high quality (success rate of innovation) For an intermediatesuccess rate the optimal is a partial disclosure policy that minimizes the probability thatthe investor is recommended to invest whenever the innovation is of low quality subjectto incentivizing innovation Full disclosure is optimal if the success rate of innovation issufficiently high while for sufficiently low success rate of innovation there is no way toincentivize innovation and ensure non-negative expected payoff to the investor

Under deregulation in addition to the success rate of innovation the platform optimaldisclosure requirement and fee structure depend on the investment level required to fund theproject and the reputation cost to the innovator Reputation cost in the model is interpretedas the difference in the innovatorrsquos payoff from the contingency when his project gets fundedand turns out to be of low quality and the contingency when the project is funded and turnsout to be of high quality

When the reputation cost is sufficiently high relative to the investment level requiredto fund the project the platform maximizes investorrsquos welfare and extracts all the surplususing the project fees The platform chooses the same disclosure requirement as the SECwould under regulation The reason is that increasing the probability of investment in thelow quality project increases the innovatorrsquos welfare less than it hurts the investor Hencedecreasing that probability enables the platform to extract higher surplus from the investorcompared to the loss in the innovator surplus extraction When the reputation cost issufficiently low relative to the investment level required to fund the project the platformsets the disclosure requirement that maximizes innovatorrsquos welfare

11 When is regulation unnecessary

Regulation is unnecessary if and only if the reputation cost is sufficiently highrelative to the cost of investment - the platform implements the same requirement

as SEC would under the regulated disclosure requirements

In addition under those conditions the optimal fee structure is such that the platformis free for the innovator and collects all profits from the investor It turns out that 95 ofreward-based platforms operating in the US are free for innovators and extract all profitsfrom investors In our framework this observation rationalizes unregulated reward-basedplatforms

12 What is the value of reputation systems to SEC

In practice disclosure requirements are not the only regulatory tool available to the SECSEC by rule can require a platform to implement certain types of online reputation systems(eg innovator rating systems)

A reputation system ties innovatorrsquos performance to his reputation A good reputationsystem induces high reputation cost for the innovator ie high payoff loss for the innovatorin case the project financed on the platform turns out to be of low quality

Reducing the reputation costs to zero (same as having no reputation system atall) is optimal for SEC under regulated disclosure requirements

The only reason SEC would implement a no full-disclosure policy is to incentivize in-novation Lowering the reputation cost enables SEC to provide better information to theinvestor about the project quality while keeping the innovatorrsquos ex-ante utility at the samelevel This increases investor welfare and does not hurt the incentives for innovation

However this does not mean that reputation systems are useless Imagine a situationwhere regulating disclosure requirements is highly costly The SEC could consider deregulat-ing disclosure requirements In order to guarantee that under deregulation the same investorwelfare is maintained SEC would need to induce sufficiently high reputation cost relativeto the cost of investment- inducing the platform to implement the SEC desired disclosurerequirements This leads to the following

Disclosure requirements and reputation systems are substitutes

13 Regulatory Experimentation

As with any other legislation setting out the rules for crowdfunding is not a one-shot prob-lem Experimentation is a key component that drives evolution of rules The SEC is involvedin continuous rulemaking while the JOBS act has been subject to several amendmentsMoreover since crowdfunding is a relatively new phenomenon there is much uncertaintysurrounding it This further stresses importance of experimentation Gubler (2013) pro-poses a regulatory experiment in order to better understand crowdfunding potential and atthe same time to adjust disclosure requirements correspondingly

rdquoSEC in adopting its rules could treat crowdfunding with a relatively lightregulatory touch for example by not requiring audited financials but specifyingthat the rule will expire after three to five years If the evidence over that periodsuggests the incidence of fraud is high then the agency might impose stricterand more permanent requirementsrdquo

At the heart of such an approach lies the task of designing disclosure requirements that ensureinvestor protection incentivize innovation and at the same time allow regulator to learnsome features of the market with the hope of designing even better disclosure requirementsas learning takes place

Based on the idea of regulatory experiment I study a dynamic version of the staticmodel with the intent to figure out optimal ways for conducting regulatory experiment Theuncertainty about the market is captured by the aggregate uncertainty on the success rateof innovation Innovators have private information about the true success rate

The optimal experimentation boils down to choosing either to learn from the observedquality of implemented projects or to learn from the actions of innovators It is a thresholdpolicy in beliefs on the success rate of innovation prescribing to set more informative disclo-sure requirements when beliefs become sufficiently in favor of high success rate - precludingentry of innovators in case there is a low success rate and thus learning the true success ratefrom the decisions of the innovators Otherwise it keeps setting less stringent requirementsand slowly learns from the market outcomes (realizations of project qualities)

The dynamics of the optimal regulatory experiment reverses Gublerrsquos proposal if we thinkabout failure instead of fraud

If the evidence suggests that the incidence of project failure is high the optimalregulatory experiment prescribes to set milder disclosure requirements

Hence there is a fundamental difference between fraud (usually due to moral hazard)and failure of a project To better understand an optimal regulatory experiment we may notignore this distinction Otherwise we may end up doing the opposite of what is optimal

Outline The remainder of the paper is organized as follows Section 2 gives an accountof related work Section 3 sets up the model Section 4 discusses the case of regulateddisclosure requirements identifies the key trade-off between incentivizing innovation andinvestor protection and includes the main result Section 5 discusses unregulated disclosurerequirements and compares regulation to deregulation Section 6 discusses the value ofreputation systems to a regulator Section 7 sets up a dynamic model and discusses optimaldynamic regulatory experimentation Section 8 provides additional results and extensionsAll formal derivations and proofs are relegated to the appendices

2 Literature Review

Most models of crowdfunding study the merits of crowdfunding that arise due to the avail-ability of pre-selling facilitating learning of consumer demand

Strausz (2016) considers a model in this spirit He studies a design of Pareto efficientcrowdfunding platform in the presence of moral hazard and asymmetric information onthe part of entrepreneur The entrepreneur once he collects funds on the platform canrun away without implementing a project He also possesses private information on hisproduction costs The principal uses a mechanism that mitigates bad incentives arisingfrom asymmetric information and moral hazard Chang (2016) and Chemla amp Tinn (2016)

compare all-or-nothing and take-it-all funding schemes also in the presence of moral hazardand demand uncertainty Ellman amp Hurkens (2015) abstract from moral hazard and interpretcrowdfunding platform as a commitment device for entrepreneurs Threshold funding rulesthat are prevalent can be thought of as entrepreneurs committing to the production only incase sufficient funds are extracted from consumers and threatening not to produce otherwise

Unlike these papers I study a less explored aspect of crowdfunding design - disclosurerequirements I abstract away from the demand uncertainty and focus on how much infor-mation should be given to investors Given their inexperience investors value informationabout the quality of projects

The paper is also related to the Bayesian persuasion literature pioneered by Kamenizaamp Gentzkow (2011) Among the descendents of that paper the most closely related to thestructure of my model are Boleslavsky amp Kim (2017) and Barron et al (2017) Those papersstudy the baseline persuasion model structure (as in Kameniza amp Gentzkow (2011)) withthe additional agent acting at the beginning of the game The twist in such classes of gamesis that the first moverrsquos decision depends on the persuasion rule rather than the realizationof posterior beliefs

The literature on two-sided markets (Rochet amp Tirole (2003)) has emphasized the impor-tance of externalities for the optimal pricing structures chosen by platforms Roughly theidea is that one should jointly take into account the demand of both sides of the market fora serviceproduct offered by a platform The intuition that my paper offers for the optimalfee structures verify some of the lessons learned from that literature in a context where theplatform has an additional tool (disclosure requirements) that could be helpful in controllingexternalities between the two sides of the market

Lastly the dynamic extension of the model is related to the papers that merge persuasionand experimentation (Kremer amp Perry 2014 Che amp Horner 2015)

3 The Model

An innovator (henceforth rdquoherdquo) has to decide whether to use the crowdfunding platform forfinancing his innovation (project) He needs an amount m gt 0 To use the platform heneeds to incur the cost c+ cE ge 0 where cE is the platform entry fee and c is the initial costof developing a product For instance c can be the cost of creating a product prototypedoing initial research or cost of some minimal level of effort that is needed to obtain theevidence that the innovator indeed has something to offer on the platform

The innovatorrsquos project can be one of two qualities - high (H) or low (L) The projectis H with probability p isin [0 1] For simplicity at the time of making the platform entrydecision the innovator does not know the quality of his project2

If the innovator does not enter the platform the game ends and he gets payoff 0 from theoutside option This normalization is without loss of generality Otherwise he must incurthe cost c+cE and in addition must comply with the disclosure requirement on the platform

2 More realistic timing where first the innovator decides on whether to do the initial development thenobserving certain signals about project quality makes the platform entry decision is developed in the section81 None of the results are altered

The set of available actions for the innovator is ENE where E denotes entry and NE -no entry

The platform is committed to a disclosure requirement that generates signals aboutprojectrsquos quality to the investor (henceforth rdquosherdquo) After the investor observes informationshared by the platform she decides whether to invest an amount m gt 0 in the project Ifinvestment takes place the innovatorrsquos payoff is 1 from the high quality project and payoffk equiv 1minus cr from the low quality project where cr captures reputation costs to the innovatorIf the project turns out to be of low quality ex post then the innovator may suffer reputationcosts or lower quality project may generate low value for him in the future Also note thatthe payoff of 1 from the high quality project is a normalization We can always normalizepayoff to the innovator from the high quality project to 1 by appropriately adjusting c

The investor gets payoff of 1 from the high quality project and 0 from the low qualityproject The set of available actions to the investor is INI where I denotes investmentand NI - no investment The investor may also have to pay fees to the platform uponinvestment (explained in more details below)

The following expression shows the payoffs to the innovator (row) and the investor (col-umn) from each combination of project qualities and actions of the investor net of thepayments to the platform and conditional on the innovator playing E

I NIH 1minus c 1minusm minusc 0L k minus cminusm minusc 0

I model a disclosure requirement as a set of signals S and a pair of conditional distribu-tions over S f(s | H) and f(s | L) I assume that S is finite Let Θ = HL and θ standfor an element from this set The set of all possible disclosure requirements is denoted Dand d = (S f(s | θ)θisinΘ) denotes a particular element from D

A given disclosure requirement can be thought of as arising from the set of auditedfinancial documentation specifications about business plan and risks questionnaires howsecurities offered are being valued and other information that signals quality of the offeringand is verifiable either by SEC the platform investors or any other credible third parties3

A disclosure requirement can also be interpreted as an experiment that provides informationabout some features of a product prototype4

The platform fee structure consists of cE ge 0 and (cI(s))sisinS such that cI(s) ge 0 foreach s isin S Here (cI(s))sisinS are the platform usage fees for the investor conditional on eachrealization of the signal I assume that the investor observes realization of the signal before

3 I do not allow the innovator to change the disclosure requirement One could argue that the innovatorcould always provide more information than required The question is whether this information would becredible The innovator could hire an audit in order to verify some additional financial information howeverSEC (or the platform) could always ignore that information by stating in the rules that such informationis not verified by a party which is trusted by SEC In practice SEC has accredited third parties and ownagencies which are eligible to check information provided by the innovator and also delegates authority ofverifying information to platforms

4 For example Kickstarter requires the projects that involve manufacturing gadgets to provide demos ofworking prototypes Photorealistic renderings are prohibited

deciding whether she wants to use the platform for making the investment The total payoffsin the game for the innovator and the investor are

U1 = 1E(1HandI + 1LandIk minus cminus cE)

U2 = 1EandIandH minus 1I(m+sumsisinS

cI(s)1s)

Let cP (d) = (cE (cI(s))sisinS) denote a fee structure for a given d and CP (d) denote allpossible fee structures for a given d Note that the set of possible fee structures depends ond only through S

Throughout the paper I assume that reputation costs are sufficiently low relative to theinnovator entry cost5

Assumption 1 k ge c

Timeline of the game

1 i) Under the regulated disclosure requirements SEC chooses d isin D

ii) Under the deregulated disclosure requirements CFP chooses d isin D

2 CFP chooses cP (d) isin CP (d) and commits to it along with d

3 Innovator chooses action from ENE not knowing the true θ

a) If NE is chosen game ends and everyone gets 0 payoff

b) If E is chosen innovator incurs c+ cE s is realized according to d

4 In case 3(b) - investor observes s updates beliefs on the project quality and choosesaction from INIa) If NI is chosen game ends

b) If I is chosen she pays cP (s) +m and project payoffs are realized

Timeline 1i234 corresponds to the game under the regulated disclosure requirementsTimeline 1ii234 corresponds to the game under the deregulated disclosure requirements

4 Regulated Disclosure Requirements

Suppose the SEC sets disclosure requirements with which the platform has to comply TheSEC wants to maximize the investorrsquos ex-ante welfare net of payments to the platform

Investor protection is mandated by the Congress to the SEC The JOBS act explicitlystates that investor protection is the objective that the SEC needs to be targeting On the

5 This assumption is made for the clarity of exposition It considerably reduces the algebraic burden ofthe proofs All qualitative features of all results still remain true for k isin (0 c) For k le 0 the analysis istrivial as investor and innovator have aligned preferences in the sense that they both prefer that L projectsnot be implemented Hence optimal disclosure (both for SEC and the platform) is always full disclosure

other hand the JOBS act was created with the aim of incentivizing innovation This meansthat investor protection and venture capital formation are the two objectives of the SECInvestor welfare net of transfers to the platform is one specification of SECrsquos payoffs thatcaptures those two objectives in my framework

More generally one could define SECrsquos payoffs as W11HandIminusW21LandI where (W1W2) isinR2

++ and 1θandI takes the value of 1 when investment is made in the project of quality θFor any such specification none of the results in this paper would be altered To save onthe notation I hereafter stick with the investor welfare net of payments to the platform asthe utility function for the SEC

The platform is profit motivated and hence maximizes the sum of expected fees collectedfrom the innovator and investor

In what follows first I discuss full disclosure and no disclosure and show how no disclosurecan mitigate investor commitment problem Then I state the main result and provide theintuition

41 Full Disclosure

Suppose the SEC sets a disclosure requirement that fully reveals the project quality to theinvestor She invests if and only if the quality of the project is high Knowing this theinnovator enters the platform if and only if p ge c (recall that p is the probability that theproject is of high quality) Hence the expected payoff to the investor Zfd

I (p) is

ZfdI (p) =

0 if p isin [0 c)

p(1minusm) if p isin [c 1]

42 No Disclosure

The investor invests and the innovator enters if and only if p ge m Investorrsquos welfare fromthe no disclosure policy is

ZndI (p) =

0 if p isin [0m)

pminusm if p isin [m 1]

Observing ZfdI (p) and Znd

I (p) more closely we see that whenever c gt m for all p isin(m c) we have Znd

I (p) gt ZfdI (p) This already highlights the commitment problem of the

investor Because the investor cannot commit to invest in a low quality project then underfull disclosure the innovator expects to be financed with probability p in which case his payoffwill be 1 Thus his expected payoff is p and since p lt c he will not innovate and the marketshuts down

No disclosure policy solves this commitment issue if p is above m - the investor getsstrictly positive expected payoff from investing Knowing this the innovator knows that hewill be financed with probability 1 His expected payoff from entry is p+ (1minus p)kminus c whichis strictly greater than 0 under Assumption 16

However we can do even better than no disclosure

6 If c gt k no disclosure would solve the commitment issue for p above maxm cminusk1minusk

43 Optimal Disclosure

Optimal disclosure is formally derived in the Appendix A Here I state the result and providethe intuition Let T opt equiv mc

(1minusm)k+m We have the following result

Proposition 1 The optimal disclosure is full disclosure for p isin [c 1] any disclosure for p isin[0 T opt) a disclosure requirement with 2 signal realizations σH σL where f(σH | H) = 1and f(σH | L) = cminusp

(1minusp)k for p isin [T opt c)

The value to the SEC and the investor (net of payments to the platform) is

ZoptI (p) =

0 if p isin [0 T opt)p[k(1minusm)+m]minusmc

kif p isin [T opt c)

The optimal fee structure is any fee structure for p isin [0 T opt) cE = 0 and cI(σL) = 0

for all p isin [T opt 1] cI(σH) = p(1 minus m) for p isin [c 1] and cI(σ

H) = p[k(1minusm)+m]minusmck

forp isin [T opt c)

Figure 1 depicts the SECrsquos payoff under the full and optimal disclosure requirements

SECrsquos Payoff

T opt c 10

Optimal and Full Disclosure

Optimal Disclosure

Full Disclosure

Figure 1

The difference in the SECrsquos payoffs between the optimal and the full disclosure occurs forp isin (T opt c) In that intermediate region partial disclosure solves the investor commitmentproblem and induces innovation The intuition for the result is as follows Suppose there isno problem in incentivizing the innovator to enter (p ge c) Then full disclosure maximizesinvestor welfare For p lt T opt there is no way to incentivize entry and at the same time to

guarantee positive expected payoff to the investor Hence any disclosure is optimal as themarket shuts down in any case and the investor gets 0 payoff from the outside option

For p isin (T opt c) we have the optimal disclosure being partial We can use a two signalmechanism where one signal recommends the investor to finance the project and anotherrecommends not to finance the project Making sure that after seeing the recommendationthe investor indeed wants to follow it we can increase the probability that the innovator isfinanced by sometimes recommending her to invest when the project is of low quality Thisrdquobadrdquo recommendation is given with the lowest probability that incentivizes the innovatorto enter

We also need to make sure that the platformrsquos optimal fee structure does not distortincentives of the players under the disclosure policy in proposition 1 Whenever p isin [0 T opt)the platformrsquos choice of fee structure is redundant as the market still shuts down for any feestructure Otherwise the platformrsquos optimal fee structure is always such that it makes theinnovator indifferent between entry and no entry and whenever the investor is recommendedto invest the platform extracts all her surplus and makes the investor indifferent betweeninvesting and not investing Thus the platformrsquos optimal fee structure does not affect theinnovatorrsquos incentive to enter and the investorrsquos incentive to invest whenever recommendedto do so

44 Discussion

Proposition 1 highlights how crowdfunding can be useful for realizing untapped potentialfor innovation If we think of the traditional venture capital investors as being effective inscreening projects then it would be likely that the innovators with intermediate potential forsuccess (p isin (T opt c)) would not be incentivized to realize their ideas If the investors areless effective in screening projects on their own (like crowdfunders) then disclosure require-ments become powerful regulatory tool with the potential for dealing with the investorrsquoscommitment problem and hence further incentivizing innovation

In practice institutional investors (eg venture capital firms) employ investment pro-fessionals and perform due diligence on entrepreneurs This process involves interviewingformer customers competitors employees experts and conducting intense financial and legalwork This level of screening would not be feasible for a small retail investor simply becauseof the costs associated with it Also a lionrsquos share of venture capital goes to Silocon ValleyNew York Boston and Los Angeles In 2016 only 22 of investment went to companies out-side those hubs7 In addition institutional investors mostly target ventures in post-startupstages8 The innovators having their projects in later development stages and coming fromthose talent hubs can be regarded as high p innovators A relatively good screening abilityof institutional investors does not disincentivize such innovators to pursue their ideas Onthe other hand innovators who are believed not to have sufficiently high success probabilitywould be disincentivized to pursue their ideas Crowdfunding could help by allowing for anew segment of investors where each investor is relatively small unable to effectively screenpotential investment opportunities and hence dependent on whatever disclosure is provided

7 httpwwwthirdwayorg8 httpswwwforbescom

by a regulator or a platform It follows that the retail investors being less experienced andless able to screen the projects compared to the institutional investors is not necessarily adisadvantage of crowdfunding On the contrary it can facilitate innovation and improve thesocial welfare if a regulator chooses correct disclosure requirements

5 Unregulated Disclosure Requirements

In this section I study the variant of the model in which the platform chooses disclosurerequirements

First recall from the previous section that if p lt mc(1minusm)k+m

then there is no way to jointlyinduce entry and investment Hence any fee structure would be optimal as the market wouldshut down for any fee structure I focus on p ge mc

(1minusm)k+m

Appendix B derives the optimal disclosure policy and fee structure Here I state theresult Recall that m is the investment required to finance the project and k is the payoff tothe innovator when he obtains the financing and his project turns out to be of low quality

Proposition 2 If m ge k the optimal disclosure is the same as in proposition 1 If m lt kthe optimal disclosure is f(σH | H) = 1 f(σH | L) = minp(1minusm)

(1minusp)m 1

Note that for m lt k the platform still cannot incentivize innovation for p lt T opt Forp gt T opt the platform sets the disclosure requirement that recommends investment in thelow quality project with strictly higher probability compared to the SEC optimal disclosurepolicy Also the platform provides worse information (in the Blackwell sense) to the investorfor higher p To understand what drives optimal behavior of the platform it is useful todistinguish among different effects of changing fees

Increasing cE has a positive first order effect on profits a negative effect on the innovatorrsquosutility and a negative effect on the investorrsquos utility The intuition for the latter is thatincreasing cE means the platform should sacrifice informativness of signal σH in order toincentivize the innovator to enter the platform but this is detrimental to the investorrsquosexpected payoff because now there is a higher chance that the investor is recommended toinvest in state L

Increasing cI(σH) has a positive first order effect on profits a negative effect on the

investorrsquos payoffs and a negative effect on the innovatorrsquos payoffs The intuition for the latteris that if the platform wants to increase cI(σ

H) then it has to provide better informationto the investor in order to increase the value of investment to the investor However thisis detrimental to the innovatorrsquos utility as he expects lower probability of his project beingfinanced

Now one could think about the following decomposition of the problem for the platformFirst for a fixed disclosure policy it is definitely optimal for the platform to increase bothcE and cI(σ

H) until the innovator keeps playing E and the investor keeps playing I whenrecommended The platform would do this for each possible disclosure policy Second giventhe fee structure associated with each disclosure policy that was found in the first stage theplatform needs to choose the optimal disclosure policy If it would like to increase cE it wouldneed to increase the probability with which the innovator is recommended to invest in state

L in order to keep inducing entry of the innovator But then the platform would also need todecrease cI(σ

H) because otherwise the investor would no longer follow the recommendationto invest Similarly if the platform decided to increase cI(σ

H) he would need to decreasecE Thus for instance if increasing cE hurts the investor more than increasing cI(σ

H) hurtsthe innovator then the platform would increase cI(σ

H) and provide better information tothe investor (decrease probability with which the investor is recommended to invest in stateL) In case m ge k this is exactly what happens and the platform chooses the disclosurerequirement that maximizes investor welfare subject to incentivizing innovation In casem lt k the opposite happens

51 Regulation vs Deregulation

Regulation is costly It is useful to understand when one could avoid those costs by deregu-lating disclosure requirements As a corollary to proposition 2 we obtain a condition guar-anteeing that the implemented disclosure requirement under regulation and deregulation arethe same

Corollary 1 Regulation is not necessary if and only if m ge k

Thus if the SEC has a reason to believe that the innovator reputation cost is sufficientlyhigh relative to the cost of investment then deregulating disclosure requirements would notthreaten investor protection

Currently the JOBS act is tailored for loan and equity based crowdfunding while reward-based crowdfunding is left relatively unregulated Because rewards are not classified asfinancial instruments or securities its remedies fall under the traditional consumer protec-tion law This gives a high degree of freedom to reward-based platforms to set their owndisclosure requirements9 Since the reward-based platforms involve pre-selling products toconsumers a candidate agency for regulating those platforms is the Consumer Product SafetyCommission which requires manufacturers to conduct certain product certifications beforeselling final good to consumers However since crowdfunders receive final product only afterthe investment takes place the certification is required only after the project is financed andproduction takes place This means that innovators are not required to do certification whenadvertising their product prototypes on the platforms This leaves disclosure requirementson the reward-based platforms unregulated

Is regulation necessary on the reward-based platformsFirst I state the result that will be the key in answering this question Recall that cE

denotes the platform entry fee for the innovator and cI(σH) gt 0 denotes payment from the

investor to the platform in case investment is recommended (in practice referred to as aproject fee)

Proposition 3 Under the deregulated disclosure requirements the platform sets cE = 0 andcI(σ

H) gt 0 if and only if m ge k and p le c

9 See httpwwweuropaeu or httpwwweuroparleuropaeu for the current regulation practices con-cerning reward based crowdfunding in EU See httpslawreviewlawucdavisedu for how the consumerprotection law has been applied in case of Kickstarter disputes

Corollary 1 and proposition 3 imply that regulation is not necessary if and only if a feestructure that sets cE = 0 and cI(σ

H) gt 0 is optimal for the platformIt turns out that most US based reward-based platforms indeed charge 0 fees to the inno-

vators and strictly positive project fees According to the data obtained from wwwcrowdsurfercomout of the 171 reward-based crowdfunding platforms that were in the active status in theUS as of April 2017 107 of them provide information on their fee structures From 107 97charge only project fees 2 charge only entry fees 5 charge both and 3 charge no fees at all

This observation along with the corollary 1 and propositions 3 imply that the modelsupports the claim that regulation of disclosure requirements on the reward-based platformsis not necessary

6 Reputation Systems

Disclosure requirements is not the only regulatory tool that is available to the SEC SEC alsocan influence innovatorrsquos reputation cost by requesting the platform to implement certainonline reputation system Indeed the SEC retains authority under the JOBS Act to requirethe platforms to rdquomeet other requirements as the Commission may by rule prescribe for theprotection of investors and in the public interestrdquo10The JOBS act (Title III) in the currentform requires equity and lending based crowdfunding platforms to obtain and publicizeinformation such as innovatorrsquos name legal status physical address and the names of thedirectors and officers In addition the SEC can by rule require publicization of innovatorrsquosonline information (eg Facebook account) or implementation of a certain type of onlinereputation system All such measures would ensure that performance of an innovator isclosely tied to his reputation11

Here I make the distinction between reputation systems and disclosure requirementsclear Disclosure requirements concern information that signals project quality to potentialinvestors Reputation systems control visibility of ex post project quality realizations to par-ties in the aftermarket (not modeled in this paper) Reputation systems also tie innovatorrsquosidentity to his project results

Several platforms are already using some forms of online reputation systems For exam-ple on Indiegogo innovator can link Facebook and Indiegogo accounts and obtain verifiedFacebook badge Feedback reputation systems similar to ones on Ebay and Amazon havealso been proposed 12

The problem with all such online reputation systems is that most innovators do not go toa platform repeatedly For example more than 90 percent of project creators propose onlyone campaign on Kickstarter 13 This weakens the effect of reputation systems on innovatorincentives

I suggest that there may not even be a need for a perfect reputation system

Proposition 4 Under regulated disclosure requirements it is optimal for SEC to set k = 1Under deregulated disclosure requirements it is optimal for the platform to set k = 1

10 See 15 USC sect 77d-1(a) (2012)11 see httpswwwsecgov12 see Schwartz (2015)13 See Kuppuswamy V Bayus BL (2013)

The intuition behind proposition 4 is straightforward The only thing that increasingk does is to relax the innovatorrsquos incentive for entry This is desirable for SEC as underregulation SEC would be able to decrease the probability of financing the project in stateL and hence would improve investor welfare Increasing k would also be beneficial for theplatform under deregulated disclosure requirements Higher k relaxes incentive constraintfor entry and guarantees weakly higher profits for the platform

But there still is a role for reputation systems

Definition 1 Reputation systems and disclosure requirements are substitutes if wheneverSEC deregulates disclosure requirements by regulating reputation systems it can guaranteeweakly higher investor welfare and strictly higher welfare for some parameter values com-pared to the case with deregulated disclosure requirements and reputation systems

Obviously the SEC can always guarantee weakly higher investor welfare when regulatingreputation systems compared to deregulated reputation systems I argue that there are caseswhere reputation systems can substitute for disclosure requirements by enabling the SEC tostrictly increase investor welfare by regulating reputation systems

Recall proposition 4 which says that under deregulated disclosure requirements it isoptimal for the platform to set k = 1 Imagine the SEC deregulating disclosure requirementsIf reputation systems are also deregulated then k = 1 gt m and proposition 2 implies thatthe platform would set the disclosure requirement that recommend the investor to invest instate L with too high probability It can be verified that if p gt c

2minusm then under the regulatedreputation systems SEC would set k = m inducing the platform to choose lower probabilityof recommending investment in state L and strictly improving investor welfare This leadsto the following

Proposition 5 Reputation systems and disclosure requirements are substitute regulatorytools

To summarize sufficiently high reputation cost (inducing m ge k) relieves the SEC fromthe need for regulating disclosure requirements on the platform - the profit maximizingplatform maximizes investor welfare and extracts surplus using project fees If the reputationcost is not sufficiently high then the platform considers extracting innovator welfare by settinga disclosure requirement that favors the innovator Such a disclosure requirement increasesthe probability of financing the project as much as possible and hence induces too highprobability of investment in state L By regulating disclosure requirements the SEC candirectly decrease the probability of financing the L project while maintaining incentivesfor innovation Alternatively the SEC could keep the disclosure requirements deregulatedand request the platform to increase the reputation cost for the innovator by implementinga more effective reputation system Which of those ways the SEC chooses would dependon the costs associated with implementing each type of regulatory tool and effectivenessof reputation systems (eg is there a reputation system that would be able to increasereputation cost)

7 Experimentation

Regulation is associated with a great deal of experimentation The SEC is involved in con-tinuous rule-making and the JOBS act has been subject to several amendments Being arelatively new phenomenon one could think about many features of the crowdfunding thatare surrounded by uncertainty One such feature is the probability of success of the averageinnovation That probability (denoted p) was assumed to be known by the regulator inthe previous sections If p were known then previous sections suffice for describing optimaldisclosure policies for each possible p under regulation and deregulation If p were unknownthen a regulator could start experimenting with one regulation learning about p and subse-quently possibly redesigning initial rules taking the updated information into account Thegoal of this section is to understand how the experimentation should be done by a regulatorwhen p is unknown

Time is infinite and discrete indexed by t There is a large number of potential innovatorsOne of those innovators has an idea at a given point in time and needs to decide whether toenter the platform There is also a representative short-lived investor who decides whetherto invest or not in a project which is posted (if posted) on the platform each period In thissection I shut down platformrsquos decision to choose a fee structure14 Otherwise the gamewithin each period with stage payoffs is the same as in the static model Once the stagegame ends the innovator quits the platform forever

There is aggregate uncertainty on p (success rate of innovation) The true p is known tothe innovators but not to the investor or the regulator True p can take one of two valuespH or pL where pH gt pL True p is distributed p sim G with probability mass function g Theaggregate uncertainty about the success rate of projects captures the fact that the market isnew it serves a new segment of potential innovators and it is not known what the averageinnovation potential is on the market

Conditional on the true p each innovatorrsquos project quality (conditional on entry) isdrawn independently of others Let atE isin AE and atI isin AI denote actions of innovator andinvestor that were taken in period t Let θt be a quality of project in period t (conditionalon atE = E)

The SEC commits to a dynamic disclosure policy on the platform Its objective is investorprotection and hence maximizes discounted payoffs of the investor with discount factor δ

[infinsumt=0

δt1atE=EatI=I(1θt=H minusm)

]All past actions and signals are publicly observable The SEC transmits information

about θt through a disclosure requirement in place in period t In addition if trade takesplace in a given period (atE = E atI = I) then θt immediately becomes public informationafter the trade

Let St be a signal realization space at t chosen by the SEC I distinguish 3 special signalsthat are necessarily part of this space - sH sL Let ht = (a0E a0I s0 atminus1E atminus1I stminus1)be a public history of previous actions and signal realizations Let H t be the set of such

14 Section 85 discusses what would happen if we also allow for a long lived platform maximizing profits

t period histories H = cupinfint=0Ht Let ht+ = (ht atE θt) and H t

+ H+ defined in an obviousmanner Let ht++ = (ht+ atI) and H t

++ H++ defined in an obvious manner

Definition 2 Dynamic disclosure policy is a pair of conditional distributions f H+ rarr Sand z H++ rarr S such that

i) f( | ht+) = 1 z( | ht++) = 1 for all ht+ isin H+ ht++ isin H++ with atE = NE

ii) z(sH | ht++) = 1 for all ht++ isin H++ with atE = E θt = H and atI = Iiii) z(sL | ht++) = 1 for all ht++ isin H++ with atE = E θt = L and atI = Iiv) z(s | ht++) = f(s | ht+) for all ht++ with atE = E atI = NI

Function z is a signal transition function from the end of period signals to the beginningof the next period signals The definition states that θt becomes public knowledge if tradetakes place if the idea was realized (entry took place) but trade failed then whatever signalwas realized according to f remains in place and if the idea was not realized then f justgenerates null signal

The functions f and z are common knowledgeLet G(ht) be the distribution of p at the beginning of period t and let ZG(ht) be its

support G(ht) is public knowledge ZG(h0) = pL pH is the support of G at the beginningof the game

Lemma 2 in the Appendix C shows that the optimal policy would involve end of exper-imentation and shutdown of the market if at some time t ZG(ht) sube [0 mc

(1minusm)k+m) Lemma

3 shows that the optimal policy would be permanent full disclosure after time t such thatZG(ht) sube [c 1] After that time beliefs would evolve without further interference of SEC inthe experimentation process

I study the non-trivial case where pH pL isin ( mc(1minusm)k+m

Let dpH = (σH σL f(σH | H) = 1 f(σH | L) = cminuspH(1minuspH)k

) and dpL = (σH σL f(σH |H) = 1 f(σH | L) = cminuspL

(1minuspL)k) Following is the main result of this section

Proposition 6 The optimal dynamic disclosure policy is a threshold policy in g with thresh-old glowast such that if g ge glowast dpH is set if g lt glowast dpL is set

Appendix C provides a formal proof of proposition 6 Here I sketch it With a slightabuse of notation let g equiv g(pH) Let also G(s) denote the posterior public belief given priorG and signal s First I consider a relaxed problem where I ignore investorrsquos constraintsie constraints that ensure that investor indeed wants to invest when receiving a signalrecommending investment This simplifies the problem because now the only thing thatdepends on the public beliefs G is the SECrsquos objective function Note that the constraintfor the innovator after history ht is

sumsisinStr

f(s | ht EH)pj + ksumsisinStr

f(s | ht E L)(1minus pj) ge c if type pj isin ZG(htE)

which does not depend on the public beliefs other than through the support of the beliefsZG(htE)

I consider the relevant dynamic programming formulation of the problem Assumingthat the value function exists I prove necessary conditions that the value function withthe associated optimal policy must satisfy Claims 12 and 3 prove those conditions - thevalue function must be convex in g and the optimal policy must be choosing disclosurerequirements from the set dpH dpL in any given period Lemma 4 shows that investorrsquosconstraints are not violated for any dynamic disclosure policy in this class

The dynamic formulation of the SECrsquos problem simplifies to

V (G) =

g [pH((1minusm)k+m)minusmc)]

k+ δ(1minus g) [pL((1minusm)k+m)minusmc)]

(1minus δ)[EG(p)(1minusm)minus cminuspL(1minuspL)k

EG(1minus p)m]+

δ[EG(p)V (G(sH)) + EG(1minus p)V (G(sL))]

For each G the SEC needs to decide whether to set f(σH | GEL) = cminuspH

(1minuspH)kor f(σH |

GEL) = cminuspL(1minuspL)k

In the former case only high type innovator (pH) enters the platformthus revealing the type and from the next period onwards static optimal policy is chosen

This generates expected discounted payoff g [pH((1minusm)k+m)minusmc)]k

+δ(1minusg) [pL((1minusm)k+m)minusmc)]k

forSEC I will call such a policy separating In the latter case the SEC sets a milder disclosurerequirement Both types of the innovator enter and learning about the success rate takesplace via observing realization of the project outcome after trade takes place I call such apolicy pooling This information comes in the form of observing θt after trade or observingonly σL if the investor does not invest Moreover since σL fully reveals that θt = L it isalways the case that optimal policy induces full revelation of θt

Using standard arguments from Stokey amp Lucas (1989) lemma 5 proves that the valuefunction satisfying above equation exists is unique continuous and strictly increasing in gI also verify that it is convex in g

Now we are ready to say more about the optimal policy The idea of the proof of theproposition 6 can be conveniently represented by the Figure 2

The yellow region depicts all possible values that V can take The lower bound curve on

these values is the function g [pH((1minusm)k+m)minusmc)]k

+ δ(1 minus g) [pL((1minusm)k+m)minusmc)]k

We know thatpoint B is the optimal value when g = 1 Suppose that we also have point A as a part ofthe value function Then because we know that V is globally convex the only way it canhappen is if it includes the entire green line Hence the proof that the optimal policy mustbe a threshold policy

If g is above the threshold glowast then the optimal thing to do is to use separating policy(dpH ) If g is below the threshold then the pooling policy is optimal

Particularly what the optimal experimentation prescribes is to set a milder disclosurerequirement when the incidence of project failure is high This is in contrast with theGublerrsquos (2013) proposal which suggests a stricter disclosure requirement when the incidenceof fraud is high

Fraud arises due to an agent taking (or not taking) some action Project failure as it isin my model does not arise due to actions taken by the agent - the probability of a projectfailure is determined by the ability of the innovator or some other exogenous factors Inpractice we usually observe unsuccessful projects and it is difficult to distinguish whether

Figure 2

no success is due to the innovatorrsquos actions or some other factors not under his control Weneed to carefully investigate the reasons behind unsuccessful projects as otherwise we mayend up implementing disclosure requirements that are opposite to what is indeed optimal

8 Extensions and Additional Results

81 Privately Informed Innovator

So far I have abstracted away from any kind of private information on the part of theinnovator Particularly it was maintained that the innovator does not learn the projectquality after incurring c and before the platform entry decision In addition he conductsrequired experiment only after entering the platform However in practice initial phase ofthe project development may provide information about the potential quality of the projectto the innovator In addition if entering the platform is not free the innovator could conductthe experiment required by the platform prior to the entry decision and then make moreinformed decision

It turns out that introducing those features do not affect qualitative features of the resultsIn particular propositions 1-8 are still true

First consider the case of regulated disclosure requirements in the static model When-ever p gt c the optimal disclosure is still full disclosure as there is no problem with incen-tivizing innovation The platform sets cE = 1 minus cp as setting higher entry fee would leadto no innovation

Whenever p lt c full disclosure does not induce entry Moreover for any disclosurerequirement set by the SEC that incentivizes innovation for cE = 0 if the platform decidesto set cE gt k then the innovator would not use the platform whenever he learns that theproject is of quality L But this would mean that ex-ante he would not be willing to incurc So the platform would be better off by setting cE le k as in this case it would at least be

getting positive fees from the investorFormally the problem is the same as formulated in the Appendix A with the addition

of 2 more constraints cE le k and pf(σH | H) + (1 minus p)f(σH | L)k minus c minus cE(pf(σH |H) + (1 minus p)f(σH | L)) ge 0 We know the solution to the relaxed problem (appendix A)Particularly pf(σH | H) + (1 minus p)f(σH | L)k minus c = 0 under the solution to the relaxedproblem Under this solution additional 2 constraints are satisfied if and only if cE = 0 butthis is exactly what the platform would do as otherwise innovation would not take placeHence proposition 1 is still true

To see that proposition 2 is still true consider the problem of the profit motivatedplatform choosing both disclosure requirements and a fee structure

maxcP (d)d

sumsisinStr(pf(s | H) + (1minus p)f(s | L))(cI(s) + cE)

f(s | H)p

f(s | H)p+ f(s | L)(1minus p)ge m+ cI(s) forall s isin Strsum

sisinStr

f(s | H)p+sumsisinStr

f(s | L)(1minus p)k ge

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

Following similar arguments as in Lemma 1 (in Appendix B) one can show that we canrestrict attention to the disclosure requirements with at most 2 signal realizations One canrewrite the problem as

maxf(σH |θL)

(p+ (1minus p)f(σH | L))cE + p(1minusm)minusm(1minus p)f(σH | L)

p+ (1minus p)f(σH | L)minusm ge 0

p+ (1minus p)f(σH | L)k minus c ge 0

cE le mink p+ (1minus p)kf(σH | L)minus cp+ (1minus p)f(σH | L)

For p ge c1minusk the right hand side of the third constraint equals k Substituting cE = k

into the objective we see that if k gt m setting f(σH | L) such that it makes investorindifferent between investing and not investing is optimal If p lt c

1minusk then we substitute

cE = p+(1minusp)kf(σH |L)minuscp+(1minusp)f(σH |L)

into the objective and again if k gt m the same disclosure would

be optimal Hence we get the same disclosure policy as in the Case 3(a) in the AppendixB (where we solve the problem 11) If k le m one can similarly verify that we get exactlysame disclosure policy as in Case 3(b) in the Appendix B Moreover it is easily verified thatwhenever k le m and p le c the platform sets 0 entry fees for the innovator Hence theproposition 2 is still true

A little inspection reveals that the propositions 34567 and 8 are also trueAlternatively one could also consider the innovator getting some signal about his project

quality before incurring cost c This case can be modeled as a special case of the dynamicmodel with T = 1 Before incurring c the innovator observes a private signal that drives hisposterior belief either to pH or to pL The innovator knows pj while the rest of the playersbelieve that with probability g the innovator gets a signal that induces pH

Recall g and gPL from the Section 85 One can verify that the SEC would like to choosethe separating policy (dpH ) if and only if g ge g and otherwise chooses the pooling policy(dpL) If the SEC chooses the pooling policy the platform sets cE = 0 if and only if g le gPLHence whenever m ge k the SEC chooses the separating policy if and only if g ge g andotherwise chooses the pooling policy

Whenever k gt m the SEC chooses the separating policy for g ge g and pooling forg le gPL For g isin (gPL g) the SEC needs to distort dpL in such a way that both types of theinnovator are incentivized to enter For this it would need to set a disclosure requirementthat induces higher probability of investment in state L compared to dpL This way theplatform would be able to set cE gt 0 without disincentivizing the entry of pL

Under the deregulated disclosure requirements the platform sets the same disclosurerequirement as the SEC would set under regulation if and only if m ge k The intuitionis the same as it was for proposition 2 The platform wants to extract as much welfareas possible from the investor For this it maximizes investor welfare and hence sets thesame disclosure requirement as the SEC would For k lt m the platform sets disclosurerequirements that induce the probability of investment in state L which is higher than theSEC would set under regulation

To summarize private information on the part of the innovator does not overturn thekey messages of the paper - partially informative disclosure requirements are optimal if rep-utation cost is sufficiently high relative to the investment cost regulation is not necessarydisclosure requirements and reputation systems are substitutes and optimal dynamic regu-lation prescribes to set milder disclosure requirements when the incidence of project failureis high

82 Not Purely Profit Motivated Platform

Suppose the deregulated platform maximizes weighted sum of the investorrsquos welfare inno-vatorrsquos welfare and profits with weights αI αE and (1minus αI minus αE) respectively

Appendix B derives the optimal fee structures and disclosure requirements for all possibleparameter values Here I discuss some special cases Recall that r equiv cI(σ

H) is the projectfee paid by the investor

The platform always chooses disclosure requirements with at most 2 signal realizationsFor any parameter values it sets f(σH | H) = 1 r cE and f(σH | L) vary with (αI αE)

821 αI = 1

The innovator and the investor are charged 0 fees and the platformrsquos optimal disclosurerequirement coincides with the SEC optimal disclosure requirement from proposition 1

822 αE = 1

The optimal policy sets f(σH | L) = minp(1minusm)(1minusp)m 1 c

E = 0 and r = 0

f(σH | L) maximizes the probability of investment in state L subject to the investorfollowing the recommendation to invest

83 Innovator Moral Hazard

Proposition 4 suggests that under the regulated disclosure requirements reputation systemsdo not have any value for the SEC Lowering the reputation cost enables the SEC to providebetter investor protection by making it easier to encourage innovation As k increasesinvestment starts taking place for lower values of p and the investor welfare strictly increasesfor all such p

One may argue that this result would not be immune to introducing moral hazard on thepart of the innovator Moral hazard would create a force in favor of lower k For instanceif we allow the innovator to control p by exerting unobserved effort we would expect thatlowering the reputation cost would weaken incentives for exerting higher effort Exertinglower effort increases the probability that a project will turn out to be of low quality but ifthe reputation cost is low payoff to the innovator in case investment takes place in the lowquality project is high and thus it is more difficult to incentivize him to exert high effort Itturns out that weakening incentives for exerting high effort need not necessarily overweightbenefits from the lower reputation cost

To make this point clear suppose that the innovator in addition chooses how mucheffort e isin [0 1] to exert on developing the idea The choice of e is private information tothe innovator The project development has two stages first the innovator needs to exerta minimum level of effort that is observable and costs c to the innovator This stage canbe interpreted as creating an actual prototype or blueprint existence of which can be easilyverified by the platform conducting a due diligence If c is incurred the innovator can exertadditional hidden effort e that is not easily verifiable

If e is exerted probability of the project ending up being H is e and the cost of effort isc(e) = e2

Using similar arguments as in the proof of proposition 1 one can verify that we canrestrict attention to the disclosure requirements with at most 2 signal realizations and thatsetting f(σH | H) = 1 is optimal The only thing we need to find is the optimal f(σH | L)We need to solve the following problem

maxf(σH |θL)

(1minusm)(1minus kf(σH | L)) + (1 + kf(σH | L))f(σH | L)m

(1 + kf(σH | L))2

The solution gives f(σH | L) = 0 for c le 14 f(σH | L) = 2

radiccminus1k

for c isin [14 (1+k)2

4] and

the market shutting down for c gt (1+k)2

Increasing k has similar implications as in the model without moral hazard It increasesrange of c for which innovation takes place and thus strictly increases the investor welfarefor all such c while not affecting the investor welfare for the rest of the values of c Settingk = 1 would still be optimal for the regulator

84 2 Period Version of the Dynamic Model

All the results from the Appendix C hold in the 2 period model Moreover we can explicitlysolve for the threshold glowast Take k = 1 Appendix D proves the following proposition

Proposition 7 Optimal policy is a threshold policy where threshold glowast is given by

glowast =(1minus pL)(pL minusmc)

(1minus pL)(pL minusmc) + (1 + δ)(pH minus pL)(1minus c)m

Comparative staticsi) partglowast

partmlt 0 Higher investment cost incentivizes learning (separation) as under no separa-

tion policy the quality of the information provided to the investors is low (higher probabilityof investment in state L) and as m increases the investor becomes more vulnerable to lowquality information

ii) partglowast

partclt 0 if m gt pL and otherwise partglowast

partcge 0

iii) partglowast

partδlt 0 Higher patience means that there is more weight on the gains from learning

through separationiv) partglowast

partpLgt 0 and partglowast

partpHlt 0 Low pL or high pH means that the first period gain from no

separation (increased probability of trade in the first period) is low

85 Dynamic Model with the Platform

In this section I discuss some implications for the optimal dynamic policy (regulated) in thepresence of the profit motivated platform The platform sets fee structure in each period andmaximizes discounted profits with the discount factor δPL The SEC commits to the dynamicdisclosure policy The platform observes same history as the SEC does In particular it doesnot know the true p I prove the following proposition

Proposition 8 For any (δ δPL) isin [0 1]2 the optimal dynamic disclosure policy is the sameas in proposition 6 if and only if m ge k

Proof Consider the threshold glowast implied by the optimal disclosure policy in proposition 6Suppose the SEC wants to implement the same policy Whenever g ge glowast there is no problemwith providing incentives for the platform the SEC separates types in the first period suchthat only pH innovates in the current period and moreover pH gets 0 ex ante payoff Hencethe platform sets cE = 0 and collects positive fees from the investor If it sets cE gt 0 noinnovation would take place and the platform would lose payments from the investor in the

current period After the initial period everyone learns true p and static optimal disclosureis set thereafter that still induces the platform to set cE = 0

Whenever g lt glowast the SEC may need to distort its pooling disclosure policy Morespecifically if SEC decides to implement the pooling policy the platform could set cE gt 0and induce only pH type to innovate However this would not be optimal for the SEC SECwould like the platform to set cE = 0 as this is the only case that would induce poolingTo mitigate such incentive for deviation the SEC could commit to shut down the marketforever if it sees that the platform sets cE gt 0 in the period when pooling is optimal Itwould simply set a sufficiently stringent disclosure requirement forever after the deviation inorder to induce shut down Then the platform would compare the expected profits of settingcE = 0 this period and continuing the business thereafter to setting cE gt 0 getting largercurrent period profit and then getting 0 profits thereafter As δPL decreases it would bemore difficult for the SEC to provide such incentives to the platform I consider the worstcase for the SEC that is when δPL = 0

In this worst case we can solve for a threshold gPL isin (0 1) such that below this thresholdthe platform would want to set cE = 0 whenever the SEC sets a pooling disclosure Abovethe threshold platform would deviate to setting cE gt 0 under the pooling disclosure Tosolve for the threshold first consider platformrsquos payoff from setting cE = 0 In this caseprofits come only through project fees which is

EG(p)(1minusm)minus cminus pL

(1minus pL)kEG(1minus p)m

This is the amount that platform extracts from investors in expectation If the platformsets cE gt 0 then the optimal cE would be such that it makes pH type indifferent betweeninnovating and not Also if innovation happens investor knows that market potential is pH

and the platform sets fees to extract entire investor surplus whenever investors have thatinformation Hence in expectation the platform gets

g[pH(1minusm)minus cminus pL

(1minus pL)k(1minus pH)m+ pH minus c+ (1minus pH)

cminus pL

(1minus pL)]

Using those two expressions I solve for the threshold

gPL =(pL((1minusm)k +m)minusmc)(1minus pL)

(pL((1minusm)k +m)minusmc)(1minus pL) + k(pH minus pL)(1minus c)If glowast le gPL then there is no problem of incentivizing the platform to set fees that comply

with the SECrsquos disclosure requirements This is because below glowast the platform sets cE = 0This is indeed what the SEC wants Above glowast the platformrsquos incentives do not matter asthe SEC is choosing separating disclosure To complete the proof I show that m ge k if andonly if glowast le gPL

To do this I provide an upper bound for glowast denoted g Recall the value function for theSEC without the presence of the platform

V (G) =

EG(1minus p)m]+

Note that g [pH((1minusm)k+m)minusmc)]

k+ (1 minus g) [pL((1minusm)k+m)minusmc)]

kge EG(p)V (G(sH)) + EG(1 minus

p)V (G(sL))] because the left hand side in the inequality is the upper bound on the expectedcontinuation value Hence if the stage payoff from choosing the separating policy is higherthan the stage payoff from choosing the pooling policy then separating policy is chosen Thisprovides the following upper bound on glowast

g =(pL((1minusm)k +m)minusmc)(1minus pL)

(pL((1minusm)k +m)minusmc)(1minus pL) +m(pH minus pL)(1minus c)Now g le gPL iff m ge k This proves the resultWhenever m lt k extracting surplus from the innovators becomes tempting for the plat-

form and the SEC may need to distort its policy

9 Conclusion

Crowdfunding is a relatively new form of venture capital formation and its regulation is stilla subject of scrutiny Crowdfunding is expected to facilitate capital formation (innovation)while protecting investors from fraud and failure A regulatory tool used to meet these objec-tives is disclosure requirements that dictate verifiable information that has to be transmittedfrom the innovators to the investors The following trade-off arises informing the investorsabout the potential risks of a given investment opportunity protects them from failure butcomes at the cost of dissuading innovation The main result of this paper is that the optimaldisclosure requirement is partial The paper exposes how partial disclosure mitigates theinvestorsrsquo lack of commitment (if the investors know that a project is of low quality theycannot commit to invest) and incentivizes innovation while maintaining a reasonable levelof investor protection I also consider the problem in the presence of a profit motivatedplatform and derive the conditions under which the platformrsquos optimal choice of the dis-closure requirement would coincide with the regulatorrsquos choice Whenever those conditionsare violated online reputation systems (eg innovator ratings) can substitute for disclosurerequirements ie decentralization of disclosure requirements would create a need for betteronline reputation systems Observing that regulation involves a great deal of experimen-tation I study the optimal dynamic regulatory experiment The results indicate that weneed to be careful in distinguishing fraud from project failure - if a regulator observes anunsuccessful project it is important to investigate whether the innovatorrsquos actions (fraud) ledto no success or other factors (innovatorrsquos ability shocks to the market) led to no successOtherwise a regulator may end up doing the opposite of whatever is optimal

References

[1] Barron Daniel George Georgiadis and Jeroen Swinkels 2016 rdquoRisk-taking and Simple Contractsrdquomimeo

[2] Boleslavsky Raphael and Kyungmin Kim 2017 rdquoBayesian Persuasion and Moral Hazardrdquo WorkingPaper

[3] Chang Jen-Wen 2016 rdquoThe Economics of Crowdfundingrdquo Mimeo UCLA working paper

[4] Che Yeon-Koo and Johannes Horner 2015 rdquoOptimal Design for Social Learningrdquo Cowles FoundationDiscussion Paper No 2000

[5] Chemla Gilles and Katrin Tinn 2016 rdquoLearning through Crowdfundingrdquo CEPR discussion paperDP11363

[6] Ellman Matthew and Sjaak Hurkens 2016 rdquoOptimal Crowdfunding Designrdquo mimeo Institute forEconomic Analysis (CSIC) and Barcelona GSE

[7] Gubler Zachary J 2013 rdquoInventive Funding Deserves Creative Regulationrdquo Wall Street Journal

[8] remer Ilan Yishay Mansour and Motty Perry 2014 rdquoImplementing the Wisdom of the CrowdrdquoJournal of Political Economy 122 no 5 988-1012

[9] Kamenica Emir and Matthew Gentzkow 2011 rdquoBayesian Persuasionrdquo American Economic Review101(6) 2590-2615

[10] Kuppuswamy Venkat and Barry L Bayus 2013 rdquoCrowdfunding creative ideas the dynamics of projectbackers in kickstarterrdquo SSRN Electronic Journal

[11] Lenz Rainer 2015 rdquoKonsumentenschutz Im Crowdfunding (Investor Protection in Crowdfunding)rdquoerschienen im rdquoJahrbuch Crowdfunding 2015rdquo hrsg Gajda o F Schwarz und K Serrar Wiesbaden2015 S 50 - 54

[12] Rochet Jean-C and Jean Tirole 2003 rdquoPlatform Competition in Two-Sided Marketsrdquo Journal of theEuropean Economic Association Vol 1 pp 990-1029

[13] Schwartz Andrew A rdquoThe Digital Shareholderrdquo 2015 Minnesota Law Review Vol 100 No 2 pp609-85

[14] Strausz Roland 2016 rdquoA Theory of Crowdfunding - A Mechanism Design Approach with DemandUncertainty and Moral Hazardrdquo CESifo Working Paper Series No 6100

A Appendix

In this part of the appendix I derive optimal disclosure policy and fee structure under the regulated disclosurerequirements

To derive the optimal disclosure policy first note that while the investorrsquos decision depends only on therealized posterior the innovatorrsquos decision depends on the entire signal rule (ie induced distribution ofposteriors) and hence we cannot readily apply Kamenica amp Genzkow (2011) concavification result

I first solve for the optimal disclosure assuming the platform sets 0 fees for the investor and innovatorfor any disclosure requirements chosen by SEC Then I will argue that even if the platform is free to chooseany cP (d) isin CP (d) the optimal disclosure requirements chosen by SEC under the relaxed problem remainsthe same

Lets first account for the expected payoffs for the innovator and investor in case the innovator plays Edenoted ZE(p) and ZI(p) respectively

ZE(p) =

minusc if p isin [0m)

p+ (1minus p)k minus c if p isin [m 1]

ZI(p) =

0 if p isin [0m)

A given disclosure requirement induces distribution over posteriors micro I argue that micro can always bereplaced by some microprime that contains at most 2 posteriors in its support To see this suppose micro contains morethan 2 posteriors Then there must be at least 2 posteriors q1 lt q2 such that they are both either on [0m)or both on [m 1] Suppose q1 lt q2 are both on [0m) We can find a new microprime such that it pools q1 q2 into a

single posterior qprime and leaving the other posteriors the same That is qprime = micro(q1)micro(q1)+micro(q2)q1 + micro(q2)

micro(q1)+micro(q2)q2 and

microprime(qprime) = micro(q1) + micro(q2) Because on [0m) we have Z1(p) Z2(p) both linear this modification of the originaldisclosure gives same expected value to both players conditional on posterior being on [0m) as the originaldisclosure requirements It is also easy to see that if a disclosure rule induces all the posteriors on either sideof m then it is equivalent to no disclosure policy

Hence to solve for the optimal disclosure requirement first we need to consider only the requirementswith exactly two signal realizations where one signal recommends the investor to invest and another recom-mends not to invest This will give us expected payoff ZlowastI (p) for the investor Then the investorrsquos expectedpayoff from the optimal disclosure requirement will be ZoptI (p) = maxZlowastI (p) ZndI (p) Recall that ZndI (p)is the value to the investor from no disclosure policy

To find ZlowastI (p) we need to solve the following linear program

ZlowastI (p) = maxf(σH |H)f(σH |L)

f(σH)(p(H | σH)minusm)

pf(σH | H) + (1minus p)f(σH | L)k ge c

1 ge p(H | σH) ge m

0 le p(H | σL) le m

σH σL are two signal realizations f(σH | H) f(σH | L) are respective probabilities of drawing thoserealizations p(H | σH) is updated probability on state H after observing σH The first constraint is tomake sure that the innovator plays E The second and third constraints make sure that the investor followsrespective recommendations Rewriting the problem

ZlowastI (p) = maxf(σH |θH)f(σH |θL)

f(σH | H)p(1minusm)minus (1minus p)f(σH | L)m

f(σH | H)f(σH | L) ge (1minus p)mp(1minusm)

(1minus f(σH | H))(1minus f(σH | L)) le (1minus p)m(1minusm)p

(f(σH | H) f(σH | L)) isin [0 1]2

Notice that all the constraints are relaxed and the objective increases if we increase f(σH | H) Hencef(σH | H) = 1 is part of the solution to the problem

Using this fact the first constraint becomes f(σH | L) ge cminusp(1minusp)k Observing that decreasing f(σH | L)

relaxes all the other constraints and increases the objective function we must have f(σH | L) = cminusp(1minusp)k If

p ge c then f(σH | L) = 0 and hence full disclosure is strictly optimal ZlowastI (p) ge 0 if and only if p ge mc(1minusm)k+m

implying that if p lt mc(1minusm)k+m then any disclosure policy leads to the innovator playing NE and the

market shutting down For p isin [ mc(1minusm)k+m c) partial disclosure is optimal where f(σH | L) = cminusp

(1minusp)k and

f(σH | H) = 1Thus we have

ZlowastI (p) =

k if p isin [T opt c)

It is straightforward to verify that ZoptI (p) = maxZlowastI (p) ZndI (p) = ZlowastI (p) for all p isin [0 1]Now we need to argue that the same policy is optimal if the platform is free to choose a fee structure

To see this consider any choice of d by SEC If d is such that it induces the innovator to play NE then thereis no fee structure that the platform could choose and induce E as cE ge 0 If d induces the innovator toplay E then it is weakly optimal for the platform to choose cE such that the innovator still wants to play EAs for the investor d induces two types of signal realizations - one that induces the investor to invest andanother that induces her not to invest For the signals that do not induce investment there is no cI(s) ge 0which would induce the investor to invest For the signals that induce investment the platform would setcI(s) such that investment is still induced Otherwise the platform would induce no investment for suchsignal and get 0 rents from the investor

Thus for any d the optimal fee structure would be such that none of the playersrsquo incentives is alteredThis completes the proof that the disclosure requirement in proposition 1 is optimal for SEC

In addition the optimal fee structure is anything for p isin [0 T opt) cE = 0 cI(σL) = 0 and cI(σ

H) =kp

cminus(1minusk)p minusm for p isin [T opt c) and cE = pminus c cI(σL) = 0 and cI(σH) = 1minusm for p isin [c 1]

B Appendix

In this part of the appendix I derive the optimal disclosure requirement and fee structure under the dereg-ulated disclosure requirements The derivation is done for the general linear preferences of the platformRecall that the platformrsquos utility is a weighted sum of investor welfare innovator welfare and profits Purelyprofit motivated platform is the special case

Fix a finite set of signal realizations S and distributions f(s | θ) Recall that (S f(s | θ)θisinΘ) is a disclosurerequirement Let Str sub S be set of signal which induce entry and investment and let Strc = SStr Recallalso that cP (d) = (cE cI(s)sisinS) is a fee structure The problem for the platform is formulated as follows

maxcP (d)isinCP (d)disinD

αI [p

sumsisinStr f(s | H)(1minusmminus c)I(s))minus

(1minus p)sumsisinStr f(s | L)(m+ cI(s)]+

αE [sumsisinStr (pf(s | H) + (1minus p)f(s | L)k)minus cminus cE ]+(1minus αI minus αE)[cE +

sumsisinStr (pf(s | H)+

(1minus p)f(s | L))(cI(s))

f(s | H)p

f(s | H)p+ f(s | L)(1minus p)ge m + cI(s) forall s isin Str (2)sum

sisinStrf(s | H)p+

sumsisinStr

f(s | L)(1minus p)k ge c+ cE (3)

Lemma 1 We can restrict attention to the disclosure requirements with at most 2 signal realizations

Proof The platformrsquos objective function can be rewritten asαI [p

sumsisinStr f(s | H)(1minusm)minus (1minus p)

sumsisinStr f(s | L)m]+

αE [sumsisinStr (pf(s | H) + (1minus p)f(s | L)k)minus cminus cE ]+

(1minus αI minus αE)cE + (1minus 2αI minus αE)sumsisinStr (pf(s | H)+

Case 1 Suppose (1 minus 2αI minus αE) le 0 Then it is optimal to set cI(s) = 0 for all s isin Str becausedecreasing cI(s) would increase objective and would relax 2 4 becomes αI [p

(1minus αI minus αE)cE

5 now depends only onsumsisinStr f(s | H) and

sumsisinStr f(s | L) Hence we can take two signal realizations

(σH σL) and set the new disclosure requirement as follows p(σH | H) =sumsisinStr f(s | H) p(σH | L) =sum

sisinStr f(s | L) The value of the objective remains the same and 3 is unaffectedTo see that 2 still holds observe that 2 implies

sumsisinStr f(s | H)p ge m

sumsisinStr (f(s | H) + (1minusp)f(s | L))

Hence by construction of our 2 signal policy this is still trueCase 2 Suppose (1minus 2αI minusαE) gt 0 This implies that an optimal policy must have 2 binding for each

s isin Str Summing over Str and rearranging 2 this implies

psumsisinStr

f(s | H)minussumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

(pf(s | H) + (1minus p)f(s | L))(cI(s))

Define r =psumsisinStr f(s|H)minus

sumsisinStr (pf(s|H)+(1minusp)f(s|L))msum

sisinStr (pf(s|H)+(1minusp)f(s|L)) 2 implies that the numerator is positive so r is

well defined Substituting this into 4 we getαI [p

sumsisinStr f(s | L)(m)]+

(1minus αI minus αE)cE + (1minus 2αI minus αE)[psumsisinStr f(s | H)minussum

sisinStr (pf(s | H) + (1minus p)f(s | L))m]

7 depends only onsumsisinStr f(s | H) and

sumsisinStr f(s | L) Hence we can apply same modification as in

case 1

r is what I refer to as project fee Given 2 signal realizations (σH σL) where σH is a recommendationfor trade the problem can now be rewritten as

αI [pf(σH | H)(1minusmminus r)minus (1minus p)f(σH | L)(m+ r)]+αE [pf(σH | H) + (1minus p)f(σH | L)k minus cminus cE ]+

(1minus αI minus αE)[cE + (pf(σH | H) + (1minus p)f(σH | L))r]

f(σH | H)p

f(σH | H)p+ f(σH | L)(1minus p)ge m + r (9)

f(σH | H)p+ f(σH | L)(1minus p)k ge c+ cE (10)

The optimal disclosure policy involves f(σH | H) = 1 as increasing f(σH | H) increases objective andrelaxes both constraint The optimization problem reduces to

maxcE rf(σH |L)

αI [p(1minusm)minus (1minus p)f(σH | L)m]+αE [p+ (1minus p)f(σH | L)k minus c] + (1minus αI minus 2αE)cE+

(1minus 2αI minus αE)(p+ (1minus p)f(σH | L))r

p+ f(σH | L)(1minus p)ge m + r (12)

p+ f(σH | L)(1minus p)k ge c+ cE (13)

cE ge 0 r ge 0 (14)

Solution to the problem 11

First recall that if p lt mc(1minusm)k+m then there is no way to jointly induce entry and investment Adding fees

to the model can only harm incentives for entry and investment The following cases focus on p ge ic(1minusm)k+m

Also recall that under assumption 1 we have k ge c

Case 1 mαIk ge αE ge max 1minusαI

2 1minus 2αIUnder this case 11 is always decreasing in cE and r Because decreasing cE and r also relaxes 12 and 13

we have cE = r = 0 mαI gt kαE also implies that 11 is decreasing in f(σH | L) Since decreasing f(σH | L)relaxes 12 and tightens 13 the optimal would be f(σH | L) = max0 cminusp

(1minusp)k Under this policy we can verify

that 12 is satisfiedHere I account for the optimal cE r f(σH | L) for case 1ccase1E = rcase1 = 0 and fcase1(σH | L) = max0 cminusp

(1minusp)k

Case 2 αE ge max 1minusαI2 1minus 2αI

We have cE = r = 0 mαI le kαE implies 11 is nondecreasing in f(σH | L) Increasing f(σH | L) tightens

12 and relaxes 13 Hence we set f(σH | L) = minp(1minusm)(1minusp)m 1 We can verify that 13 is satisfied under this

policyHere I account for the optimal cE r f(σH | L) for case 2

ccase2E = rcase2 = 0 and fcase2(σH | L) = minp(1minusm)(1minusp)m 1

Case 3 αE lt min 1minusαI2 1minus 2αI

Objective is strictly increasing in both r and cE Hence at the optimum 12 and 13 must bind We thensubstitute the constraints into the objective simplify and get the following problem

maxf(σH |θL)

(k minusm)f(σH | L)

Case 3 (a) k gt m

We can increase f(σH | L) until the first constraint binds We get f(σH | L) = minp(1minusm)(1minusp)m 1

Substituting this into second constraint we can verify that it is satisfiedHere I account for the optimal cE r f(σH | L) for case 3 (a)

fcase3(a)(σH | L) = minp(1minusm)

(1minusp)m 1 If m gt p then ccase3(a)E = p(m+(1minusm)k)

m minus c rcase3(a) = 0 If m le p

then rcase3(a) = pminusm ccase3(a)E = p+ (1minus p)k minus c

Case 3 (b) k le mWe can decrease f(σH | L) until the second constraint binds We get f(σH | L) = max cminusp

(1minusp)k 0Under this solution we can verify that the first constraint is also satisfied

Here I account for the optimal cE r f(σH | L) for case 3 (b)

fcase3(b)(σH | L) = max cminusp

(1minusp)k 0 If c gt p then ccase3(a)E = 0 rcase3(a) = kp

cminus(1minusk)p minusm If c le p then

rcase3(a) = 1minusm ccase3(a)E = pminus c

Case 4 max1minus 2αI 1minus (k+m)k αI le αE lt 1minusαI

211 is strictly increasing in cE and nonincreasing in r Hence at the optimum we must have r = 0 and

cE = p+ (1minus p)f(σH | L)k minus c ge 0 Using these we can rewrite the problem as

maxf(σH |θL)

(k(1minus αI minus αE)minus αIm)f(σH | L)

p+ f(σH | L)(1minus p)minusm ge 0

The objective is decreasing in f(σH | L) and hence we set f(σH | L) = max0 cminusp(1minusp)k One can check

that both constraints are satisfiedHere I account for the optimal cE r f(σH | L) for case 4fcase4(σH | L) = max0 cminusp

(1minusp)k If c gt p then ccase4E = rcase4 = 0 If c le p then rcase4 = 0 ccase4E = pminusc

Case 5 1minus 2αI le αE lt min 1minusαI2 1minus (k+m)

k αI11 is strictly increasing in cE and nonincreasing in r Hence at the optimum we must have r = 0 and

cE = p+ (1minus p)f(σH | L)k minus c ge 0 Using these we have similar problem as in case 4

Objective is increasing in f(σH | L) and hence we set f(σH | L) = minp(1minusm)(1minusp)m 1

Here I account for the optimal cE r f(σH | L) for case 5

fcase5(σH | L) = minp(1minusm)(1minusp)m 1 If p gt m then ccase5E = p + (1 minus p)k minus c rcase5 = 0 If p le m then

rcase5 = 0 ccase5E = p(m+(1minusm)k)m minus c

Case 6 1minusαI2 le αE lt 1minus 2αI

11 is strictly increasing in r and nonincreasing in cE At the optimum we must have cE = 0 andr = p

p+f(σH |θL)(1minusp) minusm The problem is rewritten as

maxf(σH |θL)

minus(1minus αI minus αE(1 +k

m))m(1minus p)f(σH | L)

Case 6(a) k gt m and k le m with 1minusαI1+km le αE

The objective is increasing in f(σH | L) So f(σH | L) = minp(1minusm)(1minusp)m 1 One can verify that both

constraints are satisfiedHere I account for the optimal cE r f(σH | L) for case 6(a)

(1minusp)m 1 If p le m then ccase6(a)E = 0 rcase6(a) = 0 If p gt m then rcase6(a) =

pminusm ccase6(a)E = 0

Case 6(b) k le m with 1minusαI1+km gt αE

The objective is decreasing in f(σH | L) So f(σH | L) = max cminusp(1minusp)k 0

Here I account for the optimal cE r f(σH | L) for case 6(b)

(1minusp)k 0 If p le c then ccase6(b)E = 0 rcase6(b) = pk

cminus(1minusk)p minusm If p gt c then

rcase6(a) = pminusm ccase6(a)E = 0

The figure below helps to visualize the solution For k = 1 it depicts the regions of (αI αE) for eachcase

C Appendix

I will write

Wf (G(htprime)) = (1minus δ)Efztprime

[ infinsumt=tprime

]for the average expected discounted payoffs for SEC at the beginning of period tprime under the disclosure policyf The following lemmas show how the problem can be simplified

First recall that ZG(ht) is the support of G at the history ht

Lemma 2 If ZG(ht) sube [0 mc(1minusm)k+m ) there is no policy which would avoid shutting down the market

Proof If a policy induces trade for some realizations of Str sube S sH sL it must be the case that forthose realizations the following ICs for the innovator and investor are satisfied

f(s | ht EH)EG(htE)(p)

f(s | ht EH)EG(htE)(p) + f(s | ht E L)(1minus EG(htE)(p))ge m forall s isin Str (15)

sumsisinStr

f(s | ht E L)(1minus pj) ge c iff type pj plays E (16)

15 ensures that investor invests where EG(htE)(p) is expectation of p conditional on observing innovatorplaying E in period t along with the history ht 16 says that type pj innovator would play E iff his expectedpayoff from doing this is no less than playing NE

Let pZG(ht)be an upper bound of ZG(ht) The left hand side of the inequality in 15 is nondecreasing in

EG(htE)(p) and 16 is independent of EG(htE)(p) Hence we can fix EG(htE)(p) = pZG(ht)

Then 15 and 16

are relaxed if we increase f(s | ht EH) hence we can set f(s | ht EH) = 1 for some str isin Str and take| Str |= 1

pZG(ht)(1minusm)

m(1minus pZG(ht))ge f(str | ht E L) (17)

pj + k(1minus pj)f(str | ht E L) ge c (18)

18 is further relaxed if we increase pj Since the maximum value p can take is pZG(ht)

substituting we

f(str | ht E L) gecminus pZG(ht)

(1minus pZG(ht)

)k(19)

Combining 17 and 19 we obtain that f(str | ht E L) satisfying both 17 and 19 exists iff pZG(ht)ge

mc(1minusm)k+m contradicting that ZG(ht) sube [0 mc

(1minusm)k+m )

Lemma 3 If ZG(ht) sube [c 1] full disclosure is the optimal policy

Proof If ZG(ht) sube [c 1] then fully disclosing information induces all types p isin ZG(ht) to play E ie thereis no problem of incentivizing the innovators to realize ideas The investor invests iff state is H Thismaximizes the expected stage payoffs of the investor Since this is true for all ZG(h) sube [c 1] it implies thatfor all possible beliefs paths the optimal static policy would be the same (full disclosure) Hence no matterhow beliefs evolve fully disclosing information period by period is the optimal dynamic policy as well

Now I formulate the dynamic programming problem for SEC In order to prove that the problem is welldefined and to characterize the optimal dynamic disclosure I take the following approach first I considera problem where constraints for the innovator are ignored (the relaxed problem) Assuming that the valuefunction exists I derive necessary conditions that it must satisfy (claims 1 2 and 3) Then I show that underthose necessary conditions investorrsquos constraints are satisfied This step enables me to drastically simplifythe problem In particular I reduce the space of disclosure requirements over which I need to optimize Giventhis it becomes easy to prove existence uniqueness monotonicity and continuity of the value function Inthe end I show that the optimal policy is a threshold policy in the belief about high success rate

The problem is formulated as follows

V (G(ht)) = max(Stf(middot|htEθt)θtisinΘ)

(1minus δ)sumsisinStr [f(s | ht EH)P (ZG(htE))EG(htE)(p)(1minusm)minusf(s | ht E L)P (ZG(htE))EG(htE)(1minus p)m]+

δ[sumsisinStr f(s | ht EH)P (ZG(htE))EG(htE)(p)V (G(ht E I sH))+sum

sisinStr f(s | ht E L)P (ZG(htE))EG(htE)(1minus p)V (G(ht E I sL))+[sumsisinStrc f(s | ht EH)P (ZG(htE))EG(htE)(p)+sum

sisinStrc f(s | ht E L)P (ZG(htE))EG(htE)(1minus p)]V (G(ht ENI s))+(1minus P (ZG(htE)))V (G(ht NENI))]

f(s | ht EH)EG(htE)(p) + f(s | ht E L)(1minus EG(htE)(p))ge m forall s isin Str

sumsisinStr

Here P (ZG(htE)) is the probability of ZG(htE) under G(ht)

Claim 1 V (G) is convex in the relaxed problem

Proof I consider the problem 20 without the first constraint Fix λ isin [0 1] and G = λGprime+ (1minusλ)G

primeprime I will

denote by fG a dynamic optimal policy when we start from public belief G I will use the notation VfG(G)

We know that supfλWf (Gprime) + (1minus λ)Wf (G

primeprime) le λVfGprime (G

prime) + (1minus λ)VfGprimeprime (G

primeprime) is true Hence if we show

that there exists some f such that VfG(G) le λWf (Gprime) + (1minus λ)Wf (G

primeprime) then we will obtain the convexity

I will construct such fStart from some ht such thatG = G(ht)Define f in the following manner set (Stprime f(middot | htprime E θtprime)θtprimeisinΘ) =

(SGtprime fG(middot | htprime E θtprime)θtprimeisinΘ) for all histories which proceed from ht

In particular at ht I set (St f(middot | ht E θt)θtisinΘ) = (SGt fG(middot | ht E θt)θtisinΘ) Since G = λG

prime+(1minusλ)G

primeprime

the first period expected stage payoff to the SEC is linear in G and the expression in the second constraintdoes not depend on G we must have that the first stage payoff to the SEC under fG equals first stage payoffunder f

Now consider any history htprime Let probG(ht

prime) be its probability underG I claim that probG(ht

prime)=λprobGprime(h

tprime)+(1 minus λ)probGrdquo(ht

prime) Conditional on the true market potential pj the history is determined by disclo-

sure policies and agentsrsquo actions which are the same under f and fG by construction This fact andG = λG

prime+ (1minus λ)G

primeprimeimply the claim Now we see that VfG(G) = λWf (G

prime) + (1minus λ)Wf (G

primeprime) as after any

given history fG and f are the same by construction first period expected stage payoff to the SEC is linearin G and the expression in the second constraint does not depend on G

Claim 2 In the relaxed problem the optimal policy uses at most two signal realizations (σH σL) in eachperiod and sets f(σH | ht EH) = 1 for all ht

Proof The second part follows because the second constraint in problem 20 is relaxed and stage payoffincreases if we increase f(s | ht EH) for some s isin Str In addition since V (G) is convex and increasingf(s | ht EH) for such s is the same as providing a better information (in the Blackwell sense) for thecontinuation game we have that expected continuation payoffs also go up with f(s | ht EH) for anys isin Str

Since f(σH | ht EH) = 1 for all ht the objective function and the second constraint now only dependonsumsisinStr f(s | ht E L) (not f(s | ht E L) separately) Hence we can restrict attention to two signal

realization policies

Using implications of claim 2 in the objective function we see that increasing f(σH | ht E L) strictlydecreases the objective and is only useful for incentivizing innovation (relaxing constraint 2) Hence theoptimal policy in the relaxed problem would use one of the 2 disclosure requirements in each period Onedisclosure requirement induces both pH and pL to innovate Another induces only pH to innovate Thuswe can rewrite the relaxed problem as

V (G) =

k + δ(1minus g) [pL((1minusm)k+m)minusmc)]k

maxf(σH |GEL)(1minus δ)[EG(p)(1minusm)minusf(σH | GEL)EG(1minus p)m]+

pL + kf(σH | GEL)(1minus pL) ge c

G(s) denotes posterior public belief given prior G and signal s g denotes g(pH) and 1minusg denotes g(pL)

Claim 3 f(σH | GEL) = cminuspLk(1minuspL)

Proof Continuation values are not affected by f(σH | GEL) and stage payoff strictly decreases inf(σH | GEL) Hence we make the innovatorrsquos constraint to bind

The relaxed problem can be rewritten as

V (G) = (21)

EG(1minus p)m]+

Lemma 4 The investorrsquos incentive constraints are never violated in the relaxed problem

Proof We need to verify that EG(p)

EG(p)+ cminuspL(1minuspL)k

(1minusEG(p))ge m for all G Since the expression increases

in g we take the worst case where g = 0 The condition reduces to pL ge mc(1minusm)k+m which is true for

pL isin ( mc(1minusm)k+m 1] (Otherwise apply Lemma 2)

Now we are ready to prove the existence and uniqueness of V that satisfies 21 I also show that V iscontinuous strictly increasing and convex in g

Lemma 5 There exists unique V that satisfies 21 Moreover it is continuous strictly increasing and convexin g

Proof Define operator T as follows

T (W )(g) = (22)

(1minus δ)g [pH((1minusm)k+m)minusmc)]

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

δ[EG(p)W (gH) + EG(1minus p)W (gL)]

It is straightforward to verify that T satisfies Blackwellrsquos sufficient conditions for the contraction mapping

Also it maps continuous and bounded functions to continuous and bounded functions as upper envelopeof continuous functions is continuous and current stage payoffs are bounded Also if we start with Wweakly increasing in g T will generate a function that is strictly increasing in g because stage payoffs fromchoosing either dpH or dpL are strictly increasing in g Hence we have a contraction mapping on the space ofcontinuous bounded and weakly increasing functions Moreover mapping happens from weakly increasingfunctions to the strictly increasing functions Using standard arguments from Stokey amp Lucas (1989) leadsto the result

To verify that V is convex in g one may use similar argument as in claim 1Now we are ready to conclude the proof of the proposition 6 I argue that the optimal dynamic disclosure

policy is a threshold policy in g

Proof If g = 1 then the optimal thing to do is to set static optimal policy for pj = pH This gives the

average discounted payoffs of [pH((1minusm)k+m)minusmc)]k This observation will be useful

We need to show that if for some gprimeprime lt 1 we have

V (Gprimeprime) = gprimeprime[pH((1minusm)k +m)minusmc)]

k+ δ(1minus gprimeprime) [pL((1minusm)k +m)minusmc)]

Then for all g gt gprimeprime we also have

V (G) = g[pH((1minusm)k +m)minusmc)]

k+ δ(1minus g)

[pL((1minusm)k +m)minusmc)]k

By contradiction suppose that there is some gprime gt gprimeprimesuch that V (Gprime) = (1 minus δ)[EGprime(p)(1 minus m) minuscminuspL

(1minuspL)kEGprime(1minusp)m]+δ[EGprime(p)V (G(sH))+EGprime(1minusp)V (G(sL))] gt gprime [p

H((1minusm)k+m)minusmc)]k +δ(1minusgprime) [pL((1minusm)k+m)minusmc)]

Since gprime gt gprimeprime we have that gprime [pH((1minusm)k+m)minusmc)]

k + δ(1minus gprime) [pL((1minusm)k+m)minusmc)]k

gt V (Gprimeprime) and hence V (Gprimeprime) lt V (Gprime) Since gprime gt gprimeprime V (Gprimeprime) lt V (Gprime) V (G) is convex and V (G) geg [pH((1minusm)k+m)minusmc)]

k + δ(1 minus g) [pL((1minusm)k+m)minusmc)]k for all G we must have that for all g gt gprime it is the

case that V (G) is strictly increasing Suppose this was not the case then there would be some g gt g ge gprime with

g [pH((1minusm)k+m)minusmc)]k +δ(1minusg) [pL((1minusm)k+m)minusmc)]

k le V (G) g [pH((1minusm)k+m)minusmc)]k +δ(1minusg) [pL((1minusm)k+m)minusmc)]

k leV (G) and V (G) le V (G) If we take λ isin (0 1) such that G = λGprimeprime + (1minus λ)G then V (G) gt λV (Gprimeprime) + (1minusλ)V (G) which violates convexity of V

Since for all g gt gprime it is the case that V is strictly increasing and V (G) ge g [pH((1minusm)k+m)minusmc)]k + δ(1minus

g) [pL((1minusm)k+m)minusmc)]k for all G it must be the case that V (g = 1) gt [pH((1minusm)k+m)minusmc)]

k which contradicts

that the value from optimal policy is [pH((1minusm)k+m)minusmc)]k at g = 1

D Appendix

Here I prove proposition 7 from the section 81 In the second period SEC solves maxg(pH minus mc) (1 minusg)(pL minusmc) + g((1minusm)pH minusm(1minus pH) cminuspL

(1minuspL)) I use gθ for the updated g after learning θ

SEC separates in the second period iff g ge Z0 where Z0 is

Z0 equiv(1minus pL)[pL minusmc)]

(1minus pL)[pL minusmc)] + (pH minus pL)(1minus c)m

Z0 completely pins down behavior of SEC in the second periodIf SEC chooses not to separate in the first period then in the second period there are 3 cases

Case 1 - gH gL ge Z0Case 2 - gH gL lt Z0

Case 3 - gH ge Z0 and gL lt Z0Note that gL ge Z0 and gH lt Z0 can never arise as gH gt gL

In addition I account for two more thresholds for g Z1 will be the threshold above which gH ge Z0 andZ2 threshold above which gL ge Z0 Those thresholds are

Z1 equiv pL(1minus pL)(pL minus ic)pL(1minus pL)(pL minus ic) + pH(pH minus pL)(1minus c)i

Z2 equiv (1minus pL)2(pL minus ic)(1minus pL)2(pL minus ic) + (1minus pH)(pH minus pL)(1minus c)i

Now for the three cases we account for thresholds Zcase1 Zcase2 and Zcase3 Zcase1 for instance wouldsay that if prior g is above Zcase1 and in the second period Case 1 obtains then SEC would separate Thesethresholds are calculated as follows- for each case we know what is the second period optimal policy under

each realization of θ which we substitute as a continuation value and find the optimal strategy in the firstperiod Those thresholds are given by

Zcase1 equiv (1minus δ)(1minus pL)(pL minus ic)(1minus δ)(1minus pL)(pL minus ic) + (pH minus pL)(1minus c)i

Zcase2 equiv (1minus pL)(pL minus ic)(1minus pL)(pL minus ic) + (1 + δ)(pH minus pL)(1minus c)i

Zcase3 equiv (1minus δpL)(1minus pL)(pL minus ic)(1minus δpL)(1minus pL)(pL minus ic) + (1 + δ(1minus pH))(pH minus pL)(1minus c)i

One can verify that Z2 gt Z1 gt Zcase2 gt Zcase3 gt Zcase1 Case 2 can obtain only for g isin (Z1 Zcase2]and increasing g beyond Z1 gets us to the case 3 Because Zcase3 lt Z1 CFP still separates as we increase gfurther Beyond Z2 we go to case 1 and since Zcase1 lt Z2 CFP still separates Hence we have a thresholdpolicy and the threshold is given by glowast = Zcase2 This completes the proof of proposition 8

Introduction

When is regulation unnecessary

What is the value of reputation systems to SEC

Regulatory Experimentation

Literature Review

The Model

Regulated Disclosure Requirements

Full Disclosure

No Disclosure

Optimal Disclosure

Discussion

Unregulated Disclosure Requirements

Regulation vs Deregulation

Reputation Systems

Experimentation

Extensions and Additional Results

Privately Informed Innovator

Not Purely Profit Motivated Platform

I=1

E=1

Innovator Moral Hazard

2 Period Version of the Dynamic Model

Dynamic Model with the Platform

Conclusion

Appendix

Appendix

Appendix

Appendix

Contents

1 Introduction 311 When is regulation unnecessary 412 What is the value of reputation systems to SEC 513 Regulatory Experimentation 5

2 Literature Review 6

3 The Model 7

4 Regulated Disclosure Requirements 941 Full Disclosure 1042 No Disclosure 1043 Optimal Disclosure 1144 Discussion 12

5 Unregulated Disclosure Requirements 1351 Regulation vs Deregulation 14

6 Reputation Systems 15

7 Experimentation 17

8 Extensions and Additional Results 2081 Privately Informed Innovator 2082 Not Purely Profit Motivated Platform 22

821 αI = 1 23822 αE = 1 23

83 Innovator Moral Hazard 2384 2 Period Version of the Dynamic Model 2485 Dynamic Model with the Platform 24

9 Conclusion 26

A Appendix 28

B Appendix 29

C Appendix 33

D Appendix 37

1 Introduction

2 Literature Review

3 The Model

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

1 Introduction

2 Literature Review

3 The Model

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

2 Literature Review

3 The Model

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

2 Literature Review

3 The Model

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

2 Literature Review

3 The Model

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

3 The Model

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

cI(s)1s)

Assumption 1 k ge c

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

41 Full Disclosure

I (p) is

ZfdI (p) =

0 if p isin [0 c)

42 No Disclosure

ZndI (p) =

0 if p isin [0m)

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

ZoptI (p) =

forp isin [T opt c)

SECrsquos Payoff

T opt c 10

Optimal Disclosure

Full Disclosure

Figure 1

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

44 Discussion

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

(1minusm)k+m

(1minusp)m 1

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

7 Experimentation

[infinsumt=0

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

(1minusm)k+m) Lemma

sumsisinStr

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

V (G) =

EG(1minus p)m]+

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

Figure 2

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

maxcP (d)d

f(s | H)p

sisinStr

c+ cE(sumsisinStr

f(s | L)(1minus p))

cE le k

maxf(σH |θL)

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

821 αI = 1

822 αE = 1

E = 0 and r = 0

maxf(σH |θL)

(1 + kf(σH | L))2

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

radiccminus1k

4] and

ii) partglowast

partcge 0

iii) partglowast

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

cminus pL

(1minus pL)]

V (G) =

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

EG(1minus p)m]+

9 Conclusion

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

References

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

A Appendix

ZE(p) =

ZI(p) =

0 if p isin [0m)

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

(f(σH | H) f(σH | L)) isin [0 1]2

(1minusp)k and

ZlowastI (p) =

H) =kp

B Appendix

αI [p

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

f(s | H)p

sisinStrf(s | H)p+

sumsisinStr

psumsisinStr

(pf(s | H) + (1minus p)f(s | L))m = (6)

sumsisinStr

case 1

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

f(σH | H)p

maxcE rf(σH |L)

cE ge 0 r ge 0 (14)

(1minusp)k

maxf(σH |θL)

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

Case 3 (a) k gt m

maxf(σH |θL)

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

C Appendix

I will write

[ infinsumt=tprime

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

sumsisinStr

Then 15 and 16

pZG(ht)(1minusm)

substituting we

(1minus pZG(ht)

)k(19)

(1minusm)k+m )

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

sumsisinStr

primeprime I will

prime+(1minusλ)G

primeprime

prime+ (1minus λ)G

V (G) =

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

V (G) = (21)

EG(1minus p)m]+

T (W )(g) = (22)

k + δ[gW (1) + (1minus g)W (0)]

EG(1minus p)m]+

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

k+ δ(1minus g)

k which contradicts

D Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

Introduction




Literature Review

The Model


Full Disclosure

No Disclosure

Optimal Disclosure

Discussion



Reputation Systems

Experimentation




I=1

E=1




Conclusion

Appendix

Appendix

Appendix

Appendix

Documents

Optimal Disclosure on Crowdfunding Platforms...Optimal Disclosure on Crowdfunding Platforms Giorgi Mekerishvili Economics Department Pennsylvania State University Email: [email protected]