Optimal R&D Project Selection Mechanisms�

Talia Bar

Department of Economics

Cornell University

Sidartha Gordon

Department of Economics

Université de Montréal and CIREQ

July 21, 2010

Abstract

This paper derives an optimal mechanism for a principal (a �nancing government

agency or an executive of a company) to choose up to m projects from a selection of

n R&D projects when resources are limited. R&D projects are risky and the principal

is not fully informed of each project�s prospects. Project managers hold private infor-

mation about the quality of their project. They compete for resources as they would

bene�t if their own project is �nanced. In the optimal mechanism, the principal does

not always award funding to the highest probability of success project. There is a re-

gion of high quality projects where the allocation of funds does not depend on reported

qualities. Managers have limited liability constraints, hence to provide incentives for

truth-telling project managers that were not �nanced may need to be compensated.

JEL classi�cation codes: D82, O32

Keywords: adverse selection, information acquisition, mechanism design, project selec-

tion, R&D.

�We are indebted to Dirk Bergemann for his guidance and invaluable advice. We are also grateful to

Andreas Blume, Steve Coate, Michel Poitevin, Nicolas Sahuguet, Alvaro Sandroni and seminar participants

at Cornell University, University of Bern, Université de Montréal, University of Pittsburgh, IIOA 2008 and

the Game Theory World Congress 2008 for enlightening conversations.

1

1 Introduction

Many �rms and government agencies face the need to decide how to allocate their limited

research and development (R&D) resources. R&D projects are often risky, and the like-

lihood that a project succeeds or its net expected return might not be known to one not

su¢ ciently familiar with its technical details. Project managers hold private information

about the quality of their projects, and typically prefer that their own project be �nanced.

Hence, their interests are not perfectly aligned with those of the funding decision maker,

and an agency problem arises. As Paul Sharp (vice president at SmithKline) and Tom

Keelin (1998) described:

Major resource-allocation decisions are never easy. For a pharmaceuticals

company like SmithKline Beecham, the problem is this: How do you make good

decisions in a high-risk, technically complex business when the information you

need to make those decisions comes largely from the project champions who

are competing against one another for resources?

Our paper studies a principal�s project selection problem under uncertainty and asym-

metric information. The principal can be for example an executive in a company, a �rm

or a government �nancing agency, we will refer to the principal as the �nancing company.

Facing a resource constraint, the principal can only choose to pursue a limited number of

projects (up to m projects) from a given selection of projects. Each project if �nanced

would yields some return to the principal, projects of higher quality yield higher return.

Project managers are pre-quali�ed � there is a �xed set of projects to choose from and

the principal knows the distribution of qualities for each project. Managers have private

information about the quality of their project. For example, a manager might know the

probability of success of his project. A project manager enjoys a private bene�t if his own

2

project is �nanced, and so, unless given the right incentives, may overstate the quality

of his project. Thus, to optimally select a project the principal faces costly information

acquisition.

We take a mechanism design approach and look for an optimal direct mechanism. The

mechanism determines a project selection rule (which depends on reported project quality)

and a transfer scheme that is set to provide truth telling incentives. Transfers are restricted

to be non-negative �a limited liability constraint. Our problem is similar to a design of an

optimal auction but in our problem, the �auctioneer�(the �nancing company or agency)

values the object as he will be the residual claimant of a successful project, the �bidders�

(project managers) have private information not only over their own valuations but also on

the principal�s valuation which would vary depending on his project selection and �nally,

a limited liability constraint is imposed.

We begin by allowing mechanisms that only depend on reported project qualities and

not on the �nanced project�s realized outcome. Such schemes may be appealing for R&D

projects that take a long time to be completed. For example according to Nicholas (1994)

(who interviewed Merck�s CFO Judy Lewent) it takes about 10 years to bring a drug to

market, and once there, 70% of the products fail to return the cost of capital. Moreover, in

some cases, for example in basic research projects, verifying success might prove di¢ cult.

We also characterize the optimal mechanism when transfers may be made contingent on

the realization of project selection and on realized returns of the projects. We show that in

general mechanisms that allow such state dependent transfers can yield higher pro�ts than

the simple mechanisms we discussed earlier. But, when private bene�ts of project managers

are constant, the �nancing company cannot improve its pro�ts by making transfers state

dependent.

In an e¢ cient resource allocation, under the constraint that up to m projects can

be selected, resources should be allocated to the highest quality projects. However, in

the presence of information asymmetry, the optimal mechanism is such that sometimes

3

the principal selects inferior projects. When project managers are ex ante identical (a

symmetric mode), the optimal project selection rule involves a cuto¤ quality such that if

at most m project managers report a quality above the cuto¤, the highest quality projects

would be �nanced. But if more project managers report high quality, the principal�s

allocation rule randomizes between those projects whose quality exceeds the cuto¤. Hence,

suboptimal projects may be �nanced. Similarly in the asymmetric model, there are project

speci�c cuto¤ quality levels so that pooling occurs when there are more than enough

projects with qualities above their corresponding cuto¤s.

Intuitively, the principal faces a trade-o¤between �nancing the most promising project,

and saving on costs of eliciting information. When project managers� private bene�ts

from being �nanced are very high, the cost of acquiring information may be too high,

and the optimal mechanism would be to choose a project independent of reported quality.

A mechanism that always chooses a fully e¢ cient allocation is only optimal if project

managers have no private bene�ts (or if information is complete). Between these two

extremes, there is a subset of the project qualities state space (one with multiple promising

projects) over which the principal�s selection is independent of the quality of projects in

that range. The principal must worry about low quality project managers overstating the

quality of their projects so as to get selected. By making a selection which is not necessarily

optimal in the region of high quality projects the incentive to overstate one�s quality is

reduced and thus, the costs of providing incentives to report truthfully are reduced. The

critical value that de�nes a �high� quality depends on the private bene�ts, and on the

distribution of project qualities.

The optimal mechanism involves transfers that help the principal to elicit information.

Managers of projects that were not selected may get transfers that compensate them for

truthfully reporting lower quality. While rewarding a manager of a project that is not

selected may seem surprising at �rst, such mechanism can be important for providing

incentives and can be implement in practice. Rewards for non-selected projects may be

4

monetary or of other nature such as being transferred to a (possibly better paid, more

prestigious and more secure) position involving administrative duties or even leisure rather

than R&D work. In his study of incentives to motivate exploration, Manso (2006) found

results with a similar �avor:

�...the incentive schemes that motivate innovation are fundamentally di¤erent from

standard pay-for-performance schemes. The optimal compensation scheme that motivates

innovation exhibits substantial tolerance (or even reward) for failure...�.

Manso suggests that some companies may have corporate cultures that tolerate failure

to encourage managers to take risks. We believe such a tolerance serves another purpose,

which is to screen the good from the bad projects at an early stage.

While our paper is focused on the selection of R&D projects, our model may also be

applicable to other situations. For example when a constrained employer needs to decide

who should retire. Employees bene�t from continuing to work, their bene�ts are positively

correlated with their productivity, and some aspects of it is private information (such as

the person�s health). Those who retire receive a monetary incentive to do so. Similarly an

employer who needs to decide who to promote. Or consider parents who need to decide

which of their kids to send to college, those not sent to college may receive other monetary

rewards.

The rest of the paper is organized as follows: in the remainder of this section we review

related literature; section 2 presents the model; in section 3 we consider the simple case of

deciding whether or not to �nance a single project; the optimal mechanism for the general

problem is derived in section 4. In section 5 we present examples and discuss comparative

statics and the ine¢ ciency of the optimal mechanism; in sections 6 we allow transfers to

depend on true quality or some outcome correlated with it (such as realized success or

failure of a project). Section 6 provides concluding remarks. The proofs are provided in

the Appendix.

5

1.1 Related Literature

Our work is related to a large body of literature on auctions and mechanism design and

also, in terms of our main application of interest, to a literature on R&D project selection.

Several of our derivation follow the analysis in Myerson (1981). Myerson searches for the

auction that maximizes a seller�s revenue. There are two main di¤erences between his

problem and ours. One is that he assumes an individual rationality constraint, while we

have a limited liability constraint. The second is that our principal cares about the quality

of the selected project and the identity of the winner, unlike Myerson�s auctioneer. Manelli

and Vincent (1995) study the problem of procuring a good, where the potential sellers have

private information on the quality. Like us, their auctioneer cares about quality, thus about

who wins the auction. But like Myerson, they consider an individual rationality constraint

instead of a limited liability constraint. Another key di¤erence is that in their model, it

is costly for sellers to be selected and this opportunity cost increases with quality, while

in our model, managers private bene�ts from being selected (getting their R&D project

�nanced) is weakly increasing in quality. For example, in our model managers of projects

that are more likely to succeed bene�t more if their project is �nanced. Our limited

liability constraint is essentially a budget constraint. Various authors have searched for

optimal auctions under budget constraints. La¤ont and Robert (1996) analyze a symmetric

mechanism design problem which is similar to the symmetric case in our model. One

important di¤erence is that in their model the principal maximizes revenues (not a return

which increases with �rm type). Unlike us, they use an individual rationality constraint,

which is binding at the optimal mechanism for lower types. Maskin (2000) searches for

an e¢ cient auction when sellers have a budget constraint and also cannot receive transfers

from the auctioneer. In our paper, as in these two papers, pooling is created at the top

�for high types. In Maskin (2000), this is because transfers are restricted to be between

two values (0 and the budget constraint). In our paper, transfers are constrained on

one side only (limited liability) but paying managers is costly to the �nancing company.

6

Because the constraints are di¤erent, the pooling thresholds are obtained in di¤erent ways.

Importantly, each bidder�s threshold in both Maskin�s and La¤ont and Robert�s models

depends on characteristics of the entire game. In our paper, each manager�s threshold only

depends on his own parameters.1

A mechanism design approach has been introduced to other R&D related applications.

Schotchmer (1999) takes this approach in her analysis of the patent renewal system. The

patent authority in her paper is set to design a policy to induce innovators to �nance

R&D projects when it is socially desired. Patents serve as a reward which is contingent on

success. Economists have looked at several aspects of executive selection of risky projects.

Lambert (1986) uses optimal contracting theory to study the incentives of risk-averse ex-

ecutives to choose between a risky project and a safe one. The agency problem in his

model occurs because the executive must expend e¤ort to produce additional information

about the pro�tability of the available projects. Shin (2008) analyses a model in which a

principal selects one of two projects: one familiar (on which information is common knowl-

edge), information on the other project must be acquired by the agent. His paper provides

a rationale for why an organization often generates a bias in favor of a new project. As

in our work costly information acquisition generates ine¢ ciency in project selection. Im-

portantly however, in our analysis multiple projects can be selected from a larger pool of

candidate projects. Sappington (1982) studied optimal regulation of R&D aimed at cost

reduction. The single �rm in his model has more precise information about the cost reduc-

tion technology, and also its e¤ort to reduce cost is unobserved creating a moral hazard

problem.

Our work also relates to a large management and �nance literature on capital allo-

cation. This literature usually considered the decision to �nance or not �nance a single

project, while in our paper a key component is that project managers compete for funds.

1Che and Gale (e.g. 1998, 2000) analyze auctions with budgets constraints. However, their focus is on

situations where managers hold private information on their budget, which is not the case here.

7

Asymmetry of information in this literature usually concerned with incentives to exert ef-

fort (moral hazard) while we focus on the adverse selection problem. We brie�y describe

a few of the contributions in this literature. Harris and Raviv (1996) study the capital

allocation problem within �rms. Their model consists of two players: headquarters and

a single division manager who has private information about the production technology.

In their model, an optimal scheme involves an initial capital spending limit. If the man-

ager requests additional funding he may be subject to auditing. Antle and Eppen (1985)

study budget allocation decisions when the manager has private informations regarding

the return of investment. The owner of the �rm cannot observe how much of the budget is

invested in the project and to what extent the manager is engaged in organization slack.

The owner will present the manager with a menu of returns and budget allocations. Their

model explains organization slack, and under allocation of resources. Zhang (1997) com-

pares three capital budgeting rules. Net present value, hurdle rate and capital rationing.

The paper shows that �rms may voluntarily impose capital rationing. The capital budget

serves as a device to control managerial shirking. If the �rm has a large budget and invests

in all pro�table projects, managers would under report the project quality so they can

put less e¤ort. In contrast to Zhang (1997) in which the single manager would have an

incentive to understate his success probability, in our analysis absent incentives managers

who compete over limited R&D resources might overstate the quality of their project.

2 Model

A principal faces a choice between n research projects i 2 N = f1; :::; ng. Being resource

constrained, the principal can choose to �nance up to m � n projects. Each project is

represented by an agent (a researcher or project manager). Project i�s true quality qi;

which is its net expected return for the �nancing company, is drawn from a distribution

Fi(qi) on [qi; qi] � R with a positive density function fi(qi) > 0: We assume that the qi are

independent random variables. In addition to the n projects, the �nancing company has

8

an outside option, which is a project whose quality is known to the �nancing company to

be qo 2 R. The quality of each project is private information of its manager.

Project managers enjoy a private bene�t bi(qi) � 0 if their own project is �nanced.

We assume this private bene�t is non-decreasing in the quality of the project b0i(qi) � 0:2

The private bene�t function bi(qi) represents an expected value of payo¤s contingent on

success or failure of the project.3 The focus of our paper is project selection. Thus, for

simplicity we abstract in our analysis from moral hazard considerations which may arise

after a project was selected, during the research stage.

The �nancing company decides which projects to �nance, and whether to o¤er addi-

tional payments to project managers. An allocation p = (p1; :::pn) is a probability vector

with pi representing project i�s probability to be selected.4 A transfers vector t = (t1; :::; tn)

indicates how much money each manager receives from the �nancing company. We do not

allow the �nancing company to take money from the managers, i.e. managers have limited

liability, t � 0: In a direct mechanism, each project manager reports her quality and the

�nancing company selects projects and transfers money to managers depending on the

reports. Thus it is a function q 7! (p (q) ; t (q)) :5

In the game played, the �nancing company �rst chooses and commits to a direct mech-

2Stern (2003) found, using survey data on job o¤ers made to PhD biologists, that o¤ers which contained

science-oriented provisions, were associated with lower monetary compensation and starting wages. Such

result lends support to the intuitive assumption that project managers enjoy private bene�ts from having

their own projects �nanced.3When project quality represents the probability of success of a binary random variable, we can write

b(qi) = qibsi + (1 � qi)bfi where bsi is the private bene�t in case the project succeeds and b

fi is the private

bene�t in case it fails. It is natural to assume in this example that bfi < bsi :

4Alternatively one can think of project assignments� probability vectors over subsets of N representing

the probability that each subset of projects is �nanced. It is more convenient to deal with allocations than

assignments. One can verify that every assignment induces an allocation and every allocation is induced

by at least one assignment.5 In Sections 6, we also consider more general mechanisms with transfers that depend on the true quality

of projects or on observed realized returns of selected projects.

9

anism. The managers of the n projects simultaneously report to the �nancing company

their projects�qualities and the allocation p (q) and transfers t (q) are realized. We look

for an ex-post incentive compatible mechanism. Hence, the �nancing company�s objective

is to maximize its expected pro�ts:

maxfpi(q);ti(q)gi

(Eq

"nXi=1

[qipi(q)� ti(q)]#)

(1)

subject to

feasibility :nXi=1

pi(q) � m and 0 � pj(q) � 1 for all q and j; (2)

limited liability (LLi) : ti(q) � 0 for all i and for all q; (3)

and the following incentive compatibility constraints for all q; for all i; and for all q0i,

(ICi) : pi(q)bi(qi) + ti(q) � pi(q0i; q�i)bi(qi) + ti(q0i; q�i); (4)

where q = (q1; :::qn), with its jth component being project j�s quality, and (q0i; q�i) equals

q everywhere excepts in its ith coordinate which is replaced with q0i:6

Finally, we introduce notation that would be helpful in later analyses. For any quality

of project i; qi, let

Gi (qi) = qi + bi(qi) + b0i(qi)

Fi (qi)

fi(qi)(5)

be the virtual return that is obtained from selection project i: Throughout the paper,

we assume that this function is increasing. Note that the sum of the �rst two terms is

necessarily increasing. The function is monotone as long as the derivative of the third term

is not too negative. This holds true for a wide range of parametric assumptions for example

if bi(q) is convex and Fi (qi) is a uniform or concave distribution. It also holds true for any

distribution when bi(qi) is constant.6This constraint is a dominant-strategy incentive compatibility condition, as opposed to a Bayesian

incentive compatibility constraint often used. Manelli and Vincent (2008) have shown that for problems

such as these (independent private values, transferable utilities), essentially the same allocations are im-

plementable under both types of constraints and thus the same values are reached at the optimum. Thus,

there is no fundamental di¤erence in using one constraint versus the other.

10

3 One Project

Let us �rst consider the case where n = m = 1: The derivations of this simple case

will motivate our analysis of the general problem which will be addressed in section 4.

We keep the subscript i in the notations even though i = 1 throughout this projects

so that we can use the equations we de�ne here for the general case in the next section

avoiding repetition. With only one project available, the �nancing company faces the choice

between selecting the project or the outside option. In the absence of a mechanism to elicit

quality information (no quality report and no transfers), the decision on what to select

should be made by comparing the expected quality of the project with the quality of the

outside option. If E (qi) � qo; the project should be selected with probability 1. If instead

E (qi) < qo, the �nancing company should choose its outside option with probability 1.

3.1 Threshold Mechanisms

When a quality report and a transfer is feasible, a natural mechanism to consider is to

�nance the project with probability 1 whenever its reported quality is at least a threshold

q�, when qi � q� and to select the outside option when qi < q�. To ensure truthtelling

given the suggested allocation, no compensation is needed when the project is �nanced,

but the manager of the project should be compensated whenever his project is not �nanced.

For each qi < q�; the manager should receive at least bi (qi) ; since this is the utility he

would get if he would report a high quality level in order to be selected. On the other

hand, the compensation should be the same, for all values of qi < q� or else there would

sometimes be an incentive to misreport a probability below the threshold so as to get a

higher transfer. In conclusion, the manager needs to receive a transfer of at least bi (q�) ;

whenever the project�s quality is lower than q�. We call this family of selection rules and

transfer schemes indexed by q� 2 [qi; qi] the threshold mechanisms. Clearly, �nancing the

project no matter what, without transfers, is equivalent to the threshold mechanism with

threshold qi: We refer to the mechanism of selecting the outside option no matter what,

11

without transfers, the outside option. Note that the outside option is not a threshold

mechanism. In the remainder of this section, we look for the optimal mechanism among

all threshold mechanisms and the outside option. Later we will establish that restricting

the search to this class is without loss of generality.

3.2 The Optimal Threshold Mechanism

In a threshold mechanism with threshold q�; when the project quality is qi < q� the outside

option is selected and a transfer bi(q�) is paid, hence the �nancing company�s payo¤ would

be qo� bi(q�); when the project quality is qi � q� the project is selected and no transfers are

given, the �nancing company�s payo¤ is qi. The expected pro�t for the �nancing company

who chose the threshold mechanism q� equals

V (q�) =

Z q�

qi

(qo � bi(q�)) fi(qi)dqi +Z qi

q�

qifi (qi) dqi

= qoF (q�)� bi(q�)F (q�) +Z qi

q�

qifi (qi) dqi

= qoF (q�) +

Z qi

q�

�bi (qi) fi (qi) + b

0i (qi)F (qi)

�dqi � bi (qi) +

Z qi

q�

qifi (qi) dqi:

Rearranging and using the de�nition of Gi in (5) we obtain

V (q�) =

Z qi

q�

Gi (qi) fi (qi) dqi � bi (qi) + qoF (q�) : (6)

We look for the quality threshold q�� that maximizes this value. Because Gi (�) is

increasing, the �nancing company�s pro�t V (q�) is single-peaked in q�. In an interior

solution,

V 0 (q�) = �Gi (q�) fi (q�) + qofi (q�) = 0;

and therefore V (q�) is maximized at a quality q� that equalizes the virtual social surplus

Gi (q�) with the quality of the outside option qo. The threshold q�� of the optimal threshold

12

mechanism when the outside option is qo is such that

q�� =

8>>><>>>:qi if qo > Gi (qi) ;

G�1i (qo) if qo 2�Gi�qi�; Gi (qi)

�;

qi if qo < Gi�qi�:

From these expressions, we see that the optimal threshold q�� is a nondecreasing function

of the outside option qo: Since the outside option is not a threshold mechanism itself, we

still need to compare the optimal threshold mechanism and the outside option.

3.3 Optimal Threshold Mechanism vs the Outside Option

For any qo we found a threshold q� that maximizes the �nancing company�s expected

pro�t. We now ask when is the optimal threshold mechanism better than the outside

option? If qo � Gi(qi) then q� = qi and V (q�) = qo� bi (qi) which is worse than the outside

option qo. If qo � Gi(qi) then q� = qi and V (q�) = E[qi]: This is worse than the outside

option when E[qi] < qo and better when E[qi] > qo. Consider qo 2�Gi(qi); Gi(qi)

�for

which q� 2�qi; qi

�. The pro�t di¤erence between the threshold mechanism and the outside

option is V (q�) � qo. Because we found that the optimal threshold satis�es Gi (q�) = qo;

we can write the pro�t di¤erence as:

Hi (q�) :=

Z qi

q�

[Gi (qi)�Gi (q�)] fi (qi) dqi � bi (qi) : (7)

For any quality q� 2 [qi; qi] the function Hi(q�) is strictly decreasing, and Hi(qi) < 0: Thus,

when Hi(qi) � 0; there exists a unique solution to Hi(q�) = 0; or otherwise there is no

solution. Let

a�i =

8<: the solution to Hi(a) = 0 in [qi; qi] when Hi(qi) � 0

qi when Hi(qi) < 0: (8)

Now we have that for q� < a�; the threshold mechanism with threshold q� is better than

the outside option qo = Gi (q�) ; and for q� > a�; the opposite is true.

13

The value of a threshold mechanism q� depends on the outside option. We de�ne qio

to be the outside option quality such that the best threshold mechanism (given qio) yields

the same value as the outside option V (a�i ) = qio. With this outside option, the value of

the acquired information on quality exactly balances the (incentive) costs of acquiring this

information. By (6) the outside option quality qio solvesZ qi

a�i

(Gi (qi)� qio) fi (qi) dqi � bi (qi) = 0: (9)

Using (8) we �nd that the solution to this equation is given by

qio =

8<: Gi (a�) when H(qi) � 0

E [qi] when H(qi) < 0:(10)

The �nancing company would choose the optimal threshold mechanism if qo � qio and the

outside option if qo � qio:

The threshold mechanism can be thought of as if the �nancing company o¤ers the

candidate a choice between being selected for the project or being paid a �xed monetary

amount. If the candidate takes the money, the company goes with the outside option. The

optimal monetary amount that the company should o¤er and the total expected transfer to

the candidate are nondecreasing functions of the quality of the outside option. Essentially,

the company buys information on the quality of the applicant�s project. The better the

outside option, the higher the price the company must pay to acquire this information.

If the quality of the outside option is high enough (at least qio), it is better to chose the

outside option without acquiring any information.

3.4 Financing Company�s Pro�t

We now derive a simple expression of the �nancing company�s pro�t at the optimal mecha-

nism. Consider �rst the case qo � qio; so that the optimal threshold mechanism given qo is

better than the outside option. Then the corresponding optimal threshold satis�es q�� � a�.

14

Moreover, for all qi < a� we have Gi (qi) < qio and for all qi > a� we have Gi (qi) > qi

o:

Therefore,Z qi

q��

(min fGi (qi) ; qiog) fi (qi) dqi =Z a�

q��

Gi (qi) fi (qi) dqi +

Z qi

a�qiofi (qi) dqi:

The pro�t of the �nancing company at the optimal threshold mechanism is

V =

Z qi

q��

Gi (qi) fi (eqi) deqi � bi (qi) + qoF (q�)=

Z a�

q��

Gi (qi) fi (qi) dqi +

Z qi

a�qiofi (qi) dqi +

�Z qi

a�(Gi (qi)� qio) fi (qi) dqi � bi (qi)

�| {z }

=0

+ qoF (q�)

=

Z qi

q��

(min fGi (qi) ; qiog) fi (qi) dqi + qoF (q�)

= E (max fmin fGi (qi) ; qiog ; qog) : (11)

Let us denote xi = min fGi (qi) ; qiog and refer to it as the virtual quality of the project.

Project i is selected if its virtual quality exceeds the outside option and otherwise the

outside option is chosen. Hence V which we derived in (11) can be written as

V = E (max fxi; qog) : (12)

In the case where the outside option is better than the optimal threshold mechanism,

i.e. qo � qio, the pro�t of the �nancing company at the optimal mechanism is qo; which

also equals the expression (12). Therefore, this expression equals the optimal pro�t of the

�nancing company, in all cases. In this section, we restricted attention to a limited set of

mechanism (threshold mechanisms and outside option). As we will show in a more general

setting in the next section, the optimal mechanisms we provided here are optimal among

all feasible direct mechanisms.

4 Optimal Mechanisms in the General Case

In this section we search for an optimal mechanism that is, a pro�t maximizing incentive

compatible mechanism with a feasible project selection rule and transfers that satisfy the

15

limited liability constraint. We restrict attention to mechanisms for which transfers depend

only on reported qualities. Let the payo¤ of manager i in the optimal mechanism be

Mi (q) = ti(q) + bi(q)pi(q) (13)

Applying the �Mirlees trick� (see Mirlees (1974) and Fudenberg and Tirole (1992))

we express the principal�s pro�t maximization problem de�ned by (1)-(4) in a way that

facilitates its solution. The series of transformations that lead to this expression are familiar

for this type of models, but we include them in the proof in the Appendix for completeness.

Lemma 1 The target function in (1) can be written as

Eq

"nXi=1

[Gi(qi)pi(q)�Mi(qi; q�i)]

#:

For simplicity, and without loss of generality, we will assume that the outside option has

quality qo = 0: In lemma 2 we provide an upper bound for the value that the target function

(1) can obtain with any mechanism that satis�es the constraints (2)-(4). First we introduce

some notation. Given a vector of real numbers x = (x1; :::; xn+m) ; let�x(1); :::; x(n+m)

�denote the order statistics of x, i.e. the vector obtained by sorting the coordinates of x in

a nonincreasing order, so that x(1) � ::: � x(n+m): Let Sm (x1; :::; xn+m) be the sum of the

�rst m coordinates of the order statistics vector, i.e. the sum of the m highest coordinates

of x;

Sm (x) =

mXi=1

x(i):

The �nancing company needs to choose at most m out of n projects to �nance, it

obtains its outside option of qo = 0 for a project not �nanced. Let us represent its problem

as a choice of exactly m projects from the set of n +m project that contains the original

n projects and in addition m projects that have a known quality qo = 0: For any (n-

dimensional) vector of project qualities q we de�ne an (n+m)-dimensional vector of virtual

16

project qualities

x(q) =

0@min fG1 (q1) ; qo1g ; :::;min fGn (qn) ; qong ; 0; :::; 0| {z }m times

1A : (14)

where Gi and qoi are de�ned in (5) and (10). The �rst n coordinates represent the virtual

project qualities of the n projects and the last m coordinates represent the outside option

which can replace each of the m selections.

Lemma 2 For any mechanism that satis�es (2)-(4):

Vm � Eq [Sm (x(q))] :

For every m and n the upper bound of the value of the mechanism is thus the expected

value of the sum of the highestm virtual qualities out of the n+m available virtual qualities.

We will propose a mechanism which satis�es the constraints in the original problem and

reaches the upper-bound in Lemma 2. Therefore it is optimal.

If information was symmetric then the principal would have always chosen to �nance

the m highest quality projects (or as many that are better than the outside option). How-

ever, project qualities are private information of the project managers and this information

is costly for the principal to elicit. Because each project manager bene�ts if his project

is �nanced, absent incentives, and if the principle would have selected the highest quality

projects, managers would have had an incentive to overstate the quality of their projects

so as to be chosen. In any incentive compatible mechanism the principal needs to compen-

sate project managers for truth telling. Thus, when choosing which projects to fund the

�nancing company takes into account not only the quality of projects but also the cost of

eliciting information. We show that this is done by choosing the m highest virtual quality

projects instead of simply the highest m quality projects.

For the vector of virtual qualities x(q) in (14) x(m)(q) is its m-th highest coordinate.

There may be multiple projects with this virtual quality. The allocation we propose �nances

17

with probability 1 any project that has a virtual quality which is strictly higher than the

m-th highest virtual quality (there are less than m such projects). For the remaining

selections, the mechanism chooses with equal probabilities projects from the set of (one or

more) projects that have the m-th highest virtual quality. Other projects are not �nanced.

p�i (q) =

8>>><>>>:1 if xi (q) > x(m) (q)

m�ji2fi2N :xi(q)>x(m)(q)gjjfi2N :xi(q)=x(m)(q)gj if xi (q) = x(m) (q)

0 if xi (q) < x(m) (q)

: (15)

With this allocation, x(m)(q) is the lowest virtual quality that is selected with a positive

probability. To �nd transfers that ensure this allocation is incentive compatible we �rst

�nd the quality of project i such that G(qmi ) yields the virtual quality x(m)(q) (when it

exists). For each i 2 N; let

qmi =

8>>><>>>:qi if Gi (qi) < x(m)(q)

G�1i�x(m)(q)

�if Gi

�qi�� x(m)(q) � Gi (qi)

qi if Gi�qi�> x(m)(q)

: (16)

For the allocation rule p�i (q) to be incentive compatible transfers are as follows

t�i (q) = (pi (qi; q�i)� p (q)) bi(qmi ): (17)

With these transfers, a project that is �nanced with probability pi = 1 receives no transfer

t�i = 0: For a project �nanced with probability pi = 0; the transfer equals the highest

expected bene�t from misreporting. A manager with quality lower but arbitrarily close to

qmi is not �nanced and his transfer should be arbitrarily close to pi (qi; q�i) bi(qmi ); because

he can misreport qi. Hence,pi (qi; q�i) bi(qmi ) must be the transfer of any manager with

pi = 0 who could report a quality arbitrarily close to qmi : A manager with pi 2 (0; 1)

receives a positive transfer only if pi (qi; q�i) = 1; the transfer compensates him for not

misreporting a quality qi.

Theorem 1 The project selection mechanism given by the allocation rule (15) and the

transfers (17) solves the problem stated in (1)-(4).

18

The proof follows immediately from Lemma 2 and the de�nition of the mechanism we

proposed.

De�ne a project to be admissible if G (qi) � 0: In the following Corollary we describe

the optimal mechanism when all projects are ex-ante symmetric.

Corollary 1 In the symmetric problem, qo1 = ::: = qon := q

o: (i) If qo < 0; or if none of

the projects is admissible then the optimal mechanism is not to select any project nor to

compensate any manager, i.e. ti (q) = pi(q) = 0 for all q and all i; (ii) If there are at least

m projects in the sample such that G (qi) � qo � 0; then in an optimal mechanism m of

these projects are selected at random; (iii) Otherwise, the m (or less) admissible projects

with highest quality are �nanced.

In the symmetric case, when su¢ ciently many projects have a quality that exceeds a

threshold quality, selection among these top candidates is random, and therefore a lower

quality project can be selected over a higher quality one. The mechanism we propose in

Theorem 1 is essentially the only symmetric optimal mechanism, except in the boundary

case where E (qi) = 0.7 However, even in the generic case (E (qi) 6= 0), non-symmetric gen-

eralizations of the proposed mechanism are also optimal. These are obtained by assigning

�xed (not necessarily equal) selection probabilities to agents whose virtual qualities equal

the m-th highest virtual quality.

4.1 Optimal Versus Constant Mechanisms

We compare the optimal mechanism with constant mechanisms, i.e. mechanisms that do

not depend on reported qualities (and thus optimally do not pay money to the managers,

since no information is elicited from them). The results in this section are corollaries of

7 In this case, any mechanism such that, for all q; no transfers are paid, ti (q) = 0 and the assignment

pi(q) is feasible, symmetric and does not depend on q�i is optimal. Thus in this case, there are a continuum

of optimal symmetric mechanisms.

19

Theorem 1. First we comparing the optimal mechanism to the null mechanism� selecting

none of the projects.

Corollary 2 The optimal mechanism improves upon the null mechanism if and only if

there exist i 2 N such that qio > 0:

We derived qio in (10), it depends on the distribution of project quality and on man-

agers� private bene�ts. The optimal mechanism will improve over the null when qio is

positive. This is more likely to hold for projects with more favorable distributions of

qualities and lower private bene�ts for the manager.

The optimal mechanism clearly generates at least as high a payo¤ as any constant

mechanism. We ask, when is the optimal mechanism strictly better than simply select-

ing the m projects with the highest expected value? The following corollary derives the

necessary and su¢ cient conditions.

Corollary 3 Relabel the projects so that E [q1] � ::: � E [qn] : Suppose that E [q1] � 0

and let k be the highest index in f1; :::;mg such that E [qk] � 0: The best constant mech-

anism is to select projects 1; :::; k and pay no transfers. The optimal mechanism improves

upon the best constant mechanism if and only if there are projects i; j 2 N such that the

following three conditions hold: 1. E [qi] � max f0;E [qm]g ; 2. E [qj ] � E [qk+1] ; and 3.

maxnqoj ; 0

o> qi + bi

�qi�:

The conditions in the corollary state that in the best constant mechanism project

i is �nanced and project j is not �nanced, but in the optimal mechanism (that elicits

information) it is preferred to �nance project j for some reported qualities or it is preferred

not �nanced project i for some reported qualities. When these conditions holds, the optimal

mechanism results in a higher expected return than the constant mechanism. The optimal

mechanism elicits information. This information is more valuable and allows pro�ts to

increase compared to a random selection when the quality of the worse project is low.

20

Additionally, information is more costly to elicit with larger private bene�ts and hence

the optimal mechanism will more likely improve on a random selection when the private

bene�t is lower.

Interestingly, one can construct examples where the optimal mechanism improves upon

the null mechanism, which is better than randomization. In such situations, using the op-

timal mechanism enables the �nancing company to �nance a project, although it wouldn�t

have �nanced any if only constant (independent of reported qualities) mechanisms were

allowed.

Corollary 4 The optimal constant mechanism is not to �nance any project but a project

is �nanced under the optimal mechanism (with information acquisition) if and only if

E (qi) < 0 < qo:

5 Examples and Comparative Statics

5.1 Selecting One of Two Projects

Using theorem 1 we compare project selection in a few examples with selection of m = 1

out of n = 2 projects. For these examples we will assume that [qi; qi] � R+ so selecting

a project is always better than the outside option qo = 0, and the bene�t functions are

constant bi(qi) = bi so that the virtual return has the simple formGi(qi) = qi+bi. Finally we

assume private bene�ts are such that eliciting information is better than a random selection.

In example 1, the two projects are ex ante symmetric; in example 2 the two projects di¤er

in managers�private bene�ts; in example 3 the projects di¤er in the distribution of qualities

from which they are drawn.

Example 1 (Symmetric Projects) Suppose the principal needs to choose between two ex-

ante symmetric projects. There is a cuto¤ quality a� so that when at least one project

manager reports a quality which is lower than the cuto¤ a�, the highest quality project is

21

�nanced. However, in the top region, where q1; q2 � a�; the two projects are �nanced with

equal probabilities pi = 12 , and a less e¢ cient project may be selected.

Figure 1 illustrates this symmetric mechanism for a uniform distribution on [0; 1] and

bene�ts b 2�0; 12

�. In this case, the cuto¤ quality that helps us de�ne the optimal mecha-

nism is a� = 1�p2b . Figure 1a illustrates the project selection probabilities. Transfers are

shown in �gure 1b. Compensation for the manager whose project was not �nanced is just

enough to ensure incentive compatibility. Hence, it equals zero if this manager�s reported

quality was higher than a�; otherwise, it equals b if the �nanced project had a quality below

a� and 12b if it exceeded a

�:

Example 2 (Asymmetric Bene�ts) Consider the optimal mechanism when qualities are

drawn from the same distribution F1(:) = F2 (:); private bene�ts are b2 > b1 � 0; also

assume E(q) > qi + bi so that Hi(qi) > 0 and there is a solution a�i to Hi(a�i ) = 0: These

solutions satisfy a�1 > a�2; and q

oi = Gi (a

�i ) = a

�i + bi are such that q

o1 � qo2 (this holds i¤

a�1 � a�2 � b2 � b1 which we show holds true by implicit di¤erentiation of a�i (b)): Virtual

qualities are

xi = min fqi + bi; ai + big > 0:

We observe some interesting properties of the optimal mechanism in this example. When

both qualities exceed their corresponding cuto¤s qi � ai; project 1, belonging to the manager

whose private bene�t is lower, is �nanced. This would give project manager 2 (who is most

eager to be �nanced) less of an incentive to overstate his success probability. Therefore,

there is a range of qualities for which project 1 is �nanced even though it has a lower

quality. There is also a range of probabilities where project 2 has a higher quality than

project 1 yet project 1 is �nanced. This will occur in the range where both probabilities

are below their corresponding cuto¤s, qi < a�i ; and (q1; q2) lies below the 45o but above the

G1 (q1) = G2 (q2) curve (which is given by q2 = q1 � (b2 � b1) and lies below the 45o line

and is parallel to it). In this case project 2 has a lower quality but a higher virtual quality

than project 1 q2 < q1 < q2 + b2 � b1 and it is �nanced despite the fact that project 1 has a

22

higher quality.

Example 3 (Asymmetric Distributions) Consider the optimal mechanism when project

qualities are drawn from di¤erent distribution F1(:) 6= F2 (:) but transfers are constant and

equal b2 = b1 = b: We know that Hi(a�i ) = 0; and qoi = a

�i + b: Suppose a

�1 > a

�2; as can be

the case when F1(:) �rst order stochastically dominates the distribution F2. Then qo1 > qo2:

When both qualities exceed their corresponding cuto¤s, the virtual qualities are qoi and so

project 1 (with quality drawn from the dominating distribution) is �nanced. In all other

cases the highest quality project is �nanced.

5.2 The Number of Projects

We have computed the optimal project selection mechanism for a �xed number of projects

n: As can be observed in (7) and (8) that determine the cuto¤ quality a�i which is used to

de�ne the optimal mechanism, a�i do not depend on the number of projects (only on private

bene�ts and the distribution of qualities). However, the pro�t of the �nancing company

does depend on the number of available projects to choose from. We �rst observe that

when a the set of projects the �nancing company can select from expands, pro�t (weakly)

increases. To see this, suppose the optimal pro�t with a certain selection of n projects

is V . If a �nancing company now faces a selection between the same n projects and one

additional n + 1-th project, a mechanism that treats the �rst n projects just like when

these where the only projects and ignores the last project, tn+1(q) = pn+1(q) = 0 for all q;

is feasible and will achieve the same pro�t as before. Hence, with the optimal mechanism

pro�t is at least as high.

While pro�t increases as the set of available projects expands, as long as project quality

is bounded, so is the expected pro�t. In the following proposition we derive a limit result.

Suppose that the set of projects is randomly chosen. In section 3 we derived for each

project a threshold quality qoi de�ned in (10). These thresholds determine pro�t in the

limit as the number of projects to choose from goes to in�nity.

23

Proposition 1 (i) Suppose that the vectors (qoi ; qi) of thresholds and project qualities are

independently and identically drawn from a continuous distribution.8 Let [qomin; qomax] be

the support of the marginal distribution of qoi . Suppose that the conditional cdf F (� j qoi )

are such that any distributions conditional on a higher value of qoi �rst order stochastically

dominates a distributions condition on a lower qoi . Then for a �xed m; in the limit where

n goes to in�nity, the expected pro�t of the �nancing company at the optimal mechanism

approaches mqomax:

(ii) In the symmetric case, where all thresholds equal qo; the limiting expected pro�t of

the �nancing company as the number of projects increases to in�nity is mqo:

This proposition provides an asymptotic interpretation of the thresholds qo in the sym-

metric model, qo is the limit of the average per project return when the number of available

projects approaches in�nity.

The number of projects available to a �nancing company to choose from is likely however

to be �nite. The number of skilled researchers in the relevant technology might be limited.

It is likely costly to increase the pool of potential projects identifying candidates and the

relevant distribution from which their quality is drawn, and there can be administrative

costs to consider more projects. In the following example we show how pro�ts from choosing

one project depend on the number of projects n in the symmetric model when project

qualities are uniformly distributed and bene�ts are constant. We show that this pro�t is

increasing and concave in n; hence accounting form some convex cost c(n) there would be

a unique optimal �nite number of projects.

Example 4 Suppose b (qi) = b < 12 for all qi and that the distribution is uniform on [0; 1] :

8Note that the assumption require independence across projects, but a project i�s quality and the

threshold associated with that project may be dependent.

24

The �nancing company faces a choice of m = 1 out of n projects. Then,

a� = 1�p2b (18)

qo = 1�p2b+ b: (19)

In the optimal mechanism, if at most one project has a quality that exceeds 1 �p2b; the

highest quality project would be �nanced. If two or more project exceed this cuto¤, one

project among the projects with a quality higher than 1�p2b would be selected at random.

The expected pro�t for the �nancing company is

V (n) =1

2(1 + a�2)� a

�n+1

n+ 1: (20)

The pro�t V (n) is increasing and concave in the number of projects n. As the number of

projects approaches in�nity, V (n)! 12(1 + a

�2) = 1�p2b+ b = qo:

5.3 E¢ ciency Loss Due to the Costs of Eliciting Information

E¢ cient project selection would allocate funds to the highest quality project. The optimal

(pro�t maximizing) mechanism is ine¢ cient because sometimes inferior projects are se-

lected. In particular, when the quality of more than m projects exceed their corresponding

cuto¤s, the set of projects selected under the optimal mechanism might not be the same

as the set of m highest quality projects. This range of high qualities for which selection

is ine¢ cient lowers for the �nancing company the cost of eliciting information as it re-

duces the incentive to over-report success probabilities. An additional possible source of

e¢ ciency loss results from the lower private surplus to �nanced project managers. Because

bene�ts can increase in qi; lower private bene�ts also results from ine¢ cient project se-

lection. In the two examples that follow we examine the symmetric model and consider

constant bene�t functions so that the only e¢ ciency loss results from the �nancing of lower

quality projects. Let the expected return from the pro�t maximizing mechanism be Q�

(this is the sum of the qualities of m selected projects) and let the expected return that

25

results from the choice of the e¢ cient allocation be QE (this is the sum of the qualities of

the m highest quality projects).We measure e¢ ciency loss as from the pro�t maximizing

mechanism compared to an e¢ cient allocation of the project by:

L =QE �Q�QE

:

5.3.1 Private Bene�ts and E¢ ciency Loss

In the symmetric model, we consider the e¤ect of an increase in project managers�(con-

stant) private bene�ts b. Ine¢ cient project selection occurs in the range where more than

m projects have qualities that exceed a�: The distribution is kept unchanged, hence, e¢ -

ciency loss increases as a� declines. Implicitly di¤erentiating (8) with respect to b we �nd

that

da�

db=

�1qiRa�[G0 (a)] f (eq) deq < 0:

The inequality holds because we assumed that G0() > 0: The higher the private bene�t

the researcher gets, the lower the cuto¤ a� which de�nes the range of ine¢ cient project

selection and hence, the greater the degree of ine¢ ciency in project selection.9

5.3.2 Project Quality Distribution and E¢ ciency Loss

We now ask how the degree of ine¢ ciency varies if the distribution of project qualities

improves. We compare the pro�t maximizing mechanisms with two di¤erent distributions,

one of which �rst order stochastically dominates the other. Consider two distributions F1

and F2 such that F1 �rst order stochastically dominates F2: That is, for every q; F1(q) �9Similarly, for a non constant bene�t the degree of ine¢ ciency increases with a parallel shift up b(q) =

bv(q) + b0. In this case however in addition to ine¢ ciency in project selection, there is loss in the private

bene�ts of managers.

26

F2(q): First, we show that in the more favorable distribution, the cut o¤ point a� is higher.

Let a�1 and a�2 be the cuto¤s for the two given distributions then

qZa�1

(1� F1(q)) dq = b =qZa�2

(1� F2(q)) dq

By �rst order stochastic dominance, 1 � F1(q) � 1 � F2(q) for all q: Hence, for this

equality to hold, a�1 � a�2: This means that [a�1; q] � [a�2; q] and therefore, for the dominating

distribution F1 ine¢ cient selections are made for a smaller range of qualities. However,

because Fi(:) declines with �rst order stochastic dominance, the e¤ect of stochastic domi-

nance on the probability that any project exceeds the cuto¤ 1 � Fi(a�i ) is ambiguous and

hence also the e¤ect on the expected loss in returns, L.

6 State Contingent Transfers

So far we have considered mechanisms that only depend on reported probabilities. This

is the only information available to the �nancing company when it is making a selection.

In this section we ask however whether the �nancing company can do better if it were

able to make transfers contingent on information observed at a later date. We begin with

mechanisms in which transfers can depend on true project qualities (not just reported

qualities), but the allocation rule can still only depend on reported qualities. The solution

to this problem helps us solve the more interesting problem in which true qualities are not

observed, but transfers can depend on the realized return of the projects that were selected.

For example, when project quality represents the probability of success of the project, and

the �nancing company can ex-post observe whether the �nanced project succeeded or

not. If state dependent mechanisms are available, the �nancing company would generally

achieve a higher pro�t than when transfers can only depend on reported probabilities.

27

6.1 Transfers that Depend on the True Quality

Consider the following timing: �rst, the �nancing company commits to a mechanism;

second the managers report their qualities; third, projects are selected; fourth, qualities

of all projects are observed, whether they were selected or not; �fth, transfers are paid.

Consider mechanisms such that the allocation rule is a function of reported qualities only,

while transfers depend both on the reported and true qualities of agents.

A mechanism is now a function (bq; q) 7! (p; t) ; such that p = p (bq) and ti = ti (bq; q) ;where bq and q represent respectively the vectors of reported and true qualities. Incentiveconstraints are now given by

pi(q)bi(qi) + ti (q; q) � pi(q0i; q�i)bi(qi) + ti��q0i; q�i

�; q�

(21)

Limited liability constrains are now

ti(bq; q) � 0: (22)

We can rewrite the optimization problem as follows.

maxfpi(bq);ti(bq;q)gi=1;:::;nEq

"nXi=1

[qipi(q)� ti(q; q)]#

(23)

s:t: : feasibility (2); state-by-state limited liability (22) and IC (21) (24)

Our method to solving this problem follows similar steps as in our analysis in section 4.

Let a��i be de�ned as the unique solution of the equationZ qi

a��i

[(qi + bi (qi))� (a��i + bi (a��i ))] fi(qi)dqi �Z qi

qi

bi(qi)fi(qi)dqi = 0 (25)

if there is a solution, and as qi otherwise. Also, let qooi be such thatZ qi

a��i

[(qi + bi (qi))� qooi ] fi(qi)dqi �Z qi

qi

bi(qi)fi(qi)dqi = 0:

28

Either a��i is a solution of (25), or a��i = qi: In the �rst case, qooi = a��i + bi (a��i ) : In the

second case, qooi = E (qi) : Let V be the value of the target function at some arbitrary

mechanism that satis�es all the constraints. Let

ex(q) =0@min fq1 + b1 (q1) ; qoo1 g ; :::;min fqn + bn (qn) ; qoon g ; 0; :::; 0| {z }

m times

1Abe the vector of virtual qualities, and Sm(ex(q)) the sum of the m largest virtual qualities.

We provide an upper bound on the pro�ts such mechanisms can deliver.

Lemma 3 For any mechanism with state-dependent transfers that satis�es feasibility (2),

limited-liability (22) and (21)), the value is bounded as follows.

V � Eq [Sm(ex(q)] :Under certain conditions, this pro�t can be achieved through mechanisms where the

transfers depend on the true quality in a linear way. We propose a mechanism that satis�es

the constraints in the original problem and reaches the upper-bound in Lemma 3. Therefore

it is optimal. Let ex(m) (bq) be them-th order statistics of the vector ex(bq); i.e. itsm-th highestcoordinate. Consider the following allocation rule

p�i (bq) =8>>><>>>:

1 if exi(bq) > ex(m) (bq)m�jfi2N : exi(bq)>ex(m)(bq)gjjfi2N : exi(bq)=ex(m)(bq)gj if exi(bq) = ex(m) (bq)

0 if exi(bq) < ex(m) (bq): (26)

and the following transfers.

t�i (bq; q) = (pi (qi; bq�i)� p (bq)) �bi (bqi) + b0(bqi) (qi � bqi)� : (27)

It is easy to verify that this mechanism yields a value equal to the upper bound in Lemma

3.

To ensure that the mechanism we proposed satis�es the constraints we make additional

assumptions on the bene�t functions. Assume that the bene�t functions bi (:) are convex.

29

This will imply that the incentive constraints are satis�ed. Additionally, to ensure that

the mechanism satis�es the limited liability constraint we will assume that for all bqibi (bqi)� b0(bqi) �bqi � qi� � 0 (28)

This condition holds true when bene�t functions are constant and also if qi � 0 and bi (qi)

is less-than-unit-elastic qib0i(qi)

bi(qi)� 1:

Theorem 2 Suppose that the bi (:) are convex and satisfy (28). The project selection

mechanism given by the allocation rule (26) and the transfers (27) is optimal for the problem

stated in (23)-(24).

6.2 Transfers that Depend on Observed Returns

We now address the following question. What can be achieved if transfers may depend,

not on the true quality of all projects (like in subsection 6.1), but on an observed outcome

(which is related to project quality), only for the projects that were actually selected?

More speci�cally, we assume that the �nancing company gets to observe the realization of

the return of each selected project before it pays transfers.

Let the return of each project i be a non-negative random variable Ri:We assume that,

conditional on qi; all Ri are independent and have the same support R � [0;+1). Each

project i requires an initial investment Ii that does not depend on qi: We assume that the

quality qi of project i is its expected net return conditional on qi,

E (Ri j qi)� Ii = qi:

That is, the project quality which is known to the managers is the expected return of

the project. Under these assumptions, we provide an upper bound on the value that the

principal can obtain when he can pay transfers contingent on the realization of both the

project allocation lottery, and the selected projects�outcomes. A state s = (M;R) indicates

the set of projects (if any) that were �nanced M � N and the realization of the return

30

of the projects which were �nanced R � RjM j: In this section, a mechanism is a function

(bq; s) 7! (p (bq) ; t (bq; s)) : To shorten notation let us denoteti (bq; q) = Es [ti(bq; s) j q]

Incentive constraints are given by (21) which was de�ned in the previous section. Limited

liability constrains need to hold state-by-state, i.e.

ti(bq; s) � 0: (29)

We can rewrite the optimization problem as follows.

maxfpi(bq);ti(bq;s)gi=1;:::;nEq

"nXi=1

[qipi(q)� ti(q; qi)]#

(30)

s:t: : feasibility (2); state-by-state limited liability (29) and IC (21) (31)

For any feasible mechanism in this problem, the corresponding (pi (bq) ; ti (bq; q))i=1;:::;n satis-�es the constraints (24) of Lemma 3. Therefore, the �nancing company�s pro�t is bounded

above by the upper bound in Lemma 3. Under certain conditions, this is actually the least

upper bound of the �nancing company�s pro�t, but it may not be achieved. The mecha-

nism we constructed in Theorem 2 cannot be replicated here. The reason is that when the

quality is not directly observed (only the realized return), transfers can e¤ectively depend

on the true quality of project i only if project i is selected with a positive probability, i.e.

for all bq; qi and q0i;ti (bq; q) 6= ti �bq; �q0i; q�i��) pi (bq) > 0:

This property is not satis�ed by (p�; t�) ; i.e. this mechanism cannot be replicated with

the instruments we consider in this section. However, under certain conditions, we can

construct a feasible approximation of (p�; t�) that yield a pro�t that is "-close to the bound

in Lemma 3.

Let us assume that the private bene�t function bi (qi) is convex in the expected net

return qi and that it is less than unit elastic with respect to the expected gross return

31

qi + Ii; i.e.(qi + Ii) b

0i (qi)

bi (qi)� 1;

for all qi. In particular this includes the case where the private bene�t function is linear,

�i � 0 and �i � 0; such that bi (qi) = �i + �iqi and Ii � �i.

In the feasible approximation, each project is selected with a positive probability at

any bq. The allocation rule and expected transfers are very close to p�i (bq) and t�i (bq; q):More speci�cally, let " > 0: The approximation allocates projects according to p� with

probability 1� " and selects projects at random (with equal probabilities) with probability

": Transfers are adjusted in order to satisfy the constraints. The resulting allocation rule

is

p"i (bq) = (1� ") p�i + "mn : (32)

Let X be the event where project i is not selected, which occurs with probability 1�p"i (bq).Consider the following transfers to manager i in any state in s 2 X, and in any state s0 =2 X

(i.e. when i is selected) in which i0s realized return is ri 2 Ri :

t"i ((bqi; bq�i) ; s) =[p"i (qi; bq�i)� p"i (bq)]

1� p"i (bq) �bi (bqi)� b0i(bqi) (bqi + Ii)� (33)

t"i (bqi; bq�i; s0) =[p"i (qi; bq�i)� p"i (bq)]

p"i (bq) b0i(bqi)ri: (34)

Therefore, the expected transfer to manager i; conditional on the vector of project qualities

q is

t"i (bqi; bq�i; q) = [p"i (qi; bq�i)� p"i (bq)] �bi (bqi) + b0i(bqi) (qi � bqi)� :This mechanism satis�es all the constraints and delivers a value that is "-close to the

upper bound. Denote the payo¤ of project manager i when reported qualities are (bqi; bq�i)and his true quality is qi by Ui(bqi; bq�i; qi). We now verify that incentive compatibility holds.

32

Indeed, we have

Ui((bqi; bq�i) ; q) = [p"i (qi; bq�i)� p"i (bqi; bq�i)] �bi (bqi)� bi (qi) + b0i(bqi) (qi � bqi)�+p"i (qi; bq�i) bi (qi)(35)

The �rst term of this sum is always nonnegative. Indeed, p"i (qi; bq�i)� p"i (bqi; bq�i) � 0 and,because the functions bi (�) are convex, then bi (bqi)� bi (qi) + b0i(bqi) (qi � bqi) � 0: Thus, forall true quality qi and vector of reported qualities bq;

Ui((bqi; bq�i) ; q) � p"i (qi; bq�i) bi (qi) = Ui((qi; bq�i) ; q):This ensures that the proposed mechanism is incentive-compatible. Because bi (qi) �

(qi + Ii) b0i (qi) � 0; the limited liability constraint is also satis�ed. Moreover, evaluating

the target function at the mechanism indexed by " yields

(1� ")Vmax + "m

n

NXi=1

E [qi] :

where Vmax is the upper bound in Lemma 3. Since E (qi) � Vmax; this value is lower than

but approximates Vmax; in the limit where " goes to 0:

Theorem 3 The project selection mechanism given by the allocation rule (32) and the

transfers (33)-(34) is almost optimal for the problem stated in (30).

The optimal mechanism with state dependent transfers is characterized by a similar

project selection rule as that with deterministic transfers. In particular, there is a po-

tentially ine¢ cient project selection in the region where several managers report project

qualities that exceed a cuto¤. However, while with deterministic transfers the highest

quality project is �nanced for sure outside this region, with state dependent transfers, the

�nancing company could approach maximum pro�ts if it allows low reported probability

projects to be �nanced with some small probability, and highly rewards a favorable out-

come in the unlikely event that the low quality project is selected. This can allow the

33

�nancing company to obtain a higher pro�t than with deterministic transfers. Interest-

ingly, however, when private bene�ts are constant (b0i(qi) = 0) the optimal mechanism with

state dependent transfers is identical to that when transfers can only depend on reported

probabilities. Hence, in this special case, nothing is gained from allowing the richer set of

mechanisms.

7 Concluding Remarks

Private companies as well as government agencies often face the need to decide how to

allocate limited resources to R&D projects. Due to the complexity and uncertainty involved

in R&D, the information that is needed to make decisions may be in the hands of project

managers whose interests are typically not aligned with those of the �nancing company.

Project managers have more to gain if their own project is �nanced.

We derived an optimal mechanism to acquire information and select which projects to

�nance. If the �nancing company were fully informed it would always choose the highest

quality projects, but in the presence of adverse selection, when information acquisition

is costly, this is no longer true. In the optimal mechanism there is a region of reported

probabilities in which more managers than can be selected report high qualities, such that

in this region project selection is not sensitive the project qualities. In the symmetric model

projects in this region are randomly selected. In the asymmetric model, a deterministic

selection is made. Hence, the optimal mechanism involves ine¢ cient project selection where

a lower quality project can be selected over a high quality one. This practice allows the

�nancing company to save on costs of information acquisition (the incentive transfers) as

it reduces the incentive to over state project qualities.

Information asymmetry in our analysis is of an adverse selection nature. Because

managers compete for resources and want their own project to be �nanced, the principal

must worry about managers overstating the quality of their project. In contrast in a world

with moral hazard and absent competition between managers, understating the quality of

34

the project might be a concern as it can help hide low e¤ort. In a moral hazard model, when

transfers are used to induce e¤ort, they typically reward success, while, in our setting, the

�nancing company might reward managers that are not �nanced to ensure truth-telling and

achieve better allocation of resources. The optimal mechanism in our model may reward

good outcomes of a manager who reported low quality projects under the optimal state

dependent mechanism.

A crucial feature of the analysis we made is that the �nancing company is facing a

selection from a pre-quali�ed pool of candidate projects. The �nancing company is assumed

to have some information about the distribution of qualities of each of the projects, and

the number of candidates is �xed. If anyone could apply for the R&D funds, a mechanism

like we found may attract low quality candidates who would want to enter the competition

for funds, not in hope of winning but rather with the intention of loosing and collecting a

consolation prize �the transfers that reward truthful disclose. Facing such a pool of low

expected quality candidates, not �nancing any projects might be the optimal solution. A

�xed pool of prequali�ed project managers can arise for example, in a private company

where project managers are pre-screened employees of the company; when application for

funding requires su¢ cient knowledge and there is a limited number of candidates with

the capabilities to engage in a relevant research project (as is often the case with defense

contracting); or as a �nal stage of a research funding process when the �nancing company

was able to employ a two (or more) stage procedure with early stages screening project

managers whose quality comes from unfavorable distributions.

35

8 Appendix: Proofs

Proof of Lemma 1. Step 1. Consider �rst the incentive compatibility constraints (4)

which we refer to as ICi: The utility of project manager i under a mechanism when he

reports truthfully is Mi (q) which was de�ned in (13). His utility of misreporting a type q0i

equals

ti(q0i; q�i) + bi(qi)pi(q

0i; q�i) =Mi(q

0i; q�i) +

�bi(qi)� bi

�q0i��pi(q

0i; q�i): (36)

From the incentive compatibility constraint, we know that

Mi (q) �Mi(q0i; q�i) +

�bi(qi)� bi

�q0i��pi(q

0i; q�i)

i.e.

Mi

�q0i; q�i

��Mi (q) �

�bi(q

0i)� bi (qi)

�pi(q

0i; q�i):

Using the last inequality twice (once switching the roles of qi and q0i), we get�bi(q

0i)� bi (qi)

�pi(q) �Mi

�q0i; q�i

��Mi (q) �

�bi(q

0i)� bi (qi)

�pi(q

0i; q�i): (37)

Step 2. Consider now two qualities qi; q0i such that bi (qi) < bi (q0i) and �x q�i; then

pi (qi; q�i) � pi�q0i; q�i

�holds.10 Indeed, by (37), we have [bi(q0i)� bi(qi)] [pi(q0i; q�i)� pi(q)] � 0: Because bi(q0i) �

bi(qi) > 0; this implies pi(q) � pi(q0i; q�i).

Step 3. Dividing by q0i � qi and taking the limit as q0i ! qi we �nd that

@Mi(q)

@qi= b0i (qi) pi(q

0i; q�i):

10Under the assumption that b (�) is strictly increasing, one can show that ICi holds if and only if pi (�; �)

is nondecreasing in qi and transfers satisfy the formula of Lemma 1: This result is classic for this type of

mechanism design models (e.g. see Myerson, 1981). Here, the analysis is complicated by the fact that b (�)

can be constant on a subset of its domain.

36

The equality holds almost everywhere, and the right-hand-side is continuous in qi almost

everywhere.11 From the Fundamental Theorem of Calculus, we obtain

Mi (qi; q�i) =Mi(qi; q�i)�qiZqi

b0i(eqi)pi(eqi; q�i)deqi;substituting this into (13) and rearranging yields the following expression.

ti(q) = �

0@bi(qi)pi(q)�Mi(qi; q�i) +

qiZqi

b0i(eqi)pi(eqi; q�i)deqi1A : (38)

Step 4. Fix q�i: We use integration by parts to show thatZ qi

qi

�Z qi

qi

b0i(eqi)pi(eqi; q�i)deqi� f(qi)dqi = Z qi

qi

b0i(qi)pi(qi; q�i)Fi (qi) dqi: (39)

Step 5. Now, we substitute the transfers (38) into the expected value of the pro�t made

from agent i; conditional on q�i :

Eqi [qipi(q)� ti (q)]

= Eqi

�qipi(q) +

�bi(qi)pi(q)�Mi(qi; qj) +

Z qi

qi

b0i(eqi)pi(eqi; qj)deqi��= Eqi [qipi(q) + bi(qi)pi(q)�Mi(qi; qj)] +

Z qi

qi

�Z qi

qi

b0i(eqi)pi(eqi; q�i)deqi� f(qi)dqi= Eqi [qipi(q) + bi(qi)pi(q)�Mi(qi; qj)] +

Z qi

qi

b0i(qi)pi(qi; q�i)Fi (qi) dqi

= Eqi [Gi (qi) pi(q)�Mi(qi; qj)] :

Taking the expectation over q�i and adding up over i yields the desired equality.

The following result is useful in the proofs of Lemmas 3 and 4.

11The limit of (b(q0i)�b(qi))

q0i�qipi(q

0i; q�i) exists and equals b

0 (qi) pi(q0i; q�i) for almost all qi. Indeed, at any

point of the set fqi : b0 (qi) = 0g ; the limit exists and equals 0; since pi (�) is bounded. As for the other

points, a corollary of Step 2 is that pi (�) is locally nondecreasing at any qi that is such that b0 (qi) > 0 and

thus is continuous almost everywhere in the set fqi : b0 (qi) > 0g.

37

Lemma 4 (Lemma A1) At any mechanism that satis�es the constraints, the following

holds, for each agent i 2 N .

Eqi [Gi(qi)pi(q)�Mi(qi; q�i)] � Eqi [min fGi (qi) ; qo1g pi (q)] :

Proof of Lemma A1. Let q 2 QN : First, by ICi we have ti (q) + bi (qi) pi (q) �

Mi(qi; q�i); otherwise agent i�s type qi would have an incentive to misreport as qi: By LLi;

we have ti (q) � 0; thus for all q 2 QN ;

bi (qi) pi (q) �Mi(qi; q�i) (40)

For all qi > a�i we have Gi (qi) > qo1: Therefore,(Z qi

a�i

Gi(qi)pi(q)fi(qi)dqi �Mi(qi; q�i)

)�(qo1

Z qi

a�i

pi(q)fi(qi)dqi

)

=

Z qi

a�i

(Gi(qi)� qo1) pi(q)fi(qi)dqi �Mi(qi; q�i)

�"Z qi

a�i

(Gi(qi)� qo1) fi(qi)dqi � bi (qi)#Mi(qi; q�i)

bi (qi)= 0:

The inequality between the second and third lines is an implication of the constraint (40):

The last equality holds by de�nition of qo1: ThereforeZ qi

a�i

Gi(qi)pi(q)fi(qi)dqi �Mi(qi; q�i) �Z qi

a�i

qo1pi(q)fi(qi)dqi:

which further implies Z qi

qi

Gi(qi)pi(q)fi(qi)dqi �Mi(qi; q�i)

�Z a�i

qi

Gi(qi)pi(q)fi(qi)dqi +

Z qi

a�i

qoi pi(q)fi(qi)dqi

= Eqi [min fGi (qi) ; qoi g pi (q)]

38

where the last equality holds because, for all qi < a�i ; we have Gi (qi) < qo1; and for all

qi > a�i ; we have q

o1 > Gi (qi) :

Proof of Lemma 2.

V = Eq

"nXi=1

[Gi(qi)pi(q)�Mi(qi; q�i)]

#

� Eq

"nXi=1

min fGi (qi) ; qoi g pi (q)#

� Eq

24Sm0@min fG1 (q1) ; qo1g ; :::;min fGn (qn) ; qong ; 0; :::; 0| {z }

m times

1A35= Eq [Sm (x(q))] :

where the �rst inequality holds by Lemma A1, and the second holds because of the feasi-

bility constraints (2).

Proof of Theorem 1. Under the proposed mechanism, in each case, all the weak

inequalities in Lemma A1 and in Lemma 2 hold as equalities, thus the mechanism reaches

the upper bound. But it is also feasible, therefore it is optimal.

Proof or Proposition 1. Let xi = min fGi (qi) ; qoi g : Let " > 0: We have

Pr�xi � qomax �

"

m

�= Pr

�Gi (qi) � qomax �

"

mj qoi � qomax �

"

m

�Pr�qoi � qomax �

"

m

�� Pr

�Gi (qi) � qomax �

"

mj qoi = qomax �

"

m

�Pr�qoi � qomax �

"

m

�:

The last inequality is an implication of �rst order stochastic domination. Both terms of

the product in the last line are positive. Therefore Pr�xi � qomax � "

m

�> 0: Since the xi

are iid, by the law of large numbers

limn!+1

Pr�x(m) (q) � qomax �

"

m

�= 1:

Therefore

limn!+1

Pr (Sm (x (q)) � mqomax � ") = 1:

39

Therefore

limn!+1

E [Sm (x (q))] = mqomax:

as needed.

The following result is a useful preliminary to Lemma 3.

Lemma 5 (Lemma A2) At any mechanism that satis�es the constraints (2), limited-

liability (22) and (21)) of the state dependent problem, the following holds, for each agent

i 2 N .

Eqi [(qi + bi(qi)) pi(q)� Ui (q)] � Eqi [min fqi + bi (qi) ; qoi g pi (q)] :

Proof of Lemma A2. We �rst show the following inequality:,(Z qi

a�i

(qi + bi (qi)) pi(q)fi(qi)dqi � Eqi [Ui(q)])�(qoi

Z qi

a�i

pi(q)fi(qi)dqi

)

=

Z qi

a�i

((qi + bi (qi))� qoi ) pi(q)fi(qi)dqi � Eqi [Ui(q)]

� pi(q)

"Z qi

a�[(qi + b (qi))� qoi ] fi(qi)dqi �

Z qi

qi

b(qi)fi(qi)dqi

#= 0:

Where pi(q) = supqi fpi (q)g : The inequality between the second and third lines follows

from the constraints pi (q) b (qi) � Ui (q) and pi (q) � pi (q) : The last equality holds by

de�nition of qoi : ThereforeZ qi

a�i

(qi + b (qi)) pi(q)fi(qi)dqi � Eqi [Ui(q)] �Z qi

a�i

qoi pi(q)fi(qi)dqi: (41)

which further impliesZ qi

qi

(qi + b (qi)) pi(q)fi(qi)dqi � Eqi [Ui(q)]

=

Z a�i

qi

(qi + b (qi)) pi(q)fi(qi)dqi +

Z qi

a�i

(qi + b (qi)) pi(q)fi(qi)dqi � Eqi [Ui(q)]

�Z a�i

qi

(qi + b (qi)) pi(q)fi(qi)dqi +

Z qi

a�i

qoi pi(q)fi(qi)dqi

= Eqi [min fqi + b (qi) ; qoi g pi (q)]

40

where the inequality holds by (41) and the last equality holds because, for all qi < a�i ; we

have qi + b (qi) < qoi ; and for all qi > a�i ; we have qi + b (qi) > q

o1:

Proof of Lemma 3. In the objective function (23), we substitute

ti(q; q) = Ui (q)� bi (qi) pi (q) :

Thus, the target function becomes

maxfpi(q);Ui(q)gi=1;:::;n

Eq

"nXi=1

[(qi + bi (qi))pi(q)� Ui(q)]#

(42)

The value of this target function at any mechanism that satis�es the constraints, V;

will satisfy the following

V = Eq

"nXi=1

[(qi + b (qi)) pi(q)� Ui(q)]#

� Eq

"nXi=1

min fqi + b (qi) ; qoi g pi (q)#

� Eq

24Sm0@min fq1 + b (q1) ; qo1g ; :::;min fqn + b (qn) ; qong ; 0; :::; 0| {z }

m times

1A35 :where the �rst inequality holds by Lemma A2, and the second holds because of the feasi-

bility constraints (2)

Proof of Theorem 2. We �rst claim that the proposed mechanism satis�es the

constraints. Denote the payo¤ of project manager i when reported qualities are (bqi; bq�i)and true qualities are is q by Ui((bqi; bq�i) ; q). We now verify that incentive compatibilityholds. Indeed, we have

Ui((bqi; bq�i) ; q) = [pi (qi; bq�i)� pi (bqi; bq�i)] �bi (bqi)� bi (qi) + b0i(bqi) (qi � bqi)�+pi (qi; bq�i) bi (qi)The �rst term of this sum is always nonpositive. Because, pi (qi; bq�i)� pi (bqi; bq�i) � 0 andbecause the functions bi (�) are convex, which implies that bi (bqi)�bi (qi)+b0i(bqi) (qi � bqi) � 0:Thus, for every true qualities vector q and vector of reported qualities bq;

Ui((bqi; bq�i) ; q) � pi (qi; bq�i) bi (qi) = Ui((qi; bq�i) ; q):41

This ensures that the proposed mechanism is incentive-compatible. By (28) and the as-

sumption that the bene�t function is increasing we know that [bi (bqi) + b0(bqi) (qi � bqi)] ��bi (bqi) + b0(bqi) �qi � bqi�� � 0: This implies that the limited liability constraint is also sat-is�ed.

Under the proposed mechanism, all the weak inequalities in Lemma 3 hold as equalities,

thus the mechanism reaches the upper bound. But it is also feasible, therefore it is optimal.

Proof of Theorem 3. One can easily check that the proposed mechanism is feasible.

The allocation rule p" and the expected transfers t" are convex combinations of the allo-

cation rule p� and expected transfers t�; on the one hand (with a weight of 1� ") and the

allocation rule and expected transfers of the random selection mechanism (with a weight

of ").

p"i = (1� ") p�i + "m

N

t"i = (1� ") t�i + " � 0:

The allocation rule p� and allocation transfers t� yields Vmax, by Theorem 2, whereas the

random allocation mechanism yields the value 1n

PNi=1 E [qi] : Therefore the value of the

mechanism indexed by " is (1� ")Vmax + " 1nPNi=1 E [qi] : This value is lower than Vmax;

but its limit is Vmax as " goes to 0:

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44

Figure 1a

Project selection in the optimal symmetric mechanism

Figure 1b

Transfers in the optimal symmetric mechanism with b(q)=b.

t1=0 t2=b

t1=b t2=0

t1=0 t2=0

q1

q2

0

1

1

a*

a*

t1=0 t2=½b

t1=½bt2=0

p1=1 p2=0

p1=0 p2=1

p1=½p2=½

q1

q2

0

1

1

a*

a*