# Setting Interim Deadlines to Persuade††thanks: Thanks to Pavel Kocourek,
Yiman Sun, Jan Zápal, Inés Moreno de Barreda, Jeff Ely, Jean Tirole, Péter
Esö, Margaret Meyer, Colin Stewart, Francesc Dilmé, Ansgar Walther, Ole Jann,
Maxim Ivanov, Egor Starkov, Ludvig Sinander, Maxim Bakhtin, Rastislav Rehák,
Vladimír Novák, Arseniy Samsonov, and the audiences at OLIGO 2022, SING17,
CMiD2022, EEA-ESEM Congress 2022, and EWMES 2022 for helpful comments.

Maxim Senkov Email<EMAIL_ADDRESS>CERGE-EI, a joint workplace of Center
for Economic Research and Graduate Education, Charles University and the
Economics Institute of the Czech Academy of Sciences, Politických vězňů 7,
P.O. Box 882, 111 21 Prague 1, Czech Republic.The author acknowledges
financial support from the Lumina Quaeruntur fellowship (LQ300852101) of the
Czech Academy of Sciences and from Czech Science Foundation (GACR) under grant
no. 20-27630S.

(January 25, 2022)

###### Abstract

A principal funds a multistage project and retains the right to cut the
funding if it stagnates at some point. An agent wants to convince the
principal to fund the project as long as possible, and can design the flow of
information about the progress of the project in order to persuade the
principal. If the project is sufficiently promising ex ante, then the agent
commits to providing only the good news that the project is accomplished. If
the project is not promising enough ex ante, the agent persuades the principal
to start the funding by committing to provide not only good news but also the
bad news that a project milestone has not been reached by an interim deadline.
I demonstrate that the outlined structure of optimal information disclosure
holds irrespective of the agent’s profit share, benefit from the flow of
funding, and the common discount rate.
Keywords: dynamic information design, informational incentives, interim
deadline, multistage project.
JEL Classification Numbers: D82, D83, G24, 031.

##  1 Introduction

The development of any innovation requires investment of both time and
capital, while the outcome of this investment is inherently stochastic.
Usually, the investor, being the principal, retains the option to stop funding
the innovative project if at some point it proves unsuccessful. It is widely
documented that the agent running the project tends to prefer the principal to
postpone the stopping of the funding to enjoy either the extra funds or an
additional chance to turn her research idea into a success story.111Agency
conflict in which the agent prefers the principal to postpone abandoning the
project that the agent is working on is studied in Admati and Pfleiderer,
(1994); Gompers, (1995); Bergemann and Hege, (1998, 2005); Cornelli and Yosha,
(2003). In such an agent-principal relationship, the agent’s technological
expertise and the quality of her innovative idea often allow her to manipulate
the principal by designing how and when the outcomes of the research and
development process are announced.

Recently, venture capital firms have started to pour billions into startups
focused on the development of quantum computers, which are known for their
technological complexity and difficulty of construction. The economic
viability of quantum computing is questioned by a number of experts; however,
the startups promise the investors a completed product in the foreseeable
future.222”The Quantum Computing Bubble.” _Financial Times_ , August 25, 2022.
For instance, a quantum startup PsiQuantum announced to potential investors
that it hopes to develop a commercially-viable quantum computer within five
years and managed to raise more than $\$200$ million in 2019.333”Bristol
Professor’s Secretive Quantum Computing Start-Up Raises £179m.” _The
Telegraph_ , November 16, 2019.

This paper studies the implications of the agent’s control of information
during the progress of a research and development project when the agent and
the principal disagree about the timing of when to abandon the research idea.
I ask: What is the degree of transparency to which an agent should commit
before starting to work on an innovative project? In particular, which terms
for self-reporting on the progress of the project should a startup propose
while discussing the term sheet with a venture capitalist? As I show,
depending on the properties of the project, the startup would strategically
choose both the timing for the disclosure of updates on the progress of the
project and the type of news it discloses - either good or bad.

I study a game between a startup and an investor. The startup controls the
information on the progress of the project and has the power to propose the
terms for self-reporting on it to the venture capitalist.444I discuss the
reasoning behind this assumption in Section 3.2. The startup has an
intertemporal commitment power and commits to a dynamic information policy,
which can be interpreted as designing the terms of the contract specifying how
the information on the progress of the project is disclosed over time as the
project unfolds. In return, the investor continuously provides funds and
chooses when to stop funding the project.

The project has two stages and evolves stochastically over time toward
completion, conditional on continuous investment in it. The completion of each
of the stages of project occurs according to a Poisson process. The completion
of the first stage serves as a milestone, such as the development of a
prototype, while completion of the second stage achieves the project. The
investor gets a lump-sum project completion profit if and only if he stops
investing after the project is completed and before an exogenous project
completion deadline, and the startup prefers the principal to postpone
stopping the funding.555I discuss the reasons for the presence of the project
completion deadline in Section 3.1.

As the investor receives the reward only after a prolonged period of
investment, he initially invests without being able to see if the investment
is worthwhile. Hence, it is individually rational for the investor to start
investing only if he is sufficiently optimistic regarding the future of the
project. An important feature of the setting that I consider is that the
_information is symmetric at the outset_ : not only the investor, but also the
startup is unable to find out if the project will bring profit to the investor
or not - this can be inferred only as time goes on and some evidence is
accumulated. The only tool that the startup has for persuading the investor to
start investing is the promise of future reports on the progress of the
project.

Clearly, the good news about the completion of the project is valuable to the
investor as it helps him to stop investing in a timely manner. Further, as
evidence regarding the project accumulates over time, failure to pass the
milestone in a reasonable time makes the project unlikely to be accomplished
in time - and the investor prefers to stop investing after observing such bad
news. When designing the information policy, the startup chooses optimally
between the provision of these two types of evidence in order to postpone the
investor’s stopping decision for as long as possible.

I show that the startup’s choice of information policy depends on the ex ante
attractiveness of the project for the investor. The attractiveness is captured
by the _flow cost-benefit ratio of the project_. Thus, a project is relatively
more attractive ex ante to the investor when its flow investment cost is
lower, its project completion profit is higher, or the Poisson rate, at which
completion of one stage of the project occurs, is higher.

When the project is sufficiently attractive ex ante to the investor, promises
to provide information only on the completion of the project serve as a
sufficiently strong incentive device to motivate the investor to start the
funding at the outset. Further, the future news on the completion of the
project does not harm the total expected surplus generated by the interaction
of the startup and investor, while the future news on the project being poor
decreases the surplus that the startup can potentially extract from the
investor. Accordingly, the startup commits to providing only the good news,
but not the bad news on the project in the future: it _discloses the
completion of the project and postpones the disclosure_ in order to ensure the
extraction of as much surplus as possible from the investor. In the context of
quantum computing, the startup optimally chooses and announces to the venture
capitalist the date by which it plans to have a fully developed quantum
computer. When the date comes, the startup reports completion if the quantum
computer has been completed; if not, the startup reports the completion as
soon as it occurs.

The situation changes when the project does not look promising to the investor
ex ante. In that case, if the startup commits to disclosing only the
completion of the project, the investor will not have the sufficient
motivation to start investing in it. Thus, the startup extends the information
policy to encompass not only the good news but also the bad. As in the case of
the promising project, the startup discloses the project’s completion and does
so without any postponement, thereby fully exploiting its preferred incentive
tool. In addition, the startup sets a date at which the bad news is released
if the milestone of the project has not yet been reached - this date is _the
interim reporting deadline_.

Setting the interim deadline, the startup chooses a deterministic date, which
it optimally postpones. As the startup prefers the investor to postpone
stopping the funding, it prefers the interim deadline to be at a later
expected date. Further, the completion of the stages of project according to a
Poisson process makes both the startup and the investor risk-averse with
respect to the date of the interim deadline. Thus, the startup prefers to set
the interim deadline at a deterministic date and to postpone it as late in
time as possible in order to extract all the surplus from the investor. In the
context of quantum computing, the startup optimally chooses and announces a
fixed date by which it plans to have a prototype of the quantum computer. When
the date comes, reporting having the prototype at hand convinces the investor
to continue the funding, and reporting not having the prototype leads to
termination of the project.

Finally, I demonstrate that the outlined structure of the optimal information
disclosure holds for a broad class of preferences of the startup and the
investor. I allow for profit-sharing between the startup and the investor,
varying degrees of the startup’s benefit from the flow of funding, and
exponential discounting, and show that the startup prefers not to set any
interim deadlines whenever the project is sufficiently promising to the
investor. The future disclosure of the completion of the project promises
investor profit in exchange for a prolonged investment, while the disclosure
of the stagnation of the project at the interim deadline promises investor
only saved costs, as further investment stops. Thus, when the project is
attractive, the startup can make the funding and the beneficial
experimentation relatively longer by setting no interim deadlines.

##  2 Related literature

My paper is related to the _literature on dynamic information design_. The
closest paper in this strand of literature is by Ely and Szydlowski, (2020).
Similarly to my paper, they study the optimal persuasion of a receiver facing
a lump-sum payoff to incur costly effort for a longer time. In my model, as in
theirs, the sender is concerned to satisfy the beginning-of-the-game
individual rationality constraint of the receiver when choosing the
information policy. Further, the “leading on” information policy in Ely and
Szydlowski, (2020) has a similar flavor to the “postponed disclosure of
completion” information policy in my paper: promises of news on completion of
the project serve as an incentive device sufficient to satisfy the receiver’s
individual rationality constraint.

However, there are several substantial differences between Ely and Szydlowski,
(2020) and my paper. While in their model the state of the world is static and
drawn at the beginning of the game, in my model it evolves endogenously over
time, given the receiver’s investment. As a result, the initial disclosure
used in the “moving goalposts” policy in Ely and Szydlowski, (2020) cannot
provide additional incentives for the receiver in my model. The sender in my
model uses another incentive device to incentivize the receiver to opt in at
the initial period: she commits to an interim deadline at which she discloses
that the first stage of the project is not completed.

Another closely related paper is by Orlov et al., (2020). The main similarity
to my paper lies in the sender’s incentive to postpone the receiver’s
irreversible stopping decision. The sender in their paper prefers to backload
the information provision, which is in line with the properties of the optimal
information policy in my paper. However, there are a number of substantial
differences between our papers. In Orlov et al., (2020), the sender does not
have the intertemporal commitment power. Further, the receiver obtains a
payoff at each moment of time, and thus the sender does not need to persuade
the receiver to opt in at the beginning of the game.

Ely, (2017); Renault et al., (2017); Ball, (2019) also analyze dynamic
information design models. However, their papers focus on persuading a
receiver who chooses an action and receives a payoff at each moment of time,
whereas in my paper the receiver takes an irreversible action and receives a
lump-sum project completion payoff. Henry and Ottaviani, (2019) consider a
dynamic Bayesian persuasion model in which, similarly to my model, the
receiver needs to take an irreversible decision. However, the incentives of
the sender and receiver differ from my model: the receiver wants to match the
static state of the world and the sender is concerned with both the receiver’s
action choice and with the timing of that choice. Basak and Zhou, (2020) study
dynamic information design in a regime change game. The optimal information
policy in their model resembles the interim deadline policy in my model: at a
fixed date, the principal sends the recommendation to attack if the regime is
substantially weak by that time.

My paper is also related to the _literature on the dynamic provision of
incentives for experimentation_ (Bergemann and Hege,, 1998, 2005; Curello and
Sinander,, 2020; Madsen,, 2022). The closest papers in this strand of
literature are by Green and Taylor, (2016) and Wolf, (2017). Similarly to my
model, both papers consider design of a contract regarding a two-stage
project, in which the completion of stages arrives at a Poisson rate. In Green
and Taylor, (2016), there is no project completion deadline and the quality of
the project is known to be good, while in Wolf, (2017) the quality of the
project is uncertain. In contrast to my paper, both papers focus on a
canonical moral-hazard problem and give the power to design the terms of the
contract to the investor (principal) rather than the startup (agent). In
particular, the contract in Green and Taylor, (2016) specifies the terms for
the agent’s reporting on the completion of the first stage of the project.
Similarly to my model, the optimal reporting takes the form of a deterministic
interim deadline: at a principal-chosen date, the agent truthfully reports if
she has already completed the first stage, which determines the further
funding of the project.666In a broad sense, my paper also relates to the small
strand of theoretical literature on dynamic startup-investor and startup-
worker relations under information asymmetry (Kaya,, 2020; Ekmekci et al.,,
2020). However, while these papers focus on the signaling of the type of
startup, I study the provision of information by the startup on the progress
of the project.

##  3 The model

###  3.1 Setup

I consider a game between an agent (she, sender) and a principal (he,
receiver). Time is continuous and there is a publicly observable deadline $T$,
$t\in\left[0,T\right]$.777The results for the setting without a deadline are
easily obtained by considering $T\rightarrow\infty$. They are presented in
Appendix E. For each $t$, _the principal_ chooses sequentially to invest in
the project $\left(a_{t}=1\right)$ or not $\left(a_{t}=0\right)$. The flow
cost of the investment is constant and given by $c$. The choice of $a_{t}=0$
at some $t$ is irreversible and induces the end of the game.888The absence of
the principal’s commitment to an investment policy and the irreversibility of
the stopping decision capture the venture capitalist’s option to abandon the
project, e.g., in the case of its negative net present value.

The assumption that the project needs to be completed in finite time is
natural in many economic settings. The main interpretation for $T$ is an
expected change in market conditions that renders the project unprofitable. In
the context of a research and development project, $T$ could stand for the
date at which the competitor’s innovative product is expected to enter the
market, or the date at which the competitor is expected to get a patent on the
competing innovation.

The _state of the world_ at time $t$ is captured by the number of stages of
the project completed by $t$, $x_{t}$, and the project has two stages,
$x_{t}\in\\{0,1,2\\}$. The state process begins at the state $x_{0}=0$ and,
conditional on the continuation of the investment by the principal, it
increases at a Poisson rate $\lambda>0$. Information on the number of stages
completed is controlled by _the agent_. Thus, when making investment
decisions, the principal relies on the information provided by the agent.

The project brings the profit $v$ if and only if the _second stage of the
project_ has been completed by the time of stopping, and a payoff of $0$,
otherwise. I assume that all of the profit goes to the principal. This
assumption simplifies the exposition without affecting the main results of the
paper. I relax this assumption and consider the profit-sharing between the
agent and the principal in Section 6.

There is a _conflict of interest_ between the agent and the principal as the
agent benefits from using the funds provided by the principal for running the
project, possibly diverting them for her benefit. Thus, the agent faces the
flow payoff of $c$ and wants the principal to postpone his irreversible
decision to stop as long as possible.

I study the agent’s choice of information provision to the principal. The
agent chooses an information policy to maximize her expected long-run payoff.
I assume that the agent has the power to announce and commit to a policy. An
_information policy_ $\sigma$ is a rule that for each date $t$ and for each
past history $h\left(t\right)$ specifies a probability distribution on the set
of messages $M$. The history includes all past and current realizations of the
process and all past message draws and principal’s action choices.

When choosing an information policy, the agent faces a rich strategy space.
First, she can choose if the information on the completion of the first, or
second, stage of the project will be disclosed by the policy. Second, she can
choose how to disclose the completion of a stage of the project: for instance,
to do so immediately or to postpone the disclosure.

The timing of the game is as follows. First, at $t=0$, the agent publicly
commits to an information policy $\sigma$. Second, at each $t$ the principal
observes the message generated by the information policy and makes her
investment decision. The game ends when the principal chooses to stop
investing or at $T$, if he keeps investing until $T$. I assume that whenever
indifferent about investing or not, the principal chooses to invest, and
whenever indifferent about disclosing information or not, the agent chooses
not to disclose.

Throughout the paper, I use the following intuitive notational convention: for
any two dates at which the principal stops investing, $S$ and $\tau$,

$\displaystyle S\wedge\tau$
$\displaystyle\coloneqq\min\left\\{S,\tau\right\\},$ $\displaystyle S\lor\tau$
$\displaystyle\coloneqq\max\left\\{S,\tau\right\\}.$

###  3.2 Discussion of assumptions

The main interpretation of the considered dynamic information design problem
is the contracting between the agent (startup) and the principal (investor) on
the terms of reporting on the completion of stages of the project that are not
publicly observed. The terms could take the form of a proposed formal
reporting schedule or a schedule of meetings with the investor. Non-
observability of the stage completions stems from the fact that, while the
technology is being developed in the lab, the principal either does not have
sufficient expertise to assess the progress or the full access to the lab.

I assume that the principal does not have the power to propose the terms for
reporting to the agent and, e.g., make her fully disclose the progress
achieved in the lab. The most natural interpretation of such an asymmetry in
the bargaining power is the asymmetry in the market for private equity: there
are sufficiently many investors willing to invest in a particular technology
or sufficiently few startups working on the technology.999In the alternative
interpretation of the model, contracting concerns internal corporate research
and development and takes place between the leading researcher and the
headquarters of a company. The leading researcher’s bargaining power in
proposing the terms for disclosure again stems from the market asymmetry: the
specialists having the desired level of expertise might be in a short supply.
For instance, investors’ interest in quantum computing has grown markedly in
recent years, while there are reports of a shortage of human capital in this
industry.101010”The Quantum Computing Bubble.” _Financial Times_ , August 25,
2022.111111“Quantum Computing Funding Remains Strong, but Talent Gap Raises
Concern”, a report by McKinsey Digital, https://www.mckinsey.com/business-
functions/mckinsey-digital/our-insights/quantum-computing-funding-remains-
strong-but-talent-gap-raises-concern/. Another example is the communication
software industry, which has recently experienced increased investment
activity.121212”This Is Insanity: Start-Ups End Year in a Deal Frenzy.” _Best
Daily Times_ , December 07, 2020.

As the agent enjoys the power of full control over the information on the
progress of the project, she is completely free to offer what is disclosed and
when. In particular, the contract between the agent and the principal can
specify that the completion of the second stage of the project is disclosed
with a delay rather than immediately. The agent who has an advantage in
expertise over the principal can rationalize such a condition by saying that
before the success is reported to the principal, it is worth re-checking the
data, which takes time.

Even though the principal can not dictate to the agent which information and
how she should disclose, the principal can potentially hire an external
monitor who would visit the lab and prepare an additional report on the
progress of the project. In that case, the contract signed between the agent
and the principal will account for both free information that the agent
promised to provide and additional costly information which the principal
obtains with the help of a monitor. In the baseline version of the model, I
assume that the principal can not use the help of a monitor. This can be
rationalized by the shortage of experts in the field, which makes hiring a
monitor prohibitively costly. Alternative interpretation is that the agent
restricts the principal’s access to additional information on the progress of
the project by stating that a potential information leak would put the
technology being developed at risk.131313In particular, this rationale was
used to restrict the investors’ access to information on the progress of the
project in the case of Theranos, see ”What Red Flags? Elizabeth Holmes Trial
Exposes Investors’ Carelessness.” _The New York Times_ , November 04, 2021.

The information policy relies upon the agent’s commitment power, which holds
not only within each date but also between the dates. The agent’s commitment
within each date follows from prohibitively high legal costs of cooking up the
evidence. The agent’s intertemporal commitment stems from the rigidity of
terms and form of reporting fixed in the contract that the agent and the
principal sign at the outset of the game.

##  4 No-information and full-information benchmarks

###  4.1 No-information benchmark

First, I consider the simple case when the information policy is given by
$\sigma^{NI}$: the same message $m$ is sent for all histories
$h\left(t\right)$ and all dates $t$. Thus, the agent provides no information
regarding the progress of the project. As I demonstrate, in this case the
principal starts investing in the project if and only if it is sufficiently
promising for the principal from the ex ante perspective and invests until a
deterministic interior date.

As no news arrives, the principal bases his decision about when to stop
investing on his unconditional belief regarding the completion of the second
stage of the project. I denote the unconditional belief that $n$ stages of the
project were completed by $t$, by $p_{n}\left(t\right)$, i.e.,
$p_{n}\left(t\right)\coloneqq\operatorname{P}\left(x_{t}=n\right)$. The state
of the world is fully determined by $p\left(t\right)$ given by

$\displaystyle p_{0}\left(t\right)$ $\displaystyle=e^{-\lambda t},$
$\displaystyle p_{1}\left(t\right)$ $\displaystyle=\lambda te^{-\lambda t},$
$\displaystyle p_{2}\left(t\right)$ $\displaystyle=1-e^{-\lambda t}-\lambda
te^{-\lambda t}.$

The principal’s sequential choice of $a_{t}\in\left\\{0,1\right\\}$ can be
restated equivalently as the choice of deterministic stopping time
$S^{NI}\in\left[0,T\right]$ chosen at $t=0$.141414Note that the dynamic belief
system that he faces is deterministic in a sense of being fully specified from
$t=0$ perspective. Given the principal’s continuous investment, the
probability of completion of the second stage of the project,
$p_{2}\left(t\right)$, increases monotonously over time, making obtaining the
payoff $v$ more likely. However, postponing the stopping is costly.

To decide on $S^{NI}$, the principal trades off the flow benefits and flow
costs of postponing the stopping decision, while keeping the individual
rationality constraint in mind. The flow cost of postponing the stopping for
$\Delta t$ is given by $c\cdot\Delta t$ and the flow benefit is given by
$v\cdot p_{1}\left(t\right)\lambda\Delta t$.151515To observe this, note that
the probability of the completing both the first and second stages within a
very short time $\Delta t$ is negligibly small; thus, during some $\Delta t$,
the principal receives the project completion payoff $v$ iff the first stage
has already been completed. Thus, the necessary condition for the principal’s
stopping at some interior moment of time $\left(0<S<T\right)$ is given by

$v\cdot p_{1}\left(S\right)\lambda=c.$ (1)

Let

$\kappa\coloneqq\frac{c}{v\lambda},$

the ratio of the flow cost of investment $c$ to the gross project payoff $v$
normalized using $\lambda$, the rate at which a project stage is completed in
expectation. The intuitive interpretation of $\kappa$ is the _flow cost-
benefit ratio of the project_. $\kappa$ is an inverse measure of how ex ante
promising the project is for the principal. (1) is equivalently given
by161616Here I WLOG express the flow benefits and flow costs of investing for
the principal in different units of measurement.

$\underbrace{p_{1}\left(S\right)}_{\text{flow benefit of
waiting}}=\underbrace{\kappa}_{\text{flow cost of waiting}}$ (2)

and presented graphically in Figure 1. As the state process transitions
monotonously from $0$ to $1$ and then to $2$, $p_{1}\left(t\right)$ first
increases and after some time starts to decrease. Thus, the principal has two
candidate interior stopping times satisfying (2), $\bar{S}$ and
$\bar{S}^{NI}$. The principal prefers to postpone stopping to $\bar{S}^{NI}$,
as during $\left(\bar{S},\bar{S}^{NI}\right)$ the flow benefits are higher
than the flow costs.

$0$$t$$1$ $p_{2}(t)$, probability of completion
---
of 2nd stage of project

$0$$t$$\frac{1}{e}$ $p_{1}(t)$, flow benefit
---
of waiting
$\frac{c}{v\lambda}$, flow cost
---
of waiting
optimal
---
choice
$\bar{S}$$\bar{S}^{NI}$

Figure 1: Principal’s choice under no information:
left plot: postponing stopping increases the chance of getting a project
payoff $v$;
right plot: principal trades off costs and benefits and optimally stops at
$\bar{S}^{NI}$.

The forward-looking principal can guarantee himself a payoff of $0$ if he does
not start investing at $t=0$. Thus, he will choose to start investing at $t=0$
only if his flow gains accumulated up to $T\wedge\bar{S}^{NI}$ are larger than
his flow losses, and his expected payoff is given by

$V^{NI}\coloneqq\max\left\\{0,\int_{0}^{T\wedge\bar{S}^{NI}}\left(v\cdot
p_{1}\left(s\right)\lambda-c\right)ds\right\\}.$ (3)

Geometrically, the integral in (3) represents the difference between the
shaded areas in Figure 2 that correspond to the accumulated gains and losses.
The principal starts investing at $t=0$ if, given $T$ and $\lambda$, the
normalized cost-benefit ratio $\kappa$ is low enough, so that the shaded area
of the accumulated gains is at least as large as that of the accumulated
losses. I denote such a cutoff value of $\kappa$ by
$\kappa^{NI}\left(T,\lambda\right)$ and summarize the principal’s choice
without information in Lemma 1.

$\bar{S}^{NI}$$y=x^{2}$$y=x$$\kappa\coloneqq\frac{c}{v\lambda}$$p_{1}(t)$
accumulated
---
gains
accumulated
---
losses
$t$

$T$$\bar{S}^{NI}$$y=x^{2}$$y=x$$\kappa\coloneqq\frac{c}{v\lambda}$$p_{1}(t)$
accumulated
---
gains
accumulated
---
losses
$t$

Figure 2: Principal’s choice to start investing at $t=0$ or not:
left plot: $T>\bar{S}^{NI}$; the project deadline is distant and decision-
irrelevant;
right plot: $T\leq\bar{S}^{NI}$; the project deadline is close, which leads to
lower expected benefits of investing.

In both plots the expected accumulated gains are higher than the losses, so
the principal starts to invest at $t=0$.

######  Lemma 1.

Assume no information regarding the progress of the project arrives over time.
Denote the time at which the principal stops investing by $S^{NI}$. If
$\kappa>\kappa^{NI}\left(T,\lambda\right)$, then the principal does not start
investing in the project, i.e., $S^{NI}=0$. If
$\kappa\leq\kappa^{NI}\left(T,\lambda\right)$, then the principal’s choice of
stopping time is given by

$S^{NI}=\begin{cases}\bar{S}^{NI},&\text{if }\frac{1}{\lambda}\leq T\text{ and
}\kappa\geq e^{-\lambda T}\lambda T\\\ T,&\text{otherwise ,}\end{cases}$ (4)

the closed-form expressions for $\bar{S}^{NI}$ and
$\kappa^{NI}\left(T,\lambda\right)$ are presented in the proof in Appendix C.

###  4.2 Full-information benchmark

Here, I consider the case in which the information policy is given by
$\sigma^{FI}$: $M=\left\\{m_{0},m_{1},m_{2}\right\\}$ and the message $m_{n}$
is sent for all $t$ such that $x_{t}=n$, $n\in\left\\{0,1,2\right\\}$. Thus,
the principal fully observes the progress of the project at each
$t$.171717This benchmark corresponds to equilibrium in the setting, where the
principal has the full power to propose the terms of self-reporting to the
agent. I characterize the cutoff level of the cost-benefit ratio below which
the principal is willing to start investing. Further, I show that the
principal chooses to stop when no stages of the project are completed and the
project completion deadline $T$ is sufficiently close.

At each $t$, the principal uses the information on the number of stages
completed to decide either to stop investing or to postpone the stopping. The
news on completion of the second stage of the project makes the principal stop
immediately, as this way he immediately receives the project payoff $v$ and
stops incurring the costs of further investment. If only the first stage of
the project is completed, the principal faces the following trade-off. The
instantaneous probability that the second stage will be completed during
$\Delta t$ is given by $\lambda\Delta t$, which is constant over the time.
Thus, the expected benefit of postponing the stopping for $\Delta t$ is given
by $v\cdot\lambda\Delta t$. Meanwhile, the expected cost of postponing the
stopping is given by $c\cdot\Delta t$. As a result, if $\kappa\leq 1$, then
the principal who knows that the first stage of the project has already been
completed invests until either the completion of the second stage or until the
project deadline $T$ is reached.

Consider now the case in which the principal knows that the first stage has
not yet been completed. The principal’s trade-off with respect to the stopping
decision is now more involved. Postponing the stopping for $\Delta t$ leads to
the completion of the first stage of the project with the instantaneous
probability $\lambda\Delta t$. Completion of the first stage of the project at
some $t$ implies that the principal receives the continuation value of the
game, conditional on having the first stage completed. I denote the
continuation value of the principal at time $t$ under full information and
conditional on the completion of first stage of the project by $V_{t|1}^{FI}$.
This is given by181818See the derivation in the proof of Lemma 2 in the
Appendix.

$V_{t|1}^{FI}=\left(v-\frac{c}{\lambda}\right)\left(1-e^{-\lambda\left(T-t\right)}\right).$
(5)

The principal’s expected benefit from postponing the stopping for $\Delta t$
is given by $V_{t|1}^{FI}\cdot\lambda\Delta t$ and the cost of postponing the
stopping is, as before, given by $c\cdot\Delta t$. The continuation value,
$V_{t|1}^{FI}$, shrinks over time and approaches $0$ as the project deadline
$T$ approaches. This is because the shorter the time left before the project
deadline, the less likely it is that the second stage of the project will be
completed before $T$. If at some $t$, and given that no stages are completed
yet, the expected net benefit of investing reaches $0$, it is optimal for the
principal to stop at that $t$.191919If at $t$ the expected benefit of
investing becomes lower than the cost, then, after $t$, the net expected
benefit remains negative. Thus, it is optimal for the principal to stop
investing precisely at $t$. I denote this date by $S_{0}^{P}$ and plot it in
Figure 3.

$0$$t$$S_{0}^{P}$$T$ $V_{t|1}^{FI}\lambda$, expected benefit
---
of investing at $t$
$c$, cost
---
of investing at $t$
interim
---
deadline
Figure 3: The principal optimally sets an interim deadline $t=S_{0}^{P}$ under
full information: given that the first stage of the project has not been
completed by $S_{0}^{P}$, it is optimal to stop investing at $S_{0}^{P}$.

As the principal has an incentive to stop at $S_{0}^{P}$ only if he knows that
the first stage or the milestone of the project has not been reached, the
economic interpretation of $S_{0}^{P}$ is that it is _the interim deadline_
that the principal sets for the project. If the milestone has not been reached
by the interim deadline, then it is sufficiently unlikely that the project
will be completed before the project deadline $T$. Thus, it is optimal for the
principal to “give up” on the project and stop investing at $t=S_{0}^{P}$. If
the milestone is reached by the interim deadline, then the principal has an
incentive not to stop investing until either the second stage is completed or
$T$ is hit.

Finally, given the plan to stop either at the interim deadline, or at the
completion of the second stage of the project, it is individually rational to
start investing only if the principal’s expected payoff from the $t=0$
perspective is non-negative. I denote the upper bound for the normalized cost-
benefit ratio such that the principal starts investing at $t=0$ by
$\kappa^{FI}\left(T,\lambda\right)$. Intuitively,
$\kappa^{FI}\left(T,\lambda\right)>\kappa^{NI}\left(T,\lambda\right)$:
whenever the principal is willing to start investing under no information, he
is also willing to start under the full information. I summarize the
principal’s choice under full information in Lemma 2.

######  Lemma 2.

Assume that the progress of the project is fully observable at each moment in
time. If $\kappa>\kappa^{FI}\left(T,\lambda\right)$, where
$\kappa^{FI}\left(T,\lambda\right)>\kappa^{NI}\left(T,\lambda\right)$, then
the principal does not start investing in the project. If
$\kappa\leq\kappa^{FI}\left(T,\lambda\right)$, the principal invests either
until the random date at which the second stage of the project is completed,
$t=\tau_{2}$, or until the interim deadline, $t=S_{0}^{P}$, at which he stops
if the first stage has not yet been completed. Formally, the time at which the
principal stops investing is a random variable $\tau$ given by:

$\tau=\begin{cases}\tau_{2}\wedge T,&\text{if }x_{S_{0}^{P}}\neq 0\\\
S_{0}^{P},&\text{otherwise },\end{cases}$

where
$S_{0}^{P}=T+\frac{1}{\lambda}\log\left(\frac{1-2\kappa}{1-\kappa}\right)$ and
the expression for $\kappa^{FI}\left(T,\lambda\right)$ is presented in the
proof in Appendix C.

Assume now that the agent chooses which information to provide to the
principal. As for $\kappa>\kappa^{FI}\left(T,\lambda\right)$ the principal is
not willing to start investing even under full information, there is no way in
which the agent can strategically conceal the information to her benefit. In
Section 5, I assume $\kappa\leq\kappa^{FI}\left(T,\lambda\right)$ and analyze
how the agent can strategically provide information on the progress of the
project and extract the principal’s surplus.

##  5 Agent’s choice of information policy

In this Section, I present how the agent’s choice of information policy
changes with the ex ante attractiveness of the project, which is captured by
the cost-benefit ratio $\kappa$. In Section 5.1, I start with Proposition 1
which summarizes the comparative statics result. In Sections 5.2-5.3, I
proceed with the detailed discussion of the economic mechanisms that determine
the outlined structure of the optimal information policy. Throughout Section
5, I maintain the following technical assumption:

######  Assumption 1.

$e^{\lambda T}>\lambda T\left(\lambda T+1\right)+1$.

For the sake of a clearer exposition, this assumption rules out the case in
which $T$ is so low that whenever the principal is willing to start investing
in the no-information benchmark, he invests until $T$. Relaxing this
assumption does not change the the comparative statics result in Proposition 1
qualitatively.202020I discuss the implications of relaxing this assumption in
the proof of Proposition 1.

###  5.1 The structure of optimal information disclosure

######  Proposition 1.

There exist cost-benefit ratio cutoffs
$\kappa^{ND}\left(T,\lambda\right),\kappa^{ND}\left(T,\lambda\right)<\kappa^{NI}\left(T,\lambda\right)$,
and
$\tilde{\kappa}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)<\tilde{\kappa}\left(T,\lambda\right)<\kappa^{FI}\left(T,\lambda\right)$,
such that, depending on the cost-benefit ratio of the project, the optimal
information policy has the following form:

  * 1.

when $\kappa\leq\kappa^{ND}\left(T,\lambda\right)$, the agent provides no
information and the principal invests until $T$;

  * 2.

when
$\kappa^{ND}\left(T,\lambda\right)<\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$,
the agent discloses only the completion of the second stage of the project and
does that with the postponement;

  * 3.

when
$\tilde{\kappa}\left(T,\lambda\right)<\kappa<\kappa^{FI}\left(T,\lambda\right)$,
the agent immediately discloses the completion of the second stage of the
project whenever it occurs and specifies a deterministic interim deadline, at
which it discloses if the first stage is already completed;

  * 4.

when $\kappa\geq\kappa^{FI}\left(T,\lambda\right)$, the agent provides no
information as the principal’s long-run payoff is non-positive even under full
information.

Figure 4 illustrates Proposition 1 and presents the partition of the cost-
benefit ratio space based on the corresponding forms of the optimal
information policy.

$\kappa\left(T,\lambda\right)$$0$$\tilde{\kappa}$$\kappa^{FI}$$\kappa^{NI}$$\kappa^{ND}$$\frac{1}{2}$
Postponed disclosure
---
of stage 2 completion
Immediate disclosure of
---
stage 2 completion and
interim deadline for stage 1
Non-disclosure
---
(principal
invests until $T$)
Non-disclosure
---
(principal rejects
the project)
Figure 4: Comparative statics of the form of optimal information policy with
respect to the cost-benefit ratio of the project,
$\kappa\left(T,\lambda\right)$.

The structure of optimal disclosure presented in Proposition 1 follows the
simple and intuitive pattern. The lower is the value of cost-benefit ratio,
the higher is ex ante attractiveness of the project to the principal. First,
for $\kappa\leq\kappa^{ND}\left(T,\lambda\right)$, the project is so
attractive that the principal is willing to keep investing until the project
deadline $T$ even in the no-information benchmark. Thus, there is no need to
disclose any information. For the higher values of $\kappa$, there emerges a
room for strategic disclosure, and the higher is the value of $\kappa$ (i.e.,
the lower is the ex ante attractiveness of the project), the more information
the agent has to disclose to incentivize the principal. For
$\kappa\geq\kappa^{FI}\left(T,\lambda\right)$, the project gets so
unattractive that the principal can not strictly benefit from investing even
in the full-information benchmark. In this extreme case, the agent chooses not
to disclose any information.

The most important part of the result in Proposition 1 demonstrates which
additional pieces of information the agent chooses to disclose and when she
chooses to discloses them as $\kappa$ gets higher and higher. When
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$,
the agent discloses only the completion of the second stage of the project and
does not promise any information on the completion of the first stage of the
project. Further, as $\kappa$ increases from
$\kappa^{ND}\left(T,\lambda\right)$ to $\tilde{\kappa}\left(T,\lambda\right)$,
the agent adjusts the timing of the disclosure: she postpones the disclosure
of the second stage completion less and less and discloses immediately for
$\tilde{\kappa}\left(T,\lambda\right)$. For
$\kappa\in(\tilde{\kappa}\left(T,\lambda\right),\kappa^{FI}\left(T,\lambda\right))$,
the agent not only discloses the completion of the second stage of the project
immediately, but also provides information on the completion of the first
stage at the interim deadline that she optimally chooses.

In the subsequent Sections, I provide details on the mechanisms that shape the
aforementioned comparative statics results. I omit the trivial case of non-
disclosure under $\kappa\leq\kappa^{ND}\left(T,\lambda\right)$ and start the
discussion from the optimal information policy under
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.

###  5.2 Postponed disclosure of project completion

In this Section, I restrict attention to
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$
and explain why the optimal information policy has the particular form
presented in the Proposition 1: the agent discloses only the completion of the
project and does this with the postponement.

####  5.2.1 Agent’s problem

To characterize the agent’s choice of information policy, I consider an
equivalent problem, in which the agent directly chooses the stochastic
history-contingent length of investment subject to the principal’s individual
rationality constraints that ensure optimality of such action process for the
principal. An _investment schedule_ is a random variable
$\tau:\Omega\rightarrow\ [0,T]$ defined on the probability space
$\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and adapted to the filtration
$F=(\mathcal{F}_{t})_{t\geq 0}$ generated by the stochastic process $x_{t}$.
As I demonstrate in Section 5.2.2, restricting attention to random variables
adapted to the _natural_ filtration of $x_{t}$ is without loss of generality
for the agent’s equilibrium expected payoff when
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.212121In
other words, there is no need for external randomization devices to optimally
incentivize the principal when
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.

Informally, an investment schedule $\tau$ is a random variable with support
$\left[0,T\right]$ specified by a rule that suggests when to stop investing
depending on the history of previous realizations of the number of completed
stages $x_{t}$.222222The stopping rules from the no-information and full-
information benchmarks are given in Lemmas 1 and 2, respectively. Further
examples of such rules include “stop 1 minute after the second stage of the
project is completed” and “stop at $t=S$ if only the first stage of the
project is completed by $t=S$”. The agent chooses this rule at $t=0$.
$\operatorname{P}\left(x_{\tau}=2\right)$ captures the belief about two stages
of the project completed by $\tau$, the random time of stopping in the future,
and $\operatorname{E}\left[\tau\right]$ captures the expected length of
investment from $t=0$ perspective.

Given an investment schedule $\tau$, the long-run payoff of the agent and the
principal are given, respectively, by

$\displaystyle W\left(\tau\right)$
$\displaystyle\coloneqq\operatorname{E}\left[\tau\right]c,$ $\displaystyle
V\left(\tau\right)$
$\displaystyle\coloneqq\operatorname{P}\left(x_{\tau}=2\right)v-\operatorname{E}\left[\tau\right]c.$

As an investment schedule $\tau$ is an action recommendation rule, the action
recommendations generated by this rule have to be obedient for the principal.
In other words, at each date and for each possible history the principal’s
actions suggested by $\tau$ have to be optimal for the principal. An object
useful for characterizing if an investment schedule $\tau$ generates obedient
action recommendations is given by the principal’s continuation value at some
interim date $t$ promised by the investment schedule $\tau$. This continuation
value depends on the beliefs of the principal.

As the principal does not commit to a policy at $t=0$, he rationally updates
his belief given an investment schedule $\tau$ and assesses the costs and
benefits of either further following the investment schedule $\tau$ provided
by the agent or deviating from it. The absence of stopping by some date $t$
and, thus, the fact that the game continues at $t$ serves as a source of
inference for the principal. First, he forms a belief regarding the number of
completed stages of the project by $t$, conditional on the game still
continuing at $t$, $\operatorname{P}\left(x_{t}=n|t<\tau\right)$. Second, he
forms a belief regarding the number of completed stages of the project at the
random date of stopping in the future, $\tau$,
$\operatorname{P}\left(x_{\tau}=n|t<\tau\right)$.

Given the absence of stopping by $t$, the principal’s expected payoff promised
by the schedule is given by
$\operatorname{P}\left(x_{\tau}=2|t<\tau\right)v-\operatorname{E}\left[\tau-t|t<\tau\right]c$.
The principal’s expected payoff from stopping at $t$ is given by
$\operatorname{P}\left(x_{t}=2|t<\tau\right)v$. The principal’s _continuation
value at $t$_ given the investment schedule $\tau$ is the difference between
these two expected payoffs, I denote it by $V_{t}\left(\tau\right)$:

$V_{t}\left(\tau\right)\coloneqq\left[\operatorname{P}\left(x_{\tau}=2|t<\tau\right)-\operatorname{P}\left(x_{t}=2|t<\tau\right)\right]v-\operatorname{E}\left[\tau-t|t<\tau\right]c.$
(6)

This way of formulating the continuation value is intuitive: if the
continuation value $V_{t}\left(\tau\right)$ gets negative then it is not
valuable to continue investing for the principal, and he is better-off
stopping immediately rather than following the schedule. The following Lemma
shows the necessary and sufficient conditions for an investment schedule
$\tau$ to generate obedient action recommendations for the principal.

######  Lemma 3.

An investment schedule $\tau$ is the principal’s best response to at least one
information policy $\sigma$ if and only if

$V_{t}\left(\tau\right)\geq 0,\forall t\geq 0\text{ and }V_{\tau}^{NI}<0,$ (7)

where $V_{t}^{NI}$ is the principal’s optimal continuation value in the
absence of any additional information from the agent starting from the date
$t$.

$V_{t}\left(\tau\right)\geq 0,\forall t\geq 0$ ensures that the principal does
not want to stop before the date of stopping suggested by the investment
schedule is reached, and $V_{\tau}^{NI}<0$ ensures that the principal does not
want to continue conditional on reaching the date of stopping suggested by the
investment schedule. Conditions from Lemma 3 constitute the system of
constraints for the agent’s problem.

As the agent chooses an investment schedule $\tau$ to maximize her long-run
payoff, the constraint $V_{\tau}^{NI}<0$ is inactive at
optimum.232323Otherwise, the agent can prolong the expected funding by
choosing a different $\tau$. Thus, without loss of generality, I omit this
constraint from the agent’s problem, and the problem that the agent solves at
$t=0$ is given by

$\displaystyle\max_{\tau\in\mathcal{T}}\left\\{c\cdot\operatorname{E}\left[\tau\right]\right\\}$
(8) $\displaystyle\text{s.t.}\,V_{t}\left(\tau\right)\geq 0,\forall t\geq 0,$

where $\mathcal{T}$ is the set of stopping times with respect to the natural
filtration of $x_{t}$. As the principal’s choice to postpone the stopping of
funding is costly, it is natural to interpret the system of constraints in (8)
as the _system of principal’s individual rationality constraints_.

The agent’s problem is complex, and thus I solve it in steps. First, I
characterize the investment schedule, which solves the _relaxed_ version of
(8) with the principal’s individual rationality constraints only for some
initial periods. Second, I demonstrate that there exists an investment
schedule solving the _relaxed_ agent’s problem and satisfying the _full_
system of the principal’s individual rationality constraints (7). This
investment schedule pins down optimal information policy.

####  5.2.2 Solution to the relaxed agent’s problem

In this Section, I consider the relaxed agent’s problem and discuss its
solution. This sheds light on the technical intuition behind the key
properties o the optimal information policy. The agent’s relaxed problem for
the parametric case of
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$
is given by (8) with the _principal’s individual rationality constraint only
for $t\in[0,\bar{S}^{NI}]$_. The agent’s relaxed problem for the parametric
case of
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$
is given by (8) with the _principal’s individual rationality constraint only
for $t=0$_.

Consider the agent’s long-run payoff given an investment schedule,
$W\left(\tau\right)$. This can be restated equivalently as follows:

$\displaystyle W\left(\tau\right)$
$\displaystyle=\left[W\left(\tau\right)+V\left(\tau\right)\right]-V\left(\tau\right)$
(9)
$\displaystyle=\underbrace{\operatorname{P}\left(x_{\tau}=2\right)v}_{\text{total
surplus}}\,-\underbrace{\left[\operatorname{P}\left(x_{\tau}=2\right)v-\operatorname{E}\left[\tau\right]c\right]}_{\text{principal's
surplus}}.$

The solution to the agent’s relaxed problem for both considered parametric
cases follows a simple idea: the optimal investment schedule _ensures that the
total surplus is maximal and that the principal’s surplus is minimal_.
Consider a schedule $\tau$ such that the stopping occurs after the completion
of the second stage of the project, unless the project deadline $T$ was hit,
i.e., the schedule satisfies the condition $\tau\geq\tau_{2}\land T$. Such a
schedule leads to

$\operatorname{P}\left(x_{\tau}=2\right)=\operatorname{P}\left(x_{T}=2\right).$
(10)

Given a schedule $\tau$ satisfying (10), if $\tau$ is individually rational
for the principal at date $t=0$ then the total surplus generated achieves its
upper bound and is given by $\operatorname{P}\left(x_{T}=2\right)v$, which
depends on the exogenously given project deadline $T$ and the profit $v$.
However, the stopping only after the second stage completion is not
individually rational for the principal at $t=0$ when the cost of funding is
sufficiently high, the profit is sufficiently low, or the expected time until
a project stage completion is sufficiently high.

Lemma 4 elaborates on the cost-benefit ratio cutoff value
$\tilde{\kappa}\left(T,\lambda\right)$: it distinguishes the case in which
stopping only after the second stage completion is individually rational at
$t=0$ from the case in which it is not. Based on this partition, when
$\kappa\in(\kappa^{ND},\tilde{\kappa}\left(T,\lambda\right)]$, I call the
project _ex ante promising_ for the principal.

######  Lemma 4.

For each $\left(T,\lambda\right)$ there exists
$\tilde{\kappa}\left(T,\lambda\right)$,
$\kappa^{NI}\left(T,\lambda\right)<\tilde{\kappa}\left(T,\lambda\right)<\kappa^{FI}\left(T,\lambda\right)$,
such that if $\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$
$\left(\kappa>\tilde{\kappa}\left(T,\lambda\right)\right)$ then an investment
schedule $\tau$ in which stopping after $\tau_{2}\land T$ happens with
probability one is individually rational at $t=0$ (not individually rational
at $t=0$) for the principal.

For
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$,
the schedule $\tau\geq\tau_{2}\land T$ is individually rational for the
principal at $t=0$, and it maximizes the total surplus. In addition to
choosing $\tau\geq\tau_{2}\land T$, it is optimal for the agent to choose the
investment schedule with a higher expected date of stopping the funding to
extract all the principal’s surplus subject to his individual rationality
constraints. For
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$,
the agent chooses such $\tau$ that the principal’s individual rationality
constraint at $t=0$ is binding. As a result, $V(\tau)=V^{NI}$, i.e., the
principal gets his no-information benchmark payoff given by $0$.

For
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$,
as in the no-information benchmark the principal invests until $\bar{S}^{NI}$
with certainty, the agent chooses the investment schedule as to postpone the
start of information provision at least until $\bar{S}^{NI}$. Further, the
agent chooses $\tau$ with a higher expected date of stopping so that the
principal’s individual rationality constraint at $t=\bar{S}^{NI}$ is binding.
The absence of stopping until at least $\bar{S}^{NI}$ and the fact that
individual rationality constraint binds at $t=\bar{S}^{NI}$ taken together
imply that $V(\tau)=V^{NI}$, i.e., from $t=0$ perspective, the principal gets
her no-information benchmark payoff, which is non-negative and given by (3).

The next Lemma summarizes the _necessary_ conditions for an investment
schedule to solve the agent’s relaxed problem when the project is promising.
These conditions are shared both by the relaxed problem formulated for the
case of
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$
and the relaxed problem formulated for the case of
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.
The conditions that are _both necessary and sufficient_ for an investment
schedule to solve the agent’s relaxed problem are presented in the Proof of
Lemma 5.

######  Lemma 5.

Assume $\kappa\in(\kappa^{ND},\tilde{\kappa}\left(T,\lambda\right)]$. If an
investment schedule $\tau$ solves agent’s relaxed problem, then

  * 1.

with probability one, stopping occurs after $\tau_{2}\land T$;

  * 2.

$V\left(\tau\right)=V^{NI}$, where $V^{NI}$ is the principal’s expected payoff
in the no-information benchmark, given by (3).

####  5.2.3 Optimal information policy

In this Section, I show that there exists an information policy that both
solves the agent’s relaxed problem and satisfies the full system of the
individual rationality constraints. Given this, as Lemma 5 describes the
solution to the relaxed problem, it also sheds light on the properties of the
optimal information policy for the case of a promising project. These
properties have a clear-cut economic interpretation as an investment schedule
$\tau$ can be easily interpreted in terms of action recommendations for the
principal.

An investment schedule $\tau$ can be without loss of generality implemented
using a _direct recommendation mechanism_ \- a simple policy which has
$M=\left\\{0,1\right\\}$, where $m=1$ received at date $t$ is a recommendation
to continue investing at $t$ for the principal and $m=0$ received at date $t$
is a recommendation to stop investing at $t$.242424The connection between an
investment schedule $\tau$ and a direct recommendation mechanism implementing
the schedule $\tau$ is simple: whenever, based on the evolution of the state
process, $\tau$ suggests stopping the funding, the direct recommendation
mechanism sends the message $m=0$. Keeping this in mind, it is clear from
Lemma 5 that the _optimal information policy_ has to satisfy the following
conditions. _First_ , whenever the agent recommends the principal to stop, the
second stage of the project is already completed. _Second_ , the
recommendation to stop is postponed so that the principal’s individual
rationality constraint is binding, which manifests in
$V\left(\tau\right)=V^{NI}$. The first condition presents the key feature of
the optimal information policy for the case of promising project: the agent
discloses the completion of the second stage of the project, but _stays silent
regarding the completion of the first stage of the project_. The intuition
behind the agent’s choice is simple: a recommendation to stop when no stages
of the project are completed and the project deadline $T$ is close does indeed
incentivize the principal; however, it also reduces the total surplus
generated that can be extracted via the agent’s control of information.
Meanwhile, the recommendation to stop when the two stages of the project are
completed incentivizes the principal without reducing the total surplus
generated. When $\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$, a partially
informative policy that discloses only the completion of the second stage
provides sufficient incentives to the principal, and thus the agent uses
it.252525The “leading on” information policy in Ely and Szydlowski, (2020) is
similar: the only information that the policy provides is that the task is
already completed and, thus, it is time to stop investing.

I proceed with obtaining an investment schedule that not only satisfies the
conditions in Lemma 5 and solves the relaxed problem, but also satisfies the
full system of the principal’s individual rationality constraints in Lemma 3.
Ensuring both is non-trivial. For instance, consider a mechanism that
implements an investment schedule solving the agent’s relaxed problem and
assume it recommends to continue for $t\in\left[0,S^{*}\right)$, then at
$S^{*}$ recommends stopping if the second stage is already completed, but
recommends to continue at all the subsequent dates $t\in(S^{*},T]$. A no
stopping recommendation drawn at $S^{*}$ reveals that the state is either $0$
or $1$. Clearly, after sufficient time passes after $S^{*}$, the principal
would attach a high probability to the second stage already being completed
and would potentially be tempted to deviate from the recommendation to
continue.262626In other words, $V_{t}\left(\tau\right)$ drifts down over time
and can get negative at some date. However, a _direct recommendation mechanism
that implements an optimal investment schedule exists_. I present it in
Proposition 2.

######  Proposition 2.

Assume
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.
The optimal mechanism does not provide a recommendation to stop during
$t\in[0,S^{*})$. At $t=S^{*}$, if the second stage of the project is already
completed, then the mechanism recommends the principal to stop. If the second
stage of the project is not yet completed, then the mechanism recommends the
principal to stop at the moment of its completion $t=\tau_{2}$. Formally,

$\tau=S^{*}\lor\left(\tau_{2}\land T\right),$

where $S^{*}$ is chosen such that $V\left(\tau\right)=V^{NI}$, i.e., the
respective constraint in the system of principal’s individual rationality
constraints is binding.

The recommendation mechanism starting from $S^{*}$ generates recommendations
to stop if the second stage is completed. As the recommendation to stop comes
immediately at the completion of the second stage for all $t>S^{*}$, hearing
no recommendation to stop reveals that the state is either $0$ or $1$.
Further, as time goes on, the principal attaches a higher and higher
probability to the state being $1$, which ensures obedience to the
recommendation to continue at each date. Further, the start of information
provision $S^{*}$ is sufficiently postponed to ensure that the principal’s
individual rationality constraint is binding either at $t=\bar{S}^{NI}$ or at
$t=0$.

The choice of $S^{*}$ is driven by extraction of the principal’s surplus and
depends on $\kappa$ in an intuitive way. First, consider the case
$\kappa\in(\kappa^{ND},\kappa^{NI}\left(T,\lambda\right)]$, the principal is
willing to start investing and invests until $t=\bar{S}^{NI}$ in the no-
information benchmark. The agent’s optimal choice is to set
$S^{*}>\bar{S}^{NI}$. Given such an information policy, the principal does not
stop at $\bar{S}^{NI}$, the date of stopping in the no-information benchmark,
and with probability one continues to invest during $t\in[\bar{S}^{NI},S^{*})$
even though the mechanism provides absolutely no information for all
$t<S^{*}$. This is driven by the fact that the expected benefit from stopping
at some future date $t\in[S^{*},T]$ and obtaining the project payoff $v$ with
certainty compensates the flow losses of investing during
$t\in[\bar{S}^{NI},S^{*})$.272727Similarly to the “leading on” information
policy in Ely and Szydlowski, (2020), the promises of future disclosure of the
completion of the project are used as a “carrot” to make the receiver continue
investing beyond the point at which he stops in the no-information benchmark.
Further, the agent sufficiently postpones $S^{*}$ to ensure that she extracts
the principal’s surplus and the principal gets precisely $V^{NI}\geq 0$.

In the case
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$,
the principal is not willing to start in the no-information benchmark as his
expected payoff from investing is negative. Thus, the agent chooses $S^{*}$ to
guarantee that the principal gets his reservation value $V^{NI}=0$ and is thus
willing to start investing at $t=0$. The value of $S^{*}$ is relatively lower
as compared to the previous case: as the project is less attractive, to
provide the principal sufficient incentives, the agent needs to start the
information provision regarding the completion of the project earlier.

Finally, there exist many information policies that both solve the relaxed
agent’s problem and satisfy the full system of constraints (7). This
constitutes an important advantage for the agent: she can choose an optimal
policy that is easier to implement from the real-world perspective, depending
on the particular environment. In the optimal mechanism from Proposition 2,
the recommendation to stop at some date $t$ depends only on the current state
of the world $x_{t}$. In an alternative delayed disclosure mechanism, the
recommendation to stop arrives with a pre-specified delay after the second
stage was completed. Thus, the recommendation depends only on the past history
and not on the current state of the world. In an optimal delayed disclosure
mechanism, the delay becomes smaller as more time passes. I characterize such
a mechanism in Appendix D.282828The rich set of optimal direct recommendation
mechanisms in my model encompasses both mechanisms in which the information
provision depends only on the current state, similarly to the optimal
mechanism in Ely and Szydlowski, (2020), and the mechanisms that use delay,
similarly to the delayed beep from Ely, (2017).

Recall that, as Lemma 5 suggests, the key idea of the optimal information
policy is that the agent postpones the disclosure of the completion of the
project to extract more surplus, which makes the principal’s individual
rationality constraint bind. The higher the cost-benefit ratio of the project
$\kappa$ becomes, the higher additional value the agent’s information policy
needs to provide to the principal to ensure that his active individual
rationality constraint is satisfied. The implication of this for the optimal
information policy is presented in Lemma 6.

######  Lemma 6.

Assume
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.
Given the recommendation mechanism implementing an optimal investment schedule
$\tau$, for a fixed Poisson rate $\lambda$, the expected length of investment
$\operatorname{E}\left[\tau\right]$ decreases in the cost-benefit ratio
$\kappa$.

The intuition is that the higher the cost-benefit ratio of the project
becomes, the sooner after the second stage of the project is completed the
agent recommends the principal to stop. For the cost-benefit ratio as high as
$\tilde{\kappa}\left(T,\lambda\right)$, the agent provides the recommendation
to stop immediately at the date of completion of the second stage of the
project. Further, for $\kappa>\tilde{\kappa}\left(T,\lambda\right)$, the
optimal information policy satisfying the conditions in Lemma 5 ceases to be
individually rational for the principal. As I show in the next Section, for
$\kappa>\tilde{\kappa}\left(T,\lambda\right)$, in addition to immediate
disclosure of the project completion, the agent provides the information
regarding the completion of the first stage of the project.

###  5.3 Immediate disclosure of completion and an interim deadline

When $\kappa>\tilde{\kappa}\left(T,\lambda\right)$, the project is not
promising for the principal and any investment schedule in which stopping
occurs after $\tau_{2}\land T$ with probability one violates the principal’s
individual rationality constraint. In other words, from the ex ante
perspective the future reports disclosing only the completion of the project
do not motivate the principal to start investing. Thus, an investment schedule
that provides an individually rational expected payoff to the principal should
assign a positive probability not only to stopping after the completion of the
project, but also to stopping in either state $0$, when no stages of the
project are completed, or state $1$, when only the first stage of the project
is completed. I present the necessary conditions for an investment schedule to
be optimal when the project is not promising in Lemma 7.

######  Lemma 7.

Assume
$\kappa\in(\tilde{\kappa}\left(T,\lambda\right),\kappa^{FI}\left(T,\lambda\right))$.
If an investment schedule $\tau$ solves agent’s problem, then it satisfies the
conditions

  * 1.

conditional on no completed stages of the project, stopping of the funding
happens with a positive probability;

  * 2.

conditional on one completed stage of the project, stopping of the funding
happens with probability zero;

  * 3.

conditional on two completed stages of the project, stopping of the funding
happens immediately (at $t=\tau_{2}$) and with probability one.

Stopping when only the first stage of the project is already completed is
clearly inefficient. In state $1$, the principal prefers to continue investing
until the completion of the second stage and this principal’s incentive to
wait is aligned with the agent’s incentive to postpone the stopping. Further,
stopping in state $1$ does not help overcome the problem of the violated
individual rationality constraint under
$\kappa>\tilde{\kappa}\left(T,\lambda\right)$. Meanwhile, assigning a positive
probability to stopping when no stages are completed helps to overcome the
problem of violated individual rationality constraint, as the principal
benefits from stopping at some date $t$ when the first stage of the project is
not yet completed and the project deadline $T$ is sufficiently close. Further,
the agent clearly prefers the stopping of funding after the completion of the
second stage rather than in state $0$ as the former does not harm the total
surplus generated. Thus, a schedule that is optimal assigns probability $1$ to
immediate stopping when the second stage is completed.

Lemma 7 implies that in an investment schedule, optimal for the agent,
stopping after the completion of the second stage of the project happens
immediately and stopping also happens given that no stages of the project are
completed - i.e., at the _interim deadline chosen by the agent, which I denote
by_ $S_{0}^{A}$. Thus, Lemma 7 drastically simplifies the strategy space in
the agent’s design problem: it is only left to characterize the optimal
interim deadline. At the outset of the game, the agent designs a device that
privately randomizes over the interim deadlines $S_{0}^{A}$. That is, the
agent publicly chooses a distribution $F_{S_{0}^{A}}$, then an interim
deadline is drawn according to it and privately observed by the agent. Next,
the information starts to flow. The action recommendation to stop the funding
satisfies the following investment schedule

$\tau=\begin{cases}S_{0}^{A},&\text{if }x_{S_{0}^{A}}=0\\\ \tau_{2}\land
T,&\text{otherwise},\end{cases}$ (11)

where the principal knows only the distribution $F_{S_{0}^{A}}$, but not the
draw of $S_{0}^{A}$.

Given that the completion of the second stage of the project is disclosed
immediately, stopping at the interim deadline in state $0$ leads to a loss of
expected further investment flow for the agent, and a potential savings from
abandoning a “stagnating” project for the principal. The agent’s payoff can be
without loss of generality restated as the expected loss in future investment
due to stopping at the interim deadline $S_{0}^{A}$ in state $0$ (rather than
at $\tau_{2}\land T$). Given this, the agent’s problem can be expressed as

$\min_{F_{S_{0}^{A}}}\operatorname{E}_{F_{S_{0}^{A}}}\Bigg{[}\underbrace{\operatorname{P}\left(x_{S_{0}^{A}}=0\right)\operatorname{E}\left[\tau_{2}\land
T-S_{0}^{A}|x_{S_{0}^{A}}=0\right]}_{\text{expected loss in future investment
given }S_{0}^{A}}\Bigg{]},$ (12)

subject to the system of the principal’s individual rationality constraints,
which also have a natural interpretation as the expectation of principal’s
savings on the future investment, which discontinues at $S_{0}^{A}$ in state
$0$, minus the loss in the project completion profit due to stopping the
funding at $S_{0}^{A}$ in state $0$.292929The principal’s long-run payoff is
given in (44).

Inspecting the agent’s expected loss in future investment in (12) reveals that
if the agent postpones the interim deadline $S_{0}^{A}$, then two effects
arise. First, the probability that stopping at the interim deadline will
happen decreases. Second, the expected loss in total surplus due to stopping
at the interim deadline rather than at $\tau_{2}\land T$ decreases. Thus, the
agent’s expected loss in future investment is decreasing in the date of
interim deadline and the agent prefers an interim deadline with a later
expected date.

Agent’s choice of the interim deadline distribution $F_{S_{0}^{A}}$ is
affected by the two factors. First, as the expected loss in future investment
in (12) is decreasing and convex in the date of the interim deadline, and thus
the agent is risk-averse with respect to random interim deadlines. Thus, given
some random interim deadline, the agent _directly_ benefits from inducing a
mean-preserving contraction. Second, the agent benefits from inducing a mean-
preserving contraction _indirectly_. Inspecting the principal’s long-run
payoff for some fixed $S_{0}^{A}$ reveals that the principal is also risk-
averse with respect to random interim deadlines. Thus, inducing a mean-
preserving contraction makes the principal better-off and relaxes the
individual rationality constraint at $t=0$, hence, allowing the agent to
postpone the expected interim deadline further. As a result the optimal for
the agent interim deadline takes the form of a _deterministic date_. In other
words, it is optimal for the agent to _publicly announce the interim deadline
$S_{0}^{A}$ at the outset_, so that the principal knows it.

The agent has an incentive to postpone the interim deadline and uses her
control of the information environment to postpone the deadline as much as
possible so that the principal’s individual rationality constraint at $t=0$
binds. Figure 5 demonstrates the principal’s long-run payoff as a function of
the interim deadline, which I denote by $S_{0}$. It is maximized at the
principal-preferred interim deadline $S_{0}^{P}$, which was characterized in
Lemma 2. The agent-preferred interim deadline $S_{0}^{A}$ yields the
principal’s expected payoff of $0$.

$0$$S_{0}$$V$$S_{0}^{P}$$T$$S_{0}^{A}$ Figure 5: Principal’s long-run payoff,
$V$, as a function of an interim reporting deadline chosen by the agent,
$S_{0}$.

The next Proposition summarizes the optimal investment schedule, which can be
without loss of generality implemented using a direct recommendation
mechanism:

######  Proposition 3.

Assume
$\kappa\in(\tilde{\kappa}\left(T,\lambda\right),\kappa^{FI}\left(T,\lambda\right))$.
The optimal information policy is given by a direct recommendation mechanism
that generates

  1. (a)

the recommendation to stop at the moment of completion of the second stage of
the project, $t=\tau_{2}$, and

  2. (b)

a conditional recommendation to stop at the publicly announced interim
deadline $t=S_{0}^{A}$. At $S_{0}^{A}$, stopping is recommended with certainty
if the first stage of the project has not yet been completed.

Formally,

$\tau=\begin{cases}S_{0}^{A},&\text{if }x_{S_{0}^{A}}=0\\\ \tau_{2}\land
T,&\text{otherwise},\end{cases}$

where $S_{0}^{A}$ is chosen so that the principal’s individual rationality
constraint at $t=0$ is binding, i.e., $V\left(\tau\right)=0$.

A stopping recommendation at any date other than the interim deadline
$t=S_{0}^{A}$ fully reveals that project is accomplished. Further, observing a
recommendation to stop at the interim deadline, the principal learns that the
milestone of the project has not yet been reached and becomes sufficiently
pessimistic that the project will be completed by $T$.

A notable feature of the optimal information policy when the project is ex
ante unattractive is its uniqueness. The only optimal instrument through which
the agent fine tunes the incentive provision to the principal is the choice of
interim deadline, and there is a unique optimal way to set the deadline to
make the principal’s individual rationality constraint bind.

I proceed by considering the comparative statics of the interim deadline. Both
the agent-preferred and the principal-preferred interim deadline, $S_{0}^{A}$
and $S_{0}^{P}$, respectively, increase in the exogenous deadline $T$. This is
because less time pressure relaxes the principal’s individual rationality
constraint and allows the agent to postpone the deadline further in order to
extract the principal’s surplus.

As the cost-benefit ratio increases up to $\kappa^{FI}$, the agent-preferred
deadline converges to the principal-preferred deadline. An increase in the
cost-benefit ratio of the project makes the principal’s individual rationality
constraint tighter.303030This is because the increase in $\kappa$ makes the
principal’s instantaneous benefit from waiting decrease, and the normalized
instantaneous cost of waiting becomes higher. As a result, for a higher
$\kappa$, in the absence of completion of the first stage, the principal is
willing to wait for a shorter time before stopping. Thus, both the interim
deadline preferred by the principal $S_{0}^{P}$ and the interim deadline
chosen by the agent $S_{0}^{A}$ are lower for a higher $\kappa$. Further, for
a higher $\kappa$ the agent has to choose an information policy relatively
closer to the full-information benchmark to ensure that the individual
rationality constraint at $t=0$ is satisfied. Hence, the agent-chosen interim
deadline $S_{0}^{A}$ approaches $S_{0}^{P}$, the interim deadline preferred by
the principal. The comparative statics of $S_{0}^{P}$ and $S_{0}^{A}$ with
respect to the cost-benefit ratio of the project $\kappa$ are presented in
Figure 6.

Figure 6: Interim deadline chosen by the agent $S_{0}^{A}$ (dashed) and
preferred by the principal $S_{0}^{P}$ (thick), as functions of the cost-
benefit ratio of the project $\kappa$.

##  6 General preferences

In this Section, I allow for profit-sharing between the agent and the
principal, varying degree of the agent’s benefit from the flow of funds, and
exponential discounting, and demonstrate that the optimal information policy
still has the same properties as in the baseline model.

First, I assume that the agent and the principal share the project completion
profit $v$: the principal gets $\alpha\cdot v$, while the agent gets
$(1-\alpha)\cdot v$, $\alpha\in\left(0,1\right]$. Thus, now the agent benefits
not only from the flow of funds provided by the principal for running the
project but also from the share in the profit. The assumption that the agent
gets a share in the project completion profit is natural in many situations.
In particular, the research documents that the entrepreneurs in innovative
startups are up to some extent driven by giving vent to their entrepreneurial
mindset and bringing their innovative ideas to life (Gundolf et al.,, 2017).
In such a context, a positive profit share of the agent captures that the
agent is motivated by the success of the project.

Second, I assume that given a flow cost of $c$ incurred by the principal, the
agent obtains a flow benefit $\beta c,\beta\geq 0$. $\beta$ can be interpreted
as the agent’s marginal benefit from using the funds provided by the principal
for funding the project. Alternatively, for $\beta\in\left[0,1\right]$ the
loss of $1-\beta$ of the amount of the transfer at each date can be
interpreted as the transaction costs. Finally, setting $\beta=0$ for some
$\alpha<1$ allows for abstracting from the agent’s motives for diverting the
funds and considering the agent motivated only by the success of the project.

Third, I allow for exponential discounting at a rate $r>0$. Thus, the present
value of a profit obtained at a date $t$ is given by $ve^{-rt}$ and the
present value of a stream of funding up to date $t$ is given by
$\frac{1}{r}\left(1-e^{-rt}\right)c$. The following Proposition demonstrates
that given the more general preference specification, the structure of the
optimal disclosure, present in the baseline model, preserves.

######  Proposition 4.

  * (a)

When the cost-benefit ratio of the project is low,
$\kappa\leq\tilde{\kappa}\left(T,\lambda,r,\alpha\right)$, the optimal
investment schedule $\tau$ satisfies $\tau\geq\tau_{2}\land T$, i.e., the
agent recommends the principal to stop only after the completion of the second
stage of the project.

  * (b)

When $\kappa>\tilde{\kappa}\left(T,\lambda,r,\alpha\right)$, the optimal
investment schedule $\tau$ assigns positive probability both to the stopping
in state $2$ and state $0$, i.e., the agent not only discloses the completion
of the second stage of the project, but also specifies an interim deadline for
the completion of the first stage.

Similarly to the baseline model, allowing the principal to stop after the
project completion brings profit to the principal and thus leads to a
relatively higher total surplus, which the agent can extract. Meanwhile,
allowing the principal to stop at the interim deadline does not increase total
surplus and serves solely as an expected payoff transfer from the agent to the
principal. To see that, note that stopping when the first stage of the project
is still incomplete allows the principal to save on the further costs of
funding the project when over time the project proves to be “unsuccessful”.
This can not be beneficial for the agent as she does not internalize the costs
of running the project. Further, stopping at the interim deadline is strictly
detrimental for the agent as she strictly prefers the principal to postpone
the stopping of funding when no stages of the project are completed.313131The
probability of project success and stock of obtained funds are non-decreasing
in the date of stopping irrespective of the number of the completed stages of
the project.

When the project is sufficiently ex-ante attractive, the agent can motivate
the principal to start funding the project without promising to stop the
stagnant project at the interim deadline, and this is strictly beneficial for
the agent. Thus, when the project is promising, the agent sets no interim
deadlines, which in expectation gives her more funds and more experimentation
for free.

##  7 Conclusion

A transparent flow of information is crucial for the successful management of
any innovative project. However, the researcher, who controls the information
on the progress of the project, often tends to have different motives than the
investor. This leads to the question of how a researcher chooses the
transparency of the flow of information about the progress of a project in
order to manipulate the investor’s funding decisions. I address this question
in a dynamic information design model in which the agent commits to providing
information to the principal with an incentive to postpone the principal’s
irreversible stopping of the funding.

I contribute to the dynamic information design literature by studying the
problem of the dynamic provision of information regarding the progress of a
multistage project, which evolves endogenously over time and needs to be
completed before a deadline. I show that the agent’s choice of which pieces of
information to provide and when depends on the project being either ex ante
attractive for the principal or not. In the case of a promising project, the
agent provides only the good news that the project is completed and postpones
the reports. In the case of an unattractive project, to motivate the principal
to start funding the project the agent not only reports the completion of the
project, but also helps the principal to find out when the project stagnates.
To achieve this, the agent announces an interim deadline for the project – a
certain date at which she recommends the principal to cut the funding of the
project if the milestone of the project has not been reached.

## References

  * Admati and Pfleiderer, (1994) Admati, A. R. and Pfleiderer, P. (1994).  Robust financial contracting and the role of venture capitalists.  The Journal of Finance, 49(2):371–402.
  * Ball, (2019) Ball, I. (2019).  Dynamic information provision: Rewarding the past and guiding the future.  Available at SSRN 3103127.
  * Basak and Zhou, (2020) Basak, D. and Zhou, Z. (2020).  Panics and early warnings.  PBCSF-NIFR Research Paper.
  * Bergemann and Hege, (1998) Bergemann, D. and Hege, U. (1998).  Venture capital financing, moral hazard, and learning.  Journal of Banking & Finance, 22(6-8):703–735.
  * Bergemann and Hege, (2005) Bergemann, D. and Hege, U. (2005).  The financing of innovation: Learning and stopping.  RAND Journal of Economics, pages 719–752.
  * Cornelli and Yosha, (2003) Cornelli, F. and Yosha, O. (2003).  Stage financing and the role of convertible securities.  The Review of Economic Studies, 70(1):1–32.
  * Curello and Sinander, (2020) Curello, G. and Sinander, L. (2020).  Screening for breakthroughs.  arXiv preprint arXiv:2011.10090.
  * Ekmekci et al., (2020) Ekmekci, M., Gorno, L., Maestri, L., Sun, J., and Wei, D. (2020).  Learning from manipulable signals.  arXiv preprint arXiv:2007.08762.
  * Ely, (2017) Ely, J. C. (2017).  Beeps.  American Economic Review, 107(1):31–53.
  * Ely and Szydlowski, (2020) Ely, J. C. and Szydlowski, M. (2020).  Moving the goalposts.  Journal of Political Economy, 128(2):468–506.
  * Gompers, (1995) Gompers, P. A. (1995).  Optimal investment, monitoring, and the staging of venture capital.  The Journal of Finance, 50(5):1461–1489.
  * Green and Taylor, (2016) Green, B. and Taylor, C. R. (2016).  Breakthroughs, deadlines, and self-reported progress: Contracting for multistage projects.  American Economic Review, 106(12):3660–99.
  * Gundolf et al., (2017) Gundolf, K., Gast, J., and Géraudel, M. (2017).  Startups’ innovation behaviour: An investigation into the role of entrepreneurial motivations.  International Journal of Innovation Management, 21(07):1750054.
  * Henry and Ottaviani, (2019) Henry, E. and Ottaviani, M. (2019).  Research and the approval process: The organization of persuasion.  American Economic Review, 109(3):911–55.
  * Kaya, (2020) Kaya, A. (2020).  Paying with information.  Available at SSRN 3661779.
  * Madsen, (2022) Madsen, E. (2022).  Designing deadlines.  American Economic Review, 112(3):963–97.
  * Orlov et al., (2020) Orlov, D., Skrzypacz, A., and Zryumov, P. (2020).  Persuading the principal to wait.  Journal of Political Economy, 128(7):2542–2578.
  * Renault et al., (2017) Renault, J., Solan, E., and Vieille, N. (2017).  Optimal dynamic information provision.  Games and Economic Behavior, 104:329–349.
  * Wolf, (2017) Wolf, C. (2017).  Informative milestones in experimentation.  University of Mannheim, Working Paper.

## Appendix

###  A Notational conventions

The state process is given by $x_{t},\forall t\in\mathbb{R}_{+}$, defined on
the probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$,
$t\in\mathbb{R}_{+}$. Its natural filtration is denoted by
$F=(\mathcal{F}_{t})_{t\geq 0}$. Throughout Appendices B and C, the following
notational conventions are used:

1\. I denote the random time at which the $n$th stage of the project is
completed by $\tau_{n}$. Formally, $\tau_{n}\in\mathbb{R}_{+}$ is a
continuously distributed random variable that represents the first hitting
time of $x_{t}=n$.

2\. Consider some stopping time $\tau$. Whenever “$\tau$” stands as a term in
an inequality, it stands for a realization of the stopping time $\tau$ and it
should be read as “for each $\omega\in\Omega$ and corresponding
$\tau\left(\omega\right)$”.

_Example 1._ “$\tau_{2}\land T\geq\tau$” should be read as
“$\tau_{2}\left(\omega\right)\land T\geq\tau\left(\omega\right)$, for all
$\omega\in\Omega$”.

_Example 2._ “for all $t\in[S,\tau)$” should be read as “for all
$t\in[S,\tau\left(\omega\right))$, for all $\omega\in\Omega$”.

3\. The continuation value of the agent at time $t$, given $\tau$, and
conditional on $t<\tau$:

$W_{t}\left(\tau\right)\coloneqq\operatorname{E}\left[\tau-t|t<\tau\right]c.$

4\. The total (continuation) surplus at time $t$, given $\tau$, and
conditional on $t<\tau$:

$SV_{t}\left(\tau\right)\coloneqq
W_{t}\left(\tau\right)+V_{t}\left(\tau\right).$

5\. Shorthand for posterior beliefs:

$\begin{array}[]{c}q_{n}\left(t\right)\coloneqq\operatorname{P}\left(x_{t}=n|t<\tau\right),\\\
r_{n}\left(t\right)\coloneqq\operatorname{P}\left(x_{\tau}=n|t<\tau\right).\end{array}$

###  B The principal’s continuation value

Here I present the alternative formulation of the principal’s continuation
value (6). This helps me to study some of its properties for further use in
Appendix C. The continuation value of the principal at time $t$ and given the
investment schedule $\tau$ is given by (6). Postponing the stopping for
$\Delta t$ brings a benefit in the form of project completion payoff $v$ iff
the second stage of the project is completed within $\Delta t$. As $x_{t}$
follows the Poisson process, the probability of two increments in a very short
time $\Delta t$ is negligibly small. Thus, during some $\Delta t$, the
principal gets the project completion payoff $v$ iff the first stage of the
project has already been completed at some earlier time. Thus, postponing the
stopping for $\Delta t$ brings the principal $v$ with probability $\lambda
q_{1}\left(t\right)\Delta t$. The second stage is not completed within $\Delta
t$ with the complementary probability of $1-\lambda q_{1}\left(t\right)\Delta
t$. The principal’s continuation value is thus given by

$\displaystyle V_{t}\left(\tau\right)$ $\displaystyle=\left(v\lambda
q_{1}\left(t\right)-c\right)\Delta t+\left(1-\lambda q_{1}\left(t\right)\Delta
t\right)V_{t+\Delta t}\left(\tau\right)$
$\displaystyle=v\lambda\left(q_{1}\left(t\right)-\kappa\right)\Delta
t+\left(1-\lambda q_{1}\left(t\right)\Delta t\right)V_{t+\Delta
t}\left(\tau\right).$

Differentiating both sides w.r.t. $\Delta t$ and considering $\lim_{\Delta
t\rightarrow 0}\left(.\right)$ yields

$0=v\lambda\left(q_{1}\left(t\right)-\kappa\right)-\lambda
q_{1}\left(t\right)V_{t}\left(\tau\right)+\dot{V_{t}}\left(\tau\right),$

which, after rearranging becomes

$\dot{V_{t}}\left(\tau\right)=\lambda
q_{1}\left(t\right)V_{t}\left(\tau\right)+v\lambda\left(\kappa-
q_{1}\left(t\right)\right).$ (13)

###  C Proofs

###### Proof of Lemma 1.

The beliefs regarding the number of stages of the project completed by time
$t$, $x_{t}$, evolve according to the Poisson process. The principal’s
unconditional beliefs are given by $p_{0}\left(0\right)=1$ and for any $t$
such that the game still continues,

$\displaystyle\dot{p}_{0}\left(t\right)$ $\displaystyle=$
$\displaystyle-\lambda p_{0}\left(t\right),$
$\displaystyle\dot{p}_{1}\left(t\right)$ $\displaystyle=$
$\displaystyle\lambda(p_{0}\left(t\right)-p_{1}\left(t\right)),$ (14)
$\displaystyle\dot{p}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle\lambda p_{1}\left(t\right),$

where $p_{0}\left(t\right)=e^{-\lambda t}$ and $p_{1}\left(t\right)=\lambda
te^{-\lambda t}$,
$p_{2}\left(t\right)=1-p_{0}\left(t\right)-p_{1}\left(t\right)$. The
principal’s problem is given by

$\max_{S\in\left[0,T\right]}\left\\{v\cdot p_{2}\left(S\right)-c\cdot
S\right\\}.$ (15)

I start with analyzing the choice of $S$ for the interior solution case,
$S\in\left(0,T\right)$. F.O.C. for (15) is given by

$v\cdot\dot{p}_{2}\left(S\right)=c,$ (16)

or, equivalently, $p_{1}\left(S\right)=\kappa$. There are two values
satisfying (16): $\bar{S}$ and $\bar{S}^{NI}$, $\bar{S}<\bar{S}^{NI}$. At each
$t\in\left(\bar{S},\bar{S}^{NI}\right)$ the principal receives a net positive
payoff flow. Thus, stopping at $\bar{S}$ is not optimal and the only candidate
for optimal stopping is $\bar{S}^{NI}$.323232$\bar{S}$ is a local minimum of
the objective. Further, one can obtain the closed form expression for the
interior stopping time $\bar{S}^{NI}$ from (16):

$\bar{S}^{NI}=-\frac{1}{\lambda}\mathcal{W}_{-1}\left(-\kappa\right),$ (17)

where $\mathcal{W}_{-1}(x)$ denotes the negative branch of the Lambert $W$
function. $\bar{S}^{NI}$ is well-defined for any $\kappa<e^{-1}$.

Thus, the solution to (15) could potentially be $0$, $\bar{S}^{NI}$, or $T$. I
proceed with a useful lemma.

######  Lemma 8.

The following is true regarding the principal’s continuation value in the no-
information benchmark, $\overline{V}_{t}^{NI}$: if $\overline{V}_{t}^{NI}\geq
0,\text{ for some }t\in\left[0,\bar{S}^{NI}\land T\right]$, then
$V^{NI}\left(s\right)\geq 0,\text{ for all }s\in\left[t,\bar{S}^{NI}\land
T\right]$.

###### Proof.

The principal’s continuation value in the no-information benchmark is given by

$\overline{V}_{t}^{NI}=\left[p_{2}\left(T\land\bar{S}^{NI}\right)-p_{2}\left(t\right)\right]v-\left(T\land\bar{S}^{NI}-t\right)c.$
(18)

Further,

$\dot{V}^{NI}\left(t\right)=v\lambda\left(\kappa-e^{-\lambda t}\lambda
t\right)=v\lambda\left(\kappa-p_{1}\left(t\right)\right).$

$p_{1}\left(t\right)\leq\kappa$ for all $t\in\left[0,\bar{S}\right]$ and
$p_{1}\left(t\right)\geq\kappa$ for all $t\in\left[\bar{S},\bar{S}^{NI}\land
T\right]$. Thus, $\overline{V}_{t}^{NI}$ increases for
$t\in\left[0,\bar{S}\right]$, decreases for
$t\in\left[\bar{S},T\land\bar{S}^{NI}\right]$, and
$V^{NI}\left(T\land\bar{S}^{NI}\right)=0$, which establishes the result. ∎

Lemma 8 implies that if $V^{NI}\left(0\right)\geq 0$ and the principal chooses
to opt in at $t=0$, then $\overline{V}_{t}^{NI}\geq 0$,
$t\in\left[0,\bar{S}^{NI}\land T\right]$, i.e., he invests until
$t=T\land\bar{S}^{NI}$. This implies that the solution to (15) is either
$T\land\bar{S}^{NI}$ or $0$.

Finally, at $t=0$ the principal chooses to start investing or not. The
condition for the principal to start investing at $t=0$ is given by

$V^{NI}\geq 0.$ (19)

To specify the set of parameters for which (19) is satisfied, I obtain the
cutoff value of $\kappa$ given $T$ and $\lambda$. Such a parameterization is
intuitive: $\kappa$ above the cutoff level corresponds to a project with
sufficiently high normalized cost-benefit ratio and implies that the principal
does not opt in. I denote this cutoff by $\kappa^{NI}\left(T,\lambda\right)$.
This solves (19) holding with equality. Two cases are possible.

_Case 1_ : $T\leq\bar{S}^{NI}\iff
T\leq-\frac{1}{\lambda}\mathcal{W}_{-1}\left(-\kappa\right)$. This inequality
is satisfied when either $\frac{1}{\lambda}>T$ or
$\begin{cases}\frac{1}{\lambda}&\leq T\\\ \kappa&\leq e^{-\lambda T}\lambda
T.\end{cases}$ Given $T\leq\bar{S}^{NI}$, (19) holding with equality becomes

$p_{2}\left(T\right)v-Tc=0.$

Solving it for $\kappa$ yields $\kappa=e^{-\lambda T}\left(\frac{e^{\lambda
T}-1}{\lambda T}-1\right)$.

_Case 2:_ $T>\bar{S}^{NI}$. This inequality is satisfied when
$\frac{1}{\lambda}\leq T\text{ and }\kappa>e^{-\lambda T}\lambda T.$ Given
$T>\bar{S}^{NI}$, (19) holding with equality becomes

$vp_{2}\left(\bar{S}^{NI}\right)-c\bar{S}^{NI}=0\iff
v\left(1-p_{0}\left(\bar{S}^{NI}\right)-p_{1}\left(\bar{S}^{NI}\right)\right)=c\bar{S}^{NI},$

where (recall that $\dot{p}_{2}\left(\bar{S}^{NI}\right)=\frac{c}{v}$)

$p_{0}\left(\bar{S}^{NI}\right)=\frac{1}{\lambda^{2}\bar{S}^{NI}}\dot{p}_{2}\left(\bar{S}^{NI}\right)=\frac{c}{\lambda^{2}\bar{S}^{NI}v}=\frac{\kappa}{\lambda\bar{S}^{NI}}$

and

$p_{1}\left(\bar{S}^{NI}\right)=\frac{1}{\lambda}\dot{p}_{2}\left(\bar{S}^{NI}\right)=\frac{c}{\lambda
v}=\kappa.$

Consequently,

$vp_{2}\left(\bar{S}^{NI}\right)-c\bar{S}^{NI}=v-v\cdot\kappa\left(1+\lambda\bar{S}^{NI}+\frac{1}{\lambda\bar{S}^{NI}}\right).$

Let $y\coloneqq\lambda\bar{S}^{NI}$. Note that, by definition, $y>1$. Then
$\kappa=ye^{-y}$, and so

$\left(vp_{2}\left(\bar{S}^{NI}\right)-c\bar{S}^{NI}\right)/v=1-e^{-y}\left(1+y+y^{2}\right).$

It follows that $V^{NI}\left(0\right)$ is nonnegative whenever
$\lambda\bar{S}^{NI}\geq y_{0}\doteq 1.79328$, which is equivalent to

$\kappa\leq\kappa_{0}\doteq 0.298426.$

Finally, putting the two cases together yields

$\kappa^{NI}\left(T,\lambda\right)=\begin{cases}\kappa_{0}\doteq
0.298426,&\text{if }\frac{1}{\lambda}\leq T\text{ and }\kappa\geq e^{-\lambda
T}\lambda T\\\ e^{-\lambda T}\left(\frac{e^{\lambda T}-1}{\lambda
T}-1\right),&\text{otherwise.}\end{cases}$ (20)

∎

###### Proof of Lemma 2.

The principal chooses $a_{t}\in\left\\{0,1\right\\}$ sequentially given the
observed realizations of $x_{t}\in\left\\{0,1,2\right\\}$. Whenever the
principal observes $t=\tau_{2}$, he immediately chooses $a_{t}=0$ and gets
$v$.

Consider the case $x_{t}=1,t<T$, i.e., the first stage of the project has
already been completed. As $x_{t}$ follows a Poisson process, in expectation
it would take $\frac{1}{\lambda}$ units of time for the second stage to be
completed and its completion brings the principal the value of $v$. Thus, the
necessary and sufficient condition for the principal to invest at $t$ when
$x_{t}=1,t<T$ is given by

$v-c\cdot\frac{1}{\lambda}\geq 0\iff\kappa\leq 1$

Assume that $\kappa\leq 1$ holds and $x_{t}=1$; thus, the principal chooses to
invest at $t$. In that case, the principal invests until $\tau_{2}\land T$. To
see this, recall that the only news that the principal can receive given
$x_{t}=1,t<T$ is the completion of the second stage of the project,
$\tau_{2}$, which leads to immediate stopping. At each $t<\tau_{2}\land T$
such that $x_{t}=1$, choosing $a_{t}=0$ yields an instantaneous expected
payoff of $0$, while choosing $a_{t}=1$ yields an instantaneous expected
payoff of $\lambda v\Delta t-c\Delta t$. Thus, $\kappa\leq 1$ suffices for the
principal to invest until $\tau_{2}\land T$.

Consider now the case of $x_{t}=0,t<T$, i.e., no stages of the project have
yet been completed. Postponing the stopping for $\Delta t$ brings the
instantaneous expected payoff of $V_{t|1}^{FI}\lambda\Delta t-c\Delta t$,
where $V_{t|1}^{FI}$ is the principal’s continuation value at time $t$ under
full information, conditional on the completion of the first stage of the
project. I proceed with obtaining the expression for $V_{t|1}^{FI}$. By
definition, the principal gets $v$ whenever the second stage is completed not
later than $T$. The principal invests until $\tau_{2}\land T$, and knows that
at $t$ the first stage of the project is already completed; thus,
$V_{t|1}^{FI}$ is given by

$V_{t|1}^{FI}=v\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=1\right)-c\operatorname{E}\left[\tau_{2}\wedge T-t|x_{t}=1\right].$

$\tau_{2}|x_{t}=1$ corresponds to the time between two consecutive Poisson
arrivals, and thus has exponential distribution. First, consider
$\operatorname{P}\left(\tau_{2}\leq T|x_{t}=1\right)$:

$\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=1\right)=1-e^{-\lambda\left(T-t\right)}.$

Next, consider $\operatorname{E}\left[\tau_{2}\wedge T-t|x_{t}=1\right]$:

$\displaystyle\operatorname{E}\left[\tau_{2}\land T|x_{t}=1\right]-t$ (21)
$\displaystyle=$ $\displaystyle\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=1\right)\int_{t}^{T}z\cdot\frac{\lambda
e^{-\lambda\left(z-t\right)}}{\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=1\right)}dz+\operatorname{P}\left(\tau_{2}>T|x_{t}=1\right)T-t$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\left(1-e^{-\lambda\left(T-t\right)}\right)+t-e^{-\lambda\left(T-t\right)}T+\operatorname{P}\left(\tau_{2}>T|x_{t}=1\right)T-t$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\left(1-e^{-\lambda\left(T-t\right)}\right).$

Thus,

$\displaystyle V_{t|1}^{FI}$
$\displaystyle=v\left(1-e^{-\lambda\left(T-t\right)}\right)-c\frac{1}{\lambda}\left(1-e^{-\lambda\left(T-t\right)}\right)$
(22)
$\displaystyle=\left(v-\frac{c}{\lambda}\right)\left(1-e^{-\lambda\left(T-t\right)}\right).$

From (22) one observes that $V_{t|1}^{FI}$ decreases in $t$. If the net
instantaneous benefit given by $V_{t|1}^{FI}\lambda\Delta t-c\Delta t$ gets as
low as $0$ at some $t$, then the principal chooses to stop investing at this
$t$. I denote the time at which the net instantaneous benefit reaches $0$ by
$S_{0}^{P}$. $S_{0}^{P}$ can be obtained from $\left(\lambda
V_{1}^{FI}\left(S_{0}^{P}\right)-c\right)\Delta t=0$. Thus,

$S_{0}^{P}=T+\frac{1}{\lambda}\log\left(\frac{1-2\kappa}{1-\kappa}\right).$
(23)

The principal is willing to start investing iff at $t=0$ the expected payoff
from investing at $t=0$ covers the costs of investing, i.e. $\left(\lambda
V_{1}^{FI}\left(0\right)-c\right)\Delta t\geq 0$. From (23), this corresponds
to $S_{0}^{P}\geq 0$. I denote the upper bound on the cost-benefit ratio
$\kappa$ such that the principal chooses to start investing in $t=0$ under
full information by $\kappa^{FI}\left(T,\lambda\right)$, I solve $S_{0}^{P}=0$
for $\kappa$ and obtain

$\kappa^{FI}\left(T,\lambda\right)=\frac{1-e^{-\lambda T}}{2-e^{-\lambda T}}.$
(24)

In summary, under full information, if
$\kappa\leq\kappa^{FI}\left(T,\lambda\right)$, then the principal starts
investing at $t=0$. Further, he stops at $S_{0}^{P}$ if the first stage of the
project has not been completed by that time. Otherwise, he proceeds to invest
until $\tau_{2}\land T$.

∎

###### Proof of Proposition 1.

I provide the proof for each of the four parametric cases below.

_1\. The case of $\kappa\leq\kappa^{ND}\left(T,\lambda\right)$_.

$\kappa^{ND}\left(T,\lambda\right)$ is defined as follows: for any
$\kappa\leq\kappa^{ND}\left(T,\lambda\right)$, the principal invests until $T$
in the no-information benchmark. From Lemma 1, if the principal is willing to
start investing, i.e., $\kappa\leq\kappa^{NI}\left(T,\lambda\right)$, then

$S^{NI}=\overline{S}^{NI}\land T.$

For the sake of instruction, below I consider relaxing the Assumption 1 and
demonstrate how the relation between $\kappa^{ND}\left(T,\lambda\right)$ and
$\kappa^{NI}\left(T,\lambda\right)$ changes between Case a (relaxed Assumption
1) and Case b (Assumption 1 holds).

Case a. $e^{\lambda T}\leq\lambda T\left(\lambda T+1\right)+1$. In this case,
whenever the principal is willing to start investing in the no-information
benchmark, she invests until $T$, i.e.,
$\kappa^{ND}\left(T,\lambda\right)=\kappa^{NI}\left(T,\lambda\right)$, where
$\kappa^{NI}\left(T,\lambda\right)$ is given by (20). To see that, first,
consider the extreme sub-case in which $T<\frac{1}{\lambda}$. As
$-\lambda\overline{S}^{NI}$ must belong to $-1$ axis of Lambert $W$ function,
it has a lower bound corresponding to $\frac{1}{\lambda}$. Thus,
$T<\overline{S}^{NI}$ for any $\kappa\left(T,\lambda\right)$. Second, consider
$\lambda T\in\left[1,\tilde{\lambda T}\right]$, where $\tilde{\lambda T}$
solves $e^{\lambda T}=\lambda T\left(\lambda T+1\right)+1$. In this case, from
(17), if $\kappa\left(T,\lambda\right)\leq e^{-\lambda T}\lambda T$
($\kappa\left(T,\lambda\right)\geq e^{-\lambda T}\lambda T$, respectively),
then $T\leq\overline{S}^{NI}$ ($T\geq\overline{S}^{NI}$, respectively).
However, $\kappa^{NI}\left(T,\lambda\right)\leq e^{-\lambda T}\lambda T$.
Thus, $\kappa^{ND}\left(T,\lambda\right)=\kappa^{NI}\left(T,\lambda\right)$.

Case b. $e^{\lambda T}>\lambda T\left(\lambda T+1\right)+1.$ As before, it
holds that if $\kappa\left(T,\lambda\right)\leq e^{-\lambda T}\lambda T$
($\kappa\left(T,\lambda\right)\geq e^{-\lambda T}\lambda T$), then
$T\leq\overline{S}^{NI}$ ($T\geq\overline{S}^{NI}$, respectively). Denote

$\kappa^{ND}\left(T,\lambda\right)\coloneqq e^{-\lambda T}\lambda T.$

As $\kappa^{NI}\left(T,\lambda\right)>\kappa^{ND}\left(T,\lambda\right)$, two
cases emerge. If $0<\kappa\leq\kappa^{ND}\left(T,\lambda\right)$, then
$T\leq\overline{S}^{NI}$, and from
$\kappa\leq\kappa^{NI}\left(T,\lambda\right)$, it holds that $S^{NI}=T$ and as
the agent does not strictly benefit from disclosing any information, she
chooses non-disclosure. If $\kappa>\kappa^{ND}\left(T,\lambda\right)$, then
$T>\overline{S}^{NI}$ and the agent can potentially benefit from information
disclosure.

_2\. The case of
$\kappa^{ND}\left(T,\lambda\right)<\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$_.

The result is established in Proposition 2.

_3\. The case of
$\tilde{\kappa}\left(T,\lambda\right)<\kappa<\kappa^{FI}\left(T,\lambda\right)$_.

The result is established in Proposition 3.

_4\. The case of $\kappa\geq\kappa^{FI}\left(T,\lambda\right)$_.

The principal’s long-run payoff in the full-information benchmark non-
positive. Thus, the agent can not strictly benefit from information disclosure
and chooses non-disclosure.

∎

###### Proof of Lemma 3.

_Necessity_. Assume $V_{t}(\tau)<0$ for some $t$. In that case, it is optimal
for the principal to deviate to stopping at $t<\tau$. Thus, there is no
information policy $\sigma$, for which this $\tau$ is the principal’s best
reply. Assume $V_{\tau}^{NI}\geq 0$. Thus, the principal deviates to stopping
at $t>\tau$, and there is no $\sigma$, for which this $\tau$ is the best
reply.

_Sufficiency_. Assume (7) holds. $V_{t}\left(\tau\right)\geq 0$ for all
$t<\tau$ implies that the principal prefers to continue rather than to stop
the funding for all $t<\tau$. Thus, it can not be that case that the principal
stops before $\tau$. Further, $V_{\tau}^{NI}<0$ implies that, conditional on
reaching the date of stopping $\tau$, it is better for the principal to stop
immediately rather than to stop at $t>\tau$.

Consider a direct recommendation mechanism $\sigma$ with
$M=\left\\{0,1\right\\}$ such that whenever, based on the evolution of the
state process, the considered investment schedule $\tau$ suggests stopping the
funding, the direct recommendation mechanism sends the message $m=0$ to the
principal. As it is optimal for the principal to stop at $\tau$, $\tau$ is the
principal’s best reply to $\sigma$. ∎

###### Proof of Lemma 4.

Consider the recommendation mechanism immediately disclosing the completion of
the second stage of the project; it is given by $\tau=\tau_{2}\wedge T$. There
exists such $\tilde{\kappa}\left(T,\lambda\right)$ that solves the principal’s
binding $t=0$ individual rationality constraint when $\tau=\tau_{2}\wedge T$:

$V\left(\tau_{2}\right)=0,$ (25)

where

$\displaystyle V\left(\tau_{2}\right)$
$\displaystyle=p_{2}\left(T\right)v-\operatorname{E}\left[\tau_{2}\wedge
T\right]c$ (26) $\displaystyle=v\left(1-e^{-\lambda T}-\lambda Te^{-\lambda
T}\right)-c\frac{1}{\lambda}\left(2-2e^{-\lambda T}-\lambda Te^{-\lambda
T}\right).$

The solution to equation (25) is given by

$\tilde{\kappa}\left(T,\lambda\right)=\frac{1-e^{\lambda T}+\lambda
T}{2-2e^{\lambda T}+\lambda T}.$ (27)

Further, $\kappa>\tilde{\kappa}\left(T,\lambda\right)\Rightarrow
V\left(\tau_{2}\right)<0$ and
$\kappa\leq\tilde{\kappa}\left(T,\lambda\right)\Rightarrow
V\left(\tau_{2}\right)\geq 0$. ∎

###### Proof of Lemma 5.

_Consider the case of
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$_.
The agent’s relaxed problem for this case has the individual rationality
constraints only for $t\in[0,\bar{S}^{NI}]$, and it is given by

$\displaystyle\max_{\tau\in\mathcal{T}}\left\\{c\cdot\operatorname{E}\left[\tau\right]\right\\}$
(28) $\displaystyle\text{s.t.}\,V_{t}\left(\tau\right)\geq 0,\forall
t\in\left[0,\bar{S}^{NI}\right],$

where $V_{t}(\tau)$ is given by (6) and $\mathcal{T}$ is the set of stopping
times with respect to the natural filtration of $x_{t}$.

Consider the candidate investment schedule $\tau$ such that
$\tau\geq\bar{S}^{NI}\lor\left(\tau_{2}\land T\right)$ and
$V\left(\tau\right)=V^{NI}$, where $V^{NI}$ is given by (3). I start with
arguing that the candidate $\tau$ satisfies the system of individual
rationality constraints. From Lemma 1, given candidate $\tau$, the principal
invests until $\bar{S}^{NI}$ with certainty and the constraints in (28) are
satisfied for all $t\in[0,\bar{S}^{NI})$. Further, $\tau$ implies that
$V_{\bar{S}^{NI}}(\tau)=0$, i.e., the individual rationality constraint at
$t=\bar{S}^{NI}$ is binding.

I proceed with arguing that the candidate $\tau$ maximizes the agent’s
objective function in (28). The agent’s objective can be WLOG written out as:

$\begin{array}[]{c}W\left(\tau\right)=\underbrace{\operatorname{P}\left(x_{\tau}=2\right)v}_{\text{total
surplus}}\,\,\,\,\,-\underbrace{V(\tau).}_{\text{principal's
surplus}}\end{array}$ (29)

By Lemma 4, an investment schedule $\tau$ that assigns probability one to
$\tau\geq\tau_{2}\land T$ satisfies the individual rationality constraint at
$t=0$ in (28). Note that, given $\tau\geq\tau_{2}\land T$, the total surplus
in (29) is given by $\operatorname{P}\left(x_{T}=2\right)v$, i.e., total
surplus achieves its upper bound determined by the exogenously given project
deadline $T$. The principal’s surplus in (29) is given by
$V\left(\tau\right)=V^{NI}$, i.e., principal’s surplus achieves its lower
bound specified by (3). This can be seen from the principal’s decision
problem, in which he best replies to an information policy $\sigma$. As
$\sigma$ allows the principal to condition his actions on the information
regarding the evolution of the state process, the principal’s equilibrium
payoff can not be lower than $V^{NI}$, his equilibrium payoff when he is
restricted to choosing actions without conditioning them on the information
about the state process. Thus, $\tau$ solves the relaxed problem (28).

_Consider the case of_
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$.
The agent’s relaxed problem for this case has the individual rationality
constraint only for the initial period, and it is given by

$\displaystyle\max_{\tau\in\mathcal{T}}\left\\{c\cdot\operatorname{E}\left[\tau\right]\right\\}$
(30) $\displaystyle\text{s.t.}\,V(\tau)\geq 0,$

where
$V(\tau)=\operatorname{P}\left(x_{\tau}=2\right)v-\operatorname{E}\left[\tau\right]c$.

Consider candidate investment schedule $\tau$ such that $\tau\geq\tau_{2}\land
T$ and $V\left(\tau\right)=V^{NI}$. For such $\tau$, agent’s expected payoff
(29) is given by $\operatorname{P}\left(x_{T}=2\right)v-V^{NI}$. As discussed
for the parametric case
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$,
the first term is at its upper bound. To see that the second term is at its
lower bound, note that, from Lemma 1, $V^{NI}=0$, and thus the individual
rationality constraint in (30) is binding. Hence, $\tau$ solves the relaxed
problem (30).

∎

###### Proof of Proposition 2.

The proof covers the case
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$
and the case
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$
separately.

_1\. The case of
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$_.

I start with proving the existence of $S^{*}$ such that
$V\left(\tau\right)=V^{NI}$. Assume that $S^{*}>\bar{S}^{NI}$. For all
$t\in[\bar{S}^{NI},S^{*})$, stopping never occurs, at $t=S^{*}$ it occurs if
$x_{S^{*}}=2$, and for all $t\in(S^{*},\tau)$ it occurs at $t=\tau_{2}\land
T$. For $t\in[S^{*},\tau)$, the absence of stopping induces posteriors
$q_{n}\left(t\right)$. Further, for $t\in[S^{*},\tau)$ the principal discounts
future benefits from postponing stopping using the probability of the state
being $2$. Hence, the continuation value at $t=\bar{S}^{NI}$ can be written as

$V_{\bar{S}^{NI}}\left(\tau\right)=v\lambda\left(\int_{\bar{S}^{NI}}^{S^{*}}p_{1}\left(z\right)-\kappa
dz+\int_{S^{*}}^{T}\left(q_{1}\left(z\right)-\kappa\right)\left(1-\operatorname{P}\left(x_{z}=2\right)\right)dz\right).$
(31)

The principal’s long-run payoff is given by

$V\left(\tau\right)=\int_{0}^{\bar{S}^{NI}}\left(v\cdot
p_{1}\left(s\right)\lambda-c\right)ds+V_{\bar{S}^{NI}}\left(\tau\right),$

where $\int_{0}^{\bar{S}^{NI}}\left(v\cdot
p_{1}\left(s\right)\lambda-c\right)ds=V^{NI}$. Thus, to ensure that $S^{*}$
makes the individual rationality constraint bind at $t=\bar{S}^{NI}$, i.e.,
$V\left(\tau\right)=V^{NI}$, it is necessary and sufficient that
$V_{\bar{S}^{NI}}\left(\tau\right)=0$. Using (31), this equation can be
written as

$\int_{\bar{S}^{NI}}^{S^{*}}\kappa-
p_{1}\left(z\right)dz=\int_{S^{*}}^{T}\left(q_{1}\left(z\right)-\kappa\right)\left(1-\operatorname{P}\left(x_{z}=2\right)\right)dz.$

Let $g\left(S\right)\coloneqq\int_{\bar{S}^{NI}}^{S}\kappa-
p_{1}\left(z\right)dz$ and
$k\left(S\right)\coloneqq\int_{S}^{T}\left(q_{1}\left(z\right)-\kappa\right)\left(1-\operatorname{P}\left(x_{z}=2\right)\right)dz$,
$S\in[\bar{S}^{NI},\tau)$. $q_{1}\left(t\right)\geq\kappa$, for all
$t\in\left[S^{*},T\right)$. Thus,
$g\left(\bar{S}^{NI}\right)=0,k\left(\bar{S}^{NI}\right)>0$. Further,
$p_{1}\left(t\right)<\kappa$, for all $t\in(\bar{S}^{NI},T]$. Hence,
$g\left(T\right)>0,k\left(T\right)=0$. Finally,
$p_{1}\left(t\right)\leq\kappa$, for all $t\in\left[\bar{S}^{NI},T\right]$
implies that $g^{\prime}\left(S\right)\geq 0$, for all
$s\in\left[\bar{S}^{NI},T\right]$, and $q_{1}\left(t\right)\geq\kappa$, for
all $t\in\left[S^{*},T\right]$ implies that $k^{\prime}\left(S\right)\leq 0$,
for all $s\in\left[S^{*},T\right]$. Thus, by the intermediate value theorem,
there exists $S^{*}$ solving $V_{\bar{S}^{NI}}\left(\tau\right)=0$. Thus,
there exists $S^{*}>\bar{S}^{NI}$ such that principal’s individual rationality
constraint is binding at $t=\bar{S}^{NI}$.

I proceed with proving that the investment schedule $\tau$ satisfies the
conditions in Lemma 3 and thus it is obedient.

_First, consider_ $t\leq\bar{S}^{NI}$. The principal’s continuation value for
all $t\in[0,\bar{S}^{NI}]$ can be written as

$V_{t}\left(\tau\right)=\int_{t}^{\bar{S}^{NI}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds+V_{\bar{S}^{NI}}\left(\tau\right).$

Given the binding individual rationality constraint, it becomes

$V_{t}\left(\tau\right)=\int_{t}^{\bar{S}^{NI}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds,\text{
for all }t\in[0,\bar{S}^{NI}).$

Finally, note that $V_{t}\left(\tau\right)$ above is equivalent to
$V_{t}^{NI}$ given by (18). Lemma 1 implies that given
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$,
$V^{NI}\left(0\right)=V\left(\tau\right)\geq 0$. Further, Lemma 8 implies that
$V\left(\tau\right)\geq 0\Rightarrow V_{t}\left(\tau\right)\geq 0,\forall
t\in[0,\bar{S}^{NI})$.

_Second, consider_ $t\in\left[\bar{S}^{NI},S^{*}\right]$. Given
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$,
$p_{1}\left(t\right)\leq\kappa,\forall t\in\left[\bar{S}^{NI},S^{*}\right]$.
Thus, $V_{t}^{NI}=0,\forall t\in\left[\bar{S}^{NI},S^{*}\right]$. The
principal’s continuation value is given by

$V_{t}\left(\tau\right)=\int_{t}^{S^{*}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds+V_{S^{*}}\left(\tau\right).$
(32)

As $p_{1}\left(t\right)\leq\kappa,\forall
t\in\left[\bar{S}^{NI},S^{*}\right]$,
$\int_{t}^{S^{*}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds\leq 0$ and
it is increasing in $t$. As $V_{\bar{S}^{NI}}\left(\tau\right)=0$, where
$V_{\bar{S}^{NI}}\left(\tau\right)$ is given by (31), it follows that
$V_{t}\left(\tau\right)\geq 0,\forall t\in\left[\bar{S}^{NI},S^{*}\right]$.

_Third, consider_ $t\in[S^{*},\tau)$. The absence of stopping at $t\geq S^{*}$
reveals that $x_{t}\neq 2$. Thus,
$q_{1}\left(t\right)=\frac{p_{1}\left(t\right)}{p_{0}\left(t\right)+p_{1}\left(t\right)}=\frac{\lambda
t}{1+\lambda t}$, $\forall t\in[S^{*},\tau)$, and, thus,
$\dot{q}_{1}\left(t\right)>0$. Further, $q_{1}\left(S^{*}\right)>\kappa$. The
continuation value $\forall t\in[S^{*},\tau)$ is given by

$\begin{array}[]{c}V_{t}\left(\tau\right)=\operatorname{E}\left[\int_{t}^{\tau}v\lambda\left(q_{1}\left(z\right)-\kappa\right)dz\,|\,t<\tau\right].\end{array}$

Thus, $V_{t}\left(\tau\right)\geq 0$, $\forall t\in[S^{*},\tau)$.

_2\. The case of
$\kappa^{NI}\left(T,\lambda\right)<\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$_.

I start with proving the existence of $S^{*}$ such that
$V\left(\tau\right)=0$. For all $t\in[0,S^{*})$, stopping never occurs, at
$t=S^{*}$ it occurs if $x_{S^{*}}=2$, and for all $t\in(S^{*},T]$ it occurs at
$t=\tau_{2}\land T$. The principal’s long-run payoff can be written as

$V\left(\tau\right)=v\lambda\left(\int_{0}^{S^{*}}p_{1}\left(z\right)-\kappa
dz+\int_{S^{*}}^{T}\left(q_{1}\left(z\right)-\kappa\right)\left(1-\operatorname{P}\left(x_{z}=2\right)\right)dz\right).$
(33)

To ensure that $S^{*}$ makes the individual rationality constraint bind at
$t=0$, it is necessary and sufficient that $V\left(\tau\right)=0$. The next
step of the proof consist of inspecting (33) to establish that there exists
$S^{*}$ ensuring that $V\left(\tau\right)=0$. It follows the respective part
from the proof for the parametric case
$\kappa^{ND}\left(T,\lambda\right)<\kappa\leq\kappa^{NI}\left(T,\lambda\right)$,
imposing $\bar{S}^{NI}=0$ in it everywhere; thus, I omit it for the sake of
space.

I proceed with proving that the investment schedule $\tau$ satisfies the
conditions in Lemma 3 and thus it is obedient. The principal’s continuation
value is given by (32). As
$\kappa\in(\kappa^{NI}\left(T,\lambda\right),\tilde{\kappa}\left(T,\lambda\right)]$,
it follows from Lemma 1 that $V_{t}^{NI}=0,\forall t\in\left[0,S^{*}\right]$.
_First, assume_ $S^{*}\leq\bar{S}^{NI}$. From the proof of Lemma 1, it follows
that $p_{1}\left(t\right)\leq\kappa,\forall t\in[0,\overline{S}]$, and
$p_{1}\left(t\right)\geq\kappa,\forall t\in[\overline{S},\bar{S}^{NI}]$. Thus,

$\int_{t}^{\bar{S}^{NI}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds\geq\int_{0}^{\bar{S}^{NI}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds,\forall
t[0,\bar{S}^{NI}].$ (34)

As $V_{t}(\tau)$ is given by (32), $V(\tau)=0$ and (34) imply that
$V_{t}(\tau)\geq 0,\forall t\in[0,S^{*}]$. _Second, assume_
$S^{*}\geq\bar{S}^{NI}$. As $V(\tau)=0$ and
$\int_{0}^{\bar{S}^{NI}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds<0$,
it must be that $V(\bar{S}^{NI})>0$. Further,
$\int_{t}^{S^{*}}v\lambda\left(p_{1}\left(s\right)-\kappa\right)ds$ increases
in $t$ for $t\in[\bar{S}^{NI},S^{*}]$. Thus, $V_{t}(\tau)\geq 0,\forall
t\in[0,S^{*}]$.

Finally, the proof that $V_{t}(\tau)\geq 0,\forall t\in[S^{*},\tau)$ follows
the the respective part of the proof for the parametric case
$\kappa\in(\kappa^{ND}\left(T,\lambda\right),\kappa^{NI}\left(T,\lambda\right)]$;
thus, I omit it for the sake of space.

∎

###### Proof of Lemma 6.

I provide the proof for the parametric cases
$\kappa^{ND}\left(T,\lambda\right)<\kappa\leq\kappa^{NI}\left(T,\lambda\right)$
and
$\kappa^{NI}\left(T,\lambda\right)<\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$
separately.

_1\. The case of
$\kappa^{ND}\left(T,\lambda\right)<\kappa\leq\kappa^{NI}\left(T,\lambda\right)$_.

Under any obedient optimal policy, the principal’s individual rationality
constraint is binding, thus, $V\left(\tau\right)=V^{NI}$, or equivalently
$p_{2}\left(T\right)v-\operatorname{E}\left[\tau\right]c=p_{2}\left(\bar{S}^{NI}\right)v-\bar{S}^{NI}c.$
Thus,

$\operatorname{E}\left[\tau\right]=\frac{1}{\lambda\kappa}\left(p_{2}\left(T\right)-p_{2}\left(\bar{S}^{NI}\right)\right)+\bar{S}^{NI}.$

Differentiating both sides with respect to $\kappa$ yields

$\frac{\partial\operatorname{E}\left[\tau\right]}{\partial\kappa}=\frac{e^{-T\lambda}\left(1+T\lambda\right)-e^{-\bar{S}^{NI}\lambda}-\kappa}{\kappa^{2}\lambda}.$

The equation

$e^{-T\lambda}\left(1+T\lambda\right)-e^{-\bar{S}^{NI}\lambda}-\kappa=0$

can be equivalently rewritten as

$e^{-T\lambda}-e^{-\bar{S}^{NI}\lambda}=\kappa-e^{-T\lambda}T\lambda.$

It has a unique solution corresponding to
$\kappa=\kappa^{ND}\left(T,\lambda\right)\coloneqq e^{-T\lambda}T\lambda.$ As
$\kappa>\kappa^{ND}\left(T,\lambda\right),$ it holds that
$\partial\operatorname{E}\left[\tau\right]/\partial\kappa<0$.

_2\. The case of
$\kappa^{NI}\left(T,\lambda\right)<\kappa\leq\tilde{\kappa}\left(T,\lambda\right)$_.

The principal’s long-run payoff under any obedient optimal policy is given by

$\operatorname{E}\left[\tau\right]c=p_{2}\left(T\right)v.$

Rewriting it equivalently,
$\operatorname{E}\left[\tau\right]=\frac{1}{\lambda}\frac{1}{\kappa}p_{2}\left(T\right)\Rightarrow$
$\partial\operatorname{E}\left[\tau\right]/\partial\kappa<0$.

∎

###### Proof of Lemma 7.

Lemma 4 implies that if a schedule $\tau$ assigns zero probability to stopping
in states $0$ and $1$ then $V\left(\tau\right)<0$ and the individual
rationality constraint is violated. Thus, the necessary condition for a
schedule $\tau$ to be individually rational under
$\kappa\in(\tilde{\kappa}\left(T,\lambda\right),\kappa^{FI}\left(T,\lambda\right))$
is that it assigns a positive probability to stopping not only in state $2$,
but also to stopping in either state $0$ or state $1$. Consider a schedule
$\tau$ that assigns a positive probability to stopping in state $1$. Consider
an alternative schedule $\tau^{\prime}$ which is induced by reallocating the
probability mass of stopping in state $1$ to stopping at $\tau_{2}\land T$.
Lemma 2 suggests that in state $1$ the principal strictly benefits from
postponing the stopping until the second stage of the project is completed.
Thus, $V\left(\tau^{\prime}\right)>V\left(\tau\right)$. Further, under
$\tau^{\prime}$ the principal invests strictly longer, in expectation. Thus,
$W\left(\tau^{\prime}\right)>W\left(\tau\right)$. Thus, for a schedule to be
optimal it should not assign a positive probability to stopping in state $1$.

Next, consider a schedule $\tau$ which assigns a positive probability to
stopping in states $0$ and $2$. Assume that the stopping in state $0$ happens
at date $S$, which can be either deterministic or stochastic: if $x_{S}=0$
then $\tau=S$, otherwise, $\tau\geq\tau_{2}\land T$ and there exists
$\omega\in\Omega$ such that
$\tau\left(\omega\right)>\tau_{2}\left(\omega\right)$, i.e., with a positive
probability, stopping in state $2$ happens strictly after the date of
transition to state $2$. Assume that $V\left(\tau\right)=0$. Consider the
following investment schedule $\tilde{\tau}$: if $x_{\tilde{S}}=0$ then
$\tilde{\tau}=\tilde{S}$,
$\operatorname{E}{[\tilde{S}]}>\operatorname{E}{[S]}$, otherwise,
$\tilde{\tau}=\tau_{2}\land T$, and $V\left(\tilde{\tau}\right)=0$. Further,
from (9), the agent’s objective is given by

$\displaystyle W\left(\tilde{\tau}\right)-W\left(\tau\right)$
$\displaystyle=\left(SV\left(\tilde{\tau}\right)-V\left(\tilde{\tau}\right)\right)-\left(SV\left(\tau\right)-V\left(\tau\right)\right)$
$\displaystyle=SV\left(\tilde{\tau}\right)-SV\left(\tau\right).$

The change from $\tau\geq\tau_{2}\land T$ to $\tau=\tau_{2}\land T$ induces no
loss in total surplus as the measure of $\omega\in\Omega$ satisfying the event
$\left\\{\tau_{2}\leq T\right\\}$ is equal for both schedules. Further, the
change from conditional stopping at $S$ to conditional stopping at $\tilde{S}$
induces an increase in total surplus as
$\operatorname{P}\left(x_{\tilde{S}}=0\right)<\operatorname{P}\left(x_{S}=0\right)$
and thus, in the latter case, conditional stopping happens less frequently.
Hence, $SV\left(\tilde{\tau}\right)\geq SV\left(\tau\right)$. Thus, for a
schedule that assigns positive probability to stopping in states $0$ and $2$
to be optimal, it is necessary that stopping in state $2$ happens at
$\tau_{2}$ with probability $1$.

∎

###### Proof of Proposition 3.

Given Lemma 7, the space of candidate optimal investment schedules under
$\kappa\in(\tilde{\kappa}\left(T,\lambda\right),\kappa^{FI}\left(T,\lambda\right)]$
simplifies to schedules such that stopping in state $2$ happens at $\tau_{2}$,
and also stopping in state $0$ happens with positive probability. Thus, to
characterize the optimal schedule under
$\kappa\in(\tilde{\kappa}\left(T,\lambda\right),\kappa^{FI}\left(T,\lambda\right)]$,
I need to characterize the assignment of the probability mass of stopping in
state $0$ that is optimal for the agent given the principal’s individual
rationality constraints. To do this, I consider the agent’s optimal design of
a _device that randomizes over the dates of stopping in state $0$_.

At $t=0$, the agent chooses a distribution $F_{\rho}$ on $\left[0,T\right]$,
observable to both the agent and the principal. $\rho$ stands for the random
date at which the stopping occurs if the state is $0$ by that date. $\rho$ is
drawn at $t=0$ according to $F_{\rho}$, which is independent from the state
process $x_{t}$, and the draw privately observed by the agent.

To formulate the agent’s design problem, I start with characterizing the
welfare implications of stopping in state $0$ for the agent and principal. A
few useful objects are $SV_{t|0}\left(\tau_{2}\right)$ and
$V_{t|0}\left(\tau_{2}\right)$. $SV_{t|0}\left(\tau_{2}\right)$ is the time
$t$ continuation total surplus given that $x_{t}=0$ at $t$ and completion of
the second stage of the project is immediately disclosed whenever it happens,
$\tau=\tau_{2}\land T$:

$\displaystyle SV_{t|0}\left(\tau_{2}\right)$
$\displaystyle=v\operatorname{P}\left(\tau_{2}\leq T|x_{t}=0\right)$ (35)
$\displaystyle=v\left[1-e^{-\lambda\left(T-t\right)}-\lambda\left(T-t\right)e^{-\lambda\left(T-t\right)}\right].$

$V_{t|0}\left(\tau_{2}\right)$ is the principal’s time $t$ continuation value
given that $x_{t}=0$ and completion of the second stage of the project is
immediately disclosed, $\tau=\tau_{2}\land T$:

$V_{t|0}\left(\tau_{2}\right)=v\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=0\right)-c\operatorname{E}\left[\tau_{2}\wedge T-t|x_{t}=0\right],$

where $v\operatorname{P}\left(\tau_{2}\leq T|x_{t}=0\right)$ is given by (35)
and

$\displaystyle\operatorname{E}\left[\tau_{2}\land T-t|x_{t}=0\right]$ (36)
$\displaystyle=$ $\displaystyle\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=0\right)\int_{t}^{T}z\cdot\frac{\lambda^{2}\left(z-t\right)e^{-\lambda\left(z-t\right)}}{\operatorname{P}\left(\tau_{2}\leq
T|x_{t}=0\right)}dz+\operatorname{P}\left(\tau_{2}>T|x_{t}=0\right)T-t$
$\displaystyle=$
$\displaystyle\frac{2}{\lambda}-\frac{2}{\lambda}e^{-\lambda\left(T-t\right)}-e^{-\lambda\left(T-t\right)}\left(T-t\right).$

I proceed with a useful lemma.

######  Lemma 9.

Given an investment schedule

$\tau=\begin{cases}\rho,&\text{if }x_{\rho}=0\\\ \tau_{2}\land
T,&\text{otherwise},\end{cases}$ (37)

where $\rho$ has a publicly observable distribution $F_{\rho}$ on
$\left[0,T\right]$, $\rho$ is independent of the state process $x_{t}$ and is
drawn at $t=0$, and the draw is unobservable to the players, the total surplus
at date $t$ can be written as

$SV_{t}\left(\tau\right)=SV_{t}\left(\tau_{2}\right)-\operatorname{E}_{F_{\rho}}\left[\operatorname{P}\left(x_{\rho}=0|t<\tau\right)SV_{\rho|0}\left(\tau_{2}\right)\right],$

and the principal’s expected payoff at date $t$ can be written as

$V_{t}\left(\tau\right)=V_{t}\left(\tau_{2}\right)-\operatorname{E}_{F_{\rho}}\left[\operatorname{P}\left(x_{\rho}=0|t<\tau\right)V_{\rho|0}\left(\tau_{2}\right)\right],$

for all $t\geq 0$.

###### Proof.

By construction, $SV_{t}\left(\tau\right)$ corresponds to the expected value
of the project completion payoff under stopping policy $\tau$ conditional on
stopping not having happened by $t$, i.e., $t<\tau$. Given (37), the principal
gets $v$ either if the second stage is completed before $\rho$ or if the first
stage is completed before $\rho$ and the second stage is completed before $T$.
Note that when $t<\rho$, $t<\tau$ implies that the state is either $0$ or $1$,
and, when $t\geq\rho$, $t<\tau$ implies that the state is $1$. Thus,

$SV_{t}\left(\tau\right)=v\operatorname{E}_{F_{\rho}}\bigg{[}\operatorname{P}\big{(}\left\\{x_{\rho}=1\right\\}\cap\left\\{\tau_{2}\leq
T\right\\}|t<\tau\big{)}+\operatorname{P}\big{(}x_{\rho}=2|t<\tau\big{)}\bigg{]}.$

Further, for each realization of $\rho$,

$\operatorname{P}\big{(}\left\\{x_{\rho}=1\right\\}\cap\left\\{\tau_{2}\leq
T\right\\}|t<\tau\big{)}=\operatorname{P}\left(x_{\rho}=1|t<\tau\right)\operatorname{P}\left(\tau_{2}\leq
T|x_{\rho}=1\right).$

Thus,

$SV_{t}\left(\tau\right)=v\operatorname{E}_{F_{\rho}}\bigg{[}\operatorname{P}\big{(}x_{\rho}=1|t<\tau\big{)}\operatorname{P}\big{(}\tau_{2}\leq
T|x_{\rho}=1\big{)}+\operatorname{P}\big{(}x_{\rho}=2|t<\tau\big{)}\bigg{]}.$
(38)

$SV_{\rho|0}\left(\tau_{2}\right)$ corresponds to the expected value of the
project completion payoff when $x_{\rho}=0$. In that case, $v$ is obtained
when the completion of the second stage happens not later than $T$. Thus,