# Optimal Pricing Schemes in the Presence of Social Learning and Costly
Reporting

Xi Weng111Guanghua School of Management, Peking University;
<EMAIL_ADDRESS>Xienan Cheng222Guanghua School of Management,
Peking University<EMAIL_ADDRESS>Kaiwei Zhang333Guanghua School
of Management, Peking University<EMAIL_ADDRESS>
###### Abstract

A monopoly platform sells either a risky product (with unknown utility) or a
safe product (with known utility) to agents who sequentially arrive and learn
the utility of the risky product by the reporting of previous agents. It is
costly for agents to report utility; hence the platform has to design both the
prices and the reporting bonus to motivate the agents to explore and generate
new information. We characterize the optimal bonus and pricing schemes offered
by the profit-maximizing platform. It turns out that the optimal scheme falls
into one of four types: Full Coverage, Partial Coverage, Immediate Revelation,
and Non-Bonus. In a model of exponential bandit, we find that there is a
dynamical switch of the types along the learning trajectory. Although learning
stops efficiently, information is revealed too slowly compared with the
planner’s optimal solution.

_Keywords_ : Social Learning; Costly Reporting; Dynamic Pricing.

_JEL_. D83. C02. C61. C73.

##  1 Introduction

One of the most common features of online peer-to-peer market places is that
the platform frequently does little up-front screening or certification and
instead tries to maintain quality by using reputation and feedback mechanisms
(Einav et al., 2016). In particular, for the organization of online experience
product markets such as houses on Airbnb or drivers on Uber, it crucially
requires more consumers’ reviews to relieve information asymmetry. Though the
internet has made it feasible and common for everyone to leave some feedback
after purchasing and experiencing, there is insufficient review supply in the
real economy. According to some social surveys about online reviews, nearly 53
$\%$ of all internet users worldwide don’t provide any reviews about a
product, company, or service within a month444
https://www.oberlo.com/blog/online-review-statistics; and only 5$\%$-10$\%$ of
consumers are used to writing reviews online555
https://websitebuilder.org/blog/online-review-statistics/.

Meanwhile, it is a common practice on online markets to utilize dynamic
pricing and review bonus to motivate the consumers to purchase the experience
product and leave feedback. For example, dynamic pricing is common in
experience goods markets, such as video games and lead to an increase in both
revenue and margins in the pilot
categories666https://www.chartboost.com/blog/how-to-make-dynamic-pricing-work-
for-free-to-play-mobile-games/; https://www.mckinsey.com/business-
functions/marketing-and-sales/how-we-help-clients/dynamic-pricing. Many online
platforms such as Taobao also offer bonus for consumers’ reviews. In the
literature, however, dynamic pricing (see, e.g., Bergemann and Välimäki
(1996),Bergemann and Välimäki (2000), Bergemann and Välimäki (2006),Weng
(2015), and Hagiu and Wright (2020)) and review bonus (see, e.g., Li (2010)
and Tadelis et al. (2016)) are analyzed separately. It remains unexplored how
a profit-maximizing monopolist determine the dynamic price and review bonus
together; and how the optimal bonus scheme and dynamic pricing interact with
each other.

In this paper, we study a mechanism design problem faced by a long-lived
monopoly platform who sells either a risky product (with unknown utility) or a
safe product (with known utility) to a sequence of short-lived risk-neutral
consumers. Consumers receive a private signal about the utility of the risky
product after purchasing, and incur a reporting cost to give his feedback
interpreting as publishing the signal. So a consumer will give a feedback only
when he receives a review bonus exceeding his cost, which is also the
consumer’s private information and known to the consumer before making the
purchasing decision. Once a feedback is given, both the platform and the
successors will see his private signal; hence there is incomplete social
learning in the sense that the common belief about the utility of the risky
product is updated according to the published private signals.

In this framework, we investigate how the monopolist commits to different
prices of the two products and review bonuses over time to maximize its
expected profits. It turns out that the bonus and pricing decisions are not
independent and strategically interact with each other, due to following two
reasons. First of all, when making the pricing decisions, the monopolist aims
to guide the consumers to make the appropriate purchasing decisions. Due to
the incompleteness of social learning, the bonus scheme has to be adjusted
accordingly. If the monopolist wants to facilitate the learning about the
risky product, the bonus should be set very high to guarantee that a lot of
private signals are published and vice versa. Second, the bonus scheme also
affects the pricing scheme because the bonus received from giving a feedback
is also part of the payoff from purchasing the risky product. Therefore, if
the review bonus is set very high, the monopolist does not need to lower the
price of the risky product too much to motivate the purchase of this product.

We first consider a perfect learning model where the unknown utility of the
risky product is perfectly observed after the purchase of this product. In
this case, the monopolist faces a trade-off between the value and the cost of
information: once knowing the utility of the risky product, he can better
guide the consumers to make the appropriate purchasing decisions, but the
probability of successfully getting information depends on the bonus he sets.
Intuitively, there are two obviously feasible strategies: the monopolist can
either set the highest bonus to get the information as soon as possible
(Immediate Revelation) if the information is of enough value; or never
motivate feedback at all (Non-Bonus) if he thinks the information is of no
value. Besides these two extreme strategies, we find that the monopolist may
also use a recursive bonus structure, where the optimal bonus decreases over
time as the number of remaining consumers decreases and hence the information
becomes less valuable. When using this recursive bonus strategy, the
monopolist still needs to decide how to guide the purchase the products.
Optimally, he may stick to selling the risky product until someone reports its
utility (i.e., Full Coverage) or allow the purchase of the safe product as
well (i.e., Partial Coverage). When the expected utility from the risky
product is sufficiently high but not too high to support Immediate Revelation,
the monopolist will prefer Full Coverage over Partial Coverage.

We then extend the basic model to allow imperfect learning via the exponential
bandit model (Keller et al. (2005a), Weng (2015)). In a bad state, the risky
product generates zero utility for sure, while in a good state, the risky
product generates a positive utility with a random arrival time following an
exponential distribution. We characterize the monopolist’s optimal pricing and
bonus schemes as a function with respect to the belief about the good state.
Our previous intuition still holds that the platform uses the optimal schemes
to guide consumers to purchase the appropriate products and to reveal
information in either moderate or extreme ways. The social learning trajectory
in the exponential bandit model is such that the common belief about the good
state either remains unchanged if the current agent does not purchase the
risky product or make a report, falls because of a revelation of zero utility,
or rises directly to 1 after a revelation of positive utility. Consistent with
our findings in the basic model, with the decreasing of the common belief, the
monopolist will optimally switch from Immediate Revelation to Full Coverage
and then to Partial Coverage and Non-Bonus. Interestingly, we find that the
optimal bonus is monotonically decreasing with respect to the belief, since
the marginal value of one more experiment becomes lower when the monopolist
believes that the state is more likely to be good. In this situation, although
the monopolist expects a very quick revelation of the positive utility, which
will immediately make the common belief jump to 1 and lead to even higher
profits, he is also satisfied with the current high profits by selling the
risky product, and do not want to bear the loss of bonus and the possible
decrease of belief. As a result, a higher common belief leads to a lower
optimal bonus.

We conduct welfare analysis in the imperfect learning case, where we assume a
social planner who can commit to an allocation rule and reporting rule, based
on the common belief and agents’ reporting cost. We prove that social planner
will act like the monopolist, and the optimal strategy is again one of the
possible optimal strategies used by the monopolist. A surprising result is
that unlike traditional literature, there is no inefficient earlier
termination of social learning, since the monopolist can always take advantage
of Partial Coverage by setting a low enough bonus to encourage partial agents
to reveal the information cheaply. Though stop at the same belief, we find
that the social planner may still lead a more efficient experiment than the
monopolist, in the sense that the social planner will encourage reporting with
higher probability due to lower cost of experiments. The comparative static
analysis shows that a higher learning speed will lead to more experimentation
of the risky product, and hence lower switching belief from Full Coverage to
Partial Coverage. A higher utility from the safe product or a higher discount
rate may lead to inefficiently earlier termination of the learning process.

##  2 Literature Review

The literature on informational cascades that originated with the work of
Banerjee (1992) and Bikhchandani et al. (1992) is probably the closest to the
model presented here. In these herding models, agents may directly follow
actions ahead of him without referring to his own signal, which makes public
information swamps private information and stops being updated. Bose et al.
(2006) and Bose et al. (2008) study the monopoly pricing problems in herding
situations. Our problem is different from their work because in our model,
agents are not endowed with his own signal. With this assumption, Kremer et
al. (2014) considers a social planner who can reveal information strategically
to agents, in order to persuade agents to optimally do experiments. They
assume the planner can take advantage of information asymmetry to exert
Bayesian Persuasion, hence the possiblility of exploitation makes up to the
experiment of agents’ experiment. However, like e-commerce platforms,
consumers’ reviews are public to both the seller and the buyers, which makes
Bayesian Persuasion infeasible. To our knowledge there is no relevant study,
hence we try to fill this gap.

Agents in our model must choose from two arms like in a bandit problem setting
originated with the work of Rothschild (1974). Unlike the single agent
learning problem there, we assume there are agents involved sequentially,
along the lines with works of Bolton and Harris (1999). In multi-agent setting
with public information, the positive externality often rises free-rider
problems, and social learning is always terminated insufficiently ( Bergemann
and Välimäki (1996), Bergemann and Välimäki (2000), Bergemann and Välimäki
(2006),Hagiu and Wright (2020)). Weng (2015) consider the dynamic pricing with
individual learning, where every agent may have his own type to effect his
action, though of no value to learn by others. Different to his work, our
paper considers the social learning part, where agents care about the common
utility of the risky arm.

The literature of review motivation is also highly related to our work. Though
it’s with no doubt that online reviews help consumers to decrease the
uncertainty of purchasing some certain goods, we can see that current review
provision is lacking. Some works are about dealing with missing reviews and
ratings (see Ying et al. (2006) and He et al. (2016)), while some more related
works focus on motivating more reviews from consumers (see Li (2010) and
Tadelis et al. (2016)). The idea of motivating reviews may be originated from
the work of Avery et al. (1999), where reporting cost is endogenously
generated because of externality of information. When facing reporting cost,
Krishna and Morgan (2012) find that full participation may not be optimal and
if full participation is compulsory, every one’s truthful revelation may be
not a Nash Equilibrium. Osborne et al. (2020) find that agents’ threshold
strategy limits the probability of revelation even when there are a large
number of agents. But when transferring is plausible, agents’ behavior may
make their payoff attach first best. In our work, we use the monetary
encouragement to cover agents’ reporting cost in order to complete social
learning.

Multiple studies have shown that monetary incentives are effective at
encouraging wanted behavior online. For example, experimental studies show
that financial incentives are effective in motivating consumers to write
reviews on online consuming platforms (Fradkin et al. (2015), Cabral and Li
(2015), Khern-am nuai et al. (2018)). It is important to note that financial
incentives offered in most such studies have generally been quite small, like
$ 1 or 2, or 0.5 RMB (Cabral and Li (2015)), $ 0.04 in Amazon Mechanical Turk
or 10 RMB in a Chinese online cloth retailing platform (Burtch et al. (2018)).
Empiric works also show that paying for reviews do not effect consumers’
purchasing behavior or the demand curve faced by the seller (Cabral and Li
(2015)), and the feedback motivated by financial incentives are often upward
biased since consumers feel “bribed” (Cabral and Li (2015)).

##  3 Baseline Model: Perfect Learning

In this section, we set up the baseline model in which learning is perfect,
i.e. once a consumer reports the utility of the risky arm, it is the exact
value. There is a monopoly platform selling two products $i=1,2$ (called arm 1
and arm 2 henceforth) to a sequence of risk neutral agents. The production
cost of each arm is normalized to zero. The platform’s objective is to
maximize its total profit while the agents are maximizing expected utility by
choosing one of these two arms. Each agent receives the same utility from arm
$i$ denoted by $R_{i}$. We assume that the value of $R_{2}$ is common
knowledge while the value of $R_{1}$ is unknown both to the platform and to
the agents. As in previous literature, arm 2 is called “safe arm” and arm 1 is
“risky arm”. Moreover, it is common knowledge that $R_{1}$ follows a certain
distribution $G(\cdot)$. For our further analysis, only the values of
$\mathbb{E}R_{1}$ and $\mathbb{E}\max\\{R_{1},R_{2}\\}$ matter with
$\mathbb{E}R_{1}<\mathbb{E}\max\\{R_{1},R_{2}\\}<\infty.$

A leading example fitting our model very well is the proprietary platform
operated by Amazon or JD.com. For the brands sold on such a platform, the
platform can either directly determine the price or indirectly affect the
price by offering customers coupons or discounts. Moreover, the safe arm
corresponds to the well-established big brand for which the customers are
quite certain about quality while the risky arm corresponds to the new brand
for which both the platform and the customers are uncertain about quality.

Following the social learning literature (see, e.g., Kremer et al. (2014)), we
assume that there are $T\leq\infty$ agents indexed by the integer $t$. Agents
know their positions and take actions in an exogenous order: agent $t$ can
make a decision of purchasing in period $t$ only. In our baseline model of
perfect learning, an agent can perfectly know the value of $R_{1}$ after
choosing arm 1, and then decides whether to report $R_{1}$ to the platform.
Reporting is costly, and each agent has a private reporting cost $c_{i}$
distributed in a common support interval $[0,\bar{c}]$. We assume the
distribution function $H(\cdot)$ is strictly increasing and has a smooth
density function $h(\cdot)$. Moreover, we impose the following technical
assumptions on the density function $h$:

######  Assumption 3.1.

The density function $h(\cdot)$ satisfies: $\lim\limits_{x\to 0^{+}}h(x)>0$,
and $\lim\limits_{x\to c^{-}}h(x)>0$. 777A more general assumption for part 2
in Assumption 3.1 is that there exists some order $p$ such that
$\lim\limits_{x\to\bar{c}^{-}}h^{(p)}(x)\neq 0$ if
$\lim\limits_{x\to\bar{c}^{-}}h^{(k)}(x)=0,\forall k<p$.Here if
$\lim\limits_{x\to\bar{c}^{-}}h(x)=0$, form L Hospital’s Rule
$\lambda(\bar{c})=M+\bar{c}+\lim\limits_{x\to\bar{c}^{-}}\frac{H(x)-H^{2}(x)}{h(x)}=M+\bar{c}-\lim\limits_{x\to\bar{c}^{-}}\frac{h(x)}{h^{\prime}(x)}$
so if $\lim\limits_{x\to\bar{c}^{-}}h^{\prime}(x)=0$ we can still get the same
result by using $\lim\limits_{x\to\bar{c}^{-}}h^{\prime\prime}(x)>0$ or so on.
All we need is that the limit
$\lim\limits_{x\to\bar{c}^{-}}\frac{h(x)}{h^{\prime}(x)}$ is convergent to
zero hence as long as there exist some order $p$ such that
$\lim\limits_{x\to\bar{c}^{-}}h^{(p)}(x)\neq 0$ if
$\lim\limits_{x\to\bar{c}^{-}}h^{(k)}(x)=0,\forall k<p$, which always exists
unless $h(x)$ is constant in a neighborhood of $\bar{c}$.

Technically speaking, these assumptions on the limits are needed because we
have to calculate the limit value when $x$ converges to the boundary (in
Appendix A.2), and it always holds unless $h(x)$ is constant in a neighborhood
of $\bar{c}$. In economic language, we know the first part of Assumption 3.1
restricts the marginal effect of increasing the size of bonus at the lower
bound. The second part also restricts the marginal effect of increasing the
size of bonus at the upper bound, since it can be understood as a more general
assumption than $\lim\limits_{x\to\bar{c}^{-}}h(x)>0$. For example, if
$\lim\limits_{x\to\bar{c}^{-}}h(x)=0$, from L’Hospital’s Rule
$\lambda(\bar{c})=M+\bar{c}+\lim\limits_{x\to\bar{c}^{-}}\frac{H(x)-H^{2}(x)}{h(x)}=M+\bar{c}-\lim\limits_{x\to\bar{c}^{-}}\frac{h(x)}{h^{\prime}(x)}$
so if $\lim\limits_{x\to\bar{c}^{-}}h^{\prime}(x)=0$ we can still get the same
result by using $\lim\limits_{x\to\bar{c}^{-}}h^{\prime\prime}(x)>0$ or so on.
All we need is that the limit
$\lim\limits_{x\to\bar{c}^{-}}\frac{h(x)}{h^{\prime}(x)}$ is convergent to
zero.

Define $\beta(x)=x+\frac{H(x)}{h(x)}$ as the virtue value defined in the
mechanism design literature. We impose the standard assumption on $\beta(x)$.

######  Assumption 3.2.

$\beta^{\prime}(x)$ is always positive.

In our problem, the platform commits to a sequence of bonuses
$\\{b_{1},b_{2},\cdots,b_{T}\\}$ and a sequence of prices
$\\{p_{i1},p_{i2},\cdots,p_{iT}\\}$ before the game starts, where $p_{it}$ is
the price of arm $i$ for the $t$-th agent if in period $t$ the value of
$R_{1}$ is not revealed, and $b_{t}$ is the bonus paid to agent $t$ if agent
$t$ reveals $R_{1}$ and none of the agents before $t$ report to the platform
(in perfect learning setting, once $R_{1}$ is revealed, the platform will no
longer need reports). We assume that the agent reveals her true utility if she
reports. This assumption is reasonable as there is no incentive for lying
since each agent is short-lived in our model. Each agent has a reservation
value of zero. So it is straightforward to set $p_{2t}=R_{2}$. And once
$R_{1}$ is revealed to the platform, the platform will always sell the arm
with higher value and get $\mathbb{E}\max\\{R_{1},R_{2}\\}$. The platform’s
problem is to determine $p_{1t}$ and $b_{t}$ before $R_{1}$ is revealed.

##  4 Equilibrium Analysis

This section is organized as follows. In Section 5.2 we try to do some
preliminary analysis for the optimal bonus scheming problem where some
definitions are declared, and the key conclusion is that the possible optimal
strategy is one of the following strategies: Non Bonus, Immediate Revelation,
Recursive Partial Coverage and Recursive Full Coverage. In Section 4.2 we
discussed the optimal solutions under these different strategies (mainly for
FC and PC) in both a finite setting and an infinite setting. Though the
process of solving the finite case and infinite case is different, the results
are highly similar. In Section 4.3 we describe the optimal bonus-pricing
scheme and try to do some explanations for some economics insights.

###  4.1 Preliminary Analysis

Define $t_{0}\equiv\min\\{t|\text{ agent }t\text{ reports }R_{1}\\}$. Then
firstly the monopolist can make $t_{0}\geq T$ by setting $b_{t}=0,\forall t$,
and sell safe arm or risky arm respectively by setting
$p_{1}=\mathbb{E}R_{1},p_{2}=R_{2}-\epsilon$ or
$p_{1}=\mathbb{E}R_{1},p_{2}=R_{2}+\epsilon$. In this case all the agents
purchase the corresponding arm but no information revelation are achieved. We
call these two corner solutions as “Safe Arm” strategy and “Non Bonus”
strategy respectively888The monopolist can also make $t_{0}=1$ by setting
either $b_{1}=\bar{c}$ and $p_{1,1}=\mathbb{E}R_{1}$ or $b_{1}>\bar{c}$ and
$p_{1,1}=\mathbb{E}R_{1}+b_{1}-\bar{c}$. In this case the first agent will
purchase arm 1 and report $R_{1}$ for sure. Hence we denote this strategy as
“Immediate Revelation”. We assume this stretegy out in formal context and
leave it to Appendix B.

Then given $1<t_{0}<T$, there are still two possible strategies. Agent $t_{0}$
will purchase the risky arm and agents after $t_{0}$ purchase the optimal arm
for sure. But which arm will agents before $t_{0}$ purchase? Notice that when
$R_{1}$ is unknown, agent $t$ has to choose among purchasing arm 1 and report,
purchasing arm 1 and don’t report, and purchasing arm 2 given
$\\{p_{1t},p_{2t},b_{t}\\}$. It is straightforward to see that there are two
cutoffs $\tilde{r}_{t}$, $\tilde{r}_{t}^{\prime}$ such that agent $t$ will
choose arm 1 if $c_{t}\leq\tilde{r}_{t}$ and report $R_{1}$ if
$c_{t}\leq\tilde{r}_{t}^{\prime}$. Then according to this definition we must
have

$\mathbb{E}R_{1}-p_{1t}+\max\\{b_{t}-c_{t},0\\}\geq R_{2}-p_{2},\quad\forall
c_{t}\leq\tilde{r}_{t}$

and

$b_{t}=r_{t}^{\prime}$

If we set $\tilde{r}_{t}^{\prime}<\tilde{r}_{t}$ then we must have
$b_{t}-c_{t}<0$ and set $p_{1}=\mathbb{E}R_{1}$. In this case, every buyer
purchases arm 1, and only those with $c_{t}<\tilde{r}_{t}^{\prime}$ reports
$R_{1}$.

On the other hand, if we set $\tilde{r}_{t}^{\prime}>\tilde{r}_{t}$, then we
have $p_{1t}=\mathbb{E}R_{1}+\tilde{r}_{t}^{\prime}-\tilde{r}_{t}$. In this
case, only those buyers with $c_{t}<\tilde{r}_{t}$ purchases arm1, but they
will report $R_{1}$ for sure after purchasing.

In the first case, $t_{0}=\min\\{t|b_{t}\geq c_{t}\\}$. Before $t_{0}$, the
platform sells arm 1 at price $\mathbb{E}R_{1}$ but does not pay any bonus. At
$t_{0}$, the platform gets $p_{1t}-b_{t}=\mathbb{E}R_{1}-b_{t_{0}}$. So given
$t_{0}$, the utility of the platform is:

$(1+r+...+r^{t_{0}-1})\mathbb{E}R_{1}+(r^{t_{0}}+r^{t_{0}+1}+...+r^{T-1})\mathbb{E}\max\\{R_{1},R_{2}\\}-r^{t_{0}-1}b_{t_{0}}.$
(1)

In the second case, similarly we have
$t_{0}=\min\\{t|\mathbb{E}R_{1}+b_{t}-p_{1t}\geq c_{t}\\}$. Before $t_{0}$,
the platform sells arm 2 at price $R_{2}$ and at $t_{0}$, the platform gets
$p_{1t}-b_{t}=\mathbb{E}R_{1}-b_{t_{0}}$. So the platform gets:

$(1+r+...+r^{t_{0}-2})R_{2}+r^{t_{0}-1}\mathbb{E}R_{1}+(r^{t_{0}}+r^{t_{0}+1}+...+r^{T-1})\mathbb{E}\max\\{R_{1},R_{2}\\}-r^{t_{0}-1}b_{t_{0}}.$
(2)

Hence before the information of $R_{1}$ is revealed ($t_{0}$), the monopolist
can either sell risky arm for every agent, or sell safe arm but set a bonus to
motivate someone to explore the risky arm (In our Appendix A we proved that
mixing these two strategies will never be optimal). We denote these two
strategies as “Full Coverage” and “Partial Coverage” strategy respectively. By
saying “selling the certain arm” we mean the monopolist can guide all the
agents to purchase the risky arm by pricing $\epsilon$ lower for the
corresponding arm.

$t_{0}$ is a description of equilibrium output instead of the equilibrium
strategy. After $t_{0}$ the game is trivial since the monopolist will sell the
arm with the highest utility. The following definitions are only for
strategies that may be equlibirum strategies:

######  Definition 4.1.

  1. 1.

The platform uses “Non Bonus” Strategy (NB) if the platform sets $b_{t}\leq
0,\forall t\leq T$ and keeps selling the risky arm forall $t\leq T$.

  2. 2.

The platform uses “Safe Arm” Strategy (SA) if the platform sets $b_{t}\leq
0,\forall t\leq T$ and keeps selling the safe arm forall $t\leq T$.

  3. 3.

The platform uses “Full Coverage” Strategy (FC) if the platform sets
$b_{1}\in(0,\bar{c})$ and keeps selling the risky arm before $t_{0}$.

  4. 4.

The platform uses “Partial Coverage” Strategy (PC) if the platform sets
$b_{1}\in(0,\bar{c})$ and keeps selling the safe arm before $t_{0}$.

  5. 5.

The platform uses “Immediate Revelation” Strategy (IR) if the platform sets
$b_{1}\geq\bar{c}$.

There are two points to note. Firstly we should explain the difference between
FC and PC. Under Full Coverage Strategy, the platform always sells arm 1
before $R_{1}$ is revealed but the value of $R_{1}$ may not be reported. Under
Partial Coverage Strategy, the platform sells both arm 1 and arm 2 before
$R_{1}$ is revealed, and the value of $R_{1}$ is always revealed after an
agent purchases arm 1. Though in definition we call a selling strategy Partial
Coverage whenever the platform sells the safe arm at least once, we prove in
Appendix that it’s not optimal for the platform to mix between these two arms:
in the equilibrium path, the seller either keeps selling risky arm until one
consumer reports, or he keeps selling safe arm until one consumer purchases
the risky arm and reports its utility.

Secondly it’s natural to treat NB as corner solutions of FC when $b_{t}\equiv
0$, and treat SA as corner solutions of PC when $b_{t}\equiv 0$. Here we tell
them apart to make things clear. Using the NB strategy means that the platform
believes that the expected utility of the risky arm is high enough that it
does not need to reveal information to further increase the pricing, while
using the SA strategy means that the platform believes that the risky arm is
not comparable to the safe arm, and the information is not worth revealing
since it’s too costly. IR can also be treated as a corner solution of both FC
and PC by setting $b_{1}=\bar{c}$. And using IR means the platform values this
information a lot and wants to reveal it as soon as possible even with a very
high cost.

For the subsequent analysis, we declare some notations here:

$M=\mathbb{E}max\\{R_{1},R_{2}\\}-\mathbb{E}R_{1}$
$N=M+\frac{\mathbb{E}R_{1}-R_{2}}{r}$
$\beta(x)=\frac{H(x)}{h(x)}+x,x\in[0,\bar{c}]$
$\psi(x)=\left\\{\begin{aligned} &\beta(x)+M-\frac{H^{2}(x)}{h(x)}&\text{ if
}x\in[0,\bar{c}]\\\ &x+M&\text{ otherwise }\end{aligned}\right.$

$\psi_{N}(x)=\left\\{\begin{aligned} &\beta(x)+N-\frac{H^{2}(x)}{h(x)}&\text{
if }x\in[0,\bar{c}]\\\ &x+N&\text{ otherwise }\end{aligned}\right.$
$\Psi(x)=(1-\frac{1}{r})\beta(x)+M-\frac{H^{2}(x)}{h(x)}$
$\Psi_{N}(x)=(1-\frac{1}{r})\beta(x)+N-\frac{H^{2}(x)}{h(x)}$

Finally we denote

$\Gamma(x)=\beta^{-1}(r\psi(x))$

and

$\Gamma_{N}(x)=\beta^{-1}(r\psi_{N}(x))$

for the recursive equation of bonus, and we let $b^{*},b_{N}^{*}$ stand for
the fixed point of $\Gamma(\cdot),\Gamma_{N}(\cdot)$ respectively. We can get
some immediate mathematical results based on Assumption 3.1 and 3.2:

  1. 1.

there exists $b^{*}$ if $M<(\frac{1}{r}-1)\bar{c}+\frac{1}{rh(\bar{c})}$ 999If
this condition is not satisfied, the iteration relationships in Lemma 4.2
shows for any $b_{p+1}$ we should set $b_{p}>\bar{c}$, hence an direct
conclusion is that the Immediate Revelation is always optimal when
$M>(\frac{1}{r}-1)\bar{c}+\frac{1}{rh(\bar{c})}$. To avoid too much discussion
on this trivial case, we can define $b^{*}=\infty$ for consistency in this
case, since it doesn’t matter in Proposition 4.3..

  2. 2.

if $b^{*}$ exists, it’s unique and positive.

Similar results about $b_{N}^{*}$ can be found in Appendix Fact A.6 and A.7.

According to our Definition 4.1 there are five possible optimal strategies in
the equilibrium. In order to focus our analysis on the parts relevant to FC
and PC, we ignore the possibility of IR in the main text and take care of IR
in Appendix B.1. The following assumption requires a large enough $\bar{c}$ in
order to rule out the case when Immediate Revelation is optimal. Assumption
4.1 also guarantees the existence of $b^{*}$.

######  Assumption 4.1.

$\Psi(\bar{c})<0$ and $\Psi_{N}(\bar{c})<0$

In the following analysis we consider four different strategies: FC and PC,
and their corresponding corner solutions NB and SA.

###  4.2 Optimal Solutions under Different Strategies in Finite Case

In Section 4.2, we consider the cases of finite cases (i.e. $T<\infty$) and
infinite cases (i.e. $T=\infty$) simultaneously. Though strictly speaking the
proof process is different, our results show a highly similarity of solutions
under different strategies or different cases.

####  4.2.1 Full Coverage Strategy (including corner case Non Bonus)

In the full coverage case, holding price for arm 2 $R_{2}$, the optimal price
of arm 1 is always $\mathbb{E}R_{1}-\epsilon$ before $R_{1}$ is fully revealed
to guide consumers to purchase arm 1. So we only need to characterize the
optimal bonus scheme. The following lemma is the first step of fully
characterizing the optimal solution to maximization problem generated by
equation (1):

######  Lemma 4.2.

Suppose that there exists $1\leq p<T$ such that the corresponding optimal
bonus scheme satisfies $b_{p+1}<b^{*}$. Then we have the following recursive
relationship for any $1\leq T\leq+\infty$:

$b_{p}=\Gamma(b_{p+1}),$ (3)

where $M=\mathbb{E}max\\{R_{1},R_{2}\\}-\mathbb{E}R_{1}$ and $\psi(\cdot)$ as
denoted ahead. Given $0<b_{p+1}<b^{*}$, the solution of equation 3, $b_{p}$
exists and $b_{p+1}<b_{p}<b^{*}$

Lemma 4.2 actually describes the necessary condition of a recursive (interior)
bonus scheme when using Full Coverage strategy. In the proof process we can
find that the results shown in Lemma 4.2 does not depend on $T<\infty$ or not.
The monotonicity of above functions guarantees the existence and uniqueness of
the solution $b_{p}$ given $b_{p+1}$. It also guarantees the existence and
uniqueness of the fixed point, $b^{*}$, which is the root of $\Psi(x)$.
Actually equation (3) defines an iteration constructed by monotone increasing
function:

$b_{p}=\Gamma(b_{p+1})\equiv\beta^{-1}(r*[\beta(b_{p+1})+M-\frac{H^{2}(b_{p+1})}{h(b_{p+1})}])$

Moreover, Lemma 4.2 implies that the optimal interior bonus for agent $p$ only
depends on agent $p+1$’s bonus plan in the backward induction sense.
Therefore, the platform’s optimal bonus scheme is time consistent, i.e., if
the platform designs the optimal bonus scheme at period $p$, the optimal
solution should be the same as the initial commitment solution.

Here we characterize the optimal bonus scheme where $T<\infty$. The recursive
relationship given in Lemma 4.2 allows us to use a backward induction
procedure to calculate the optimal bonus scheme. First of all, it is easy to
see that $b_{T}=0$ since the platform no longer care about the knowledge of
$R_{1}$ when no agent is left. Then, Equation (3) allows us to solve the
optimal bonus scheme $b_{t}$ backwardly. Assumption 4.1 allows us to rule out
the corner solution when $b_{t}>\bar{c}$, and the following proposition
describes the optimal FC strategy.

######  Proposition 4.3.

Under our Assumption 3.1, Assumption 3.2 and Assumption 4.1, the optimal Full
Coverage (containing Non Bonus) strategy is to set:

If $T=1$, the optimal full coverage strategy is to set $b_{1}=0$ (degenerates
into corner solution NB)

For $\forall T\geq 2$, NB is never optimal. Construct the sequence
$\\{b_{t}\\}$ as follows:

  1. 1.

if $T<\infty$, let $b_{t}=\Gamma^{(T-t)}(0)$ for $1\leq t\leq T$.

  2. 2.

if $T=\infty$, let $b_{t}=b^{*}$ for $1\leq t\leq T$ ($b^{*}$ is the fixed
point of function $\Gamma(\cdot)$ as declared in section 4.1).

Proposition 4.3 shows the algorithm of optimal bonus structure in finite case.
Sequence $\\{d_{t}\\}$ reveals the recurrence structure of the bonus scheme
(recursive FC). If $T\geq 2$, $d_{1}$ is positive for sure since $d_{1}$ is
the result of $T-1$ times of increasing iteration starting from 0, hence we do
not need to concern whether $d_{1}\leq 0$ in this case. In other words,
information about $R_{1}$ is always worth exploring as long as there is left
agents. Or equivalently, NB is always dominated by the optimal FC when $T\geq
2$.

Results in infinite case are also presented in Proposition 4.3. Different from
finite case, we can not exert backward induction, hence we have to perform the
induction from the beginning by using $b_{p+1}=\Gamma^{-1}(b_{p})$ given in
Lemma 4.2. The existence and the uniqueness of $b_{p}$ given $b_{p-1}$ can be
inherited from Equation (3). A constant bonus scheme is intuitive given the
fact that the platform faces the same designing condition and the same stage
game where ever the game starts. Still, we know the fixed point here is
positive and unique, which means the intuition before still holds.

####  4.2.2 Partial Coverage Strategy (including corner case Safe Arm)

If the platform chooses partial coverage strategy, he sells the safe item
before $R_{1}$ is revealed. In other words, the platform will optimally set
$p_{2}=R_{2}$, $p_{1}=\mathbb{E}R_{1}+\epsilon$, hence for agents whose
$c_{t}>b_{t}$, he purchases $R_{1}$ get surplus $-\epsilon<0$ hence he will
choose $R_{2}$. But for agents $c_{t}<b_{t}$, he will still choose $R_{1}$ and
get surplus $\mathbb{E}R_{1}+b_{t}-c_{t}-p_{1}$. We should notice that
$p_{1}-b_{t}$ will always be shown as a part, hence we can set any
$p_{1}>\mathbb{E}R_{1}$ and make $b_{t}$ an upward translation with equal
distance. Hence in the following analysis we just set
$p_{1}=\mathbb{E}R_{1}+\epsilon$ and let $\epsilon\to 0^{+}$. The mathematical
structure is similar to Full Coverage case, and Lemma A.5 (omitted here) is
similar to 4.2 which describes the recursive structure of possible optimal
strategies.

######  Proposition 4.4.

For $\forall T\geq 1$, construct the sequence $\\{d_{t}\\}$ as follows:

  1. 1.

let $d_{T}$ to be the root of equation $\beta(d_{T})=\mathbb{E}R_{1}-R_{2}$

  2. 2.

if $T<\infty$, let $d_{t}=\Gamma_{N}^{(T-t)}(d_{T})$ for $1\leq t\leq T$.

  3. 3.

if $T=\infty$, let $d_{t}=b_{N}^{*}$ for $1\leq t\leq T$ ($b_{N}^{*}$ is the
fixed point of function $\Gamma_{N}(\cdot)$ as declared in section 4.1).

Under our Assumption 3.1, Assumption 3.2 and Assumption 4.1, the optimal
Partial Coverage (including Safe Arm) strategy is to set:

  1. 1.

$(b_{N1},b_{N2},...b_{NT})=(d_{1},d_{2},...d_{T})$, if $0\leq d_{1}$ (PC).

  2. 2.

$(b_{N1}=0,b_{N2}=0,b_{N3}=0,..b_{NT}=0)$ if $d_{1}<0$ (SA).

The bonus scheme is almost the same despite that the platform may have a
negative $d_{1}$ in the iteration sequence. The reason is that the starting
point of the iteration, $d_{T}$ may be negative, unlike in Full Coverage
Strategy where $d_{T}=0$ for sure. This phenomena is a key difference in PC
and FC. Intuitively, a negative $d_{1}$ shows that the platform acts as if
it’s “punishing” consumers who purchase arm 1, though it’s not realistic since
consumers can always purchase arm 1 and leaves no feedback to avoid this
negative bonus. However, it seems that this is the best the platform can do in
order to de-motivate consumers from buying arm 1. Actually, it’s equivalent to
set $(0,0,0...0)$ and set $p_{1}=\mathbb{E}R_{1}+\epsilon$ and $p_{2}=R_{2}$
to guide consumers to purchase arm 2. A straightforward explanation is, in
this case, the platform do not want any consumers to purchase from arm 1.
Though knowing the exact value of $R_{1}$ is beneficial, it may not be
worthwhile to let one consumer to exploit arm 1, meaning the platform has to
bear the expected loss $R_{1}-\mathbb{E}R_{1}$ to do so. In other words,
recursive Partial Coverage (PC) may degenerate into Safe Arm (SA) in this
case.

As for the infinite case, a certain problem occurs. It’s possible that the
fixed point in PC, $b_{N}^{*}$, may be negative. Actually, $b_{N}^{*}$ is the
root of the following equation:

$(\frac{1}{r}-1)[\frac{H(x)}{h(x)}+x]+\frac{H^{2}(x)}{h(x)}=N,$ (4)

Obviously, the left hand side of Equation 4 is monotonically increasing with
respect to $x$, hence it values from 0 to
$(\frac{1}{r}-1)[\frac{1}{h(\bar{c})}+\bar{c}]+\frac{1}{h(\bar{c})}$. Hence
when $N<0$, using the interior expression of $H(\cdot)$ results in a negative
$b_{N}^{*}$. Hence we say in this case we have $b_{N}^{*}<0$, by which we mean
Equation 4 has no reasonable solution in interval $[0,\bar{c}]$ since $N<0$.

###  4.3 Optimal Bonus Scheme

To find out the optimal bonus scheme, we calculate the payoffs first. We have
the following lemma, whose illustration is in Figure 1.

######  Lemma 4.5.

For $\forall 0\leq T\leq\infty$ if $b_{1}$ and $b_{N_{1}}$ both lie in
$[0,\bar{c}]$, then:

$\pi_{L}=\left\\{\begin{aligned}
&\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{1-r^{T}}{1-r}-\psi(b_{1})&\text{ if
}L=\text{Full Coverage}\\\
&\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{1-r^{T}}{1-r}-\psi(b_{N_{1}})&\text{ if
}L=\text{Partial Coverage}\\\ &R_{2}\frac{1-r^{T}}{1-r}&\text{ if
}L=\text{Safe Arm}\\\ &\mathbb{E}R_{1}\frac{1-r^{T}}{1-r}&\text{ if
}L=\text{Non Bonus}\end{aligned}\right.$

We use Figure 1 as a numerical illustration101010In this numerical case we
assume $T=3,\bar{c}=1.1,r=0.95,R_{1}\sim U[0,4]$, hence $\mathbb{E}R_{1}=2$.
We let $R_{2}$ ranges from 1.1 to 3.1 hence the max possible $M$ values around
1.2 (in this case $M=\frac{R_{2}^{2}}{8}$). $\Psi(\bar{c})\leq-0.166<0$, and
$\Psi_{N}(\bar{c})<0$ since $N<0$ in this case. Hence Assumption 4.1 is always
satisfied. . The illustration shows FC tends to NB when R2 is low enough, and
PC tends to SA when R2 is high enough. In other words, there exists cutoffs of
$R_{2}$ such that SA dominates PC when $R_{2}>\hat{R}_{2}$, and PC dominates
FC when $R_{2}>\bar{R}_{2}$. Given the explicit expression of our Lemma 4.5,
we know $\bar{R}_{2}=\mathbb{E}R_{1}$ at which FC=PC and SA=NB. $\hat{R}_{2}$
is the cutoff when $b_{N1}=0$ where PC=SA. Moreover we have
$\hat{R}_{2}>\bar{R}_{2}$ given $T\geq 2$ for sure since $R_{2}=\bar{R}_{2}$
leads to $b_{NT}=0$ and therefore a positive $b_{N1}$. When Assumption 4.1
fails, i.e. we have to consider the Immediate Revelation case, an illustration
and analytical solution can be found in Appendix B.1 with Figure
LABEL:3totalUR2T8pic and LABEL:3totalUR2T8barc01pic. When $T\to\infty$ the
similar illustration can be found in Figure 3.

Figure 1: Profit-$R_{2}$ Under Different Strategies

Given Lemma 4.5, then the following proposition is quite clear:

######  Proposition 4.6.

Under our Assumption 3.1, Assumption 3.2 and Assumption 4.1, if $T\geq 2$,
there exists two cutoffs of $R_{2}$: $0<\bar{R}_{2}<\hat{R}_{2}$ such that

The optimal selling strategy is FC if $0<R_{2}<\bar{R}_{2}$

The optimal selling strategy is PC if $\bar{R}_{2}<R_{2}<\hat{R}_{2}$

The optimal selling strategy is SA if $\hat{R}_{2}<R_{2}$

If $T=1$, there exists one cutoff of $R_{2}$: $0<\bar{R}_{2}$ such that

The optimal selling strategy is NB if $0<R_{2}<\bar{R}_{2}$

The optimal selling strategy is SA if $\bar{R}_{2}<R_{2}$

Proposition 4.6 concludes the optimal strategy when we can ignore IR. And it
works for both $T<\infty$ and $T=\infty$. Then we emphasize some trade offs
here.

####  4.3.1 PC v.s. FC

We should point out the key difference between FC and PC strategy and we take
the finite case as an example. The starting point of backward induction may be
different. In FC case, arm 1 is always purchased no matter whether the bonus
is taken, hence by backward induction the starting point is always setting
$b_{T}=0$. But in PC case, if the bonus is not taken, arm 2 will be purchased.
Hence when $R_{2}$ is extremely small the platform would have motivation to
set some bonus for agents to purchase arm 1 even when he is the only one agent
left and information of $R_{1}$ is of no value. Otherwise when $R_{2}$ is
extremely large, the platform may prefer not to motivate agents to purchase
arm 1 by setting zero bonus, and that’s the reason why Safe Arm strategy
appears. Figure 1 describes this insight, since we can find that when $R_{2}$
is small, FC dominates PC, and when $R_{2}$ is large, PC is more attractive
(the platform will decrease the bonus to 0 when $R_{2}$ is large enough, which
is the when PC degenerates into SA).

A seemingly surprising conclusion is that, in both strategies the $t_{0}$ has
the same expectation value. Though it seems that in PC we define
$t_{0}=\min\\{t|c_{t}<r_{t}+\mathbb{E}R_{1}-p_{1}\\}$, but in equilibrium the
platform will increase the bonus for the agent to compensate the utility loss
$p_{1}-\mathbb{E}R_{1}$ for agents hence no matter how large $p_{1}$ is, it
will be return back in form of bonus. Hence in our equilibrium analysis we
directly fix $p_{1}=\mathbb{E}R_{1}$. In other words, the motivation to
encourage information (which is revealed by $b_{i}$ and accordingly,
$\mathbb{E}t_{0}$), is only affected by
$\mathbb{E}\max\\{R_{1},R_{2}\\}-\max\\{\mathbb{E}R_{1},R_{2}\\}$, hence the
$\mathbb{E}t_{0}$ is the same no matter we use PC or FC.

Fix $\mathbb{E}R_{1}$ and let $R_{2}$ increase rapidly, from Figure 1 we can
see there are two important thresholds for $R_{2}$.

The first one is $\tilde{R}_{2}=\mathbb{E}R_{1}$ or 2.0 in Figure 1. At this
point $\mathbb{E}R_{1}=R_{2}$,$M=N$, and $b_{T}=0$ in PC case. When
$R_{2}<\mathbb{E}R_{1}$, we know if the platform insists on using PC we have
$\beta(b_{T})=\mathbb{E}R_{1}-R_{2}>0$, which means the platform would still
like to use a positive bonus to encourage the last agent to purchase arm 1
even there is only one agent left. Hence it’s obviously dominated by FC, a
more direct way to guide agents purchase arm 1. Hence $\forall
R_{2}\in[0,\mathbb{E}R_{1}]$, the platform should use Full Coverage
Strategy111111strictly speaking, Full Coverage and Non-Bonus dominates Partial
Coverage and Safe Arm.. When $R_{2}>\mathbb{E}R_{1}$, in both strategies
reporting agent get $\mathbb{E}R_{1}$ and yields the same profit. But agents
who arrives before $t_{0}$ and does not report will get $R_{2}$ in Partial
Coverage instead of $\mathbb{E}R_{1}$ in Full Coverage. Hence when
$R_{2}>\mathbb{E}R_{1}$ the platform should use Partial Coverage Strategy.

The second one is $\hat{R}_{2}$ which makes $N=0$ or 3.26 in Figure 1. At this
point $N=0$ and $b_{N}^{*}=0$. When $R_{2}>\hat{R}_{2}$, $N<0$ hence the
interior solution sequence $\\{b_{t}\\}$ can never be positive (see Figure A2.
More details can be found in Appendix A.3). Hence the total bonus structure is
equivalent to $(0,0,...0)$, meaning the RPC completely degenerates into a safe
arm structure. When $\tilde{R}_{2}<R_{2}<\hat{R}_{2}$, though $b_{T}$ is
negative, the interior solution sequence $\\{b_{t}\\}$ may finally get
positive since $b_{N}^{*}>0$ (see Figure A2), meaning that when there are
still enough agents left the platform should use a positive bonus to encourage
reporting, otherwise it will set a zero (negative is equivalent to zero) bonus
to prevent loss of $R_{2}-\mathbb{E}R_{1}$ caused by reporting.

####  4.3.2 Finite v.s. Infinite

The platform’s optimal selling strategy and bonus method in infinite case are
also contained in the corresponding propositions or lemmas, though simply
taking the limit is not strict though it yields the right answer and it’s more
straightforward. Hence in the Appendix A.2 and A.3 we focus on the proof of
finite cases, and all the proof in infinite cases are placed in Appendix A.4
since they are based on a different solving process.

Mathematically speaking, the only difference between finite case and infinite
case is the discrimination condition of corner solution. In finite case, take
FC as an example, we construct sequence $\\{d_{t}\\}$ and use
$d_{1}=\Gamma^{(T-1)}(0)$ to compare with $\bar{c}$ for corner solution. In
infinite case, we use $d^{*}$, which is the fixed point of the iteration
equation and can be understood as $d^{*}=\Gamma^{(\infty)}(0)$. When
$T=\infty$, our insights before may be better understood because the
discrimination conditions and the bonus can be expressed explicitly. We
provide a numerical illustration in Figure 3.

Figure 2: Profit-$R_{2}$ When $T=\infty$

Figure 3: Bonus-$R_{2}$ When $T=\infty$

####  4.3.3 $R_{2}$ and $\mathbb{E}R_{1}$ Are Not Enough to Pin Down
Everything

One may think our results are trivial since we are simply comparing $R_{2}$
and $\mathbb{E}R_{1}$ for choosing FC and PC. However we have to point out
that fixing $R_{2}$ and $\mathbb{E}R_{1}$, the optimal bonus strategy will not
be fixed accordingly. The reason is that the gap between
$\mathbb{E}\max\\{R_{1},R_{2}\\}$ and $R_{2}$ will affect the incentive to
motivate social learning under Partial Coverage. The same logic works for the
Full Coverage region. We may have different $\mathbb{E}\max\\{R_{1},R_{2}\\}$
even though we pin down $\mathbb{E}R_{1}$ and $R_{2}$. A simple example can be
constructed as follows. We assume $R_{1}=\rho x+(1-\rho)y$ where
$\rho\in(\frac{3}{4},1)$ is a given constant, $x\sim U(0,4)$ and $y\sim
U(1,3)$. Let $R_{2}=3$ then we can calculate $\mathbb{E}\max\\{R_{1},R_{2}\\}$
explicitly (more details can be found in Section A.5.2).

Hence $\mathbb{E}\max\\{R_{1},R_{2}\\}$ is continuous, smooth for
$\rho\in[0,1]$, and it’s monotonically increasing with respect to $\rho$. As a
result, bonus will increase if $\rho$ increases. Hence we should expect bonus
will increase no matter the strategy is FC or PC. In this case
$R_{2}=3>\mathbb{E}R_{1}=2$, hence we should use PC. Instead, if we set
$R_{2}=1$, the calculation should be similar, and we will see the bonus is
increasing with respect to $\rho$ in FC, too. An illustration can be found in
Figure 4.

Figure 4: $\rho$ Affects Bonus

##  5 Imperfect Learning

In this section, we consider a situation where each agent cannot perfectly
learn $R_{1}$. We use the exponential bandit to model this imperfect learning
(see, e.g., Keller et al. (2005b) and Weng (2015)).

###  5.1 Model Set Up

As in the previous model, we assume that $R_{2}$ is a safe product with known
utility $s$ and $R_{1}$ is a risky product modeled as an exponential bandit
arm. When purchasing $R_{1}$, it can generate either a lump-sum value $z$ or
zero utility. There are two states of the world, good or bad. When the state
is good, the arrival rate of realizing value $z$ follows $EXP(\lambda)$
distribution or the probability density function is $\lambda e^{-\lambda t}$.
When the state is bad, $R_{1}$ always leads to zero utility.

Firstly we assume that the agent will live for duration $\Delta$. The price
$p_{it},i\in\\{1,2\\}$ is paid at the beginning of this period, where we
denote $p_{1t}$ as the price for the risky arm at time t, and $p_{2t}$ as the
price for the safe arm. If he chooses the risky arm he gets the lump sum
utility $z$ which may realize at any time within $\Delta$. If he reports, he
immediately gets bonus $b_{t}$ and bears cost $c_{t}$, otherwise he gets no
bonus and no cost. If the buyer chooses the safe arm, he gets a deterministic
flow payoff $s>0$ in period $\Delta$. We further consider the discount factor
is $r$ and the form is exponential form, i.e. a payoff $v$ arriving after time
$t$ generates utility $e^{-rt}v$ at time 0.

Moreover, we want our analysis to be meaningful, i.e. the bandit problem is
non trivial. If the platform has already known the state is good and keeps
selling $R_{1}$, the total utility will be:

$\tilde{V}=\int_{0}^{\Delta}e^{-rt}(\lambda e^{-\lambda
t})zdt+e^{-r\Delta}\tilde{V}$ (5)

Let $\Delta\to 0$ we can get $\tilde{V}=\frac{\lambda z}{r}$, and we denote

$g\equiv\lambda z$

Similarly, keeping selling $R_{2}$ leads to payoff $\tilde{U}=\frac{s}{r}$. We
make a key assumption here to guarantee our discussion is meaningful:

######  Assumption 5.1.

$g\equiv\lambda z>s>0$

As before, there are four possible strategies: two iteration strategies Full
Coverage, Partial Coverage and two extreme strategies, None Bonus and Safe Arm
(Immediate Revelation in Appendix). None Bonus here again stands for the case
that platform is unwilling to do more experiments to update the belief and
keep selling risky arm only. And as before, Non Bonus can be treated as a
special case of Full Coverage when setting $b=0$. Similarly Safe Arm means to
do no more experiments and keep selling safe arm. As for the Immediate
Revelation, we should expect it’s nothing but setting an upper truncation of
our value function when $\alpha$ is large enough, since it should be an
extreme case for Full Coverage when belief is high enough. Of course when
$\bar{c}$ is large enough, using Immediate Revelation is too expensive to be
the optimal strategy. Hence in the text, for simplicity we make Assumption 5.2
in order to ignore the possibility of Immediate Revelation, and leave the more
general cases with possibility of Immediate Revelation to Appendix B.2.
Assumption 5.2 requires a large enough $\bar{c}$, and intuitively, the
monopolist will have less motivation to use Immediate Revelation when
$\bar{c}$ is larger. The Assumption 5.2 is a sufficient but not necessary
condition of our claim “Immediate Revelation is never the optimal strategy” we
make assumptions in this form for a explicit mathematical expression. The
equivalent condition is also discussed in Appendix B.2.

######  Assumption 5.2.

$\bar{c}>(\frac{\lambda}{r}+1)(g-s)$

###  5.2 Preliminary Analysis

In imperfect learning setting, the state variables of the platform is his
belief (which is also the common belief that holds for consumers) of the
possibility that the state is good. At time $t$ with belief
$\alpha\circ\theta_{G}+(1-\alpha)\circ\theta_{B}$ the platform can choose one
possible strategy $L$ from the set $\\{$ NB(Non Bonus), SA(Safe Arm), FC(Full
Coverage), PC(Partial Coverage) $\\}$ which affects his value function by
affecting immediate payoff (which item will be sold) and can set a bonus $b$
which influences the value function by affecting the probability of belief
updating. In this case, it’s obvious that the monopolist should use a cutoff
strategy: when $\alpha=0$ there is no Bayesian updating hence the monopolist
will always use SA, and when $\alpha=1$, there is no need to do any test,
hence with assumption 5.1 the platform should use NB. Since SA (NB) can be
achieved from PC(FC) by setting non bonus, we know that the monopolist is
actually using a cutoff strategy: when $\alpha$ is higher, he is more likely
to use FC and NB, and less likely to use SA or PC. Moreover, if the belief
$\alpha$ is higher, the value function should always be (weakly) higher, since
even though we use the same strategy, the immediate payoff would always
increase if there is positive probability to sell the risky arm. Our question
is how to design the optimal switching among these strategies.

As before we can always let the seller to set price equal to the expected
utility. So the price for safe arm should be

$p_{2t}=\int_{0}^{\Delta}e^{-rt}sdt=\frac{s(1-e^{-r\Delta})}{r}$

Under Full Coverage, the seller will set $p_{1t}$ as the expected surplus for
purchasing the risky arm

$p_{1t}=\int_{0}^{\Delta}e^{-rt}\lambda e^{-\lambda t}\alpha
zdt=\frac{\lambda}{r+\lambda}(1-e^{-(r+\lambda)\Delta})\alpha z$

and the agent will report if and only if

$b_{t}-c_{i}\geq 0$

Hence for the seller the immediate surplus is then

$\pi_{1,FC}(\Delta)=p_{1t}-\int_{0}^{\Delta}e^{-rt}\lambda e^{-\lambda
t}H(b_{t})b_{t}=\frac{\lambda}{r+\lambda}(1-e^{-(r+\lambda)\Delta})[\alpha
z-H(b_{t})b_{t}]$

from which we can get

$\lim_{\Delta\to 0}\frac{\pi_{1,FC}(\Delta)}{\Delta}=\lambda[\alpha
z-H(b_{t})b_{t}]$

Under Partial Coverage, the agent will purchase the risky arm and report if
and only if the expected surplus is positive:

$-p_{1t}+\int_{0}^{\Delta}e^{-rt}\lambda e^{-\lambda t}[\alpha
z+b_{t}-c_{i}]dt>0$

We can always set $p_{1t}$ to be the expected utility of risky arm, i.e.
$p_{1t}=\int_{0}^{\Delta}e^{-rt}\lambda e^{-\lambda t}\alpha zdt$. However as
we have shown in the discrete model before, the seller can set any large
$p_{1t}$ and compensate through $b_{t}$. Hence we still allow this setting,
and use $R$ instead of $b$ for the motivation of providing feedback. To do so,
denote $k(\Delta)=\frac{\lambda}{r+\lambda}(1-e^{-(r+\lambda)\Delta})$,
$R_{t}(\Delta)=\frac{-p_{1t}}{k(\Delta)}+\alpha z+b_{t}$ and
$R_{t}=\lim\limits_{\Delta\to 0}R_{t}(\Delta)$. Then there is probability
$H(R_{t}(\Delta))$ such that the current consumer will purchase the risky arm
and report. This will lead to immediate payoff $\pi_{1,PC}(\Delta)$ for the
platform 121212Since every function in our model is continuous we can exchange
the order of function and limits:

$\pi_{1,PC}(\Delta)=H(R_{t}(\Delta))[p_{1t}-\int_{0}^{\Delta}e^{-rt}\lambda
e^{-\lambda t}b_{t}]+[1-H(R_{t}(\Delta))]p_{2t}$ (6)

hence

$\displaystyle\lim\limits_{\Delta\to 0}\frac{\pi_{1,PC}(\Delta)}{\Delta}=$
$\displaystyle H(R_{t})\lim_{\Delta\to 0}\frac{k(\Delta)}{\Delta}[\alpha
z-R_{t}(\Delta)]+[1-H(R_{t})]\lim_{\Delta\to 0}\frac{p_{2t}}{\Delta}$
$\displaystyle=$ $\displaystyle H(R_{t})\lambda[\alpha z-R_{t}]+[1-H(R_{t})]s$

As a conclusion, $\pi_{1,L}(\Delta)$ stands for the expectation of immediate
net profit within period $\Delta$ using strategy $L$, and they fit

$\lim_{\Delta\to 0}\frac{\pi_{1,L}(\Delta)}{\Delta}=\left\\{\begin{aligned}
&\lambda[\alpha z-H(b_{t})b_{t}]&\text{ if }L=\text{FC}\\\
&H(R_{t})\lambda(\alpha z-R_{t})+[1-H(R_{t})]s&\text{ if }L=\text{PC}\\\
&\alpha z\lambda&\text{ if }L=\text{NB}\\\ &s&\text{ if
}L=\text{SA}\end{aligned}\right.$ (7)

We can characterize $P_{L}(b_{t},R_{t})$ as the probability of purchasing
risky arm and reports its utility131313Actually in FC case the seller can
choose optimal $b_{t}$ since we pin down the price $p_{1t}$. And in PC case
the seller can simultaneously choose $b_{t}$ and $p_{1t}$, but it’s equivalent
to choose the single variable $R_{t}$.

$P_{L}(b_{t},R_{t})=\left\\{\begin{aligned} &H(b_{t})&\text{ if
}L=\text{FC}\\\ &H(R_{t})&\text{ if }L=\text{PC}\\\ &0&\text{ if }L=\text{NB
or SA}\end{aligned}\right.$

Hence his value function $\Pi(\cdot)$ follows the dynamic process:

$\Pi(\alpha)=\max\limits_{L,b,R}\pi_{1,L}(\Delta)+P_{L}(b,R)e^{-r\Delta}\gamma(\alpha,\Delta)\Pi(\alpha^{\prime})+P_{L}(b,R)e^{-r\Delta}(1-\gamma(\alpha,\Delta))\tilde{V}+(1-P_{L}(b,R))e^{-r\Delta}\Pi(\alpha)$
(8)

where

$\gamma(\alpha,\Delta)=1-\alpha(1-e^{-\lambda\Delta})$

stands for the probability that in time period $\Delta$ the current agent
reports a zero utility and then leads to belief updating to $\alpha^{\prime}$,
and $1-\gamma(\alpha)$ stands for the probability that the agent reports a
utility $z$ which makes the common belief jump to 1 immediately. $P_{L}(b)$
stands for the probability of motivating the current agent to report
successfully under strategy $L$.

We have to mention that here we assume there exists a monotonically increasing
value function $\Pi(\alpha)$ with continuous and smooth derivative function
$\Pi^{\prime}(\alpha)$, and then this value function must satisfies equation
8, since equation 8 is not the strict definition because the right hand side
also has $\Pi(\cdot)$ term. Moreover, this value function has lower bound
$\tilde{U}$ and upper bound $\tilde{V}$, since by simply setting
$P_{L}=\gamma=0$ and $L=SA$, we can get $\Pi(\alpha)=\tilde{U}=\frac{s}{r}$.
Similarly, if we let
$\pi_{1,L}(\Delta)=\frac{\lambda}{r+\lambda}(1-e^{-(r+\lambda)\Delta}z)$
(which is only possible in the most ideal case when $\theta$ is known to be
$\theta_{G}$) and accordingly, optimally choose $P_{L}=1$ and $\gamma=0$ we
can get $\Pi(\alpha,\Delta)=\tilde{V}$.

Combining with Bayesian updating rule

$\alpha^{\prime}=\frac{\alpha e^{-\lambda\Delta}}{\alpha
e^{-\lambda\Delta}+1-\alpha}$

and Taylor expansion (for more details, see Appendix C.1), we can get
$\Pi(\alpha)$ actually follows the ordinary differential equation based on
equation (8):

$[r+P_{L}(b)\alpha\lambda]\Pi(\alpha)=\lim_{\Delta\to
0}\frac{\pi_{1,L}(\Delta)}{\Delta}+P_{L}(b)(\alpha^{2}-\alpha)\lambda\Pi^{\prime}(\alpha)+P_{L}(b)\frac{\alpha\lambda^{2}z}{r}$
(9)

In the left of this section, we characterize the optimal pricing and bonus
scheme for the monopolist at first. To do so we first derive the value
function of monopolist who can not use Partial Coverage (i.e. Full Coverage
only) or monopolist who can not use Full Coverage (i.e. Partial Coverage only)
respectively, and combine these two results to get the optimal switching
condition to describe the value function of a monopolist. Then we characterize
the optimal pricing and bonus scheme for a social planner for welfare analysis
who cannot do Partial Coverage in our model setting.

###  5.3 “Naive” Monopolist

The monopolist can optimally choose one of three possible strategies in Full
Coverage, Partial Coverage, Safe Arm and Non Bonus based on the common belief,
and it’s quite intuitive that when the belief is high enough, the platform is
willing to choose Full Coverage or Non Bonus, and when it’s low , Safe Arm or
Partial Coverage would be better. How to use Non Bonus and Safe Arm is quite
clear: we know when the belief is extremely high (low), NB (SA) is optimal for
sure. We also know that since NB (SA) can be considered as a corner solution
of FC (PC), there should be a natural transition from FC (PC) to NB(SA).

The only question is whether the platform may choose Partial Coverage as a
transition between Full Coverage and Safe Arm when the belief is decreasing,
or switch from Full Coverage to Safe Arm directly (similarly between PC, FC
and NB). The only way to answer this question is to compare the cutoff
switching belief of switching from Full Coverage to Safe Arm and the cutoff
switching belief of switching from Partial Coverage to Safe Arm. Hence, we
should characterize the optimal strategy of “Non optimal switching Monopolist”
or “Naive Monopolist”, by which we mean the monopolist can only use part of
the strategies and switch among them.

Firstly we consider “Full Coverage Only” monopolist (i.e. the platform can
only use Full Coverage or Non bonus). We denote $V(\alpha)$ as the value
function of using Full Coverage and plug results in equation (7) using Full
Coverage into equation (9), we can get

$[r+H(b_{t})\lambda\alpha]V(\alpha)=\max\limits_{b}\alpha
z\lambda[1+H(b_{t})\frac{\lambda}{r}]-\lambda
H(b_{t})b_{t}+H(b_{t})(\alpha^{2}-\alpha)\lambda V^{\prime}(\alpha)$

If we only consider the possibility that the monopoly can stop doing Full
Coverage experiments and switch to the safe arm for constant value, then the
optimal stopping problem is when to switch from $V(\cdot)$ to $\tilde{U}$. The
following lemma describes the strategy under this setting:

######  Lemma 5.1.

If the monopolist can only use “Full Coverage Strategy”, then he would set
prices $p_{1}=\alpha z\lambda-\epsilon$ and $p_{2}=R_{2}$, and would set bonus
$b_{t}$ for agent $t$ to report the utility of the risky arm. Agents would try
the risky arm when $\alpha>\underline{\alpha}_{1}$ and safe arm otherwise.
Denote $\underline{b}_{1}$ as the reporting cut-off at belief
$\underline{\alpha}_{1}$, then $b_{t}(\alpha),V(\alpha)$ and
$\underline{\alpha}_{1},\underline{b}_{1}$ fit the following relationships:

  1. 1.

First order condition:

$\lambda\beta(b)=\frac{\alpha z\lambda^{2}}{r}+(\alpha^{2}-\alpha)\lambda
V^{\prime}(\alpha)-\alpha\lambda V(\alpha)$

  2. 2.

Bellman equation:

$rV(\alpha)=\alpha z\lambda+\lambda\frac{H^{2}(b)}{h(b)}$

  3. 3.

Optimal Stopping:

$\left\\{\begin{aligned}
&\lambda\beta(\underline{b}_{1})=\frac{\underline{\alpha}_{1}\lambda}{r}(z\lambda-s)\\\
&s=\underline{\alpha}_{1}z\lambda+\lambda\frac{H^{2}(\underline{b}_{1})}{h(\underline{b}_{1})}\end{aligned}\right.$

  4. 4.

Optimal bonus scheme $b(\alpha)$ is monotonically decreasing with respect to
$\alpha$.

  5. 5.

There exists $\underline{\alpha}_{1}<\hat{\alpha}_{FC}^{NB}<1$ such that the
bonus is zero for $1>\alpha>\hat{\alpha}_{FC}^{NB}$, i.e. Full Coverage
degenerates into Non-Bonus case.

It’s helpful to do some analysis for Lemma 5.1 before starting to consider the
“Partial Coverage Only” monopolist. As we assumed in Assumption 3.2, we know
$\beta(b)=\frac{H(b)}{h(b)}+b$ is monotone increasing hence the uniqueness of
the solution $b^{*}(\alpha)$ can be guaranteed. Another important fact is that
the monopolist is less willing to motivate agents’ learning when belief is
higher. Intuitively, there are two powers affecting bonus setting. When the
belief is high, the monopolist may think it’s more possible to jump to 1 and
hence get higher profit, which provides high incentive to set high bonus.
However, if the belief is always high enough, the seller may face the risky to
lose the continue high price $p_{1}$ if it’s proven that the state is bad.
This concern provides low incentive to set high bonus. Our result shows the
latter power overcomes the first one. Actually as we will see later, the first
concern can be solved by using PC.

A not surprising result is

$s>\underline{\alpha_{1}}z\lambda$

which means the full coverage only monopolist would like to encourage agents
to test the risky arm even when it yields lower expected utility than safe
arm. But keep in mind that in this section we consider the case when the
monopolist can not use the Partial Coverage Strategy to sell the risky arm, so
he can only switch to selling safe arm when belief is low. In the formal
model, the seller may switch from FC to PC (instead of SA), which will be
discussed in section 5.4.

Last but not least, we should emphasis the possibility of zero bonus under
Full Coverage. From interior solution got from part 1 in Lemma 5.1, Full
Coverage will generate zero bonus when

$\frac{\alpha z\lambda^{2}}{r}+(\alpha^{2}-\alpha)\lambda
V^{\prime}(\alpha)-\alpha\lambda V(\alpha)\leq 0$

Combined with part 2 in Lemma 5.1,it’s equivalent (in the interior solution
part) to

$\alpha>\hat{\alpha}^{NB}_{FC}$ (9)

where $\hat{\alpha}^{NB}_{FC}$ is the solution of $V(\alpha)=\alpha z\lambda$.
Equation 9 shows that the monopolist would like to set $b\equiv 0$ for
$\forall\alpha\in(\hat{\alpha}^{NB}_{FC},1)$. In other words, if the temporary
common belief $\alpha_{t}\in(\hat{\alpha}^{NB}_{FC},1)$, the monopolist will
sell risky arm to each agent and won’t motivate them to report the utility. As
a result, the common belief would never change since no social signals are
generated in this case. As mentioned before, the monopolist would set lower
bonus when common belief is higher, given the common belief
$\alpha<\hat{\alpha}^{NB}_{FC}$. In this period, the monopolist is motivating
agents to do more experiments). Otherwise when
$\alpha\in(\hat{\alpha}^{NB}_{FC},1)$, the monopolist would not set any belief
though it’s possible that some agents come and report a positive utility and
push the common belief directly to 1, since he can sell the risky arm and take
advantage of the pretty high common belief. Here we use
$\hat{\alpha}^{NB}_{FC}$ since it’s not the true switching cutoff for the
optimal switching monopolist, since the value function for the optimal
switching monopolist has the different boundary condition, even though they
fit the same relationship between $V^{\prime}$ and $V$ mentioned in 5.1.

A main take away here is that when common belief is high enough (i.e.
$\alpha>\hat{\alpha}^{NB}_{FC}$ which is a sufficient condition for the
monopolist to sell risky arm), the monopolist faces a trade-off between the
following two extreme strategies: doing experiments at extremely high speed
(i.e. using Immediate Revelation) or motivating no experiments to keep the
high belief (i.e. using Full Coverage while set $b=0$) . The reason why we
need not to care about recursive Full Coverage (i.e. Full Coverage with a
positive bonus) is that given in Lemma 5.1, where we find at very high speed
it is dominated by setting $b=0$ within the right field of
$\hat{\alpha}^{NB}_{FC}$. However when the belief is higher, exploration may
leads to higher profits because the probability of pushing common belief to 1
may be high enough to cover the exploration cost. Hence when $\bar{c}$ is
small enough, we can find Immediate Revelation will be the optimal strategy in
this case (more detailed discussions in Appendix B.2).

Secondly, we consider Partial Coverage Only Monopolist setting. In this case
for agents with reporting cost $c$, if $R\equiv\lambda\alpha z+b-p_{1}>c$ this
agents will use arm 1 and report for sure no matter how large actual $R_{1}$
is. Hence the value function $U(\alpha)$ must fit the following ordinary
differential equation:

$[r+H(R)\alpha\lambda]U(\alpha)=[H(R)(\lambda\alpha z-\lambda
R)+(1-H(R))s]r+H(R)(\alpha^{2}-\alpha)\lambda
U^{\prime}(\alpha)+H(R)\frac{\alpha\lambda^{2}z}{r}$

And the following lemma characterize the solution of optimal stopping problem
of switching from $U(\cdot)$ to $\tilde{U}$

######  Lemma 5.2.

If the monopolist can only use “Partial Coverage Strategy”, then he would set
prices $p_{1}=\alpha z\lambda$ and $p_{2}$, and would set bonus $b_{t}$ for
agent $t$ to report the utility of the risky arm. It’s equivalent to set
$R=\lambda z\alpha+b_{t}-p_{t}$. Agents would try the safe arm when
$\alpha>\underline{\alpha}_{1}$ unless his own reporting cost
$c_{t}<\lambda\alpha z+b_{t}-p_{1}$ when he would try the risky arm and report
the utility. Denote $\underline{b}_{2}$ as the reporting cut-off at belief
$\underline{\alpha}_{2}$, then $b_{t}(\alpha),U(\alpha)$ and
$\underline{\alpha}_{2},\underline{R}_{2}$ fit the following relationships:

  1. 1.

First order condition:

$\lambda\beta(R)=\lambda\alpha z-s+\frac{\alpha
z\lambda^{2}}{r}+(\alpha^{2}-\alpha)\lambda U^{\prime}(\alpha)-\lambda\alpha
U(\alpha)$

  2. 2.

Bellman equation:

$rU(\alpha)=s+\lambda\frac{H^{2}(R)}{h(R)}$

  3. 3.

Optimal Stopping:

$\left\\{\begin{aligned} &\underline{R}_{2}=0\\\
&\underline{\alpha}_{2}=\frac{rs}{\lambda(g-s)+rg}\end{aligned}\right.$

  4. 4.

$R(\alpha)$ is monotonically increasing with respect to $\alpha$.

Lemma 5.1and 5.2 describes the value function of the Naive Monopolist. A quick
illustration is in 5 141414Here we use hollow circles for $\alpha_{FC}$ and
$\alpha_{PC}$ since they are the cutoffs of naive monopolist. As comparison we
use star marks for smooth pasting cutoffs in Figure 7. (to emphasis
$V(\alpha)$ got here is not the optimal strategy of the sophisticated
monopolist, we call it “original”). A similar version considering Immediate
Revelation can be found in Figure D1 in Appendix B.2.

Figure 5: Value Functions of the Naive Monopolist

Moreover, a noticeable fact is that in Partial Coverage the net bonus $R$ is
monotonically increasing with respect to $\alpha$, unlike Full Coverage. A
more detailed discussion about this fact is illustrated in the Figure 8, where
we translate the bonus scheme of the monopolist into a mechanism designed
based on the current belief and the reporting cost.

###  5.4 Optimal Switching Monopolist

In this section we consider whether the monopolist should switch from Full
Coverage into Partial Coverage, or directly into Safe Arm when the belief is
decreasing. As claimed before, comparing the cutoff believes got from analysis
of “Naive Monopolists” is helpful. Obviously from lemma 5.1 we know

$\displaystyle rs$
$\displaystyle=\underline{\alpha}_{1}gr+\underline{\alpha}_{1}\lambda(g-s)H(\underline{b}_{1})-r\lambda\underline{b}_{1}H(\underline{b}_{1})$
$\displaystyle<\underline{\alpha}_{1}gr+\underline{\alpha}_{1}\lambda(g-s)$

while from lemma 5.2 we know

$rs=\underline{\alpha}_{2}gr+\underline{\alpha}_{2}\lambda(g-s)$

hence we always have

$\underline{\alpha}_{1}>\underline{\alpha}_{2}$

This fact shows that, $V(\alpha)$ will intersect with $U(\alpha)$ in interval
$(\underline{\alpha}_{2},\underline{\alpha}_{1})$. Combined with convexity, we
know $V(\alpha)$ and $U(\alpha)$ is single-crossing in non trivial part. In
other words, there should be an optimal switching between FC and PC. And PC
should optimally switching to Safe Arm. Hence we denote

$\alpha^{PC}_{SA}=\underline{\alpha}_{2}$

Which shows that the ODE (together with boundary condition) described in lemma
5.2 pin down the value function of optimal switching monopolist using PC.

We denote the switching belief as $\alpha^{PC}_{FC}$. Then the optimal
switching condition (boundary condition for the ODE described in lemma 5.1 )
should be changed as

$V(\alpha_{PC}^{FC})=U(\alpha_{PC}^{FC}),V^{\prime}(\alpha_{PC}^{FC})=U^{\prime}(\alpha_{PC}^{FC})$
(9)

An intuitive explanation may be that Partial Coverage here can be understood
as a mixed strategy of using Full Coverage and selling safe arm. A numerical
illustration can be found in comparison between figure 5 and figure 6, where
it looks as if we try to connect $V(\alpha)$ and $U(\alpha)$ smoothly when the
first optimal switching occurs. More over we can prove:

$\alpha_{PC}^{FC}=\frac{s}{z\lambda}$

where the right hand side is nothing but the cut-off of the myopic monopolist
(i.e. the monopolist who can only consider the present profit instead of the
value function including discounted future profits). Base on Lemma 5.1, 5.2
and equation 9, we use the following proposition to state the optimal strategy
of the monopolist completely.

We can also prove $\alpha_{FC}^{NB}>\alpha_{PC}^{FC}$, since we have $b>0$ at
$\alpha_{PC}^{FC}$, and $b=0$ at $\alpha_{FC}^{NB}$.

######  Proposition 5.3.

There exists cutoffs $0<\alpha_{SA}^{PC}<\alpha_{PC}^{FC}<\alpha_{FC}^{NB}<1$,
where $\alpha_{SA}^{PC}=\frac{sr}{\lambda(g-s-
zr)},\alpha_{PC}^{FC}=\frac{s}{\lambda z},\text{ and }\alpha_{PC}^{FC}\text{
solves }V(x)=xz\lambda$, such that:

  * •

if $0<\alpha<\alpha_{SA}^{PC}$, the social planner would use Safe Arm.

  * •

if $\alpha_{SA}^{PC}<\alpha<\alpha_{PC}^{FC}$, the social planner would use
Partial Coverage.

  * •

if $\alpha_{PC}^{FC}<\alpha<\alpha_{FC}^{NB}$, the social planner would use
Full Coverage.

  * •

if $\alpha_{FC}^{NB}<\alpha<1$, the social planner would use Non Bonus.

To illustrate our results, a numeric example is provided whose details can be
found in Appendix C.7. Figure 6 is a numeric result based on the same
parameters in Figure 5. The horizontal axis stands for $\alpha$ and the
vertical axis stands for the value function $V(\alpha)$ or $U(\alpha)$.

Moreover, we should notice that the optimal switching belief between FC and PC
is $\alpha_{PC}^{FC}=\frac{s}{z\lambda}$, which is the exact belief when the
one shot payoff of selling the risky arm, $\alpha z\lambda$ is equal to the
one shot payoff of selling the safe arm $s$. According to our bullet points 4
in Lemmas 5.1 and 5.2, we know the optimal reporting bonus should be
monotonically decreasing with respect to $\alpha$ under FC but increasing
under PC. This is to say, at the switching belief $\alpha_{PC}^{FC}$, the
reporting bonus attaches the local maximum. Intuitively, the value of
information attaches the maximum when the belief is at its vaguest state. An
illustration is at Fig 7.

Figure 6: Optimal Switching Monopolist

###  5.5 Welfare Analysis

####  5.5.1 Social planner’s problem : a mechanism design way

We consider such a setting, where a social planner, commits an allocation rule
$p(\alpha,\cdot)$, a transfer rule $t(\alpha,\cdot)$ (negative for agents’ net
gain) and a reporting rule $q(\alpha,\cdot)$. Then the current agent with true
cost $c_{i}$ arrives and reports his own reporting cost $\tilde{c}_{i}$, and
the social planner will act using $\alpha$ and $\tilde{c}_{i}$.

Allocation rule stands for the probability of delivering the risky arm to the
current agent, and probability $1-p(\alpha,\cdot)$ to deliver the safe arm.
The agent will be charged a transfer $t(\alpha,\tilde{c}_{i})$ no matter which
arm is purchased. Then conditional on getting the risky arm, the agent has
probability $q(\alpha,\tilde{c})$ to be obliged to report the true utility and
$1-q(\alpha,\tilde{c})$ to do nothing. Then for the agent with true cost
$c_{i}$, reporting $\tilde{c}$ leads to utility151515In duration $\Delta$, the
agents’ utility can be written as
$\mathbb{E}(Surplus)=-t(\alpha,\tilde{c})+p(\alpha,\tilde{c})\int_{0}^{\Delta}\lambda
e^{-\lambda t}e^{-rt}(\alpha
z-q(\alpha,\tilde{c})c)dt+[1-p(\alpha,\tilde{c})]\int_{0}^{\Delta}se^{-rt}dt$
. We denote $U(c,\tilde{c},\alpha)=\lim\limits_{\Delta\to
0}\frac{\mathbb{E}(Surplus)}{\Delta}$ and re-write
$t(\alpha,\tilde{c})=\lim\limits_{\Delta\to
0}\frac{t(\alpha,\tilde{c})}{\Delta}$

$U(c,\tilde{c},\alpha)=-t(\alpha,\tilde{c})+p(\alpha,\tilde{c})[\alpha
z\lambda-\lambda q(\alpha,\tilde{c})c]+(1-p(\alpha,\tilde{c}))s$

and IC constraint hence requires

$c\in\arg\max_{\tilde{c}}U(c,\tilde{c},\alpha)$

For the social planner, in period $\Delta$, if the current agent reports
$\tilde{c}$ (we use $c$ to replace $\tilde{c}$ since in equilibrium we have
truthfully reporting), his immediate gain from total social surplus, should be

$\tilde{\pi}_{1}(\triangle,c)=\int_{0}^{\Delta}\lambda e^{-\lambda
t}e^{-rt}p(\alpha,\tilde{c})[\alpha
z-q(\alpha,\tilde{c})c]+e^{-rt}(1-p(\alpha,\tilde{c}))sdt$ (10)

Then we define the value function, which is based on $\alpha$ only and stands
for the expected total utility when holding belief $\alpha$. According to our
definition of $\tilde{\pi}(\Delta,c)$, we have (The formation details can be
found in Appendix)

$W(\alpha)=\mathbb{E}_{c}(\pi(\alpha,c))=\int_{c}\pi(\alpha,c)dH(c)$

Then we apply the operator $\int_{c}\cdot dH(c)$ for equation (10), (i.e. take
the expectation with respect to $c$) for both sides, we can execute the limit
calculation by using L’Hospital’s Rule:

$[r+\lambda\alpha\mathbb{E}(pq)]W(\alpha)=r\lambda\alpha
z\mathbb{E}(p)+[1-\mathbb{E}(p)]sr-r\mathbb{E}(pqc)+\lambda^{2}z\alpha\mathbb{E}(pq)+(\alpha^{2}-\alpha)\lambda
W^{\prime}(\alpha)\mathbb{E}(pq)$

And the social planner should set the optimal $(t,q,p)$ to maximize
$W(\alpha)$. Proposition 5.4 describes the optimal strategy for the social
planner. Denote
$B(\alpha)\equiv\frac{\alpha\lambda^{2}z}{r}+(\alpha^{2}-\alpha)\lambda
W^{\prime}(\alpha)-\lambda\alpha W(\alpha)$,
$C_{1}(\alpha)\equiv\frac{B(\alpha)+\lambda z\alpha-s}{\lambda}$ and
$C_{2}(\alpha)\equiv\frac{B(\alpha)}{\lambda}$, then we have:

######  Proposition 5.4.

There exists optimal mechanism $(p,q,t)$ to guarantee that every agent will
truthfully report their cost. There exists cutoff $\bar{\alpha}_{SA}^{PC}$,
$\bar{\alpha}_{PC}^{FC}$ and $\bar{\alpha}_{FC}^{NB}$ and cutoff
$C_{1}(\alpha)$ and $C_{2}(\alpha)$ for agents’ cost, such that:

  * •

if $0<\alpha<\bar{\alpha}_{SA}^{PC}$, the social planner would set $p=q=0$
(i.e. using Safe Arm).

  * •

if $\bar{\alpha}_{SA}^{PC}<\alpha<\bar{\alpha}_{PC}^{FC}$, the social planner
would set $q=1$ for sure and $p=1$ if $c<C_{1}(\alpha)$ (i.e. using Partial
Coverage).

  * •

if $\bar{\alpha}_{PC}^{FC}<\alpha<\bar{\alpha}_{FC}^{NB}$, the social planner
would set $p=1$ for sure and $q=1$ if $c<C_{2}(\alpha)$ (i.e. using Full
Coverage).

  * •

if $\bar{\alpha}_{FC}^{NB}<\alpha<1$, the social planner would set $p=1,q=0$
for sure (i.e. using Non Bonus).

Moreover, the social planner will switch from Full Coverage to Partial
Coverage, and switch from Partial Coverage and Non Bonus at the same believes
as the monopolist (i.e. $\bar{\alpha}_{SA}^{PC}=\alpha_{SA}^{PC}$, and
$\bar{\alpha}_{PC}^{FC}=\alpha_{PC}^{FC}$).

Above proposition 5.4 shows that social planner’s decision is also based on
the same optimal switching. Like before, the social planner is more likely to
use Full Coverage when the common belief is high (and Non Bonus is the common
belief is even higher), use Safe Arm when belief is low, and use Partial
Coverage as a transition when belief is moderate. More specifically, when
$\alpha>\bar{\alpha}_{PC}^{FC}$, $p=1$ for sure, which means the social
planner will always ask the agent to try the risky arm and report if the
current agent is with a low enough $c$ such that $q=1$, which is like using
Full Coverage. Otherwise when
$\bar{\alpha}_{SA}^{PC}<\alpha<\bar{\alpha}_{PC}^{FC}$, $p$ is possible to be
zero when $c$ is large. The social planner will only ask agents with low
enough costs to test the risky arm, hence it’s equivalent to Partial Coverage.
What’s still important here is that the once the social planner ask an agent
to test the risky arm, the agent will be obliged to report(for more details
please check Appendix C.5). In a more intuitive way, these agents whose
reporting cost is small enough to let $p=1$ for them are asked to use risky
arm to explore rather than exploit the risky arm since myopically safe arm is
better. When $\alpha<\bar{\alpha}_{SA}^{PC}$, the social planner will set
$p=0$ for sure, which means the information aggregation is terminated, and the
social planner sets Safe Arm. As before we can still characterize $W(\alpha)$
via a ODE system, which can be found in section C.5.2.

A key message is that the switching cutoff belief is irrelevant with the
decision maker. To be more specific:

  1. 1.

The social planner will switch to Partial Coverage at the same belief cutoff
as the monopolist. At the cutoff belief, the decision maker should feel
indifferent myopically between safe arm and risky arm, since the reporting
probability is continuous at this cutoff.

  2. 2.

The social planner will stop at the same belief cutoff as the monopolist.
Though surprising at the first sight, actually the switching condition
guarantees this equivalence. $\underline{\alpha}_{2}$ actually stands for the
switching belief between Partial Coverage and Non Bonus, at which in the long
run the decision maker feels indifferent between doing more experiment or not.
One more experiment is of no value when the common belief is low enough, and
it’s irrelevant with the cost of experiment. Intuitively speaking, because of
the existence of Partial Coverage, if the information is valuable for the
social planner, it’s also valuable for the monopolist since the monopolist can
always set a low enough bonus and use Partial Coverage, and keep waiting until
agents with low enough cost arrive, try the risky arm and report.

####  5.5.2 Comparison of social planner’s strategy and monopolist’s strategy

We measure the efficiency of the learning process led by the monopolist in two
ways. Firstly, will the monopolist stop experiments at a higher belief? And
secondly, will the monopolist attach the termination sooner than the social
planner? The following proposition answers the first question:

######  Proposition 5.5.

At any belief, the social planner will do experiments with (weakly) higher
probability. As a result $\bar{\alpha}_{FC}^{NB}>\alpha_{FC}^{NB}$.

Proposition 5.5 that the monopolist’s willingness do experiments is less than
the social planner, hence every stage there is lower probability to report.
Since the social planner bears less cost than the monopolist
($\mathbb{E}(c|c<b)<b$), a quick thought should be that the social planner
will encourage agents with higher reporting cost to report than the
monopolist. In other words, the learning speed of the social planner should be
higher. “Learning speed” actually means reporting probability in our model
(instead of arrival rate $\lambda$). We should realize that here social
planner can naturally do better since he can design the reporting probability
on $\alpha,c$ while the monopolist can only set it on $\alpha$.

A further illustration is in Figure 7. Both using Full Coverage, when the
current agent’s reporting cost $c$ is lower than the red solid line, the
social planner is willing to set $p=q=1$, while the monopolist is only willing
to set $p=q=1$ when $c$ is lower than the red dotted line. We can find the
monopolist will have lower probability to do experiments than the social
planner. Similarly, blue lines stands for the Partial Coverage, and our
observations still hold.

Figure 7: Mechanisms Comparison

####  5.5.3 Social Surplus under Monopolist’s Setting

Section 5.5.2 shows that there is inefficiency of the experiment process led
by the monopolist since the learning speed is lower, and because of lower
learning speed, the social surplus is also lower under the monopolist setting.
However one may think this result is trivial since the bonus higher than
reporting cost is not taken into consideration for the social planner’s
surplus. In this section we will show that this is not the only reason why the
social planner is efficient. In other words, the social planner is still more
efficient even we don’t consider the bonus higher than the reporting cost(see
figure 8).

We can calculate the social surplus under our monopolist setting to show the
social surplus is also lower than the social planner to illustrate that it’s
the learning speed which makes the social planner more efficient. To do so, we
define value function under this setting as $\Gamma(\alpha,c)$ and modify the
immediate surplus as

$\lim_{\Delta\to
0}\frac{\pi_{1_{,}L}(\Delta,c)}{\Delta}=\left\\{\begin{aligned} &\alpha
z\lambda-\lambda H(b)c&\text{ if }L=\text{FC}\\\
&[H(R)\lambda(z\alpha-c)+[1-H(R)]s&\text{ if }L=\text{PC}\\\ &s&\text{ if
}L=\text{SA}\\\ &\alpha z\lambda&\text{ if }L=\text{NB}\end{aligned}\right.$

and use first order conditions in Lemma 5.1and 5.2, and original boundary
conditions we get the following iteration relationships for value function
$\Gamma(\alpha,c)$ conditional on current reporting cost $c$ and common belief
$\alpha$:

$[r+P_{L}(b)\alpha\lambda]\Gamma(\alpha,c)=\lim_{\Delta\to
0}\frac{\pi_{1,L}(\Delta,c)}{\Delta}+P_{L}(b)(\alpha^{2}-\alpha)\lambda\Gamma_{\alpha}(\alpha,c)+P_{L}(b)\frac{\alpha\lambda^{2}z}{r}$

and let

$\Lambda(\alpha)=\mathbb{E}_{\alpha}[\Gamma(\alpha,c)|c<b(\alpha)]\text{ in
Full Coverage }$

or

$\Lambda(\alpha)=\mathbb{E}_{\alpha}[\Gamma(\alpha,c)|c<R(\alpha)]\text{ in
Partial Coverage }$

to stand for the value function of the social surplus under monopolist
setting, then combine with FOC gotten before we can get the following rule of
motion for $\Lambda(\alpha)$. Then for
$\alpha_{SA}^{PC}<\alpha<\alpha_{PC}^{FC}$ we have

$\left\\{\begin{aligned} &\lambda\beta(R)=\lambda\alpha z-s+\frac{\alpha
z\lambda^{2}}{r}+(\alpha^{2}-\alpha)\lambda\Lambda^{\prime}(\alpha)-\lambda\alpha\Lambda(\alpha)\\\
&\Lambda(\alpha)=-\lambda H(R)\mathbb{E}(c|c<R)+s+\lambda H(R)\beta(R)\\\
&r\Lambda(\alpha_{SA}^{PC})=\frac{s}{r},\Lambda^{\prime}(\alpha_{SA}^{PC})=0\end{aligned}\right.$

and for $\alpha_{PC}^{FC}<\alpha<\alpha_{FC}^{NB}$ we have

$\left\\{\begin{aligned} &\lambda\beta(b)=\frac{\alpha
z\lambda^{2}}{r}+(\alpha^{2}-\alpha)\lambda\Lambda^{\prime}(\alpha)-\lambda\alpha\Lambda(\alpha)\\\
&r\Lambda(\alpha)=-\lambda H(b)\mathbb{E}(c|c<b)+\alpha z\lambda+\lambda
H(b)\beta(b)\\\ &\Lambda(\alpha)\text{ is smooth at
}\alpha_{PC}^{FC}\end{aligned}\right.$

and for $\alpha_{FC}^{NB}<\alpha<1$ we have
$\Lambda(\alpha)=\frac{\alpha\lambda z}{r}$.

Figure 8: Social Welfare Comparison

It’s quite straight forward to see that $\Lambda(\alpha)>\Pi(\alpha)$ since
they two share the same first order condition but the right hand side of the
second line of $\Lambda(\alpha)$ is larger than the right hand side of the
Bellman Equation for $\Pi(\alpha)$. What’s important is that
$\Lambda(\alpha)<W(\alpha)$, as we mentioned before, the whole society suffer
from lower learning speed caused by the monopolist.

###  5.6 Comparative Statics

Here we do some comparative statics. We have discussed the effects of
monopolist’s behavior caused by discount rate $r$, relative utility $R_{2}$ ,
$\mathbb{E}R_{1}$ and $\mathbb{\max}\mathbb{E}\\{R_{1}.R_{2}\\}$. Here we add
some new intuitions: how $\lambda$ (the perfect level of experiment) affects
the monopolist’s behavior in section 5.6.1, and how discount rate affects the
social planner’s behavior in section 5.6.2.

####  5.6.1 Comparative Statics for $\lambda$

In this case we focus on discussing how the arrival rate $\lambda$ affects our
results. Actually the larger arrival rate stands not only the risky arm is
more attractive when the state is known to be good (since selling the risky
arm leads to immediate profit $\alpha\lambda z$ ), but also the measure of the
imperfect learning (since after purchasing the risky arm, there is probability
$\alpha(1-e^{-\lambda\Delta})$ to get a positive feedback after an experiment
with duration $\Delta$). So if we fix $s$, a larger $\lambda$ will make FC and
NB more attractive hence we will see $\alpha_{SA}^{PC}$ and $\alpha_{PC}^{FC}$
will shrink to zero. A more meaningful way is to focus on the second power:
how will the results change if the learning is “less imperfect”. To do so we
can re-define $s=\lambda z\rho$ where $\rho<1$ is a fixed constant who
guarantees that the safe arm and risky arm is comparable in stage utility
(i.e. to rule out the first effect). The intuition should be that if the
learning is “less imperfect”, then the seller will be more willing to do the
experiment, hence we should expect a smaller $\alpha_{SA}^{PC}$ (and tends to
0 if $\lambda\to\infty$), and a larger $\alpha_{SA}^{PC}$ (and tends to 1 if
$\lambda\to\infty$). The first intuition is easy to check since we have the
analytical solution for $\alpha_{SA}^{PC}$. As for the $\alpha_{FC}^{NB}$, we
have the following illustration Figure 9 by letting $\rho=0.5,r=0.5,z=7$ and
we let $\lambda$ to grow from 0.2 to 2.

(a) $\lambda=0.3$

(b) $\lambda=0.8$

(c) $\lambda=1$

(d) $\lambda=2$

Figure 9: $\lambda$ Affects $\alpha_{FC}^{NB}$

####  5.6.2 Comparative Statics for $r$

We do comparative statics here with respect to $r$, the discount factor of the
decision maker. A direct idea is value functions should be monotonically
decreasing with respect to the discount rate $r$. Moreover, with higher $r$,
the decision maker will be less willing to do experiments, no matter in
Monopolist case or Social Planner case. An illustration (take social surplus
as an example) is like Figure 10:

(a) $r=0.2$

(b) $r=0.5$

(c) $r=0.9$

(d) $r=1.2$

Figure 10: $r$ Affects Experiment Incentive

$r$ here stands for the patience of the decision maker, hence more efficient
experiment will be allowed when he is more patient. The decision whether to
switch from Full Coverage to Partial Coverage seems like a myopic decision
since $\underline{\alpha}_{3}$ is not affected by $r$: the monopolist stops
selling risky arm and starts to sell safe arm when risky arm makes less
profit. However, it’s actually not the case. As we have mentioned before,
Partial Coverage has the same learning speed with Full Coverage. Switching to
Partial Coverage will not slow down the speed of learning the state, hence $r$
will not affect the switching belief here.

##  6 Conclusion

The seller can always set Non-Bonus or Immediate Revelation in extreme cases.
Besides, there are two possible selling strategies: Full Coverage Strategy and
Partial Coverage Strategy. In Full Coverage Strategy, the platform keeps
selling the risky arm and combine it with a bonus scheme of recursive
structure. In Partial Coverage Strategy the platform set $p_{2t}\equiv s$. He
can set an arbitrary $p_{1t}$ because at time $t$ with belief $\alpha_{t}$ he
has to choose optimal $R_{t}(\alpha_{t})$ and set bonus
$b_{t}=p_{1t}+R_{t}(\alpha_{t})-\lambda\alpha_{t}z$. In this case agents
purchasing arm 1 is a stochastic process but once agent $t$ chooses to buy arm
1 he will report for sure, unlike in the Full Coverage where agents reporting
$R_{1}$ is a stochastic process instead. The linkage of pricing scheme and
bonus scheme actually reflects how the seller can take advantage of leading
the social experiment. The monopolist may switch from Full Coverage to Partial
Coverage optimally, and when the belief is low enough, he may stop doing
experiments and setting Non-Bonus. Since the monopolist can always take
advantage of Partial Coverage by setting a low enough bonus to encourage
partial agents to reveal the information cheaply, the monopolist can bear the
same experiment extent as the First Best. However, to save costs, the
monopolist will attach the termination of experiments at a lower speed.

## References

  * Avery et al. (1999) Avery, C., P. Resnick, and R. Zeckhauser (1999).  The market for evaluations.  American economic review 89(3), 564–584.
  * Banerjee (1992) Banerjee, A. V. (1992).  A simple model of herd behavior.  The quarterly journal of economics 107(3), 797–817.
  * Bergemann and Välimäki (1996) Bergemann, D. and J. Välimäki (1996).  Learning and strategic pricing.  Econometrica: Journal of the Econometric Society, 1125–1149.
  * Bergemann and Välimäki (2000) Bergemann, D. and J. Välimäki (2000).  Experimentation in markets.  The Review of Economic Studies 67(2), 213–234.
  * Bergemann and Välimäki (2006) Bergemann, D. and J. Välimäki (2006).  Dynamic pricing of new experience goods.  Journal of Political Economy 114(4), 713–743.
  * Bikhchandani et al. (1992) Bikhchandani, S., D. Hirshleifer, and I. Welch (1992).  A theory of fads, fashion, custom, and cultural change as informational cascades.  Journal of political Economy 100(5), 992–1026.
  * Bolton and Harris (1999) Bolton, P. and C. Harris (1999).  Strategic experimentation.  Econometrica 67(2), 349–374.
  * Bose et al. (2006) Bose, S., G. Orosel, M. Ottaviani, and L. Vesterlund (2006).  Dynamic monopoly pricing and herding.  The RAND Journal of Economics 37(4), 910–928.
  * Bose et al. (2008) Bose, S., G. Orosel, M. Ottaviani, and L. Vesterlund (2008).  Monopoly pricing in the binary herding model.  Economic Theory 37(2), 203–241.
  * Burtch et al. (2018) Burtch, G., Y. Hong, R. Bapna, and V. Griskevicius (2018).  Stimulating online reviews by combining financial incentives and social norms.  Management Science 64(5), 2065–2082.
  * Cabral and Li (2015) Cabral, L. and L. Li (2015).  A dollar for your thoughts: Feedback-conditional rebates on ebay.  Management Science 61(9), 2052–2063.
  * Einav et al. (2016) Einav, L., C. Farronato, and J. Levin (2016).  Peer-to-peer markets.  Annual review of economics 8, 615–635.
  * Fradkin et al. (2015) Fradkin, A., E. Grewal, D. Holtz, and M. Pearson (2015).  Bias and reciprocity in online reviews: Evidence from field experiments on airbnb.  EC 15, 15–19.
  * Hagiu and Wright (2020) Hagiu, A. and J. Wright (2020).  Data-enabled learning, network effects and competitive advantage.  In working paper.
  * He et al. (2016) He, X., H. Zhang, M.-Y. Kan, and T.-S. Chua (2016).  Fast matrix factorization for online recommendation with implicit feedback.  In Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval, pp. 549–558.
  * Keller et al. (2005a) Keller, G., S. Rady, and M. Cripps (2005a).  Strategic experimentation with exponential bandits.  Econometrica 83(1), 39–68.
  * Keller et al. (2005b) Keller, G., S. Rady, and M. Cripps (2005b).  Strategic experimentation with exponential bandits.  Econometrica 73(1), 39–68.
  * Khern-am nuai et al. (2018) Khern-am nuai, W., K. Kannan, and H. Ghasemkhani (2018).  Extrinsic versus intrinsic rewards for contributing reviews in an online platform.  Information Systems Research 29(4), 871–892.
  * Kremer et al. (2014) Kremer, I., Y. Mansour, and M. Perry (2014).  Implementing the “wisdom of the crowd”.  Journal of Political Economy 122(5), 988–1012.
  * Krishna and Morgan (2012) Krishna, V. and J. Morgan (2012).  Voluntary voting: Costs and benefits.  Journal of Economic Theory 147(6), 2083–2123.
  * Li (2010) Li, L. (2010).  Reputation, trust, and rebates: How online auction markets can improve their feedback mechanisms.  Journal of Economics & Management Strategy 19(2), 303–331.
  * Osborne et al. (2020) Osborne, M. J., J. S. Rosenthal, and C. Stewart (2020).  Information aggregation with costly reporting.  The Economic Journal 130(625), 208–232.
  * Rothschild (1974) Rothschild, M. (1974).  A two-armed bandit theory of market pricing.  Journal of Economic Theory 9(2), 185–202.
  * Tadelis et al. (2016) Tadelis, S., X. Zhou, et al. (2016).  Buying reputation as a signal of quality: Evidence from an online marketplace.  Technical report, National Bureau of Economic Research.
  * Weng (2015) Weng, X. (2015).  Dynamic pricing in the presence of individual learning.  Journal of Economic Theory 155, 262–299.
  * Ying et al. (2006) Ying, Y., F. Feinberg, and M. Wedel (2006).  Leveraging missing ratings to improve online recommendation systems.  Journal of marketing research 43(3), 355–365.

##  Appendix A Appendix A: Discrete Case

###  A.1 Proof of Full Coverage: Lemma 4.2 and Proposition 4.3

####  A.1.1 Proof of Lemma 4.2

###### Proof.

If the platform uses full coverage strategy the goal is:

$\begin{split}\max\limits_{b}\mathbb{E}R_{1}(1+r+..+r^{t_{0}-1})+\mathbb{E}\max\\{R_{1},R_{2}\\}(r^{t_{0}}+..+r^{T-1})-\mathbb{E}\text{cost}\\\
\Leftrightarrow\max\limits_{b}\mathbb{E}R_{1}\frac{r^{t_{0}}-1}{r-1}+\mathbb{E}\max\\{R_{1},R_{2}\\}r^{t_{0}}\frac{r^{T-t_{0}}-1}{r-1}-\mathbb{E}\text{cost}\end{split}$

where $t_{0}=\min\\{t|H(b_{t})>c_{t}\\}$.

Hence we know for each agent he has probability $S_{i}=H(b_{i})$ to report his
utility. We let
$a_{i}=Pr(t_{0}=i)=S_{i}\prod\limits_{j=1}^{i-1}(1-S_{j}),\forall i\leq T-1$
for the probability that agent $i$ reports, and let
$a_{T}=Pr(t_{0}=T)+Pr(\text{no reports})=\prod\limits_{j=1}^{T-1}(1-S_{j})$
stand for the probability that no agents before $T$ reports. Using this
notation makes $\sum\limits_{1}^{T}a_{i}=1$. By this definition we have

$\displaystyle
profit=\sum\limits_{i=1}^{T}Pr(t_{0}=i)[\mathbb{E}R_{1}\frac{1-r^{i}}{1-r}+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{i}-r^{T}}{1-r}-b_{i}r^{i-1}]+Pr(\text{no
reports})\mathbb{E}R_{1}\frac{1-r^{T}}{1-r}$
$\displaystyle=\sum\limits_{i=1}^{T-1}Pr(t_{0}=i)[\mathbb{E}R_{1}\frac{1-r^{i}}{1-r}+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{i}-r^{T}}{1-r}-b_{i}r^{i-1}]+a_{T}\mathbb{E}R_{1}\frac{1-r^{T}}{1-r}-Pr(t_{0}=T)b_{T}r^{T-1}$
$\displaystyle=\sum\limits_{i=1}^{T-1}a_{i}[\frac{\mathbb{E}R_{1}-\mathbb{E}\max\\{R_{1},R_{2}\\}r^{T}}{1-r}+\frac{Mr^{i}}{1-r}-b_{i}r^{i-1}]+a_{T}\mathbb{E}R_{1}\frac{1-r^{T}}{1-r}-Pr(t_{0}=T)b_{T}r^{T-1}$
$\displaystyle=(1-a_{T})\frac{\mathbb{E}R_{1}-\mathbb{E}\max\\{R_{1},R_{2}\\}r^{T}}{1-r}+\sum\limits_{i=1}^{T-1}a_{i}r^{i}\frac{M-b_{i}(1-r)/r}{1-r}+a_{T}\mathbb{E}R_{1}\frac{1-r^{T}}{1-r}-Pr(t_{0}=T)b_{T}r^{T-1}$
$\displaystyle=\sum\limits_{i=1}^{T-1}a_{i}r^{i}\frac{M-b_{i}(1-r)/r}{1-r}-Pr(t_{0}=T)b_{T}r^{T-1}+a_{T}\frac{Mr^{T}}{1-r}+constant$

Obviously optimal $b_{T}$ should be 0. Then we can combine two parts together.
Notice $a_{T}=1-\sum\limits_{i=1}^{T-1}a_{i}$. Which is to say, above problem
is equivalent to:

$\max\limits_{b}\sum\limits_{i=1}^{T-1}a_{i}\\{r^{i}[M+\frac{r-1}{r}b_{i}]-Q\\}$

where $Q=Mr^{T}$. For $\forall p\leq T-1$, we have:

$\sum\limits_{i=1}^{T-1}a_{i}\\{r^{i}[M+\frac{r-1}{r}b_{i}]-Q\\}=\sum\limits_{i=1}^{p-1}a_{i}\\{r^{i}[M+\frac{r-1}{r}b_{i}]-Q\\}+\prod\limits_{j=1}^{p-1}(1-S_{j})[S_{p}\\{r^{p}[M+\frac{r-1}{r}b_{p}]-Q\\}+(1-S_{p})L_{p}],$

where

$L_{p}=\sum\limits_{i=p+1}^{T-1}\\{r^{i}[M+\frac{r-1}{r}b_{i}]-Q\\}S_{i}\prod\limits_{k=p+1}^{i-1}(1-S_{j})$

(In the case $N_{1}>N_{2}$, let $\sum\limits_{N_{1}}^{N_{2}}=0$ and
$\prod\limits_{N_{1}}^{N_{2}}=1$.) F.O.C requires the following equation must
hold for the optimal bonus structure ($L_{p}$ is irrelevant with $b_{p}$):

$h(b_{p})[r^{p}(M+\frac{r-1}{r}b_{p})-Q]+H(b_{p})r^{p}\frac{r-1}{r}-h(b_{p})L_{p}=0$

which can be simplified as:

$\frac{r-1}{r}\beta(b_{p})+M=\frac{L_{p}+Q}{r^{p}}$

We can check the second order condition, the left hand side should be

$[r^{p}(M+\frac{r-1}{r}b_{p})-L_{p}-Q]h^{\prime}(b_{p})+2h(b_{p})r^{p}\frac{r-1}{r}$

and if it’s minimum solution we want it to be positive. Combine with the FOC
we know it’s equal to

$-[H(b_{p})r^{p}\frac{r-1}{r}]h^{\prime}(b_{p})+2h^{2}(b_{p})r^{p}\frac{r-1}{r}$

By Assumption 3.2 we know the fact that $\beta(x)$ is increasing implies
$2h^{2}(x)-H(x)h^{\prime}(x)>0$ for sure. Hence SOC is satisfied. Moreover,
from the definition of $L_{p}$, we know that for $p\leq T-2$:

$\begin{split}L_{p}=\sum\limits_{i=p+1}^{T-1}\\{r^{i}[M+\frac{r-1}{r}b_{i}]-Q\\}S_{i}\prod\limits_{k=p+1}^{i-1}(1-S_{j})\\\
=(1-S_{p+1})\sum\limits_{i=p+2}^{T-1}\\{r^{i}[M+\frac{r-1}{r}b_{i}]-Q\\}\prod\limits_{k=p+2}^{i-1}(1-S_{j})+\\{r^{p+1}[M+\frac{r-1}{r}b_{p+1}]-Q\\}S_{p+1}\\\
=(1-S_{p+1})L_{p+1}+\\{r^{p+1}[M+\frac{r-1}{r}b_{p+1}]-Q\\}S_{p+1}\\\
\end{split}$

which is to say

$L_{p}+Q=(1-S_{p+1})(L_{p+1}+Q)+r^{p+1}[M+\frac{r-1}{r}b_{p+1}]S_{p+1}$

Then the F.O.C implies

$\frac{1}{r}[\frac{r-1}{r}\beta(b_{p})+M]=(1-S_{p+1})[\frac{r-1}{r}\beta(b_{p+1})+M]+[\frac{r-1}{r}b_{p+1}+M]S_{p+1}$

which is equivalent to

$\beta(b_{p})=r\psi(b_{p+1})$ (11)

And for $p=T-1$ we can still expand $a_{T-1}$. We are trying to maximize
$a_{T-1}[r^{T-1}(M+\frac{r-1}{r}b_{T-1})-Mr^{T}]$, which is equivalent to

$\max_{b_{T-1}}H(b_{T-1})[(1-r)M+\frac{r-1}{r}b_{T-1}]$

hence the FOC for $b_{T-1}$ is like

$b_{T-1}+\frac{H(b_{T-1})}{h(b_{T-1})}=rM$

since $b_{T}=0,\text{ and }\psi(0)=M$, we can rewrite it as
$\beta(b_{T-1})=r\psi(b_{T})$. Hence equation (11) holds for $\forall p\leq
T-1$ And SOC holds too. Combining the above two equations we can finish the
proof of this recursive structure for $\forall p\leq T-1$. We need the
following facts to guarantee the sequence $\\{b_{t}\\}$ is monotone:

######  Fact A.1.

$\beta(x)-\frac{H^{2}(x)}{h(x)}$ is monotone increasing

###### Proof.

Its first order derivation is
$\frac{[1-H(x)][2h^{2}(x)-H(x)h^{\prime}(x)]}{h^{2}(x)}>0$ ( from Assumption
3.2) for any $x\in[0,\bar{c})$. ∎

######  Fact A.2.

$(\frac{1}{r}-1)\beta(x)+\frac{H^{2}(x)}{h(x)}$ is monotone increasing

###### Proof.

Its first order derivation is
$[\frac{1}{r}-1+H(x)]\frac{2h^{2}(x)-H(x)h^{\prime}(x)}{h^{2}(x)}>0$ ( from
Assumption 3.2, and $r<1$) for any $x\in[0,\bar{c})$. ∎

######  Fact A.3.

$b_{p+1}<b_{p}<b^{*}$ given $0<b_{p+1}<b^{*}$.

###### Proof.

$\beta(b_{p})-\beta(b_{p+1})=r[M-\frac{H^{2}(b_{p+1})}{h(b_{p+1})}+\beta(b_{p+1})]-\beta(b_{p+1})>r[M-\frac{H^{2}(b^{*})}{h(b^{*})}+\beta(b^{*})]-\beta(b^{*})=0$
hence $b_{p}>b_{p+1}$.
On the other hand
$\frac{\beta(b_{p})}{r}=M-\frac{H^{2}(b_{p+1})}{h(b_{p+1})}+\beta(b_{p+1})<M-\frac{H^{2}(b^{*})}{h(b^{*})}+\beta(b^{*})=\frac{\beta(b^{*})}{r}$
hence $b_{p}<b^{*}$ ∎

Then we finish the proof of Lemma 4.2. ∎

###  A.2 Proof of Proposition 4.3

###### Proof.

Given Lemma 4.2, we only need to take care of the existence of $b_{p}$ and
whether it’s valid (i.e. lies in interval $[0,b^{*}]$). The following fact is
enough to finish our proof for the finite case.

######  Fact A.4.

$b_{T-1}<b^{*}$ given $b_{T}=0$.

###### Proof.

Fact A.2 guarantees $\beta(x)-\frac{H^{2}(x)}{h(x)}$ is monotone increasing.
Then we know:
$\frac{\beta(b_{T-1})}{r}=M+\beta(0)-\frac{H^{2}(0)}{h(0)}=M=\frac{H^{2}(b^{*})}{h(b^{*})}+(\frac{1}{r}-1)\beta(b^{*})<\beta(b^{*})+(\frac{1}{r}-1)\beta(b^{*})=\frac{\beta(b^{*})}{r}$
, hence $b_{T-1}=\Gamma(0)<b^{*}$ ∎

Hence starting from $b_{T}=0$ we get the $b_{t}$ sequence and it’s
monotonically increasing with respect to $t$, i.e.
$b^{*}>b_{1}>b_{2}>\cdots>b_{T-1}>b_{T}=0$. So if $T=1$ we have $b_{T}=0$
hence it’s actually a Non-Bonus Strategy. Otherwise we have $b_{1}>0$ hence
it’s a Full Coverage Strategy. Up to now we have provided the proof for finite
part of Proposition 4.3.

When $T=\infty$, actually Lemma 4.2 still works hence we know the optimal
$b^{*}$ is the solution for equation $\frac{\beta(x)}{r}=\psi(x)$. However a
more safe way is to use the value function like

$V=\max_{x}H(x)[\mathbb{E}R_{1}-x+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r}{1-r}]+(1-H(x))[\mathbb{E}R_{1}+rV]$

hence F.O.C. is

$\beta(x)=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r}{1-r}-rV$

Using F.O.C to replace $V$ in goal function we get

$[1-r+rH(x)][\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r}{1-r}-\beta(x)]=r\mathbb{E}R_{1}+rH(x)[\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r}{1-r}-x]$

hence for the solution $b^{*}=x$ we have:

$\beta(x)=r\psi(x)$

and

$V=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{1}{1-r}-\psi(b^{*})$

As before we can guarantee that $b^{*}$ lies in the interval $(0,\bar{c})$
hence when $T=\infty$, the optimal Full Coverage (including Non Bonus)
strategy is to set $b_{i}\equiv b^{*}$. ∎

###  A.3 Proof of Partial Coverage: Proposition 4.4

####  A.3.1 Proof of Proposition 4.4

###### Proof.

If the platform uses partial coverage strategy and the reporting happens at
period $t_{0}\leq T$, then the profit is:

$R_{2}(1+r+..+r^{t_{0}-2})+\mathbb{E}R_{1}r^{t_{0}-1}+\mathbb{E}\max\\{R_{1},R_{2}\\}(r^{t_{0}}+..+r^{T-1})-\mathbb{E}cost$

with the same notation $a_{i}\forall 1\leq i\leq T$ defined in the proof of
Lemma 4.2, we have profit=

$\displaystyle\sum\limits_{i=1}^{T}Pr(t_{0}=i)[R_{2}\frac{1-r^{t_{0}}}{1-r}+(\mathbb{E}R_{1}-R_{2})r^{t_{0}-1}+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{t_{0}}-r^{T}}{1-r}-b_{t_{0}}r^{t_{0}-1}]$
$\displaystyle+Pr(\text{no reports})R_{2}\frac{1-r^{T}}{1-r}$
$\displaystyle=\sum\limits_{i=1}^{T-1}Pr(t_{0}=i)[R_{2}\frac{1-r^{t_{0}}}{1-r}+(\mathbb{E}R_{1}-R_{2})r^{t_{0}-1}+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{t_{0}}-r^{T}}{1-r}-b_{t_{0}}r^{t_{0}-1}]$
$\displaystyle+a_{T}R_{2}\frac{1-r^{T}}{1-r}+Pr(t_{0}=T)((\mathbb{E}R_{1}-R_{2})r^{T-1}-b_{T}r^{T-1})$
$\displaystyle=\sum\limits_{i=1}^{T-1}a_{i}[\frac{R_{2}-\mathbb{E}\max\\{R_{1},R_{2}\\}r^{T}}{1-r}+\frac{(\frac{(\mathbb{E}R_{1}-R_{2})(1-r)}{r}+\mathbb{E}\max\\{R_{1},R_{2}\\}-R_{2})r^{i}}{1-r}-b_{i}r^{i-1}]$
$\displaystyle+a_{T}R_{2}\frac{1-r^{T}}{1-r}+Pr(t_{0}=T)((\mathbb{E}R_{1}-R_{2})r^{T-1}-b_{T}r^{T-1})$
$\displaystyle=(1-a_{T})\frac{R_{2}-\mathbb{E}\max\\{R_{1},R_{2}\\}r^{T}}{1-r}+\sum\limits_{i=1}^{T-1}a_{i}r^{i}\frac{N-b_{i}(1-r)/r}{1-r}+$
$\displaystyle+a_{T}R_{2}\frac{1-r^{T}}{1-r}+Pr(t_{0}=T)((\mathbb{E}R_{1}-R_{2})r^{T-1}-b_{T}r^{T-1})$
$\displaystyle=\sum\limits_{i=1}^{T-1}a_{i}r^{i}\frac{N-b_{i}(1-r)/r}{1-r}+Pr(t_{0}=T)((\mathbb{E}R_{1}-R_{2})r^{T-1}-b_{T}r^{T-1})+$
$\displaystyle
a_{T}\frac{(\mathbb{E}\max\\{R_{1},R_{2}\\}-R_{2})r^{T}}{1-r}+constant$

Where we denote $N=M+\frac{\mathbb{E}R_{1}-R_{2}}{r}$. Notice
$a_{T}=1-\sum\limits_{i=1}^{T-1}a_{i}$, the above problem is equivalent to:

$\max\limits_{b_{T}}H(b_{T})(\mathbb{E}R_{1}-R_{2}-b_{T})$

and

$\max\limits_{b}\sum\limits_{i=1}^{T-1}a_{i}\\{r^{i}[N+\frac{r-1}{r}b_{i}]-W\\}$

where $W=(\mathbb{E}\max\\{R_{1},R_{2}\\}-R_{2})r^{T}$.

We first solve the optimal $b_{T}$. We should notice that unlike $M$ is always
positive, $N$ may be negative. Unlike the case in full coverage, it’s possible
that $b_{T}$ is positive or negative. The optimal $b_{T}$ satisfies F.O.C:

$\beta(b_{T})=ER_{1}-R_{2}$

And S.O.C. satisfies according to assumption 3.2 This result shows that when
$R_{2}$ is very large, the optimal $b_{T}$ may be negative, and the consumers
would choose arm 2 to get utility 0 instead of a negative utility $-\epsilon$
by purchasing arm1 since he will never report to get a negative bonus and pay
a reporting cost.

As for the $b_{t},\forall t\leq T-1$, we claim the recursive relation for
interior solution using such a lemma:

######  Lemma A.5.

Suppose that there exists $1\leq p<T$ such that the corresponding optimal
bonus scheme satisfies $0<b_{p}<\bar{c}$. Then we have the following recursive
relationship:

$\frac{\beta(b_{p})}{r}=\psi_{N}(b_{p+1}),$ (12)

where $N=M+\frac{\mathbb{E}R_{1}-R_{2}}{r}$ and
$M=\mathbb{E}\max\\{R_{1},R_{2}\\}-\mathbb{E}R_{1}$ and $\psi_{N}(\cdot)$ as
denoted ahead.

###### Proof.

Like the proof of Lemma 4.2 before, for $\forall p\leq T-1$, we have:

$\sum\limits_{i=1}^{T-1}a_{i}\\{r^{i}[N+\frac{r-1}{r}b_{i}]-W\\}=\sum\limits_{i=1}^{p-1}a_{i}\\{r^{i}[N+\frac{r-1}{r}b_{i}]-W\\}+\prod\limits_{j=1}^{p-1}(1-S_{j})[S_{p}\\{r^{p}[N+\frac{r-1}{r}b_{i}]-W\\}+(1-S_{p})L_{p}]$

where

$L_{p}=\sum\limits_{i=p+1}^{T-1}\\{r^{i}[N+\frac{r-1}{r}b_{i}]-W\\}S_{i}\prod\limits_{k=p+1}^{i-1}(1-S_{j})$

Notice it’s only valid for $p\leq T-2$. F.O.C requires the following equation
must hold for the optimal bonus structure ($L_{p}$ is irrelevant with
$b_{p}$):

$h(b_{p})[r^{p}(N+\frac{r-1}{r}b_{p})-W]+H(b_{p})r^{p}\frac{r-1}{r}-h(b_{p})L_{p}=0$

which can be simplified as:

$\frac{r-1}{r}\beta(b_{p})+N=\frac{L_{p}+W}{r^{p}}$

We can check the second order condition, the left hand side should be

$[r^{p}(N+\frac{r-1}{r}b_{p})-L_{p}-W]h^{\prime}(b_{p})+2h(b_{p})r^{p}\frac{r-1}{r}$

and if it’s minimum solution we want it to be positive. Combine with the FOC
we know it’s equal to

$[H(b_{p})r^{p}\frac{r-1}{r}]h^{\prime}(b_{p})+2h^{2}(b_{p})r^{p}\frac{r-1}{r}$

By Assumption 3.2 we know the fact that $\beta(x)$ is increasing implies
$2h^{2}(x)-H(x)h^{\prime}(x)>0$ for sure. Hence SOC is satisfied.

Moreover, from the definition of $L_{p}$, we know that for $p\leq T-2$:

$\begin{split}L_{p}=\sum\limits_{i=p+1}^{T-1}\\{r^{i}[N+\frac{r-1}{r}b_{i}]-W\\}S_{i}\prod\limits_{k=p+1}^{i-1}(1-S_{j})\\\
=(1-S_{p+1})\sum\limits_{i=p+2}^{T-1}\\{r^{i}[N+\frac{r-1}{r}b_{i}]-W\\}\prod\limits_{k=p+2}^{i-1}(1-S_{j})+\\{r^{p+1}[N+\frac{r-1}{r}b_{p+1}]-W\\}S_{p+1}\\\
=(1-S_{p+1})L_{p+1}+\\{r^{p+1}[N+\frac{r-1}{r}b_{p+1}]-W\\}S_{p+1}\\\
\end{split}$

which is to say

$L_{p}+W=(1-S_{p+1})(L_{p+1}+W)+r^{p+1}[N+\frac{r-1}{r}b_{p+1}]S_{p+1}$

hence

$\frac{1}{r}[\frac{r-1}{r}\beta(b_{p})+N]=(1-S_{p+1})[\frac{r-1}{r}\beta(b_{p+1})+N]+[\frac{r-1}{r}b_{p+1}+N]S_{p+1}$

which is equivalent to

$\beta(b_{p})=r\psi_{N}(b_{p+1})$

∎

However, unlike the case in Full Coverage, where we have the recursive
structure and guarantee the interior solutions are valid hence the optimal
structure is clear. In Partial Coverage case, having Lemma A.5 is not enough
to get the final PC (containing SA) solutions. The reason is there may be
negative interior bonus. But given $b_{p+1}<0$, which we should treat as
$b_{p+1}=0$, $b_{p}$ will solve the F.O.C. for $b_{p+1}=0$ hence breaks the
recursive rule.

Then we discuss the optimal solution given above interior recursive structure.
Unlike Full Coverage case, we can’t guarantee $b_{t}\geq 0$ here. Hence, for
simplicity, we use sequence $\\{d_{t}\\}$ for the interior solution, allowing
distribution functions have similar functional form outside the support, and
$\\{b_{t}\\}$ for the final results, taking validity (i.e. valid $b$ should be
positive) into consideration. we should notice the following facts:

######  Fact A.6.

If $N>0$, then equation

$(\frac{1}{r}-1)\beta(x)+\frac{H^{2}(x)}{h(x)}=N$

has solution $b_{N}^{*}>0$

If $\mathbb{E}R_{1}-R_{2}>0$, the equation

$\beta(x)=\mathbb{E}R_{1}-R_{2}$

have solution $b_{T}>0$. Otherwise denote $b_{T}=0$.

###### Proof.

We have shown in Fact A.2 that the left hand side is monotonically increasing
with respect to $x$ and gets 0 when $x=0$. So given a positive $N$, we get
$d_{N}^{*}>0$. Otherwise we can not find a valid $d_{N}^{*}>0$ (However, in
numerical solutions we can still solve a negative one if we let
$\beta(x)=2x,\forall x$).

Similarly, $b_{T}$ solves

$\beta(x)=\mathbb{E}R_{1}-R_{2}$

and the same logic works for $\beta(\cdot)$. ∎

######  Fact A.7.

There are three contingencies for the sequence $\\{b_{t}\\}$. An numerical
illustration for the second and the third case are shown in Figure A2 and
Figure A2, if we simply let $H(x)=x(\text{ instead of }x\mathbb{I}[0,1])$ and
$h(x)=1(\text{ instead of }1\mathbb{I}[0,1])$, i.e. ignore the fact that
$h(x)$ and $H(x)$ is not defined outside the interval [0,1].

  * •

If $N>0$ and $d_{T}>0$, we have
$d_{N}^{*}>d_{1}>d_{2}>d_{3}>...>d_{T-1}>d_{T}>0$. Hence in this case
$b_{t}=d_{t}$.

  * •

If $N>0$ and $d_{T}<0$, we have
$d_{N}^{*}>d_{1}>d_{2}>d_{3}>...>d_{T-1}>0>d_{T}$. Hence in this case
$b_{t}=d_{t}$ for $t\leq T-1$ and $b_{T}=0$.

  * •

If $N<0$, we have $0>d_{N}^{*}>d_{1}>d_{2}>d_{3}>...>d_{T-1}>d_{T}$. Hence in
this case $0=b_{N}^{*}=b_{1}=b_{2}=b_{3}=...=b_{T-1}=b_{T}$.

###### Proof.

From the definition of $b_{N}^{*}$ we know
$(\frac{1}{r}-1)\beta(b_{N}^{*})+\frac{H^{2}(b_{N}^{*})}{h(b_{N}^{*})}=N=M+\frac{\mathbb{E}R_{1}-R_{2}}{r}$,
and
$(\frac{1}{r}-1)\beta(b_{T})+\frac{H^{2}(b_{T})}{h(b_{T})}=(\mathbb{E}R_{1}-R_{2})(\frac{1}{r}-1)+\frac{H^{2}(b_{T})}{h(b_{T})}=\frac{H^{2}(b_{T})}{h(b_{T})}+R_{2}-\mathbb{E}R_{1}+\frac{\mathbb{E}R_{1}-R_{2}}{r}$.

In the first case, if $b_{T}>0$, we have
$\beta(b_{T})-\frac{H^{2}(b_{T})}{h(b_{T})}>\beta(0)-\frac{H^{2}(0)}{h(0)}=0$,
hence
$\frac{H^{2}(b_{T})}{h(b_{T})}+R_{2}-\mathbb{E}R_{1}+\frac{\mathbb{E}R_{1}-R_{2}}{r}<\beta\left(b_{T}\right)+R_{2}-\mathbb{E}R_{1}+\frac{\mathbb{E}R_{1}-R_{2}}{r}=0+\frac{\mathbb{E}R_{1}-R_{2}}{r}<M+\frac{\mathbb{E}R_{1}-R_{2}}{r}=N$,
hence $b_{T}<b_{N}^{*}$. Then with the same logic above, this inequality can
be iterated, hence we finish the proof of the first part here.

When $b_{T}<0$, the direction of inequality changes. Actually above function
is not defined when $b_{T}<0$. However, with our basic definition,
$\psi_{N}(x)=\beta(x)-\frac{H^{2}(x)}{h(x)}+N$, and when $x<0$ it degenerates
into simply $N$. Hence in the first step we have $\beta(b_{T-1})=rN$, which
will lead to a positive $b_{T-1}$. In this case, we have
$(\frac{1}{r}-1)\beta(b_{N}^{*})+\frac{H^{2}(b_{N}^{*})}{h(b_{N}^{*})}=N=M+\frac{\mathbb{E}R_{1}-R_{2}}{r}$,
and
$(\frac{1}{r}-1)\beta(b_{T-1})+\frac{H^{2}(b_{T-1})}{h(b_{T-1})}=rN(\frac{1}{r}-1)+\frac{H^{2}(b_{T-1})}{h(b_{T-1})}=\frac{H^{2}(b_{T-1})}{h(b_{T-1})}+rN(\frac{1}{r}-1)$.
Since $b_{T-1}>0$, we have
$\beta(b_{T-1})-\frac{H^{2}(b_{T-1})}{h(b_{T-1})}>\beta(0)-\frac{H^{2}(0)}{h(0)}=0$,
hence
$\frac{H^{2}(b_{T-1})}{h(b_{T-1})}+rN(\frac{1}{r}-1)<\frac{H(b_{T-1})}{h(b_{T-1})}+b_{T-1}+rN(\frac{1}{r}-1)=0+N$,
hence $b_{T-1}<b_{N}^{*}$. And with the same logic in the first part, we have
$b_{N}^{*}>b_{1}>b_{2}>\cdots>b_{T-1}>0$.

Moreover, we should notice that when $x<0$, $\beta(x)$ degenerates to
$\beta(x)=x$ and $\psi_{N}(x)$ degenerates to $\psi_{N}(x)=x+N$. Hence given
$N<0$, $\beta(b_{T-1})=\psi_{N}(b_{T})=b_{T}+N<0$ so we should set
$b_{T-1}=0$. The same logic guarantees $b_{i}=0,\forall i$. ∎

Figure A1: $b_{T}<0$ and $N>0$

Figure A2: $b_{T}<0$ and $N<0$

Last but not least, we should point out that the second part in Assumption 4.1
actually rules IR out in PC cases, since the same logic works in the proof of
Lemma 4.2. Lemma A.5, combined with Fact A.6 and A.7 finishes proof of finite
part in Proposition 4.4.

When $T\to\infty$, we again use the value function way.

$V=\max_{x}H(x)[\mathbb{E}R_{1}-x+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r}{1-r}]+(1-H(x))[R_{2}+rV]$

hence F.O.C. is

$\beta(x)=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r}{1-r}-rV+\mathbb{E}R_{1}-R_{2}$

Using F.O.C to replace $V$ in goal function we get

$r\mathbb{E}\max\\{R_{1},R_{2}\\}+(\mathbb{E}R_{1}-R_{2}-\beta(x))(1-r)-r\frac{H^{2}(x)}{h(x)}=rR_{2}$

hence

$\beta(b_{N}^{*})=r\psi_{N}(b_{N}^{*})$

and

$V=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{1}{1-r}-\psi_{N}(b_{N}^{*})-\frac{\mathbb{E}R_{1}-R_{2}}{r}$

Then from Fact A.6 we know exists $b_{N}^{*}>0\Leftrightarrow N>0$, and
Assumption 4.1 guarantees that $b_{N}^{*}<\bar{c}$. Hence we should use
Partial Coverage when $N>0$ and otherwise Safe Arm. ∎

###  A.4 Proof of Lemma 4.5(total utility)

###### Proof.

If $T=\infty$, we have got the expression of its utility in the proof of
proposition 4.3 and proposition 4.4.

When $T$ is finite, we denote $\eta_{p}$ to be the total utility of bonus
structure

$(0,0,...0,\Gamma^{(p)}(b_{T}),\Gamma^{(p-1)}(b_{T}),..\Gamma^{(1)}(b_{T}),b_{T}).$

Hence obviously we have the following equation by denoting
$x=\Gamma^{(p)}(b_{T})$:

$\begin{split}\eta_{p}=\mathbb{E}R_{1}(1+r+..r^{T-p-1})-H(x)xr^{T-p-1}+H(x)\mathbb{E}\max\\{R_{1},R_{2}\\}(r^{T-p}+r^{T-p+1}+..r^{T-1})\\\
+[1-H(x)][\eta_{p-1}-\mathbb{E}R_{1}(1+r+..r^{T-p-1})]\\\
=H(x)\mathbb{E}R_{1}\frac{r^{T-p}-1}{r-1}-H(x)xr^{T-p-1}+H(x)\mathbb{E}\max\\{R_{1},R_{2}\\}r^{T-p}\frac{r^{p}-1}{r-1}+(1-H(x))\eta_{p-1}\end{split}$

We have known $x=\Gamma^{(p)}(b_{T})$ maximizes $\eta_{p}$ given $\eta_{p-1}$
fixed. Hence F.O.C. must hold for $x=\Gamma^{(p)}(b_{T})$, thus we can get the
expression of $\eta_{p-1}$ (we just use the it as a necessary condition here
so we don’t need to check S.O.C., which has been satisfied before):

$\eta_{p-1}=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{T}-r^{T-p}}{r-1}+\mathbb{E}R_{1}\frac{r^{T-p}-1}{r-1}-r^{T-p-1}\beta(x)$

hence

$\eta_{p-1}=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{T}-r^{T-p}}{r-1}+\mathbb{E}R_{1}\frac{r^{T-p}-1}{r-1}-r^{T-p}\psi(\Gamma^{(p-1)}(0))$

Hence
$\eta_{T-1}=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{T}-1}{r-1}-\psi(b_{1})$.
Similarly, if the platform uses Partial Coverage Strategy, we have

$\begin{split}\eta_{p}=H(x)[R_{2}(1+r+..r^{T-p-2})+\mathbb{E}R_{1}r^{T-p-1}-xr^{T-p-1}+\mathbb{E}\max\\{R_{1},R_{2}\\}(r^{T-p}+r^{T-p+1}+..r^{T-1})]\\\
+[1-H(x)]\eta_{p-1}\\\
=H(x)[R_{2}\frac{r^{T-p-1}-1}{r-1}+R_{1}r^{T-p-1}-xr^{T-p-1}+\mathbb{E}\max\frac{r^{T}-r^{T-p}}{r-1}]+[1-H(x)]\eta_{p-1}\end{split}$
(13)

From above analysis we know the platform chooses the optimal $x$ to maximize
$\eta_{p}$ given $\eta_{p-1}$, hence the F.O.C must hold at $x$ :

$\eta_{p-1}=-\beta(x)r^{T-p-1}+R_{2}\frac{r^{T-p-1}-1}{r-1}+\mathbb{E}R_{1}r^{T-p-1}+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{T}-r^{T-p}}{r-1}$

This equation should hold for $\forall 1\leq p\leq T-1$, hence let $p=T-1$,
$\eta_{T-2}=-\beta(x)+\mathbb{E}R_{1}+\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{T}-r}{r-1}$
where $x=b_{N_{1}}$. Plug this result in to the recursive equation (13) at
$p=T-1$, we can get the final $\eta_{T-1}$:

$\eta_{T-1}=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{r^{T}-1}{r-1}-\psi(b_{N_{1}})$
(14)

where $b_{N_{1}}=\Gamma_{N}^{(T-1)}(b_{T})$. Notice it’s $\psi(b_{N_{1}})$
instead of $\psi_{N}(b_{N_{1}})$. To ensure, consider the case when $T=1$.
F.O.C. is $\beta(b_{T})=ER_{1}-R_{2}$, and the value function is

$\displaystyle
H(b_{T})(\mathbb{E}R_{1}-b_{T})+(1-H(b_{T}))R_{2}=H(b_{T})[\beta(b_{T})-b_{T}]+R_{2}$
$\displaystyle=H(b_{T})\frac{H(b_{T})}{h(b_{T})}+R_{2}-\mathbb{E}R_{1}+\mathbb{E}R_{1}$
$\displaystyle=H(b_{T})\frac{H(b_{T})}{h(b_{T})}-\beta(b_{T})+\mathbb{E}R_{1}+\mathbb{E}\max\\{R_{1},R_{2}\\}-\mathbb{E}\max\\{R_{1},R_{2}\\}$
$\displaystyle=\mathbb{E}\max\\{R_{1},R_{2}\\}-\psi(b_{T})$

Here is an example to show that this result is robust when sequence
$\\{b_{t}\\}$ contains negative items (Actually the negative item also fits
equation (13), when $b_{p}<0$, $\eta_{p}=\eta_{p+1}$). For example, only
$b_{1}>0$ and $b_{2},b_{3},..b_{T}=ER_{1}-R_{2}<0$. The recursive relation for
$t\geq 2$ is
$\frac{\beta(b_{p})}{r}=\psi_{N}(b_{p+1})\Rightarrow\frac{b_{p}}{r}=N+b_{p+1}$.
Hence $b_{t}+\frac{rN}{r-1}=r^{T-t}(b_{T}+\frac{rN}{r-1})$. The above result
shows
$\eta_{T-2}=-\beta(\Gamma^{(T-1)}(b_{T}))+R_{1}+Emax\\{R_{1},R_{2}\\}\frac{r^{T}-r}{r-1}$.
Combine these two results together we can get
$\eta_{T-2}=\frac{r^{T}-1}{r-1}R_{2}$, which fits the direct calculation given
structure $b_{2},b_{3},..b_{T}=ER_{1}-R_{2}<0$. ∎

###  A.5 Numeric Examples in Section 4

####  A.5.1 Details for Figure 1

We consider $R_{1}\sim U[0,4],c\sim U[0,1],T=2$. Hence
$M=Emax\\{R_{1},R_{2}\\}-ER_{1}=\frac{R_{2}^{2}}{8}$. In RFC, given $b_{2}=0$,
we know $\beta(b_{1})=2b_{1}=r\psi_{M}(b_{2})=rM$ so
$b_{1}=\frac{rM}{2}=\frac{rR_{2}^{2}}{16}$. The root of $b_{1}=\bar{c}=1$ is
$\frac{4}{\sqrt{r}}$, which is just the $\bar{R}_{2}$. In this case
$\bar{R}_{2}>\bar{c}$ hence IRM should not be optimal, but we can make it real
by choosing a large $T$. In RPC, if $ER_{1}-R_{2}>0,b_{2}>0$,
$\beta(b_{2})=2b_{2}=ER_{1}-R_{2}$, hence
$\beta(b_{1N})=2b_{1}=\psi_{N}(b_{2})$ which will lead to
$b_{1N}=\frac{r}{2}[ER_{1}-R_{2}+N-\frac{(ER_{1}-R_{2})^{2}}{4}]=\frac{r}{2}+1-\frac{R_{2}}{2}-\frac{rR_{2}^{2}}{16}$.
The root of $b_{1N}=\bar{c}=1$ is our $\underline{R}_{2}$. Hence
$b_{1}-b_{1N}=(R_{2}-2)(\frac{r}{8}R_{2}+\frac{2+r}{4})$. We predict before
that $\hat{R}_{2}=ER_{1}$ and we can see it’s true here.

$N=\frac{R_{2}^{2}}{8}-\frac{2-R_{2}}{r}$ hence the root of which is
$\frac{4(1\pm\sqrt{1-r})}{r}$. The smaller one, $\frac{4(1-\sqrt{1-r})}{r}<4$
is $\tilde{R}_{2}$ as we analyzed before.

####  A.5.2 Details for Figure 4

To be more specific, if $x<3$ then $\rho x+(1-\rho)y<3\rho+3(1-\rho)=3$.
Otherwise for any $x\in(3,4)$, we have $\rho x+(1-\rho)y>3$ requires
$y>\frac{3-\rho x}{1-\rho}$. Then there are two cases. If $\frac{3-\rho
x}{1-\rho}>1\forall x\in[3,4]$ (i.e. $\rho<\frac{2}{3}$), then we know

$8\mathbb{E}\max\\{R_{1},R_{2}\\}=\int_{0}^{3}\int_{1}^{3}3dydx+\int_{3}^{4}\int_{1}^{\frac{3-\rho
x}{1-\rho}}3dydx+\int_{3}^{4}\int_{\frac{3-\rho x}{1-\rho}}^{3}\rho
x+(1-\rho)ydydx$

which is equal to

$18+\frac{12-15\rho}{2(1-\rho)}+\frac{\rho(\rho+9)}{6(1-\rho)}$

whose differential with respect to $\rho$ is equal to

$\frac{(2-\rho)\rho}{6(1-\rho)^{2}}>0$

In the other case, we have

$\displaystyle
8\mathbb{E}\max\\{R_{1},R_{2}\\}=\int_{0}^{3}\int_{1}^{3}3dydx+\int_{3}^{\frac{2+\rho}{\rho}}\int_{1}^{\frac{3-\rho
x}{1-\rho}}3dydx+$
$\displaystyle\int_{3}^{\frac{2+\rho}{\rho}}\int_{\frac{3-\rho
x}{1-\rho}}^{3}\rho
x+(1-\rho)ydydx+\int_{\frac{2+\rho}{\rho}}^{4}\int_{1}^{3}\rho
x+(1-\rho)ydydx$

which is equal to

$\frac{58}{3}+\frac{13\rho}{3}+\frac{4}{3\rho}$

Again, it’s differential for $\rho$ is equal to

$\frac{13}{3}-\frac{4}{3\rho^{2}}>0$

We should notice that these two expressions meet at $\rho=\frac{2}{3}$
smoothly.

####  A.5.3 To Provide a Comparison with Figure 5

We assume $R_{1}\sim U[0,X]$ with $X\geq R_{2}=2$. In this case we have

$\mathbb{E}\max\\{R_{1},R_{2}\\}=\int_{0}^{2}2\frac{1}{X}dx+\int_{2}^{X}x\frac{1}{X}dx=\frac{X^{2}+4}{2X}$

Using the technical results discussed before we can get such a illustration in
figure A3. The following figure carries the same message as shown in figure 3
(infinite case) or figure 1 (finite case), but in a opposite direction: there
exists two cutoffs which stands for the optimal switching between PC and FC
(when $R_{2}=\mathbb{E}R_{1}$, and between PC and SA (when $N=0$). Again, $NB$
is never optimal. Figure A3 is only for comparison with figure 6.

We emphasize here that $NB$ is never optimal in perfect learning cases, and we
will show that in imperfect learning cases, the region where NB is optimal
will shrink when $\lambda\to\infty$ (i.e. when learning tends to be perfect).
An illustration can be found in Figure 9 .

Figure A3: Profit-X When $T=\infty$

##  Appendix B Appendix B: When Assumption4.1 or 5.2 Fails

###  B.1 Discrete Immediate Revelation

In this subsection we consider the case when Assumption 4.1 is not satisfied,
i.e. there is possibility that $d_{1}$ or $d_{N,1}$ is higher than $\bar{c}$
and hence the interior solution leads to the corner case. We should first
notice that Assumption 4.1 requires a large enough $\bar{c}$ to rule out IR,
and it’s actually equivalent to $b^{*}<\bar{c}$. A modified version of
Proposition 4.3 and 4.4 should be as follows:

######  Proposition B.1.

For $\forall T\geq 2$, construct the sequence $\\{d_{t}\\}$ as follows:

  1. 1.

if $T<\infty$, let $d_{t}=\Gamma^{(T-t)}(0)$ for $1\leq t\leq T$.

  2. 2.

if $T=\infty$, let $d_{t}=b^{*}$ for $1\leq t\leq T$ ($b^{*}$ is the fixed
point of function $\Gamma(\cdot)$ as declared in section 4.1).

Then $d_{1}>0$ if $T>1$. Under our Assumption 3.1 and Assumption 3.2, the
optimal full coverage strategy is to set:

  1. 1.

$(b_{1},b_{2},...b_{T})=(d_{1},d_{2},...d_{T})$, if $0<d_{1}<\bar{c}$ (i.e.
using recursive structure).

  2. 2.

$(b_{1}=\bar{c},b_{2}=0,b_{3}=0,..b_{T}=0)$, if $d_{1}>\bar{c}$ (i.e., using
immediate revelation) .

######  Proposition B.2.

For $\forall T\geq 2$, construct the sequence $\\{d_{t}\\}$ as follows:

  1. 1.

let $d_{T}$ to be the root of equation $\beta(d_{T})=\mathbb{E}R_{1}-R_{2}$

  2. 2.

if $T<\infty$, let $d_{t}=\Gamma_{N}^{(T-t)}(d_{T})$ for $1\leq t\leq T$.

  3. 3.

if $T=\infty$, let $d_{t}=b_{N}^{*}$ for $1\leq t\leq T$ ($b_{N}^{*}$ is the
fixed point of function $\Gamma_{N}(\cdot)$ as declared in section 4.1).

Under our Assumption 3.1, Assumption 3.2 and Assumption 4.1, the optimal full
coverage strategy is to set:

  1. 1.

$(b_{N1},b_{N2},...b_{NT})=(d_{1},d_{2},...d_{T})$, if $0<d_{1}<\bar{c}$ (i.e.
using recursive structure).

  2. 2.

$(b_{N1}=\bar{c},b_{N2}=0,b_{N3}=0,..b_{NT}=0)$, if $d_{1}>\bar{c}$ (i.e.,
using immediate revelation) .

  3. 3.

$(b_{N1}=0,b_{N2}=0,b_{N3}=0,..b_{NT}=0)$ if $d_{1}<0$ (i.e., setting non-
bonus).

Then back to the optimal strategy, by setting IR the platform can get profit
$\pi_{L}=\mathbb{E}\max\\{R_{1},R_{2}\\}\frac{1-r^{T}}{1-r}-M-\bar{c}\text{ if
}L=\text{Immediate Revelation}\\\ $ Combine this results with Lemma 4.5, we
have a more complex comparative relationships like Proposition 4.6:

######  Proposition B.3.

The comparison relationships between these four possible optimal strategies
are as follows:

$PC\prec FC\Leftrightarrow R_{2}<\bar{R}_{2}$

$FC\prec IR\Leftrightarrow R_{2}<\tilde{R}_{2}$

$PC\prec IR\Leftrightarrow R_{2}>\tilde{R}^{\prime}_{2}$

$PC\prec SA\Leftrightarrow R_{2}>\hat{R}_{2}$

An illustration of Proposition B.3 can be found in Figure B6 where we set
$\bar{c}=0.1$ and $T=3$. We can still guarantee that $\bar{R}_{2}<\hat{R}_{2}$
but it’s not sure where the $\tilde{R}_{2}$ or $\tilde{R}^{\prime}_{2}$ lies.
In other words, IR may be or may not be the optimal solution. Besides what we
saw in figure 1 that FC tends to NB when $R_{2}$ is small, we can see FC tends
to IR when $R_{2}$ is large, and PC tends to IR when $R_{2}$ is small.

Then we try to reveal some intuition by comparing IR vs FC. Intuitively we
know there are four incentives for the platform to use the highest bonus at
the beginning (IR) :

  1. 1.

when $T$ is large (many agents)

  2. 2.

when $M=\mathbb{E}\max\\{R_{1},R_{2}\\}-\mathbb{E}R_{1}$ is large (valuable
information)

  3. 3.

when $\bar{c}$ is small (cheap process of information)

  4. 4.

when $r$ is large (future value matters)

Above are four straightforward conditions which help $b_{1}>\bar{c}$ to hold,
and our result fits these four insights well:

Firstly since $b_{1}=\Gamma^{(T-1)}(0)$, whether $b_{1}$ exceeds a fixed value
$\bar{c}$ is actually a symbol of the size of $T$. We know the iteration of
function $\Gamma$ is increasing from our Fact A.3. Moreover, it’s convergent
from our proof of Lemma 4.2. Hence when $T$ is large enough, $b_{1}$ can get
arbitrarily close to $b^{*}$, which will make it get over $\bar{c}$ as long as
$b^{*}>\bar{c}$. The platform hence should use the upper bound bonus,
$\bar{c}$, to get the information immediately.

Points 2 and 3 are easy to understand. Holding everything else constant, a
higher $M$ will lead to a higher $b_{1}$, hence makes $b_{1}<\bar{c}$ less
important to hold hence provides more incentives to use “Immediate revelation”
strategy. A smaller $\bar{c}$ directly makes $b_{1}<\bar{c}$ less important to
hold (see figure B5 and B5 where $T=8,r=0.95,\mathbb{E}R_{1}=2$ in both cases,
a lower $\bar{c}$ makes IR possible to be optimal).

Figure B4: $\bar{c}=0.4,T=8$

Figure B5: $\bar{c}=1,T=8$

Figure B6: Profit-$R_{2}$ Under Different Strategies with IR

####  B.1.1 Proof of Proposition B.1 and B.2

###### Proof.

######  Fact B.4.

For any $\\{b_{i}\\}$ and $x>0$, setting $b_{p}=x>0$ and $b_{p-1}=0$ is always
(strictly) dominated by setting $b_{p}=0$, $b_{p-1}=x$ and remaining others.

###### Proof.

$u(r_{1},..r_{p}-1,x,0,r_{p+2,..})-u(r_{1},..r_{p}-1,0,x,r_{p+2,..})=r^{p-1}\prod\limits_{j=1}^{p-1}(1-H(r_{j}))H(x)[Mr-x(1-r)]\geq
r^{p-1}\prod\limits_{j=1}^{p-1}(1-H(r_{j}))H(x)[Mr-\bar{c}(1-r)]\geq 0$ ∎

Given Fact B.4, we know it’s impossible to set $b_{p-1}=0$ before any positive
$b_{p}$. Hence we have to set $b_{p-1}=\bar{c}$ when corner solution is
necessary. The next problem is given $b_{p}=\bar{c}$, what should $b_{p-1}$
be. Based on the fact that the recurrence relationship is monotone increasing
we know the interior solution is even larger than $\bar{c}$ hence corner
solution is again needed, combine with Fact B.4 we know the only possible
solution is to use $\bar{c}$. With the same logic we should set
$b_{p-2}=...=b_{1}=\bar{c}$, then we can get our lemma , which is a key step
in forming the optimal structure.

######  Lemma B.5.

The bonus structures where $\exists p,s.t.b_{p}=\bar{c}$, is always dominated
by setting $b_{1}=\bar{c}$.

Lemma B.5 and Lemma 4.2 are enough to prove our proposition B.1. With the same
logic we can prove our proposition B.2.

∎

###  B.2 Continuous Immediate Revelation

Assumption 5.2 is trying to simplify our analysis since in the text we want to
focus on the Full Coverage and Partial Coverage. However, we can characterize
the value function satisfied under Immediate Revelation.

$(r+\alpha\lambda)\Pi_{IR}(\alpha)=(\alpha
zr\lambda-r\bar{c})+(\alpha^{2}-\alpha)\lambda\Pi_{IR}^{\prime}(\alpha)+\alpha\lambda^{2}z$

Now we prove Assumption 5.1 is a sufficient but not necessary condition to
ignore the existence of Immediate Revelation.

###### Proof.

We consider the “Immediate Revelation Only” monopolist who can only switch
from Immediate Revelation to Safe Arm. We assume the switching cutoff is
$\alpha_{IR}$, then similar as stated in section 5.2, the monopolist face a
smooth pasting problem where there should exist cutoff belief $\alpha_{IR}$
such that

$\Pi_{IR}(\alpha_{IR})=\tilde{V},\Pi_{IR}^{\prime}(\alpha_{IR})=0$

combined with equation (IR-ODE) we get

$\alpha_{IR}=\frac{r(\bar{c}+s)}{\lambda(g-s)+rg}$

∎

Again, there should be a switching problem between Immediate Revelation and
Full Coverage. We denote the switching point as $\underline{\alpha}_{0}$.
Unlike before, we can not know for sure the optimal switching condition when
using IR. We know $\alpha_{IR}>\alpha_{PC}$, but it’s not sure whether
$\alpha_{IR}>\alpha_{FC}$. Actually given
$\min\\{\alpha_{FC},\alpha_{IR}>\alpha_{PC}\\}$, we know both IR curve and the
FC curve will optimally switch into PC instead of SA. Hence even
$\alpha_{IR}>\alpha_{FC}$, there is still possibility that
$\alpha_{IR}^{PC}>\alpha_{FC}^{PC}$ (switching goal is denoted here).
Obviously the smaller the $\bar{c}$, the higher probability that IR performs
better than FC.

Otherwise, we should let FC to smooth paste to PC, and let the IR to smooth
paste to FC (or NB). If the cutoff happens before $\alpha_{FC}^{*}$, the
smooth pasting condition can be written as

$\Pi_{IR}(\underline{\alpha}_{0})=V(\underline{\alpha}_{0}),\Pi_{IR}^{\prime}(\underline{\alpha}_{0})=V^{\prime}(\underline{\alpha}_{0})$

which gives the boundary condition of $\Pi(\cdot)$ here.

There is also probability that the cutoff happens after $\alpha_{FC}^{*}$. In
this case there is no smooth pasting condition, which means IR curve will meet
NB curve in the form of intersect rather than tangent, since if we let
$\Pi_{IR}=\alpha z\lambda$ we get

$(\alpha^{2}-\alpha)(z\lambda^{2}-\lambda\Pi^{\prime}(\alpha))=-r\bar{c}$

Hence we must have $\Pi^{\prime}(\alpha)<\lambda z$.

It’s quite counter-intuitive to find there is a discontinuity on $C(\alpha)$
in Monopolist setting. Mathematically,two reasons result in the jump from
$\bar{c}$ to 0. Firstly, in this monopolist setting when Assumption 5.2 is not
satisfied, the monopolist will choose Immediate Revelation when $\alpha\to 1$