# Auctions with Dynamic Scoring

Martino Banchio  Aranyak Mehta
Google Research  Andres Perlroth

###### Abstract

We study the design of auctions with dynamic scoring, which allocate a single
item according to a given scoring rule. We are motivated by online advertising
auctions when users interact with a platform over the course of a session. The
platform ranks ads based on a combination of bids and quality scores, and
updates the quality scores throughout the session based on the user’s online
activity. The platform must decide when to show an ad during the session. By
delaying the auction, the auctioneer acquires information about an ad’s
quality, improving her chances of selecting a high quality ad. However
information is costly, because delay reduces market thickness and in turn
revenue. When should the auctioneer allocate the impression to balance these
forces?

We develop a theoretical model to study the effect of market design on the
trade-off between market thickness and information. In particular, we focus on
first- and second-price auctions. The auctioneer can commit to the auction
format, but not to its timing: her decision can thus be cast as a real options
problem. We show that under optimal stopping the first-price auction allocates
efficiently but with delay. Instead, the second-price auction generates more
revenue by avoiding delay. The auctioneer benefits from introducing reserve
prices, more so in a first-price auction.

##  1 Introduction

Most users of online content providers (platforms) consume a variety of
digital media during a single session. For example, the average YouTube
session lasts 30 minutes and consists of multiple
videos,111https://www.statista.com/statistics/1257254/youtubecom-time-spent-
per-visit/ and users spend an average of over 10 minutes per session on
TikTok, a platform where the best-performing videos range between 15 - 60
seconds.222https://www.statista.com/statistics/579411/top-us-social-
networking-apps-ranked-by-session-length/ As the user dynamically engages with
the platform within the session, he provides a more precise indication of his
intentions and interests. Understanding a user’s interest is a key for
successful online advertising, and longer sessions allow the platform to
better match users with relevant ads.

As a measure of relevance, the platform estimates _quality scores_. In our
model of sponsored ads, we identify quality scores with the probability that
users will click on an ad — the ad’s “click-through rate”. Historically, in
online auctions for sponsored advertising, ads are ranked by both bids and
quality scores and advertisers pay only for ads that receive a click, in what
is called the “pay-per-click” model. Weighing bids by precise quality scores
helps the platform select highly relevant ads. While there are several
considerations involved in the allocation decision in an ads context (e.g., ad
match quality, ad load, or ad diversity), in this work we focus on the problem
of a revenue-maximizing platform that refines its quality score estimates
dynamically throughout a user session.

When selecting a winning ad with an auction, the platform faces a decision.
She can improve her estimates of the quality of an ad the more a session
progresses, but extra precision requires delaying the selection of the winning
ad. The ad eventually displayed will be of high-quality, which guarantees a
good chance of a click. However, the high-quality inventory is typically
limited to few competitors, which brings down the expected payment the
platform will receive. In contrast, the platform can decide not to improve her
quality estimates and select an ad early in the session from a larger
inventory. A larger inventory generates stronger competition and auction
pressure, which generally increases the expected payment. However, the
probability of a click is lower when quality estimates are more uncertain. In
other words, the platform must decide how much information to acquire, and the
cost of information is market thickness: additional information reduces market
thickness, degrading expected payments conditional on a click but increasing
the click probability. In this paper we examine how the two most common
auction formats, first- and second-price auction , balance these forces.

Formally, we study the trade-off between market thickness and estimate
precision in a stylized model of auctions with dynamic quality scores. A
platform (the auctioneer) wants to sell a single ad to $2$ advertisers
(buyers) over the course of a session, indexed by $t\in[0,\infty)$. An
advertiser’s value of showing the ad is the product of two components. First,
$v_{i}$, is the valuation for an ad that receives a click. Second,
$\theta_{i}\in\\{0,1\\}$, a binary component that represents the quality score
(e.g., click-through rate). We say that an ad is a good or bad match if
$\theta_{i}=1$ or $0$, respectively. In practice, the advertiser values the
click at $v_{i}$, but his value from showing an ad to an unknown user is
$\theta_{i}v_{i}$ because the user may not click on the ad. At time $0$, each
advertiser draws an i.i.d. valuation $v_{i}$ and a Bernoulli quality score
$\theta_{i}$. The advertiser observes his own valuation $v_{i}$ but not his
match quality $\theta_{i}$, which is unknown to everyone. As the session
progresses over time, the platform learns more about the quality of a given
ad, because the content consumed by the user communicates his intent. In the
model, the platform observes the realization $\tau_{i}$ of an exponentially
distributed clock with parameter $\lambda\cdot(1-\theta_{i})$. A tick of the
clock guarantees that the ad is a bad match, but if the clock hasn’t ticked at
time $t$ the auctioneer forms beliefs about the realization of $\theta_{i}$.
In this model, all news is bad news, and no news is good news.333In Section 5
we discuss other information acquisition processes and how the results adapt
to such models. We allow the user to leave the platform at any time, and we
model his departure as an exponential clock of rate $r\geq 0$.

We assume that the length of a session is shorter than the average response
time of an advertiser. In particular, this implies that the advertisers cannot
adjust their bid dynamically as the session progresses.444Note that if
advertisers were able to adjust their bids at such low latency, the platform
would need to decide whether to share or not the realizations of $\tau_{i}$.
Even without sharing, the auction format may allow bidders to make inference
about the realization of $\tau_{i}$, complicating the auction choice. We
choose to ignore these incentives by shutting down all advertiser learning. At
time $0$ the platform announces an auction format and solicits bids. With all
the bids in hand, she then faces a real option problem: she needs to decide
when to execute the auction, i.e. to "exercise her option". The platform here
cannot commit to an auction timing, but instead optimizes conditional on the
auction format she announced and the bids reported by the advertisers. At the
time of exercise, the winning advertiser pays _if and only if_ he has a true
quality score of $\theta_{i}=1$. Notice that the platform generates revenue
only when showing high-quality ads.

Our main result is that first- and second-price auction can have diametrically
opposite effects on the resolution of the trade-off between market thickness
and quality score’s precision. In contrast to the static setting, first- and
second-price auction under their respective optimal exercise policy are not
equivalent. The first-price auction is most efficient: the auctioneer delays
its exercise until she has precise information about the quality scores. She
allocates to the buyer with the highest valuation among those that are a good
match. Of course, buyers come to expect the efficient allocation, and they
shade their bids aggressively. Instead, we show that if the auctioneer runs a
second-price auction, she does so with less delay, when information about the
quality scores is imperfect. The auction is truthful but inefficient, because
it may sometimes allocate to a buyer who is not a good match ($\theta=0$), in
which case she receives no payment.

Let us describe the intuition behind these results in the simpler case in
which the user does not leave the platform ($r=0$). To understand why the
first-price auction waits until quality scores are precise, while the second-
price auction allocates the item earlier, consider the incentives of the
auctioneer in each auction format. In a second-price auction the price paid by
the winner is determined by his closest competitor. The upside of delaying the
auction is allocating to buyers who are good matches. However, the downside is
that the market is thin with large delay, which reduces the expected price.
The auctioneer allocates the item at the point in time when the benefit of
waiting (more accurate estimates of the quality scores) is exactly offset by
its loss (softer competition). In a first-price auction the winner instead
sets his own price. Thus, conditional on the bids the auctioneer simply wants
to allocate to a good match. There is no downside from delaying the auction as
much as possible. Of course, in equilibrium the buyers will shade their bids,
to account for the fact that, when they win, their quality is high, so they
are competing with fewer buyers. Although efficient, the first-price auction
leads to lower revenue. In a second-price auction the auctioneer directly
controls the effect of competition on pricing. Her exercise decision affects
the quality of the allocation as well as the price, and this additional
control allows her to trade off lower efficiency for higher expected revenue.

We wish to highlight here the role played by commitment power. The first-price
auction suffers greatly from the lack of time commitment, because bidders
anticipate excessive exploration by the auctioneer and bid shade aggressively.
This poses in sharp contrast with a situation in which the auctioneer could
commit to a time rule in advance. There, the auctioneer can implement the
optimal mechanism via a first-price auction . Surprisingly, when the
auctioneer can commit to timing, our results flip: the first-price auction
performs better than the second-price auction .555In our online advertisement
motivation, sustainable time commitment seems implausible. For example, for
such commitment to be verifiable the auctioneer would need to disclose the
realizations of the uncertainty to all bidders.

The auctioneer may contemplate solving the thinning market problem by setting
a reserve price. We then show that a first-price auction with optimal reserve
price implements the optimal mechanism. Setting a reserve price limits the bid
shading at time $0$, which is when pricing decisions are made. Instead, in a
second-price auction pricing is determined during the session, which limits
the effect of reserve prices on revenue. Again, the second-price auction with
reserves under-performs the first-price auction .

We are motivated by the monetization problem of content providers, and our
model tightly reflects our fundamental question. However, it turns out that
the trade-off between competition and quality scores appears in a number of
interesting economic settings. We propose three examples below of auctions
with _dynamic_ scoring, where the winner selection procedure relies on factors
other than the buyers’ values, and such factors are dynamically updated.

#### Monetizing Generative AI models.

Applications based on generative AI models, such as personal assistants,
chatbots, and search engines, have recently grown in popularity, following
quick advances in the fields of Machine Learning and Artificial Intelligence.
While some applications offer subscriptions and per-query API pricing,666See
https://openai.com/pricing for OpenAI’s ChatGPT, and https://www-
files.anthropic.com/production/images/model_pricing_july2023.pdf for
Anthropic’s Claude model. many other are offered free of charge: for example,
at the time of writing the conversational search engine Bing Chat does not
require a login, much less a
subscription.777https://www.zdnet.com/article/now-you-can-access-bing-chat-
without-a-microsoft-account/ In the same way that online search is monetized
with advertising, we envision that sponsored ads could play a salient role for
making free generative AI models cost-effective. This is a special case of our
motivating example: the provider of the generative AI chat model can
dynamically estimate the quality scores, paying for information with market
thickness.

#### Mergers & Acquisitions.

Multiple firms are interested in purchasing a small business. In most
competitive M&A processes, buyers submit bids to acquire the business with
some contingencies that will be resolved during the due diligence period.888
See Marquardt and Zur (2015) and Wangerin (2019). Such contingencies may
result in contract termination. While due diligence is costly, a natural
question is whether bidders should be encouraged to perform due diligence
_before or after_ the winner selection. Selecting a winner before due
diligence makes the process more competitive, but exposes the business to the
risk of a failed M&A deal. Viceversa, allowing due diligence to take place
before selecting the winner may thin out the pool of interested parties,
thereby reducing competitive pressure.

#### Public Procurement.

Government agencies procure goods and services from suppliers in an auction.
Regulators require that suppliers pass certain probity measures – for example,
money-laundering and criminal record checks. Should the agencies front-load or
back-load compliance verification when procuring from industry suppliers? That
is, should the agencies run their auction and then audit the winner, or should
they audit all participants and only after choose the winner? EU directives
only require a bidder self-declaration at auction time, and then demand full
compliance checks to be performed on the winner.999Directive 2014/24/EU, art.
57-61. Again, by performing compliance checks after the auction, the agencies
take advantage of market thickness to procure at low prices, at the risk of
unfulfilled contract. Instead, running the auction after thorough checks would
ensure contract fulfillment but would possibly increase prices.

The paper is structured as follows. In Section 2 we formally describe the
model. Section 3 presents the main results we described in this introduction:
first, we compare first- and second-price auction without reserve prices, then
with reserves. We study two extensions in Section 4, and then we discuss our
assumptions in Section 5.

###  1.1 Literature Review

We connect with a large literature on online advertising auctions starting
with Edelman et al. (2007) and Varian (2009). While early papers study the
static problem (Athey and Nekipelov (2010), Börgers et al. (2013)), the
literature has moved on to the repeated auction problem (Balseiro et al.
(2015), Chen (2017)). We abstract away from cross-auction incentives and focus
instead on a single dynamic auction. The key force we study is intrinsically
dynamic, and dynamics are incorporated in the auction mechanism.

A large literature studies dynamic auctions; for a non-exhaustive list of
reference see Bergemann and Said (2010). The literature mainly focuses on two
sources of dynamics. The first source of dynamics is an evolving population of
agents, each having fixed private information. The second source of dynamics
is instead a fixed population of agents whose private information evolves
dynamically. In our paper we have a fixed population of agents, but private
information is also fixed. The only source of dynamics is in the ad quality
scores, which the auctioneer exploits to optimize her revenue. Related to our
paper is Chaves and Ichihashi (2024), which also considers an auctioneer who
chooses the timing to run the auction optimizing for market thickness. In our
setting delay reduces market thickness, whereas there it is to increase market
thickness that the auctioneer delays the auction. Importantly, instead of
focusing on finding the optimal delay, we compare auction formats.

There is a large literature on real options in finance and economics that
deals with the revenue maximizer’s optimal stopping problem; there are many
reviews of this literature, such as Sick (1995). In our setting, solving the
optimal stopping problem is instrumental but not sufficient to characterize
the auctioneer’s problem, because advertisers optimally respond to the
exercise timing. The equilibrium is a fixed-point problem of advertisers and
auctioneer’s decisions. In this sense, we connect to the literature on game-
theoretic real option problems, such as those described in Grenadier (2000).
Methodologically we also adopt techniques from the continuous-time literature
(see Hörner and Skrzypacz (2017) for some excellent examples), and in
particular we mainly adopt the “bad news” Poisson model studied in Keller and
Rady (2015).

Finally, our motivation connects us to a nascent literature on Large Language
Models (LLM), chatbots, and game theoretic models. As we mentioned, our
problem can be thought of as the advertising decision of a LLM-based chatbot
provider. Some recent papers in the literature have started thinking about the
effect of LLMs on auctions and auction design, for example Duetting et al.
(2023) which studies how mechanism design should aggregate several LLM-
generated input in an incentive compatible way for online advertising. Feizi
et al. (2023) proposes a setup for online ads on LLMs. They mention that the
system for predicting quality scores can “update the estimate in (almost) real
time, which will increase the accuracy of the prediction”. We consider the
effect of the prediction dynamics on the auctioneer’s revenue. An iterative
refinement of beliefs similar to the one we model here appears in Harris et
al. (2023).

##  2 Model

Consider a user of a content platform with a private type
$\theta=(\theta_{1},\theta_{2})$, where each component $\theta_{i}$ is drawn
independently from a Bernoulli distribution with parameter $p$. The type
$\theta_{i}$ represents the user’s interest in the ad of advertiser
$i$.101010In Section 5 we consider a model with more than $2$ advertisers.
Specifically, the user clicks on ad $i$ if and only if $\theta_{i}=1$; in this
case we say that ad $i$ is a _good quality ad_. Each advertiser derives a
value $v_{i}\sim F(v)$ from a successful click, where $F(v)$ is a regular
distribution over $[0,\overline{v}]$.

An auctioneer, who controls the platform, determines which advertiser’s ad is
displayed. She receives a payment from the winning advertiser if and only if
the user clicks on the ad, and she can choose whether to allocate the ad
according to a first- or second-price auction.

The user interacts with the platform in a _session_ which unfolds over a
continuous time $t\in\mathbb{R}_{+}$. During the session, the auctioneer
gradually gleans insights about the quality of the ads. While good ads
($\theta_{i}=1$) offer no direct signal of the user’s interest, bad ads
($\theta_{i}=0$) progressively reveal themselves through the user’s
interaction with the platform. Formally, we model the arrival of news for ad
$i$ as an exponential clock $\tau_{i}$ with rate $\lambda\cdot(1-\theta_{i})$.
When the clock of ad $i$ ticks (i.e. at time $\tau_{i}$), the auctioneer
learns that such ad is of bad quality. Absence of news is good news — as time
passes, an ad becomes more likely to be of good quality if its clock has not
ticked yet.111111In the terminology of Keller and Rady (2015) this is a model
of “breakdowns”. Formally, the auctioneer forms beliefs
$\mu_{t}=(\mu^{1}_{t},\mu^{2}_{t})$ about the value of the ad qualities
$\theta$. If the exponential clock of ad $i$ ticks, the belief jumps to
$\mu^{i}_{t}=0$. Instead, when no news arrive, the belief drifts upward
according to the following law of motion:

$\begin{cases}\dot{\mu}^{i}_{t}=\lambda\mu^{i}_{t}(1-\mu^{i}_{t})\\\
\mu^{i}_{0}=p\end{cases}$ (1)

where the prior $p$ is the parameter of $\theta_{i}$’s Bernoulli distribution.

Finally, the user may interrupt his session on the platform at any time. This
means that the advertising opportunity may disappear, making learning about
qualities explicitly costly. We assume that the agent leaves at an
instantaneous rate $r>0$, so that conditional on the user not having left yet
he will leave at an exponentially distributed time with rate $r$.121212An
alternative interpretation of the rate of departure of the agent $r$ is that
both the advertisers and the auctioneer discount the future at rate $r$.

###  2.1 The Auctioneer’s Problem

The auctioneer strategically chooses the ad to display according to either a
first- or second-price auction without reserves, and then she decides at what
time
$\tau\in\operatorname*{\overline{\mathbb{R}}}\coloneqq\mathbb{R}_{+}\cup\\{+\infty\\}$
to run such auction. The timing of the allocation problem is as follows.

  * •

First, before the session begins, the auctioneer announces an auction format.
She commits to running either a first- or second-price auction, but she cannot
commit to a time $\tau$ at which such format will be run.

  * •

Then, the auctioneer solicits bids $b_{1},b_{2}$ from the bidders
(advertisers). All of the advertisers’ strategic bidding decisions take place
before the session begins.

  * •

The session starts at time $0$, and the auctioneer begins learning about the
user’s type according to the beliefs in Equation 1.

  * •

At some time $\tau\in\operatorname*{\overline{\mathbb{R}}}$, which may depends
on the advertisers’ bids and the realizations of the uncertainty the
auctioneer decides to “stop”, at which point she runs the auction format she
committed to.

If the winner is advertiser $i$, under a first-price auction she collects
revenue $\pi_{1P}=e^{-r\tau}b_{i}\theta_{i}$, while in a second-price auction
she collects
$\pi_{2P}=e^{-r\tau}\frac{b_{-i}\mu^{-i}_{\tau}}{\mu^{i}_{\tau}}\theta_{i}$.

We make two main assumptions with respect to timing. First, advertisers cannot
adjust their bids dynamically, but instead make all their strategic
considerations at time $0$. Latency considerations alone justify such an
assumption. Additionally, it seems unrealistic for a platform to require that
advertisers monitor the news arrival process and update their bids based on
the arrival or absence of news. Even more simply, allowing advertisers to
dynamically adjust their bids based on calendar time would require a complex
infrastructure on both the platform and the advertiser’s end. Second, we
assume that the auctioneer cannot commit to a time to run the auction.131313We
will relax this assumption in Section 4.2. Then, once she receives the bids of
the advertisers, the auctioneer faces a real options problem. She can hold off
on running the auction to learn more about the ad qualities, because she
profits only when she displays a good quality ad. On the other hand,
information is costly. For one, the user may leave the platform at any point
in time, making the profit opportunity vanish. More subtly, delaying the
auction erodes the competitive landscape. Auctions rely on competition to
price the ad opportunities, but delay reduces market thickness and thus
greatly diminishes competition in the auction. We show that the auction format
plays an important role in the resolution of this trade-off between
information and market thickness.

Figure 1: This figure depicts a sample path taken by the belief process
$\mu_{t}$. The beliefs drift upwards in the absence of news (as is the case
for the process $\mu^{1}_{t}$). Instead, news about advertiser $2$ arrived at
time $\tau_{2}$, and thus $\mu^{2}_{t}=0$ from $\tau_{2}$ onwards.

##  3 Auction formats, market thickness, and information

In order to study the auctioneer’s problem, we need to characterize the
optimal “exercise policy”, i.e. the optimal time in the session to stop and
run the auction from the auctioneer’s perspective. To this end we solve a
model in which the user never leaves the platform. That is, we assume $r=0$.
We shut down the direct costs of information acquisition, leaving only the
indirect cost of a market that becomes thinner the more information is
acquired.

###  3.1 Auctions without reserves

Suppose an auctioneer is facing bidders whose types come from unknown
distributions. Instead of guessing the reserve price, she may simply run an
auction without reserves. In this subsection we thus focus our attention on
first- and second-price auction without reserve prices.

Consider an auctioneer who committed to running a second-price auction. She
observes some reports $b_{1},b_{2}$ and then must decide when to stop to
maximize her expected revenue. Her problem can be written as

$\max_{\tau}\mathbb{E}_{0}\Big{[}\min\big{\\{}\mu^{1}_{\tau}b_{1},\mu^{2}_{\tau}b_{2}\big{\\}}\Big{]}$

We are ready to prove our first result.

######  Lemma 1.

The optimal exercise policy for the second-price auction is $\tau^{*}_{2P}=0$.

###### Proof.

Following the usual Bayesian arguments, notice that the belief process
$\mu^{i}_{t}$ is a martingale: $\mathbb{E}_{t}[\mu_{t^{\prime}}]=\mu_{t}$ for
any $t^{\prime}\geq t$. Then, by definition

$\mathbb{E}_{t}\Big{[}\min\big{\\{}\mu^{1}_{t^{\prime}}b_{1},\mu^{2}_{t^{\prime}}b_{2}\big{\\}}\Big{]}\leq\mathbb{E}_{t}\Big{[}\mu^{i}_{t^{\prime}}b_{i}\Big{]}=\mu^{i}_{t}b_{i}$

for all $i=1,2$ and for all $t^{\prime}\geq t$. This implies that

$\mathbb{E}_{t}\Big{[}\min\big{\\{}\mu^{1}_{t^{\prime}}b_{1},\mu^{2}_{t^{\prime}}b_{2}\big{\\}}\Big{]}\leq\min\big{\\{}\mu^{1}_{t}b_{1},\mu^{2}_{t}b_{2}\big{\\}}.$

Thus, the revenue process is a super-martingale. Doob’s Optional Stopping
Theorem tells us that a super-martingale is optimally stopped at $0$. ∎

Lemma 1 says that the auctioneer stops at time $0$ _uniformly_ over the
reports of the advertisers. In particularly, she will not use time as a
screening device, and the timing of the auction will not affect the incentives
of the advertisers, who then bid truthfully. The second-price auction is thus
a truthful auction in this setting, and it is executed at time $0$.

The auctioneer is willing to allocate to the bidder with the highest value
despite having a low chance of receiving payment. Recall that the auctioneer
will profit only if the ad she displays is of good quality. Here the
auctioneer profits only with probability $\mu_{0}$, while by delaying the
auction she could increase her chances of profiting from the displayed ad.
However, the probability that any given advertiser drops out of the race
($\mu^{i}_{t}=0$) increases with delay. The second-price auction relies on
competition to price the ad opportunity, but as soon as the belief about one
advertiser drops to $0$ competition vanishes, leaving a perfectly good
opportunity to be sold for free to the remaining advertiser. The auctioneer
runs the auction immediately to avoid such a catastrophic outcome.

Intuitively, it seems that the second-price auction suffers from lack of
competition more than the first-price auction. That is because even if one
advertiser drops out, in the first-price auction the other advertiser will
purchase the ad opportunity at the bid he chose at time $0$, before realizing
his competition disappeared. We show next that the first-price auction delays
the exercise of the auction until the auctioneer is sure to allocate
efficiently. Consider then an auctioneer who committed to running a first-
price auction. She observes bids $b_{1},b_{2}$ and then decides when to stop
to maximize her expected revenue, written as

$\max_{\tau}\mathbb{E}_{0}\Big{[}\max\big{\\{}\mu^{1}_{\tau}b_{1},\mu^{2}_{\tau}b_{2}\big{\\}}\Big{]}$

Using this expression, we can characterize the optimal exercise policy for the
first-price auction.

######  Lemma 2.

The optimal exercise policy for the first-price auction is
$\tau^{*}_{1P}=\infty$.

###### Proof.

By definition

$\mathbb{E}_{t}\Big{[}\max\big{\\{}\mu^{1}_{t^{\prime}}b_{1},\mu^{2}_{t^{\prime}}b_{2}\big{\\}}\Big{]}\geq\mathbb{E}_{t}\Big{[}\mu^{i}_{t^{\prime}}b_{i}\Big{]}=\mu^{i}_{t}b_{i}$

for all $i=1,2$ and for all $t^{\prime}\geq t$. This implies that

$\mathbb{E}_{t}\Big{[}\max\big{\\{}\mu^{1}_{t^{\prime}}b_{1},\mu^{2}_{t^{\prime}}b_{2}\big{\\}}\Big{]}\geq\max\big{\\{}\mu^{1}_{t}b_{1},\mu^{2}_{t}b_{2}\big{\\}}.$

Thus the revenue process of the first-price auction is a sub-martingale until
the clock of an advertiser ticks, if ever. A sub-martingale’s expected value
increases over time, so the auctioneer will never stop before one of the
advertisers’ clock ticks. If it does, the revenue process turns into the
(bounded) martingale $\mu^{i}_{t}\cdot b_{i}$ for some $i$, and the auctioneer
is indifferent between stopping and continuing at any time $t$. We say the
auctioneer "stops at $\infty$", where the value of $\mu^{i}_{\tau}b_{i}$ at
$\tau=\infty$ is given by the pointwise limit of the process
$\mu^{i}_{t}b_{i}$ which exists almost surely because $\mu^{i}_{t}b_{i}$ is
bounded. ∎

The first-price auction is efficient: the advertiser that displays his ad is
always the one with the largest $b_{i}\cdot\theta_{i}$. This auction is not
truthful however: the advertisers at time $0$ foresee that, conditional of
having a good quality ad, they will have no competition with probability
$1-\mu_{0}$. Only with probability $\mu_{0}$ they will need to outbid their
opponent, and their shading will reflect the probability of this
event.141414We will show that there is a unique symmetric equilibrium in
Proposition 3.

How does the bid shading affect the revenue to the auctioneer? The following
proposition responds to this question.

######  Theorem 1.

The revenue $\pi^{*}_{2P}$ generated by the truthful optimally-stopped second-
price auction dominates the revenue $\pi^{*}_{1P}$ generated by an optimally-
stopped first-price auction for any value distribution $F$. In particular,

$\frac{\pi_{2P}^{*}}{\pi_{1P}^{*}}=\frac{1}{\mu_{0}}$

###### Proof.

To prove the first part of the theorem, notice that the second-price auction
is truthful regardless of its stopping time, as long as the stopping time does
not depend on the values reported by the advertisers. In particular, consider
a second-price auction stopped at $\tau^{*}_{1P}$. This stopping time depends
solely on the path of the beliefs $\mu_{t}$, but cannot be influenced by the
reports $v_{1},v_{2}$, as shown in Lemma 2. Then, the static Revenue
Equivalence Theorem (RET) applies to auctions that run at time
$\tau^{*}_{1P}$: the first- and second-price auction that stop at
$\tau^{*}_{1P}$ allocate identically, and therefore generate the same revenue.
But the auctioneer receiving truthful reports prefers to stop at time
$\tau^{*}_{2P}<\tau^{*}_{1P}$, which implies that her revenue must be (weakly)
larger by stopping earlier than by delaying the auction until $\tau^{*}_{1P}$.
Therefore, the second-price auction generates weakly higher revenue than the
first-price auction.

To prove the second part of the theorem, we leverage the virtual value
characterization of the auctioneer’s revenue. The expected revenue of an
optimally-exercised first-price auction is given by
$\mu_{0}^{2}\mathbb{E}_{v}\big{[}\max\\{\phi(v_{1}),\phi(v_{2})\\}\big{]}+2\mu_{0}(1-\mu_{0})\mathbb{E}_{v}\big{[}\phi(v)\big{]}$
where $\phi(v)$ is the virtual value function. Intuitively, the first-price
auction allocates efficiently: with ex-ante probability $2\mu_{0}(1-\mu_{0})$
there is only one ad which is a good match, and with probability $\mu_{0}^{2}$
the auctioneer can choose the good match with the largest virtual value. The
usual integration by part argument shows that $\mathbb{E}_{v}[\phi(v)]=0$,
thus implying that
$\pi_{1P}^{*}=\mu_{0}^{2}\mathbb{E}_{v}\big{[}\max\\{\phi(v_{1}),\phi(v_{2})\\}\big{]}$.
Similarly, the expected revenue of the optimally-exercised second-price
auction is
$\pi_{2P}^{*}=\mu_{0}\mathbb{E}_{v}\big{[}\max\\{\phi(v_{1}),\phi(v_{2})\\}\big{]}$.
Intuitively, the second-price auction allocates immediately, with probability
$1$ to the bidder with the largest virtual value. She will collect her revenue
only when that bidder is a good match, which happens with probability
$\mu_{0}$. Therefore,

$\frac{\pi_{2P}^{*}}{\pi_{1P}^{*}}=\frac{\mu_{0}\mathbb{E}_{v}\big{[}\max\\{\phi(v_{1}),\phi(v_{2})\\}\big{]}}{\mu_{0}^{2}\mathbb{E}_{v}\big{[}\max\\{\phi(v_{1}),\phi(v_{2})\\}\big{]}}=\frac{1}{\mu_{0}}$

∎

The bid shading induced by the first-price auction is detrimental to revenue,
so much so that the auctioneer would rather implement a second-price auction
and allocate inefficiently. In fact, regardless of the value distribution, the
revenue obtained by a first-price auction are exactly a fraction $\mu_{0}$ of
the revenue obtained by a second-price auction . Thus the benefit from
adopting an optimally-exercised second-price auction increases as the
probability of encountering a good-quality ad decreases.

The case in which the user does not leave a session leads to a crisp, stark
result. When information would otherwise be costless, the only disincentive to
information acquisition is market thickness. Different auction formats
leverage market thickness differently: a second-price auction requires
competition to price the good for sale. A first-price auction instead requires
competition to limit the extent of the bidders’ equilibrium shading. Because
shading takes place ex-ante, this limits the ability of prices to react to a
changing competitive landscape. The second-price auction retains this
flexibility, so that the auctioneer can commit in a sequentially rational way
not to learn anything about the competitive landscape. These intuitions carry
over to the world in which information is costly, i.e. when $r>0$.

###  3.2 Reserve Prices

Suppose the auctioneer knows the distribution $F$ of the bidders’ values.
Here, we’ll analyze such an auctioneer’s strategy in determining when to stop
a first-price or second-price auction while using optimal reserve prices.

Our previous findings illustrate that the auctioneer’s timing decision heavily
depends on market thickness. Both first- and second-price auctions leverage
market thickness for revenue generation, albeit through distinct methods. An
alternative strategy to address market thickness involves setting reserve
prices. In this section, we’ll examine the design of first- and second-price
auctions with reserve prices under the assumption of costless information
(r=0). With a reserve price, the auctioneer mitigates the concern of a
thinning market, relying on this price floor to maintain revenue.

First, the reserve price does not affect the information acquisition strategy
of the auctioneer in a first-price auction.

######  Lemma 3.

The optimal exercise policy for the first-price auction with reserve price $R$
is $\tau^{*}_{1P,R}=\infty$.

###### Proof.

The auctioneer’s expected revenue when $b_{1},b_{2}>R$ can again be written as

$\max_{\tau}\mathbb{E}_{0}\Big{[}\max\big{\\{}\mu^{1}_{\tau}b_{1},\mu^{2}_{\tau}b_{2}\big{\\}}\Big{]},$

and we know from Lemma 2 that the stopping time takes the form
$\tau^{*}_{1P}=\infty$ Instead, if $b_{i}>R>b_{-i}$, the auctioneer’s revenue
at time $t$ is the martingale $\mathbb{E}_{t}[\theta_{i}b_{i}]$, so she is
indifferent between stopping and continuing at all times — in particular, we
can assume she will stop at infinity with a similar argument to Lemma 2.
Finally, if $b1,b_{2}<R$ the auctioneer’s revenue is $0$ regardless of her
stopping decision, so she is again indifferent between all stopping times, and
we assume she stops at infinity. ∎

Bidders can no longer shade as aggressively because the reserve price acts as
a backstop. The first-price auction is no longer efficient, because of the
introduction of a reserve price, but it maximizes the revenue for the
auctioneer. To see this, notice that the direct revelation mechanism that
maximizes revenue in this context is one such that

$x(v_{1},v_{2}|\theta_{1},\theta_{2})\in\operatornamewithlimits{argmax}_{x}\big{\\{}x\theta_{1}\phi(v_{1}),x\theta_{2}\phi(v_{2}),0\big{\\}}$

That is, the mechanism maximizes revenue if it allocates to the bidder with
the highest _quality-weighted_ virtual value, provided it is positive.
Quality-weighted virtual values can be negative only when the virtual values
are negative, thus the auctioneer can implement such an allocation with an
auction with a reserve price $r$ such that $\phi(r)=0$. In particular, the
allocation function of the first-price auction when there exists a symmetric
equilibrium in increasing bidding functions corresponds to $x(v_{1},v_{2})$,
and thus the first-price auction maximizes the auctioneer’s revenue.

Perhaps surprisingly, in general the second-price auction instead cannot
implement the optimal auction. This is because an auctioneer running a second-
price auction cannot commit to stopping at infinity. Instead, the auctioneer
sometime will have an incentive to stopping early, to capitalize on
competition.

######  Lemma 4.

Suppose that the auctioneer is running a second-price auction with reserve
price $R$. If $b_{1},b_{2}>2R$, then $\tau^{*}_{2P,r}(b_{1},b_{2})=0$.

###### Proof.

Let without loss bidders submit bids $b_{1}>b_{2}>2R$. Then the auctioneer’s
expected revenue from stopping at time $t$ (provided that no clock has ticked
so far) is $\mu_{t}b_{2}$. Her value function at belief $\mu_{t}$ satisfies

$V(\mu)=\max\big{\\{}\mu_{t}b_{2},V(\mu_{t+\Delta})\big{\\}}$

for some small $\Delta>0$. Suppose by contradiction that the auctioneer
decided to continue at belief $\mu_{t}$. Then her value function would
satisfy151515See Appendix A for a full derivation.

$\displaystyle V^{\prime}(\mu)\mu$ $\displaystyle=2\big{(}V(\mu)-\mu
R\big{)}.$

If the auctioneer decided to stop at a belief $\overline{\mu}>\mu_{t}$, by
continuous pasting at such a belief it must be that

$\overline{\mu}b_{2}=2\overline{\mu}b_{2}-2\overline{\mu}R\iff b_{2}=2R.$

When $b_{2}>2R$, there is no such belief $\overline{\mu}$. Thus, either the
value function $V(\mu)$ is larger than $\mu b_{2}$ for all $\mu$, or
$V(\mu)=\mu b_{2}$ and the auctioneer stops immediately. We know that
$V(1)=b_{2}$, and therefore by continuity $\lim_{\mu\to 1}V(\mu)=b_{2}$. This
holds when $V(\mu)=\mu b_{2}$ but cannot possibly hold when $V(\mu)$ solves
the ODE $V^{\prime}(\mu)\mu=2(V(\mu)-\mu R)$: if that was the case, then

$V^{\prime}(\mu)=2(b_{2}-R)\geq 2b_{2}-b_{2}=b_{2}$

which by continuity implies that $V(\mu)<\mu b_{2}$ for $\mu$ sufficiently
close to $1$. This is impossible, and concludes the proof. ∎

The auctioneer’s incentive to stop early improved her revenue with respect to
a first-price auctionwithout reserve prices; instead, with reserves the first-
price auction dominates the second-price auction.

######  Theorem 2.

The optimal mechanism can be implemented as a first-price auction with
reserves. Instead, there exist distributions $F$ such that no second-price
auction with reserve implements the optimal mechanism.

###### Proof.

We have already shown that the first-price auction with reserve $R$ such that
$\phi(R)=0$ implements the optimal mechanism. We are left to prove that the
second-price auction cannot possibly implement the optimal mechanism for some
distribution $F$. To do this, take any distribution $F$ over the support
$[0,\overline{v}]$ such that $\overline{v}>2R$ where $\phi(R)=0$, and fix
$\varepsilon<\frac{\overline{v}-2R}{2}$. We can restrict attention to the
second-price auction with reserve $R$, since any other second-price auction
cannot implement the optimal mechanism.161616To see this, note that the
reserve cannot be lower because types $R$ may then get an allocation, and
cannot be higher because the auctioneer must allocate to types immediately
above $R$.

First, note that the second-price auction with a reserve $R$ is not truthful.
To see this, consider the incentives of the first advertiser with type
$2R+\varepsilon$ if the other advertiser bids truthfully. The auctioneer will
allocate at infinity if the other advertiser’s bid is lower than $2R$, in
which case bidding truthfully is be optimal. However, if the other
advertiser’s bid is larger than $2R$, the auctioneer stops immediately, at
time $0$. Then, the advertiser has a profitable deviation from truth-telling:
she can willfully shade her report as to induce the auctioneer to wait and
allocate the item at infinity, once she fully learns the quality scores. This
is a profitable deviation because with probability $1-p$ the competitor’s
quality score will be $\theta_{2}=0$, and the advertiser wins the item, pays
$R$, and collects an expected payoff
$(1-p)(2R+\varepsilon-R)=(1-p)(R+\varepsilon)$ which is larger than the payoff
she could have obtained by instead reporting truthfully and getting allocated
the object with probability $F(2R+\varepsilon)-F(2R)$ and paying
$\mathbb{E}[v|v\in(2R,2R+\varepsilon)]>2R$.

Naturally there is no mixed strategy equilibrium that implements the optimal
mechanism, so we focus our attention on symmetric equilibria in increasing
bidding functions $\beta(-)$. If such an equilibrium implemented the optimal
mechanism, it may not stop before fully learning at least one of the quality
scores. This can only happen if all bids are uniformly bounded above by $2R$.
But then, an advertiser with type $\overline{v}-\varepsilon$ has an incentive
to bid $x>2R$ instead of $\beta(\overline{v}-\varepsilon)\leq 2R$: the
auctioneer will run the auction at infinity, but the advertiser will now win
the item even when his opponent of type $\overline{v}$ has quality score
$\theta=1$. He will pay $\beta(\overline{v})<2R$ and make a profit of
$\overline{v}-\varepsilon-\beta(\overline{v})>0$. ∎

##  4 Extensions

In this section we explore two natural extensions to the baseline model.
First, we show that our main insights carry over to a model where information
acquisition has a direct cost, in addition to the indirect, market-thickness,
cost. Second, we study how could the auctioneer gain from being able to
contract on time. Naturally the auctioneer would be better off, but we show
that first- and second-price auction remain different even with full
commitment.

###  4.1 Costly Information Acquisition ($r>0$)

The problem of the auctioneer becomes more complex when she faces two separate
costs of information. As before, she pays for information indirectly, by
reducing market thickness through delay. Additionally, now information is
costly in and of itself, because the user may leave the platform at a rate
$r>0$.

This additional force seems to push the auctioneer toward less delay. However,
the same discounting applies to the advertisers, who may be tempted to shade
their bids less. Receiving the allocation in the future is less valuable in
this world, so they may want to try and influence the timing of allocation by
reporting different bids. We begin by showing that such incentives are not
sufficient to change the auctioneer’s decision in the optimally-stopped
second-price auction.

######  Proposition 1.

The optimal exercise policy for the second-price auction is $\tau^{*}_{2P}=0$.

###### Proof.

For any pair of reports $v_{1},v_{2}$ the auctioneer faces the following
optimization problem:

$\max_{\tau}\mathbb{E}_{0}\Big{[}e^{-r\tau}\min\\{\mu^{1}_{\tau}b_{1},\mu^{2}_{\tau}b_{2}\\}\Big{]}$

We know from Lemma 1 that $\min\\{\mu^{1}_{\tau}b_{1},\mu^{2}_{\tau}b_{2}\\}$
is a super-martingale, and the auctioneer here is trying to maximize a
discounted super-martingale, which is itself a super-martingale. The result
then follows exactly as in Lemma 1. ∎

Because this result again holds uniformly over reports, the second-price
auction simply allocates at time $0$ to the highest bidder. This result is
independent of the discount factor, which is particularly striking: the
auctioneer allocates immediately, and is unwilling to take any risk with
respect to the realization of the quality process. In a world in which
competition may suddenly collapse (as it happens when one of the clocks ticks)
the second-price auction hedges against such risk by sacrificing efficiency.
With probability $1-\mu_{0}$ the auctioneer will not generate any revenue, but
with probability $\mu_{0}$ she will do so at a time in which competition is
strong. Finally, note that because the second-price auction allocates at a
time independent of the reports of the advertisers, advertisers have no
incentive to misreport, and the auction is truthful.

Next, we describe the optimal exercise policy of the first-price auction as a
function of the advertisers’ bids. First note that the auctioneer will never
stop after $\min\\{\tau_{1},\tau_{2}\\}$. In particular, suppose the
auctioneer has not stopped yet at $\tau_{1}<+\infty$, when the clock of
advertiser $1$ ticks. Then the auctioneer will stop exactly at $\tau_{1}$,
because the auctioneer’s expected revenue at time $t>\tau_{1}$ is given by
$\mathbb{E}_{\tau_{1}}\big{[}e^{-rt}\mu_{t}^{2}b_{2}\big{]}$ (which is a
super-martingale) since advertiser $1$ is out of the race. Thus, we only need
to consider the case in which no news has arrived yet. In this case, the state
space is completely characterized by a single number $\mu$, because
$\mu_{t}^{1}=\mu_{t}^{2}$ before the arrival of news. The auctioneer’s value
at beliefs $\mu^{1}_{t}=\mu^{2}_{t}=\mu$ solves the Hamilton-Jacobi-Bellman
equation below. Its derivation is in Appendix A.

$\displaystyle\underbrace{V^{\prime}(\mu)\lambda\mu(1-\mu)}_{\text{gain from
delay}}=\overbrace{rV(\mu)}^{\text{direct cost of
delay}}+\underbrace{(1-\mu)\lambda}_{\text{clock
ticks}}\Big{(}\overbrace{V(\mu)-\mu b_{1}}^{\text{cost if ad $2$ is
bad}}+\overbrace{V(\mu)-\mu b_{2}}^{\text{cost if ad $1$ is bad}}\Big{)}$

The terms in the HJB highlight the balance between benefits and costs of
information. The left-hand side represents the increase in continuation value
induced by a marginal delay —since beliefs are drifting upwards, if the clock
deosn’t tick, the expected revenue increases. The right-hand side represents
the decrease in continuation value over the marginal delay — first because of
information cost (in terms of discounting) and second because a tick of the
clock shatters competition. To find out when the auctioneer will run the
auction, without loss we set $b_{1}\geq b_{2}$. By requiring continuous
pasting on the stopping boundary we find that, if no clock has ticked yet, the
auctioneer will stop and run the auction when the beliefs
$\mu^{1}_{t}=\mu^{2}_{t}$ reach the threshold
$\overline{\mu}=\max\big{\\{}1-\frac{r}{\lambda}\frac{b_{1}}{b_{2}},\mu_{0}\big{\\}}$.
That is, unless a clock ticked the auctioneer runs the auction at

$\tau^{no-
news}(b_{1},b_{2})=\begin{cases}\frac{1}{\lambda}\log\Big{(}\frac{\lambda}{r}\frac{\min_{i}b_{i}}{\max_{i}b_{i}}-1\Big{)}\quad\text{
if }1-\frac{r}{\lambda}\frac{\max_{i}b_{i}}{\min_{i}b_{i}}>\mu_{0}\\\ 0\hskip
87.00009pt\text{otherwise.}\end{cases}$

and allocates to the highest bidder. From this equation we glean some insight
into the trade-offs faced by the auctioneer. First, the auctioneer runs the
auction with the largest delay when both bidders’ bids coincide. Intuitively,
when the bids are close to each other, the auctioneer has little to lose from
one of the advertisers dropping out of the race. Instead, she wants to
anticipate the auction when one bid is substantially larger than the other,
because her revenue is now more at risk. This can be seen by noting that the
threshold $\overline{\mu}$ is decreasing in the ratio $\frac{b_{1}}{b_{2}}$.

This concludes the characterization of the optimal stopping time for the
first-price auction , which we report below.

######  Proposition 2.

The optimal exercise policy for the first-price auction is

$\tau^{*}_{1P}(b_{1},b_{2})=\min\Big{\\{}\tau^{no-
news}(b_{1},b_{2}),\tau_{1},\tau_{2}\Big{\\}}$

Unfortunately, computing the equilibrium of the first-price auction requires
aggregating exercise policies over the set of possible reported pairs of bids.
This is a largely intractable problem, even for simple value distributions. We
prove however that when the probability of the user leaving is sufficiently
small, the revenue from a second-price auction dominates the revenue from a
first-price auction . In other words, when the advertisers and the auctioneer
become more patient (or the beliefs converge sufficiently fast to the truth)
the first-price auction allocates almost efficiently, and thus its revenue
cannot possibly be close to the revenue of the inefficient second-price
auction .

######  Proposition 3.

For every discount rate $r\geq 0$, there exists a pure strategy equilibrium
for the first-price auction game. Moreover, for $r=0$ the unique equilibrium
is symmetric.

The key idea of the existence proof relies on the continuity of
$\tau^{*}_{1P}(b_{1},b_{2})$ on the bids for a fixed discount rate $r$. In
some sense the dynamic first-price auction then inherits the same continuity
properties of the static first-price auction , which is sufficiently
continuous to apply the machinery of Reny (1999).171717See for instance the
paper’s Example 5.2, that shows that for static first-price auctions there
exists a pure strategy equilibrium with strictly increasing bidding
strategies.

To obtain equilibrium uniqueness for the case when $r=0$, we leverage Theorems
3.1 and 4.6 in Chawla and Hartline (2013). However, a key assumption for their
results is that the allocation depends only on the ranking of the bids. When
$r>0$ instead the auction timing is bid-dependent, and thus from an ex-ante
perspective bids affect the auction rule. This is not the case when $r=0$,
allowing us to conclude uniqueness and symmetry of equilibrium in that case.

With the existence of equilibrium at hand and the fact that the optimal
stopping time $\tau^{*}_{1P}$ is also continuous on the discount rate, we show
in Appendix B that the equilibrium bidding strategies are continuous on the
discount rate. Therefore we can show a generalized version of Theorem 1 for
the case where the information cost – measured by the discount rate – is
small.

######  Theorem 3.

Fix a value distribution $F$ and a discount rate $\underline{r}>0$. Then for
any $r<\underline{r}$ the revenue generated by the second-price auction
dominates the revenue generated by any equilibrium of the first-price auction.

###  4.2 Contractibility of time

Suppose now that the auctioneer can commit to an auction timing. This
commitment power allows the auctioneer to optimize information acquisition
without concerns about dynamic inconsistency.

The bidding strategies of advertisers at time $0$ hinge on the auctioneer’s
commitment to a mechanism and its timing. For example, bids will differ if the
auctioneer commits to an immediate allocation versus allocating only after
identifying a low-quality ad. In this section, the auctioneer can commit to a
random variable $\tau$ which denotes the time at which she will run the
auction. We focus on how first- and second-price auctions drive a balance
between acquiring information (by delaying allocation) and ensuring market
thickness and robust competition.

######  Theorem 4.

Let the auction designer commit to the optimal stopping for each format. Then,
a first-price auction generates more revenue than a second-price auction.

We build up to this theorem by proving several results of independent
interest. We do this in steps:

  * •

First, we characterize the timing of the optimal mechanism.

  * •

Second, we show that the optimal mechanism can be implemented as a first-price
auction.

  * •

Third, we prove that there is no implementation of the optimal mechanism as a
second-price auction .

###  4.3 Ideas of the proof

#### Step 1: Optimal Mechanism

Given a general mechanism $x(v,\mu),p(v,\mu)$ the incentive compatibility
constraint can be written as

$v\in\operatornamewithlimits{argmax}_{z}\mathbb{E}_{0}\Big{[}e^{-r\tau}\mu_{t}\big{(}v_{i}x(z,v_{-i},\mu_{\tau})-p(z,v_{-i},\mu_{\tau})\big{)}\Big{]}\qquad\forall
i,v_{i},v_{-i}$

If the allocation rule is monotone, we can rewrite the utility of the bidder
as

$U(v)=\int_{0}^{v}\mathbb{E}_{0,v_{-i}}\big{[}e^{-r\tau(z,v_{-i},\mu_{\tau})}x(z,v_{-i},\mu_{\tau})\mu_{\tau}\big{]}dz$

and therefore the constraint becomes

$\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu^{i}_{\tau}p(v_{i},v_{-i},\mu_{\tau})\big{]}=\mathbb{E}_{0,v}\Bigg{[}e^{-r\tau}\mu^{i}_{\tau}v_{i}x(v_{i},v_{-i},\mu_{\tau})-\int_{0}^{v_{i}}e^{-r\tau(z,v_{-i},\mu_{\tau})}\mu^{i}_{\tau}x(z,v_{-i},\mu_{\tau})dz\Bigg{]}$

With the usual Myersonian steps, one can characterize the optimal mechanism as
the solution to the following problem:

$\begin{gathered}\max_{x,p}\Bigg{[}\max_{\tau}\mathbb{E}_{0}\Big{[}\sum_{i=1}^{n}e^{-r\tau}\mu_{\tau}x(v_{i},\mu_{\tau})\phi(v_{i})\Big{]}\Bigg{]}\\\
\text{subject to}\\\
\begin{array}[]{l>{\displaystyle}lr}&\sum_{i=1}^{n}x(v_{i},\mu)\leq 1&\forall
v,\mu\end{array}\end{gathered}$

where $\phi(v)$ is the virtual value function associated with the distribution
$F$. If both bidders submit values $v_{1},v_{2}$ such that their virtual
values $\phi(v_{1}),\phi(v_{2})$ are negative, the optimal mechanism does not
allocate the item. If one of the bidders has a positive virtual value, the
auctioneer runs the auction at time $0$. This is a direct consequence of the
fact that the auctioneer’s problem is a discounted martingale (and thus a
supermartingale) in this case. Finally, if both bidders report positive
virtual values, the auctioneer’s value at given beliefs $\mu^{i}=\mu^{j}=\mu$
solves the same HJB equation as the one reported in Section 4.1, with bids
replaced by the virtual values. Then, unless a clock ticked, the auctioneer
runs the auction at

$\tau^{*}(v_{1},v_{2})=\begin{cases}\frac{1}{\lambda}\log\Big{(}\frac{\lambda}{r}\frac{\min_{i}\phi(v_{i})}{\max_{i}\phi(v_{i})}-1\Big{)}\quad\text{
if
}1-\frac{r}{\lambda}\frac{\max_{i}\phi(v_{i})}{\min_{i}\phi(v_{i})}>\mu_{0}\\\
0\hskip 97.0001pt\text{otherwise.}\end{cases}$

and allocates to the highest-virtual-value bidder. This completes the
description of the optimal mechanism.

#### Step 2: The first-price auction implements the optimal mechanism

We show that the auctioneer can commit to a stopping time such that the first-
price auction achieves the maximum revenue possible. This is the revenue
generated by the optimal mechanism, and we leverage the optimal mechanism’s
characterization to construct such a stopping time.

######  Proposition 4.

There exists a stopping time $\tau^{*}_{1P}$ such that the first-price auction
implements the optimal mechanism.

Intuitively, $\tau^{*}_{1P}$ replicates the stopping of the optimal mechanism.
However, the first-price auction without reserves must allocate the item even
when the virtual values are negative. To replicate the allocation rule of the
optimal mechanism, when virtual values are negative the auctioneer will choose
$\tau^{*}_{1P}=+\infty$, because even if both advertisers have good quality
ads, running the auction “at infinity” discounts the contribution of the
allocation to the optimization problem. Of course, we still need to show that
a first-price auction can indeed implement the allocation for positive virtual
values. A complete proof of this result is in Lemma 6 in Appendix A.

#### Step 3: The second-price auction cannot implement the optimal mechanism

Consider the first-price implementation of the optimal mechanism. Discounting
and commitment power allow the auctioneer to replicate a reserve price with
the timing rule. Moreover, because the optimal $\tau^{*}$ has to solve the HJB
problem, if the second-price auction could implement the optimal mechanism
then it would have to be the case that $\tau^{*}_{2P}=\tau^{*}_{1P}$. Suppose
by contradiction that there exists a bidding equilibrium of the second-price
auction that implements the optimal mechanism according to $\tau^{*}_{1P}$.
Since the optimal allocation is different for each of the valuations
$v\geq\hat{v}$, where $\phi(\hat{v})=0$, in equilibrium there is no pooling
across types $v\geq\hat{v}$ . Second, because $\tau^{*}_{1P}(\hat{v},\cdot)=0$
(i.e., SPA runs at time $0$ if one bidder reports $\hat{v}$) and in the
optimal mechanism $v_{i}=\hat{v}$ wins whenever $v_{-i}<\hat{v}$, we have that
$\mathbb{E}[b(\hat{v})]>\mathbb{E}[b(v_{-i})|v_{-i}]=\hat{v}$. We show instead
that $\mathbb{E}[b(\hat{v})]=\hat{v}$, which completes the contradiction.

Indeed, notice that for any $\epsilon$ there exists a sufficiently small
$\varepsilon$ such that
$\tau^{*}_{1P}(\hat{v}+\varepsilon,\cdot)\leq\epsilon$. That is, the stopping
time is sufficiently continuous, so the auction runs close to time $0$ (a.s.)
when the type lies between $\hat{v}$ and $\hat{v}+\varepsilon$. Suppose by
contradiction that $\mathbb{E}[b(\hat{v})]>\hat{v}+\delta$, for some
$\delta>0$, then by the monotonicity of the equilibrium bids we must have that
$\mathbb{E}[b(v)]>\hat{v}+\delta$. But then, types
$v\in(\hat{v},\hat{v}+\epsilon)$ have an incentive to mimic type $\hat{v}$,
because they would rather pool with type $\hat{v}$ than book a loss whenever
their opponent’s type lies in $(\hat{v},\hat{v}+\varepsilon)$. This proves
that $\mathbb{E}[b(\hat{v})]=\hat{v}$, which contradicts the assumption and
shows that the second-price auction cannot implement the optimal mechanism.

##  5 Discussion

In this section we extend the insights we derived in the previous sections to
more general models.

###  5.1 The Information Acquisition Process

All the analysis so far focused on the exponential “bad news” model, in part
because it delivers tractability to this complex setting, and in part because
it represents quite a good model of dynamic quality scores for online
advertising. A platform is unlikely to discover whether an ad will receive a
click _before_ showing the ad, but it may instead become obvious that the ad
will not receive a click, maybe because the user’s queries steer the session
away from the advertiser’s products. Fortunately, some of our results hold for
a general information acquisition process.

Let the news learning process be a (possibly multidimensional) stochastic
process $(X_{t})_{t=0}^{\infty}$ with law $\nu^{\theta}$ which depends on the
user’s type $\theta$.181818Formally, let $\Omega$ be a Polish metric space and
$\mathcal{F}$ a $\sigma$-algebra over $\Omega$. For each $\theta\in\Theta$ let
$P^{\theta}$ be a probability measure over $(\Omega,\mathcal{F})$. The
collection
$X=\big{(}X_{t}\colon\Omega\to\mathbb{R}\big{)}_{t\in{\mathbb{R}_{+}}}$ is
such that every $X_{t}$ is measurable with respect to the Borel
$\sigma$-algebra on $\mathbb{R}$. We denote by
$\nu_{t}^{\theta}=P^{\theta}\circ X_{t}^{-1}$ the law of the random variable
$X_{t}$ for a given parameter $\theta$.

The auctioneer’s information is given by the natural filtration of $X$,
denoted by $\mathbb{F}=(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}$. She makes
Bayesian inference about the value $\theta$. That is, the auctioneer’s belief
at time $t$ about the quality of advertiser $i$’s ad is
$\mu^{i}_{t}=\mathbb{E}_{t}\big{[}\theta_{i}\big{]}\coloneqq\mathbb{E}\big{[}\theta_{i}|\mathcal{F}_{t}\big{]}$.
We drop the script $i$ when we refer to vectors. For example, the vector of
beliefs is denoted by $\mu_{t}=(\mu_{t}^{1},\mu_{t}^{2})$. The Bayesian prior
$\mu^{i}_{0}=\mathbb{E}_{0}[\theta_{i}]$. We require that a long enough
conversation will identify the qualities $\theta_{i}$.

######  Definition 1.

A belief $\mu_{t}$ is _consistent_ if
$P\big{(}\lim_{t\to\infty}\mu_{t}=\theta\ |\ \theta\big{)}=1$ almost surely in
$\theta$.

Definition 1 says that the beliefs should converge in probability to the true
quality $\theta$. Assumption 1 ensures that consistency is guaranteed.

######  Assumption 1.

Assume

  1. 1.

For any $t>0$ such that $\mathcal{F}_{t}\subset\mathcal{F}$, there exists a
$t^{\prime}>t$ such that $\mathcal{F}_{t}\subset\mathcal{F}_{t^{\prime}}$.

  2. 2.

$\theta\to P^{\theta}(A)$ is measurable with respect to the Borel
$\sigma$-algebra on $\Theta$ for every $A\in\mathcal{F}$.

  3. 3.

There exists a measurable function $f\colon\Omega\to\Theta$ such that
$f(\omega)=\theta$ almost surely with respect to $P^{\theta}$.

The first condition ensures that the stochastic process is truly informative
by requiring that the filtration expand over time. The second condition is
technical, and it guarantees that we can compute probabilities with respect to
the user’s type. The third condition ensures identifiability: for any point in
the sample space there can only be one parameter $\theta$ that generates such
sample.

Under these assumptions, we can prove a version of Doob’s consistency theorem
(see Schwartz (1965)):

######  Proposition 5.

If Assumption 1 is satisfied, then the beliefs
$\mu_{t}=\mathbb{E}[\theta|\mathcal{F}_{t}]$ are consistent.

Assumption 1 is satisfied in various common settings. For example, the
exponential clock process we studied in earlier sections fulfills all
requirements and is indeed consistent. The assumptions are similarly verified
by a $2$-dimensional Brownian motion with drift parameter $\theta$, in which
case the beliefs take the form of Kalman filters. All results in Section 3.1
generalize immediately to this alternative model.

######  Theorem 5.

Let the news process $(X_{t})_{t}$ satisfy Assumption 1, and let $r=0$. Then,

  * (Lemma 1)

The optimal exercise policy for the second-price auction without reserves is
$\tau^{*}_{2P}=0$

  * (Lemma 2)

The optimal exercise policy for the first-price auction without reserves is
$\tau^{*}_{1P}=\infty$

  * (Theorem 1)

The revenue $\pi_{2P}^{*}$ generated by an optimally-stopped second-price
auction dominates the revenue $\pi_{1P}^{*}$ generated by an optimally-stopped
first-price auction for any value distribution $F$. In particular

$\frac{\pi_{2P}^{*}}{\pi_{1P}^{*}}=\frac{1}{\mu_{0}}.$

The only subtle part of the proof here is that the Revenue Equivalence Theorem
no longer holds for all $t>0$. This is because allocations now depend on the
quality scores, which can be asymmetric between the two bidders. Nonetheless,
the Revenue Equivalence Theorem holds both at time $0$ and in the limit, which
allows us to conclude. Some of the results in Section 3.2 are more complex to
derive in this setting — for example, Lemma 4 relies on a form of the value
function specific to the bad news model. The failure of the RET highlights
also the difficulty in proving the results in Section 4.1: aside from
characterizing optimal stopping, which requires a multidimensional state
space, there is no simple way to compare the revenue from first- and second-
price auction without explicitly computing the equilibrium of the former.
However, if the process has enough regularity similar continuity results can
prove that when the discount rate $r$ is small enough, the revenue under
second-price auction dominates the revenue under first-price auction.

###  5.2 The Number of Advertisers

Another direction in which we could generalize the results is the number of
advertisers. Consider again the exponential news model with discount rate
$r=0$, and let there be $n>2$ advertisers competing for the ad slot. We can
show the following:

######  Lemma 5.

For any realization of the exponential clocks $\tau_{1},\dots,\tau_{n}$ the
second-price auction optimally stops earlier than the first-price auction,
i.e.

$\tau^{*}_{2P}\leq\tau^{*}_{1P}\quad\text{ pointwise.}$

###### Proof.

To see this, first note that the revenue from the first-price auction is a
maximum of $n$ martingales, and thus is a sub-martingale. Let $K_{t}$ be the
number of advertisers $i$ such that $\mu_{t}^{i}\neq 0$. Then,

$\tau_{1P}^{*}=\inf_{t}\\{K_{t}|K_{t}=1\\}.$

That is, the auctioneer stops in the limit when there are at least two
advertisers who are a good match. She stops earlier when there are less than
two good matches and only one competitor remains in the race.

Instead, the revenue from a second-price auction is no longer a super-
martingale. The auctioneer’s optimal stopping can be characterized via the
following HJB:

$nV(\mu)=V^{\prime}(\mu)\mu+\mu\Big{(}(n-1)b^{(1)}+b^{(2)}\Big{)}.$

Continuous-pasting techniques imply that the auctioneer never stops, so long
as $n\geq 3$. When $n=2$ instead the problem becomes a super-martingale, and
the auctioneer ends the auction. Therefore,

$\tau^{*}_{2P}=\inf_{t}\\{K_{t}|K_{t}=2\\}.$

The lemma’s statement follows immediately from this description of the
stopping times. ∎

From Lemma 5 and the Revenue Equivalence Theorem (which again holds because of
the exponential news assumption) immediately follows that the revenue
generated by the second-price auction is larger than the revenue generated by
a first-price auction.

##  6 Conclusion

In this paper, we investigate how dynamic scoring influences optimal auction
design. Two key countervailing incentives for the auctioneer determine when
the auction should be held:

  * •

Information Acquisition: Over time, signals about ad quality improve, allowing
the auctioneer to make better allocation decisions.

  * •

Market Thickness: As quality signals improve, advertisers separate between
more and less competitive. As the market thins, revenue deteriorates.

We prove that the second-price auction is most sensitive to market thickness:
the auction happens early when quality scores are very uncertain. In the
first-price auction instead, the auctioneer is most sensitive to information.
She wants to learn as much as possible before running the auction, but the
bidders anticipate the auctioneer’s behavior and shade their bids
aggressively. The auctioneer generates more revenue by running a second-price
auction. Interestingly, the introduction of reserve prices wipes the second-
price auction’s advantage. The first-price auction implements the optimal
auction, while the second-price auction is not truthful and sometimes ends too
early.

Our theory has interesting implications for ad auctions. In light of the rapid
adoption of interactive systems where platforms can dynamically learn the
preferences of users, platforms that are planning to monetize using ad-
auctions should think carefully about the dynamic forces we highlight when
deciding their auction formats.

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##  Appendix A Calculations for Poisson Model

The evolution of beliefs in a Poisson model follow the derivation below.

$\displaystyle\mu_{t+\Delta}^{i}$
$\displaystyle=\mathbb{P}(\theta_{i}=1|\text{ no news before }t+\Delta)$
$\displaystyle=\frac{\mathbb{P}(\text{no news in
}[t,t+\Delta))\mathbb{P}(\theta_{i}=1|\text{ no news before
}t)}{\mathbb{P}(\text{no news in }[t,t+\Delta|\text{ no news before }t)}$
$\displaystyle=\frac{\mu^{i}_{t}}{\mu^{i}_{t}+(1-\mu^{i}_{t})e^{-\lambda\Delta}}$

Then

$\displaystyle\dot{\mu}^{i}_{t}=\lim_{\Delta\to
0}\frac{\mu_{t+\Delta}^{i}-\mu^{i}_{t}}{\Delta}$
$\displaystyle=\lim_{\Delta\to
0}\frac{\mu^{i}_{t}}{\Delta}\bigg{(}\frac{1}{\mu^{i}_{t}+(1-\mu^{i}_{t})e^{-\lambda\Delta}}-1\bigg{)}$
$\displaystyle=\lim_{\Delta\to
0}\frac{\mu^{i}_{t}}{\Delta}\bigg{(}\frac{1-\mu^{i}_{t}-(1-\mu^{i}_{t})e^{-\lambda\Delta}}{\mu^{i}_{t}+(1-\mu^{i}_{t})e^{-\lambda\Delta}}\bigg{)}$
$\displaystyle=\lambda\mu^{i}_{t}(1-\mu^{i}_{t})$

Fix $b_{1}\geq b_{2}$. The value function for non-zero beliefs is given by

$\displaystyle V(\mu_{t},\mu_{t})=\max$
$\displaystyle\bigg{\\{}b_{1}\mu_{t},e^{-r\Delta}V(\mu_{t+\Delta},\mu_{t+\Delta})(\mu_{t}^{2}+2\mu_{t}(1-\mu_{t})e^{-\lambda\Delta}+e^{-2\lambda\Delta}(1-\mu_{t})^{2})$
$\displaystyle+e^{-r\Delta}(\mu_{t}(1-\mu_{t})(1-e^{-\lambda\Delta})(b_{1}\mu_{t}+b_{2}\mu_{t})+(1-\mu_{t})^{2}(1-e^{-2\lambda\Delta})\big{(}\frac{b_{1}\mu_{t}+b_{2}\mu_{t}}{2}\big{)}\bigg{\\}}$

At a belief where the auctioneer wants to continue,

$\displaystyle V(\mu_{t})$
$\displaystyle=e^{-r\Delta}V(\mu_{t+\Delta})\big{(}\mu_{t}+(1-\mu_{t})e^{-\lambda\Delta}\big{)}^{2}$
$\displaystyle+e^{-r\Delta}(b_{1}\mu_{t}+b_{2}\mu_{t})\bigg{(}\mu_{t}(1-\mu_{t})(1-e^{-\lambda\Delta})+\frac{(1-\mu_{t})^{2}}{2}(1-e^{-2\lambda\Delta})\bigg{)}$
$\displaystyle 0$
$\displaystyle=\frac{e^{-r\Delta}V(\mu_{t+\Delta})\big{(}\mu_{t}+(1-\mu_{t})e^{-\lambda\Delta}\big{)}^{2}-V(\mu_{t})}{\Delta}$
$\displaystyle+\frac{e^{-r\Delta}}{\Delta}(b_{1}\mu_{t}+b_{2}\mu_{t})\bigg{(}\mu_{t}(1-\mu_{t})(1-e^{-\lambda\Delta})+\frac{(1-\mu_{t})^{2}}{2}(1-e^{-2\lambda\Delta})\bigg{)}$

Taking the limit for $\Delta\to 0$, and writing $\rho=\frac{r}{\lambda}$ we
get

$\displaystyle V(\mu_{t})\big{(}r+2\lambda(1-\mu_{t})\big{)}$
$\displaystyle=V^{\prime}(\mu_{t})\dot{\mu}_{t}+(b_{1}+b_{2})\mu_{t}\Big{(}\mu_{t}(1-\mu_{t})\lambda+(1-\mu_{t})^{2}\lambda\Big{)}\iff$
$\displaystyle\iff V^{\prime}(\mu)\mu(1-\mu)$ $\displaystyle=\rho
V(\mu)+(1-\mu)\Big{(}2V(\mu)-\mu(b_{1}+b_{2})\Big{)}$

Note that if $r=0$ the ODE becomes

$V^{\prime}(\mu)\mu=2V(\mu)-\mu(b_{1}+b_{2}),$

as we used in Lemma 4. The solution to the general ODE is

$V(\mu)=\frac{\mu(1-\mu)}{\rho+1}(b_{1}+b_{2})+K\mu^{2}\bigg{(}\frac{\mu}{1-\mu}\bigg{)}^{\rho}$

where

$K=\frac{b_{1}}{(1+\rho)}\bigg{(}\frac{b_{2}}{\rho b_{1}}-1\bigg{)}^{-\rho}$

which results in

$V(\mu)=\frac{\mu(1-\mu)}{\rho+1}(b_{1}+b_{2})+\frac{b_{1}}{(1+\rho)}\bigg{(}\frac{b_{2}-\rho
b_{1}}{\rho
b_{1}}\bigg{)}^{-\rho}\mu^{2}\bigg{(}\frac{\mu}{1-\mu}\bigg{)}^{\rho}$

We find this value of $K$ by requiring continuous pasting at the threshold
$\bar{\mu}$ such that $\bar{\mu}b_{1}=V(\bar{\mu})$. This threshold turns out
to be such that $1-\bar{\mu}=\frac{\rho b_{1}}{b_{2}}$. Then we can rewrite

$K=\frac{b_{1}}{(1+\rho)}\bigg{(}\frac{1}{1-\bar{\mu}}-1\bigg{)}^{-\rho}=\frac{b_{1}}{(1+\rho)}\bigg{(}\frac{1-\bar{\mu}}{\bar{\mu}}\bigg{)}^{\rho}$

and

$V(\mu)=\frac{\mu(1-\mu)}{\rho+1}(b_{1}+b_{2})+\frac{b_{1}}{(1+\rho)}\mu^{2}\bigg{(}\frac{\mu(1-\bar{\mu})}{\bar{\mu}(1-\mu)}\bigg{)}^{\rho}$

###  A.1 Optimal Mechanism

The optimal mechanism problem is the solution to the following problem:

$\begin{gathered}\max_{x,p}\Bigg{[}\max_{\tau}\mathbb{E}_{0}\Big{[}\sum_{i=1}^{n}e^{-r\tau}\mu_{\tau}p(v_{i},\mu_{\tau})\Big{]}\Bigg{]}\\\
\text{subject to}\\\
\begin{array}[]{l>{\displaystyle}lr}&\sum_{i=1}^{n}x(v_{i},\mu)\leq 1&\forall
v,\mu\\\
\text{(DC)}&\tau\in\operatornamewithlimits{argmax}_{t}\mathbb{E}_{s}\Big{[}\sum_{i=1}^{n}e^{-rt}\mu_{t}p(v_{i},\mu_{t})\Big{]}&\forall
v,s,\mu_{s}\\\
\text{(IC)}&v\in\operatornamewithlimits{argmax}_{z}\mathbb{E}_{0}\Big{[}e^{-r\tau}\mu_{t}\big{(}vx(z,v_{-i},\mu_{\tau})-p(z,v_{-i},\mu_{\tau})\big{)}\Big{]}&\forall
i,v_{i},\hat{v},v_{-i}\\\
\text{(IR)}&\mathbb{E}_{0}\Big{[}e^{-r\tau}\mu_{\tau}\big{(}v_{i}x(v_{i},v_{-i},\mu_{\tau})-p(v_{i},v_{-i},\mu_{\tau})\big{)}\Big{]}\geq
0&\forall i,v_{i},v_{-i}\end{array}\end{gathered}$

Using the envelope formula (and assuming that the allocation is monotone) we
can rewrite the utility of a bidder as

$U(v)=\int_{0}^{v}\mathbb{E}_{0,v_{-i}}\big{[}e^{-r\tau(z,v_{-i},\mu_{\tau})}x(z,v_{-i},\mu_{\tau})\mu_{\tau}\big{]}dz$

and by substituting and integrating by parts, we get

$\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu_{\tau}p(v,\mu_{\tau})\big{]}=\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu_{\tau}x(v,\mu_{\tau})\phi(v)\big{]}$

Thus, the optimal mechanism problem reduces to

$\begin{gathered}\max_{x,p}\Bigg{[}\max_{\tau}\mathbb{E}_{0}\Big{[}\sum_{i=1}^{n}e^{-r\tau}\mu_{\tau}x(v_{i},\mu_{\tau})\phi(v_{i})\Big{]}\Bigg{]}\\\
\text{subject to}\\\
\begin{array}[]{l>{\displaystyle}lr}&\sum_{i=1}^{n}x(v_{i},\mu)\leq 1&\forall
v,\mu\\\
\text{(DC)}&\tau\in\operatornamewithlimits{argmax}_{t}\mathbb{E}_{s}\Big{[}\sum_{i=1}^{n}e^{-rt}\mu_{t}x(v_{i},\mu_{t})\phi(v_{i})\Big{]}&\forall
v,s\\\ \end{array}\end{gathered}$

Therefore, we can use the mathematics we developed earlier in this section to
describe the optimal mechanism. If both bidders submit values such that
$\phi(v_{1}),\phi(v_{2})<0$ then the auctioneer commits to a stopping time
$\tau=\infty$. This is equivalent to not allocating the object at all when the
discount factor is positive. If instead one of the bidders has a positive
virtual value, the auctioneer stops immediately and allocates to that bidder.
This is a direct consequence of the fact that the auctioneer’s problem in this
case is a discounted martingale. Finally, if both agents have positive virtual
values the auctioneer may wait: her decision depends on the ratio
$\frac{\rho\phi(v_{1})}{\phi(v_{2})}$ where $\phi(v_{1})\geq\phi(v_{2})$.

######  Lemma 6.

The first-price auction implements the optimal mechanism for positive virtual
values.

###### Proof.

We use the envelope formula to produce the first-price auction bids that
implement the optimal mechanism. Recall that the expected payment at time $0$
for the advertisers is given by

$\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu_{\tau}p(v,\mu_{\tau})\big{]}=\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu_{\tau}x(v,\mu_{\tau})\phi(v)\big{]}$

We define

$p(v_{i})=\frac{\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu_{\tau}p(v,\mu_{\tau})\big{|}v_{i}\big{]}}{\mathbb{E}_{0,v}\big{[}e^{-r\tau}\mu_{\tau}\big{|}v_{i}\big{]}}$

These payments are increasing, which can be seen by taking a simple
derivative. Thus, they are an equilibrium implementation of the optimal
mechanism through a first-price auction. ∎

##  Appendix B Proofs of Section 4.1

###### Proof of Proposition 3.

Observe that the our dynamic first-price auction , can be summarized as pay
your bid auction where the payoff for bidder $i$ with valuation $v_{i}$ and
given bids $b_{1},b_{2}$ is
$u_{i}(v_{i},b_{i},b_{-i}=\mu_{0}\cdot(v_{i}-b_{i})x_{i}(b_{i},b_{-i}|r)$
where

$\displaystyle x_{i}(b_{i},b_{-i}|r)=\mathbb{E}_{0}\big{[}$
$\displaystyle\mathbf{1}_{\\{\tau_{-i}\geq\tau^{no-
news}(b_{i},b_{-i}|r)\\}}e^{-r\tau^{no-
news}(b_{i},b_{-i}|r)}\big{(}\mathbf{1}_{\\{b_{i}>b_{-i}\\}}+\frac{1}{2}\cdot+\mathbf{1}_{\\{b_{i}=b_{-i}\\}}\big{)}$
$\displaystyle+\mathbf{1}_{\\{\tau_{-i}\geq\tau^{no-
news}(b_{i},b_{-i}|r)\\}}e^{-r\tau_{-i}}\big{]}.$

Observe that for bids $b_{i}\neq b_{-i}$ the payoff is continuous on $b_{i}$
as the clock $\tau_{1},\tau_{2}$ are independent of the bids and the optimal
policy $\tau^{no-news}(b_{i},b_{-i}|r)$ is continuous on $b_{i}$. Therefore,
the same proof of the multi-unit pay your bid auction in Example 5.2 in Reny
(1999) applies to our setting which guarantees the existence of a pure
strategy equilibrium.

For $r=0$, notice that $\tau^{no-news}(b_{1},b_{2}|r)=\infty$, and therefore
the allocation rule of bidder $i$ only depends on the ranking of the bids but
not on the value of the bids. Thus, using the language of Chawla and Hartline
(2013), the auction is a rank-based allocation rule. Moreover, the first price
nature of the auction implies that it satisfies the bid-based payment and win-
vs-tie-strict properties required for Theorems 3.1 and 4.6 in Chawla and
Hartline (2013) which imply that there is a unique equilibrium for the game
and bidders use the same pure strategy bidding function. ∎

To show that that the equilibrium is continuous as function of the discount
rate, we consider the mixed extension of the auction game.191919We refer to
Chapter 2 of (Parthasarathy, 2005) for textbook treatment on the space of
Borel measures and the weak topology on it. We denote the Borel measures on
$[0,1]^{2}$ by $\mathcal{B}([0,1]^{2})$ so that a strategy
$B\in\mathcal{B}([0,1]^{2})$ corresponds to a probability measure $(v,b)\sim
B$ with $\mathbb{P}_{B}[v^{\prime}\leq v]=F(v)$ for all $v\in[0,\overline{v}]$
(almost surely). For example, a pure strategy $b(v)$ has mixed strategy
representation $B_{b}$ where $\mathbb{P}_{B_{b}}[(v,b):b=b(v)]=1$.

We endow $\mathcal{B}([0,1]^{2})$ with the weak topology, so that
$B_{n}\Rightarrow B$ if for all continuous function $f:[0,1]^{2}\to R$ we have
that $\int fdB_{n}\to\int fdB$. Because $\mathcal{B}([0,1]^{2})$ is compact
and the subsspace of mixed strategies is closed, we conclude that the subspace
of mixed bidding strategies is compact.

######  Claim 1.

Given a discount rate $r\geq 0$, consider an equilibrium
$(B^{*}_{1}(r),B^{*}_{2}(r))$. Then the probability of having a tie on the
equilibrium is zero.

###### Proof.

Suppose that by contradiction that there are ties on the equilibrium. So
consider a type $v_{i}$ on where there is a tie. By equilibrium optimality we
must have that $\mathbb{P}_{B^{*}_{i}(r)}[(v_{i},b):b\geq v_{i}]=0$: else, the
bidder obtains a payoff of zero while by bidding small $\epsilon$ obtains a
positive payoff since it wins the object with positive probability whenever
the other bidder has a valuation $v_{-i}\leq\epsilon$. Since type $v_{i}$ is
bidding less that its value, if there is a tie it can raise its bids by a
small $\epsilon$. By doing that it removes the ties and gets a mass increase
on their payoff, and because $\tau^{*}_{1P}$ is continuous the time allocation
changes smoothly so that would be a profitable deviation. Therefore, there are
no ties on the equilibrium. ∎

######  Claim 2.

Consider $r_{n}\to 0$ and equilibrium bids
$(B^{*}_{1}(r_{n}),B^{*}_{2}(r_{n}))$ then
$(B^{*}_{1}(r_{n}),B^{*}_{2}(r_{n}))\Rightarrow(B_{b^{*}},B_{b^{*}})$, where
$b^{*}(\cdot)$ is the symmetric bidding strategy in the first-price auction
game when $r=0$. Furthermore, if $B^{*}_{i}(r_{n})$ is a a pure strategy for
all $n$, then $b^{*}(v|r_{n})\to b^{*}(v|0)$ for all $v\in[0,\overline{v}]$
(a.s.).

###### Proof.

Because the space of actions is compact, consider a subsequence of
$(B^{*}_{1}(r_{n_{k}}),B^{*}_{2}(r_{n_{k}}))\Rightarrow(B^{*}_{1}(0),B^{*}_{2}(0))$.
We assert that $(B_{1}(0),B_{2}(0))$ is a nash equilibrium in the mixed
extension of the game, which by Proposition 3 implies that
$B_{i}(0)=B_{b^{*}}$. This further implies that is the unique accumulation
point of the sequence $(B^{*}_{1}(r_{n}),B^{*}_{2}(r_{n}))$ which, given that
the space is compact, implies that $B^{*}_{i}(r_{n})\Rightarrow B_{b^{*}}$.

Next, observe that $e^{-r\tau^{no-news}(b_{i},b_{-i}|r_{n})}$ is increasing as
$n\to\infty$ and converges pointwise to 1, thus by Dini’s Theorem we have that
it converges uniformly to $1$. The same convergence result can be used for the
indicator functions $\mathbf{1}_{\\{\tau_{-i}\geq\tau^{no-
news}(b_{i},b_{-i}|r)\\}}$, $\mathbf{1}_{\\{\tau_{-i}\geq\tau^{no-
news}(b_{i},b_{-i}|r)\\}}$ which implies that $x_{i}(b_{i},b_{-i}|r_{n})$
converges uniformly to $x_{i}(b_{i},b_{-i}|r_{n})$ and therefore that
$u_{i}(v_{i},b_{i},b_{-i}|r_{n_{k}})$ converges uniformly to
$u_{i}(v_{i},b_{i},b_{-i}|r_{n_{k}})$.

Since there are no ties on the equilibrium (Claim 1) so that the payoff
functions are continuous when there are no ties (see proof of Proposition 3),
and the payoff function has uniform convergence, we have that
$U_{i}(B^{*}_{1}(r_{n_{k}}),B^{*}_{2}(r_{n_{k}})|r_{n_{k}})\to
U_{i}(B_{1}(0),B_{2}(0)|r=0)$ where

$U_{i}(B_{1}(r),B_{2}(r),r)=\int\int
u_{i}(v_{i},b_{i},b_{-i}|r)dB_{-i}(v_{-i},b_{-i})dB_{i}(v_{i},b_{i}).$

To show that $(B_{1}(0),B_{2}(0))$ is a Nash Equilibrium, consider a pure
strategy $b$. First, suppose that $b$ is increasing on $v$. Observe that
$U_{i}(B^{*}_{i}(r_{n_{k}}),B^{*}_{-i}(r_{n_{k}})|r_{n_{k}})\geq
U_{i}(B_{b},B^{*}_{-i}(r_{n_{k}})|r_{n_{k}})$ since
$(B^{*}_{i}(r_{n_{k}}),B^{*}_{-i}(r_{n_{k}}))$ is a Nash Equilibrium when the
discount rate is $r_{n_{k}}$. Because $b$ is increasing we know that there are
no ties so the payoff function is continuous and has uniform converge as
$n_{k}\to\infty$. Therefore, by taking the limit we conclude that
$U_{i}(B_{1}(0),B_{2}(0)|r=0)\geq U_{i}(B_{b},B_{2}(0)|r=0)$. If the function
$b$ is non-decreasing, then for every $\epsilon>0$ an increasing function
$\tilde{b}\geq b$ exists such $U_{i}(B_{b},B_{2}(0)|r=0)\leq
U_{i}(B_{\tilde{b}},B_{2}(0)|r=0)+\epsilon$. We conclude by taking
$\epsilon\to 0$, that for all feasible bidding strategies
$U_{i}(B_{1}(0),B_{2}(0)|r=0)\geq U_{i}(B_{b},B_{2}(0)|r=0)$. Therefore, the
uniqueness of equilibrium when $r=0$ implies that $B_{i}(0)=B_{b^{*}}$.

Finally, if $B^{*}_{i}(r_{n})$ is a sequence of pure strategy equilibria.
Consider $f_{\epsilon}$ a continuous approximation of the indicator function
$\mathbf{1}_{[v-\epsilon,v+\epsilon]}$. Because $B^{*}_{i}(r_{n})\Rightarrow
B_{b^{*}}$ we have that $\int f_{\epsilon}dB^{*}_{i}(r_{n})\to\int
f_{\epsilon}dB_{b^{*}}$. Thus, by taking $\epsilon$ we get the desired
pointwise convergence result. ∎

###### Proof of Theorem 3.

From Proposition 1, we have that the optimal stopping is independent of the
discount rate $r=0$, which in turn implies that the auctioneer’s revenue when
optimally running a second-price auction is independent of $r$. Thus,
$\pi^{*}_{2P}(r)=\pi^{*}_{2P}(0)$.

For the first-price auction case, we want to claim that for $r_{n}\to 0$, and
any mixed equilibrium of the game when $r_{n}>0$ we have that
$\pi^{*}_{1P}(r_{n})\to\pi^{*}_{1P}(0)$. Indeed, consider a sequence of mixed
bidding equilibrium $(B^{*}_{1}(r_{n}),B^{*}_{2}(r_{n}))$. From Claim 2, we
have that $B^{*}_{1}(r_{n})\Rightarrow B_{b^{*}}$, where $b^{*}$ correspond to
the symmetric bidding strategy in the unique equilibrium of the first-price
auction game when $r=0$ (Proposition 3). Fixing $\tau_{1},\tau_{2}$, we have
that
$\tau^{*}_{1P}(b1,b2|r_{n})\to\min\\{\tau_{1},\tau_{2}\\}:=\tau_{1}\wedge\tau_{2}$
converges pointwise and $\tau^{*}_{1P}(b1,b2|r_{n})$ is increasing on $r_{n}$.
Then, Dini’s Theorem implies that the convergence is uniformly on
$b_{1},b_{2}$. Therefore, the following limit holds

$\displaystyle\lim_{n\to\infty}\int\int
e^{-r\tau^{*}_{1P}(b_{1},b_{2}|r_{n})}\min\\{\mu^{1}_{\tau}b_{1},\mu^{2}_{\tau}b_{2}\\}dB^{*}_{1}(r_{n})dB^{*}_{2}(r_{n})$
$\displaystyle=\int\int\min\\{\mu^{1}_{\tau_{1}\wedge\tau_{2}}b_{1},\mu^{2}_{\tau_{1}\wedge\tau_{2}}b_{2})\\}dB^{*}_{b^{*}}dB^{*}_{b^{*}}$
$\displaystyle=\min\\{\mu^{1}_{\tau_{1}\wedge\tau_{2}}b^{*}(v_{1}),\mu^{2}_{\tau_{1}\wedge\tau_{2}}b^{*}(v_{2})\\}$

By taking expectations on $\tau_{1},\tau_{2}$, we conclude that
$\pi^{*}_{1P}(r_{n})\to\pi^{*}_{1P}(0)$.

Finally, we invoke Theorem 1 that shows that $\pi^{*}_{1P}(0)<\pi^{*}_{2P}(0)$
and conclude that a $\underline{r}>0$ exists such that for
$r\leq\underline{r}$, $\pi^{*}_{1P}(r)<\pi^{*}_{2P}(r)$. ∎


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