Aston University
Final Year Project
Revenue Equivalence and the
Importance of Auction Design
Author:
Supervisor:
Sulman Khan
Prof. David Lowe
119209086
Course:
BSc Mathematics with Computing
April 30, 2015
Contents
Abstract
i
Acknowledgements
ii
List of Tables and Figures
iii
1 Introduction
1.1
1.2
1.3
1
Online Advertising
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Game Theory and Ad Auctions
. . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.1
Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.2
Risk Neutrality
6
1.2.3
Designs of Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Revenue Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Auctions
2.1
2.2
2.3
2.4
9
Common Auction Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
Open Bid Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2
Sealed Bid Auctions
3.2
3.3
9
11
Importance of Auction Design . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Bidding Functions in First Price Auction . . . . . . . . . . . . . . . . . . . .
14
2.3.1
2 Bidders
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.2
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Bidders
Expected Payos in a Second Price Auction
. . . . . . . . . . . . . . . . . .
Expected Revenue
18
22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1.1
First Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1.2
Second Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Violating Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2.1
Risk Approach
30
3.2.2
Auction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Example of Auctions Violating Equivalence . . . . . . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Conclusion
4.1
9
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Revenue Equivalence Theorem
3.1
1
34
Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendices
34
36
1
Basic Statistical Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2
Proof of Revenue Equivalence Theorem . . . . . . . . . . . . . . . . . . . . .
39
References
41
Abstract
This project explored the concepts studied in the AM30MR Mathematics Report and
develops on the mathematical foundations of the research. It started with a discussion
of how auctions have revolutionised the online advertising medium over the last decade.
Four common auction designs were discussed; First Price, Second Price, English and the
Dutch auction. An important concept from game theory called the Revenue Equivalence
Theorem was then explored. Initially derived by William Vickrey in 1961, the theorem
states that given bidders are risk neutral with their valuation only known to themselves
and that the object is allocated to the bidder with the highest valuation, a seller can
expect to gain the same revenue regardless of the auction design.
The research concluded that the same revenue can indeed be obtained through various
auction mechanisms, veried with the First Price and Second Price designs. Examples using the Uniform distribution on [0,100] were given to illustrate this in practice,
followed by a general proof for each mechanism. Scenarios were also proposed where
the Revenue Equivalence Theorem may not hold given violation of one or more of the
conditions. These included deviating risk attitudes from risk neutrality and breaking
of auctions rules such as collusion of bidders, resulting in valuations that are no longer
independently distributed. eBay was used as a prime example to show violation of
the Revenue Equivalence Theorem, since bidders can publicly view previous bidding
information about an item which may aect their bidding behaviour in the remainder
of the auction.
Primary source reference materials were used when conducting this research, predominantly books, journals and original sources where possible to form basic opinions and
this was augmented with examples of my own.
i
Acknowledgements
I take this opportunity to thank my supervisor, Professor David Lowe at Aston University, for his support this year. He has been a great supervisor to me when completing
my Final Year Mathematics Report as well as my Final Year Project, which is greatly
appreciated. I would also like to thank my examiner Dr. Juan Neirotti for his time
and eorts in examining my work during this research project and in the AM30MR
Mathematics Report.
The support of my family and friends throughout my studies at Aston is what has
helped me to progress. I have developed some great relationships both professionally
and personally with students and sta. I stress a big thank you especially to my mother
and father for their unreserved support over the last 22 years of my life.
ii
List of Figures
1
Comparison of Percentage Global Advertising Spend Between 2013-2018 . . .
2
Global Internet Advertising Spend by Category
3
Risk-Utility Indierence Curves
4
Hierarchy of Auction Designs
5
First Price Bidding Prole . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Bidding in the Second Price Auction
. . . . . . . . . . . . . .
7
Overbidding in the Second Price
8
Second Price Payos
9
Second Price Auction Payos
10
Second Price Auction Payos
11
Bid History in eBay Auctions
12
Uniform Probability Distribution Function (f (x))
. . . . . . . . . . . . . . .
37
13
Uniform Cumulative Distribution Function (F (x)) . . . . . . . . . . . . . . .
38
bi > v
2
. . . . . . . . . . . . . . . .
3
. . . . . . . . . . . . . . . . . . . . . . . . .
6
. . . . . . . . . . . . . . . . . . . . . . . . . .
10
b<v . . . . .
Auction bi < v < b
17
19
. . . . . . . . . . . . . .
19
. . . . . . . . . . . . . . . . . . . . . . . . . . .
20
b<v
b<v
. . . . . . . . . . . . . . . . . . . . . . .
20
. . . . . . . . . . . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . . . . . . . . . . . .
32
iii
1
Introduction
The AM30MR Mathematics Report introduced the concept of online advertisement (ad)
auctions and their relation to mathematical auction theory.
Concepts such as the Nash
Equilibrium were discussed with an analysis into why search engines preferred the Generalised Second Price auction to the Generalised First Price.
This coincided with Google's
unique choice of auction style involving a factor known as quality score when ranking advertisements in search results. This research project is about a game theoretical concept that
shows dierent auction design types gives the same return to the seller. When designing auctions, sellers want to gain the highest revenue possible from their sales. Using the theory of
auctions and a brief overview of four of the commons designs used today, the concept known
as the Revenue Equivalence Theorem (RET) will be explored. Specic examples implementing two dierent auction designs known as the First Price and Second Price mechanisms
are used to prove that the RET holds provided particular conditions are fullled. Cases are
discussed where designs can violate these conditions to show that the revenue would not be
equivalent, concluding with ndings, reection and developments to this research.
1.1 Online Advertising
[30]
Online advertising has matured dramatically over the past decade
, from selling adver-
tisement spaces, to the introduction of clickability moving towards pop up adverts
then the paid placement model (PPM)
[4]
[16]
and
, an approach that allowed advertisers to purchase
more desirable advertising slots on search engines
[13]
.
Why online?
User behaviour and
[27]
surng habits can be measured more eectively online compared to oine
.
1998,
In
a
company called GoTo.com introduced the rst auction feature where online adverts were
[13]
ranked subject to bids submitted by advertisers
1999,
new economic revenue stream and by
billion industry
[13]
. Known as ad auctions, they became a
the online advertising medium had become a $1
. Today this form of advertising is closing in to become the largest en-
2018[30] .
tertainment and media advertising segment, forecast to be worth $194.5 billion by
Combined with television advertising, online advertising occupied a joint share of almost
50% of the advertising market in 2013
[4]
. Figure 1 shows that by 2018, this is forecast to
rise to 55% at the expense of newspaper advertising. As of May
2006,
the combined value
of Google and Yahoo!'s revenue from online advertising exceeded $150 billion
[10]
.
Vieria and Camilo (2014) discuss how the ranking of webpages shares a similar principle
[40]
to that of supermarket shelf stacking
stacked on prime shelves i.e.
. Product manufacturers pay for their items to be
[40]
at eye level or within arm's reach of customers
.
These
items tend to be of a higher price than those on the lowest shelf or topmost shelf, resulting
in greater sales and maximum prots
[40]
.
There are limited items that can be stacked on
prime shelves. Likewise, a limited number of adverts can be displayed on a page, ranked
[13]
in order of prominence
. Each position has a desirability unique to advertisers
[10]
. A user
[10]
is more likely to click a link at the top of their search rather than further down the page
Hence, a website more likely to receive a click has a greater chance of improving business
1
.
[4]
Figure 1: Comparison of global percentage categoric advertising spend between 2013 and 2018. Television
and internet occupied 49% of the advertising market in 2013 and is forecast to reach 55% by 2018 with
newspapers making the largest cuts in advertising spends. (*CAGR: Compound Annual Growth Rate)[30] .
making the top advertising space more demanding to certain advertisers based on their needs.
Today companies utilise search engines for advertisement purposes. There are three common forms of online advertising; display adverts, contextual adverts and sponsored search
adverts.
Display adverts come in dierent forms of banners such as animations or videos
often delivered by famous brands
[14]
.
Prices can be xed, negotiated in advance or even
decided through auctions similar to those used for sponsored search ad auctions
[27]
. Con-
textual adverts are customised and based on the content a user is currently viewing or has
previously viewed.
For web search, advertisers submit their maximum bids for particular
keywords to the search engine provider
[10]
[38]
tiser's keyword, an auction commences
[1]
the ads, has fully loaded
. When a user's search query matches the adver-
. The auction concludes when the page, including
. The advertiser with the highest bid has their webpage ranked
rst in the search results, with advertisers of successive bids ranked one after the other
[10]
.
On clicking these links, the user is navigated to the company's website. In turn, the company
[38]
pays the search engine a fee. This approach is called pay per click (PPC)
. When multiple
companies require the same keyword(s), the ranking of these links and the amount payable
to the search engine (per click) invokes the need of auctions. Easley and Kleinberg (2010)
discuss how bids for particular queries such as loan consolidation, mortgage renancing,
[8]
and mesothelioma can reach an excess of $50 per click, for every user that visits the site
.
Sponsored search advertising is similar to that of contextual advertising because adverts are
targeted specic to a users behaviour and in sponsored search, adverts are targeted according
[27]
to key words that are searched
. A report by PricewaterhouseCoopers LLP discusses how
[30]
search will retain its dominant position in online advertising
spending on various online advertising mechanisms.
2
. Figure 2 shows the global
Figure 2: Global percentage internet advertising spend by category. Sponsored search advertising has been
the persistent leading form of advertising since 2009, followed by display advertising. (*CAGR: Compound
Annual Growth Rate)[30] .
Internet auctions play an important contribution in electronic commerce
[4]
. For many web-
sites providing free content such as videos, news and blogs, advertising is their primary source
of revenue
[1]
. Recent developments in targeting consumer behaviours has increased the eec-
[1]
tiveness of these adverts
. With the rise of social media and online trac, it has become a
simple process for advertisers to purchase online advertising space. Jason Knapp developed
the Real Time Bidding (RTB) Exchange, similar to the stock exchange, but primarily for
the use of advertising on websites
[16]
.
Analysing and characterising online consumer behaviour requires data, formulas, statistics,
analytics, correlations, patterns, predictive modelling and testing
mathematics to ensure an advert reaches its target audience.
[16]
- all of which rely on
Auction theory and game
theory play a signicant role when a company is trying to ensure they attract business
[16]
customers over their competitors
.
The next section introduces auction theory and game theory and the signicant role they
play when a company is bidding for online advertisements, including concepts such as the
Nash Equilibrium and designs of auctions.
3
1.2 Game Theory and Ad Auctions
In an auction, every bidder has their own valuation
v
and place a bid
b
on the item for sale.
In an online advertising auction (for web search), the item is an advertising slot
may be multiple slots
si = s1 , s2 , ..., sn
s.
There
available in which case advertisers prefer the upper
most slot, since this will be the link more users are likely to click on, hence the prominence
of
s1 > s2 > ... > sn .
A game involves
n
[3]
The auction can be represented as a game
number of players where
n < ∞.
.
Each player naturally forms a set of
strategies to win over their competitors, known as the dominant strategy - one that will make
a player successful in the game. There also needs to be a quantitative means of describing
[2]
the winnings or losses for each player in every game
. Strategies can be very simple or vastly
complicated as in the game of chess, for example. In advertising, there are many dierent
moves that can be taken, implying the many possible (although nite) strategies
[2]
.
A simultaneous game is one where players make a move at the same time. This means each
player uses their best strategy every turn, as they do not know what the other player(s)
are going to do. An example of this is rock-paper-scissors
[3]
. Each player pursues a strategy
they think will help them win. Where a player does have knowledge of other players previous
or future strategies, this is known as perfect information
[3]
. (This is not to be confused with
complete information where all players are aware of each others moves). Conversely, games
where players make a move one after another are referred to as sequential games
[2]
.
Cases where players are unaware of the strategies of other players are known as imperfect
information
[2]
. The relevance of game theory in auctions can be seen here. Ad auctions are in
fact games of imperfect information. Each player (each bidder of the auction) is unaware of
the strategy of the other players in the game. Players have information private to themselves
that other bidders are not aware of
[3]
.
All players know the rules of the auctions and therefore a distribution of values for the
object(s) to be won is known. A company's valuation for their desired advertising slot may
vary subject to the time of day, holiday period, location and the time of year which may be
[28]
appropriate to their business
1.
b > v,
2.
b < v,
3.
b = v.
. There are three possibilities:
When multiple companies submit a bid for the same keyword (bi , i
= 1, ..., n),
an auction is
run as soon as the user searches for a query and terminates once the page has fully loaded.
Whichever of the above possibilities is attributed to a bidder, depends on the design of the
auction (see Section 1.2.1 on Nash Equilibrium).
Companies are ranked in order of the
highest bid, which takes the top slot, followed by the ranking of successive bids.
case of Google, a quality score is combined with
4
b
(In the
to decide on the ranking. This prevents
large companies with excessive budgets from dominating the prominent slots whilst allowing
adverts to be displayed according to relevance)
[13]
.
The next section introduces a concept called Nash Equilibrium - a state when companies
[19]
reach their optimum strategies and are unable to progress or improve any further
1.2.1
.
Nash Equilibrium
Fundamental concepts of game theory and auctions include Nash Equilibrium. When bidders
partake in an auction and apply their dominant strategies (regardless of what other players
[19]
do), optimal bidding behaviour is reached which is known as a Nash equilibrium
.
A
Nash Equilibrium is a steady, self-reinforcing state where no further strategy can benet any
[28]
player
. We start by dening a game environment as
• N = 0, ..., n
where:
represents the list of participants in the game,
• Ω = {ω1 , ...ωn }
• Θ = {θ1 , ...θn }
• {vi }
Γ = {N, Ω, Θ, {vi }},
represents the strategy space of all game players,
represents the payo functions for each player,
is the valuation for each bidder
[28]
.
The utility function for the players can be expressed as Ui (Ω) = U1 (ω1 , ..., ωn ), ..., Un (ω1 , ..., ωn ).
∗
∗
The strategy prole {ω1 , ..., ωn } is a Nash Equilibrium if every agent maximises their utility
[28]
given every other player sticks to their equilibrium strategy
:
∗
∗
Ui (ωi∗ , ω−i
) ≥ Ui (ωi , ω−i
).
(1)
If a change is made to the equilibrium status, it could result in the player bearing responsi-
[19]
bility for heavy costs as well as negative consequences
. Each players strategy is only the
`best response' to the other players' strategies if they are in equilibrium
[28]
. Otherwise, there
will be other strategies available which provide a more optimal utility. Winning allocations
are known to be Pareto-optimal if there is no state available where all bidders can be bet-
[35]
ter o, without taking a strategy that makes at least one of the other bidders worse o
.
Section 2.1 discusses various auctions designs as well the Nash Equilibrium strategies for the
First and Second Price auctions.
[42]
Game theoretic analysis of auctions assume bidders are rational
. A rational bidder is one
who bids according to the Nash Equilibria of the auction assuming that their utilities are of a
von Neumann-Morgenstern nature, i.e. utility functions are linear to valuations. Equilibrium
values only operate if it is assumed game players are rational and that they are playing the
game to maximise their utility, although in practice there is evidence to suggest bidders are
not always rational. For example, bidders may choose their best response strategies based
on historic information or knowledge from previous auctions they may have participated in
[33]
to provide them with an advantage
.
5
1.2.2
Risk Neutrality
A risk neutral bidder is one which is indierent to the choices between risk and return
available to them, since the expected value of each choice is identical. A bidder who is risk
averse prefers not to take any risk in increasing their utility i.e. given two choices with the
same expected return, they will take the less risky option
[19]
.
On the other hand, a risk
seeking bidder would take any risk involved to increase their return.
Figure (3) helps to
visualise the utility of these bidders with respect to their risk proles
.
[28]
Utility
Risk neutral
Risk averse
Risk seeker
Risk, σ
Figure 3: Dierence in utility for bidders with unique risk attitudes. The Revenue Equivalence Theorem
requires bidders to be of a risk neutral nature i.e. they are indierence to choices between greater risk or
return and would obtain the same utility for either choice. A risk seeking bidder would take a greater risk
to improve their utility whilst a risk averse bidder would gain a higher utility from a choice with certainty
of winning rather than a higher expected value of winning i.e. they take a less risky choice.[42]
1.2.3
Designs of Auctions
v
2
v . Whichever tactic bidders employ during the bidding process is subject to the
During an auction, bidders may choose to bid a fraction of their valuation for example
or even
b=
b=
auction design. (Optimal bidding strategies are known as a Nash Equilibrium which will be
proven for both First Price and Second Price in Section 2). Each bidder determines the best
point they would be prepared to submit their bid to obtain their maximum expected gain
given the information they are aware of or have assumed of other bidders, such as probable
bids
[39]
.
An example of where the importance of auction theory and game theory were highlighted
was through the design failings of 3G spectrum auctions in the Netherlands and in Italy
in 2000, where both led to revenue signicantly less than forecasted due to shortcomings
in auction design. Due to the rules of another 1990's auction in New Zealand, only $5,000
was raised despite the winner bidding $7 million
6
[19]
. An auction in Germany for 3 spectrum
blocks only sold at the reserve price and several other US auctions backred, again due
to design aws and collusion of bidders through predetermined signals
[19]
. Academics Ken
Binmore and Paul Klemperer had addressed these issues in their auction earlier in the year
1
for 5 telecomms licenses, which led to a return of $34 billion (≡ £22 billion) for the UK
2
[17]
government
, through utilising fundamental concepts of auction theory and game theory.
The cumulative return of over $100 billion from worldwide 3G auctions has since provoked
[19]
great interest and developments in these areas
.
This section discussed auction design using the success of the 3G spectrum auctions as an
example. The next section introduces the mathematical concepts surrounding the Revenue
Equivalence Theorem.
1.3 Revenue Equivalence Theorem
There are many dierent auction designs used today
[26]
. When an item is being sold, the seller
aims to gain the maximum possible revenue from their sale, hence they prefer an auction
design that provokes this
[26]
. They may prefer a Second Price to First Price mechanism for
example, because bidding is truthful and bidders submit bids representing their maximum
valuations.
Yet there is a theory which suggests that given certain conditions, it can be
proven that these designs will yield the same revenue for the seller
[19]
. Known as the Revenue
Equivalence Theorem (RET), the theory was rst developed by William Vickrey in 1961
[39]
for a simple case using the Uniform distribution
, later developed in the 1980's by other
academics. The theorem arose from a study of optimal auctions; those designed to maximise
a sellers expected revenue
[34]
.
Myerson (1981) and Riley & Samuelson (1981) discuss the
more generic cases of RET with
n
bidders and various distributions throughout bidders.
The RET could potentially take away the `stress' of designing an auction to provide the
best return for a seller, since they no longer need to tailor designs to nd the best revenue.
[8]
However, the theorem only holds provided the general conditions below have been satised
Conditions for Revenue Equivalence
Assume potential buyers have a signal about the value of an object, then any auction
mechanism in which the below conditions are satised yields the same expected revenue
[19]
to the seller (and each bidder making the same expected payment):
∗
∗
All players are risk neutral.
Each player's valuation
v
is private information (only known to themselves)
and is independently drawn from a continuously dierentiable function
∗
∗
The object is allocated to the player with the highest bid.
A player with
v=0
does not make a prot from the auction
7
[12]
.
F (v).
.
This was used as a motivation to study auction designs, raising questions as to why multiple
designs existed if they could all achieve the same revenue.
A number of analyses were
spawned in the 1980's which resulted in `tweaks' to the model to explain the preference of
[24]
particular auctions in practice
. The next section begins with a discussion of 4 common
designs implemented today and areas where they are used. Bidders payos are calculated
for two auctions mechanisms, First Price and Second Price. The mathematical workings of
the RET are discussed in Section 3 using examples to illustrate how it works with a Uniform
distribution on the interval [0,100]. The examples show that the same expected revenue for
the seller can be achieved (provided the above conditions are satised for the auction). A
proof with a general distribution is also done to demonstrate this concept. (A general proof
for the RET is stated in Appendix C). The study then proceeds to discuss cases where the
RET may not hold, concluding with ndings of this research and an evaluation.
8
2
Auctions
To motivate why we study the RET later in this research, this chapter explores what an
auction is and the types of auction designs that can be used. Although bidders may prefer
one auction style over another, this section will discuss and calculate how much a bidder can
expect to pay during an auction and how this payo is subject to the auction design being
used.
An auction is a sale of real or personal property through open public bidding
[11]
. According
to Chaey et al (2009), a bid is a nancial commitment made by a trader under conditions
generally agreed in advance of the auction. An auction is complete when a bid is accepted
[11]
by the seller, creating a binding contract
.
The advancement of internet and technology now allows buyers to bid in an auction from
anywhere around the world especially on websites such as eBay.
Auctions are used for
dierent purposes, with many new markets being introduced innovatively for example for
mobile-phone licenses, sales of electricity and pollution permits
[19]
.
A large quantity of
products, services, nancial instruments and property are sold through auctions and the
dierent variants of auctions are suited to individual industries
[19]
.
Easley and Kleinberg
(2010) discuss how auctions are generally used by sellers in situations where they do not
have a good estimate of the buyers' true values for an item and where buyers do not know
each other's values
[16]
. Hence they induce competition between interested buyers serving
[19]
as a mechanism for price discovery
.
Popular auction designs are discussed in the next
section.
2.1 Common Auction Designs
There are 4 common auction designs used today
[8]
. These include sealed bid auctions known
as First Price and Second Price, and open-bid auctions known as the Dutch auction and
English auctions
[19]
. Figure (4) shows a hierarchy of these auctions. Each have their own
advantages and disadvantages which are briey discussed.
2.1.1
Open Bid Auctions
This section considers auction designs of an open nature, where bidders participate in an
open area and are subject to observations of their bidding practice by other bidders in real
time
[8]
.
2.1.1.1
Ascending Auctions
The ascending bid auction is a common mechanism used today. These traditional auction
models assume that each bidder bears an intrinsic value,
not exceeding their valuation of the product being sold
the case
[6]
. Bidders tend to adjust their idea of
9
v
[8]
v,
in mind i.e. they bid up to but
. In a typical auction, this is rarely
whilst learning more about the product
Auction
Designs
Open Bid
Sealed Bid
Auctions
Auctions
Ascending
Descending
(English)
(Dutch)
First Price
Second Price
Figure 4: Hierarchy of auction designs discussed in this research project.
being sold as well as the habits of other bidders
[6]
. This, along with the rules of the auction,
provokes a game between bidders.
Known also as English auctions,
[19]
they are conducted in real-time with bids made either
electronically or in person. Each bidder increments their bid, whilst other bidders exit the
[8]
auction once their intrinsic value has been reached
.
Cars and houses are sold using this method, particularly online. Today companies like eBay
use the ascending auction mechanisms to allow goods to be sold, since they serve as a reliable
process of price discovery, creating open competition
information which inates the value of the item
the bidding process, further bids are provoked
[6]
[15]
[15]
. Often, bidders may hold crucial
. As more information is revealed during
. Another advantage to ascending auctions
is that they allow for the opportunity to exceed the highest bid, but the loss of the auction
would constitute to a bidders' assessment of the items true value, particularly if they consider
selling the item at a later stage
[6]
.
Cramton (1998) states experiments have shown that ascending auctions perform well because the incentive for playing the dominant strategy is clearer to bidders
for an item with
well before
v
v = £10,
[6]
. For example,
it would be bad practice for a bidder to exit an ascending auction
has been reached or even bid once
v
eciency as well as participation of more bidders
has exceeded
[6]
[6]
. This strategy encourages
.
However, ascending auctions are vulnerable to bidders `spying' on each other during the
bidding process.
This allows them to establish whether or not their own strategy will be
[19]
successful, before placing their bid
.
10
2.1.1.2
Descending Auction
After the established tradition to use this mechanism for the sale of owers in the Nether-
[8]
lands, descending auctions were branded Dutch auctions
. As the name suggests, the seller
starts from a high price and decreases the bid sequentially. In the Dutch ower auctions,
[19]
potential buyers are seated at desks in the same room with buzzers connected to a clock
.
The clock displays information about the item being sold and when the auction begins, lights
around the clock illustrate the percentage decrease from the start price until a bidder buzzes
to accept and pay their bid. Where there are multiple items available, the buyer can choose
how many lots they intend on buying and any remaining lots will be re-auctioned.
This
auction is particularly useful for sellers who intend on nding the optimal market value for
their product. Fish are sold using a similar mechanism in Israel as are tobacco products in
[19]
Canada
.
The Dutch auction is equivalent to the First Price auction (discussed later in this section).
The seller lowers the price of the item from an initial high price and no bidder accepts a
[19]
bid until someone's valuation is reached
. Bidders do not learn any new information about
other bidders during the course of the auction, except that no one's
bidder has their own bid
bi
v has been reached.
Each
they are willing to accept when the price reaches so. Similarly
with the First Price auction, we will see how this price plays the role of bidder
i's
bid,
bi ,
such that the item goes to the bidder with the highest value and this bidder also pays the
[8]
value of their bid
2.1.2
.
Sealed Bid Auctions
This section considers auction designs of a closed nature. The First Price and Second Price
mechanisms are discussed along with advantages and disadvantages of each.
2.1.2.1
First Price Auction
In the First Price auction, bidders each submit their bids in private. The bidder holding the
highest bid pays their price and wins the item. This style of auction can be ecient and
convenient for both the buyers and vendors when buying a property for example. Bids can
exceed the guide price, landing sellers with a sale larger than they may have rst anticipated.
Bidders are not actively submitting multiple bids in competition with each other and their
[8]
`sealed' bids cannot be adjusted
.
Bidders are all unaware of what each other are thinking in the process
[3]
.
Each bidders
valuation of the product is only known to themselves (imperfect information) and they do
not want to bid at or above
v
[19]
otherwise this would result in a zero or negative prot
.
The environment is of the nature that bidders `fabricate' their bid, resulting in an inecient
bidding process and detrimental eects to the seller's revenue.
Equilibrium or optimal strategy to bid truthfully (i.e.
b = v)
Hence it is not a Nash
in the First Price auction
[28]
.
These types of auctions are commonly used for submitting government contracts and selling
[19]
real estate property
.
11
One major disadvantage to the First Price auction is collusion between bidders. As bidders
[29]
discuss their perceptive valuations of the product, bids can be rigged
procurement auctions for school milk has resulted in
being convicted with nes totalling
$46
million
[29]
45
.
Bid rigging in
corporations and
49
individuals
.
We next introduce the Second Price auction with advantages and disadvantages to this type
of design.
2.1.2.2
Second Price Auction
The Second Price auction allows the highest bidder to win the item but instead pay the price
of the second highest bidder,
bi+1 .
Whilst Second Price auctions are dierent to English auc-
tions, the approach of bidders bidding up to
v
(their optimum value) remains the same.
[28]
Hence it is a Nash Equilibrium to bid truthfully
because
v
. If a bidder quits the auction (perhaps
has been exceeded) they are not allowed to return
[19]
. The sealed-bid environment
means bids are submitted at once, making it a time ecient strategy, whilst encouraging simple bidding strategies. Klemperer (2004) states that the U.S. Treasury Department trialled
this auction type when trying the sell the national debt during the
1970's[19] .
When comparing to the First Price sealed-bid auction and English auction, the second price
auction reveals to other bidders, the demand of a product
[41]
. A disadvantage, particularly
in some online cases, allows sellers to spy on the auction and produce false bids to increase
the payment of the nal bidder
[41]
. Collusion amongst bidders, as in the case with rst-price
auctions, is another disadvantage to this system
[36]
.
Search engines adopted this mechanism prominently after it was designed by Yahoo! to overcome the issues faced with the Generalised First Price mechanism. In
2002,
Google adopted
and modied Yahoo!'s second price design to create today's online advertising auction which
takes place everytime a search is queried through their search engine
[13]
. This overcame some
of the issues faced with the First Price mechanism such as inecient allocations and volatile
prices
[16]
.
Other auction designs exists such as `all pay' auctions where all bidders, whether they win
[8]
or lose, have to pay their bid
.
These are not commonly used except in some contests.
`Penny' auctions are another uncommon example where bidders have to pay for submitting
a bid during the auction
[8]
. Madbid is an example of this where bidders purchase tokens to
participate in the auction. The next section discusses auction design in more detail.
12
2.2 Importance of Auction Design
Designing the `perfect' auction can be a dicult task due to great uncertainty and dynamics
of the environment. Sellers prefer a design that would bring them greatest return. However,
changing the auction design invokes changes in bidder behaviours and hence equilibrium
strategies
[21]
.
Designs also have to compensate for bidders' strategic behaviours in trying
to obtain items at the lowest price possible by adjusting their bids accordingly
[21]
. For this
reason, ascending auctions are particularly vulnerable to bidders reluctant to participate.
This can be the case with other auction mechanisms with high entry costs, inecient reserve
[18]
bids and signicantly large asymmetry between bidders
.
Sellers can design auctions in any way to bring them the best revenue, however they need
to remember that participating bidders are going to be producing that revenue.
If the
auction rules are such that bidders are deterred or put o from the auction because of fear
of losing signicant amounts of money, their expected revenue would decline and the auction
[18]
would be rendered inecient
.
It is in the interest of the seller to use a design which
mitigates the challenge of ensuring competition between bidders and preventing collusion
[18]
between bidders
. One means of enforcing this could be through punishments
[18]
unreasonable reserve bids provoke predation and collusion between bidders
[18]
. In fact,
.
If a seller designs their auction to satisfy the conditions of the Revenue Equivalence Theorem,
then they should be aware of the fact that mathematically, whichever mechanism they choose
would produce the same revenue. From a bidders perspective, with all conditions satised,
[13]
auctions should be designed to be appealing and robust. Auctions may have entry fees
.
This means regardless of whether a bidder wins or loses an item, they would be subject to
these fees. It is also assumed that the seller has complete commitment to their auction once
the rules have been dened. Bidders should have complete condence that the rules will be
honoured otherwise their bidding behaviours would be inuenced, should they become aware
that the seller may demand a higher price or become dishonest
[31]
. Factors like these can
inuence a bidders decision to even participate in the auction since they could potentially
result with a negative prot.
If the mechanism is unnecessarily complex, or there is an
unusually high reserve price, bidders may hesitate to participate altogether, proving only
[29]
disadvantageous to the seller
.
Comparably, an auction which doesn't require entry fees or bidders to pay their bid regardless
of whether they win the auction would be more appealing psychologically, since bidders do
not lose any money if they don't win the item.
the risk attitude of the bidder.
It can be argued that this is subject to
If a bidder is `risk seeking', they may be willing to join
such `high price' auctions because it suits their needs
[18]
. (From a sellers perspective, they
are still earning the same revenue as the previously discussed design since they have met
the conditions of Revenue Equivalence.
This denes the importance of auction design in
Revenue Equivalence).
We now move on to calculate expected payos for bidders under the First Price auction.
13
2.3 Bidding Functions in First Price Auction
This section mathematically considers bidders expected payos and optimum bidding strategy in the First Price auction. We start with a simple case of 2 bidders and then study a
more complex case with
n
bidders, using a Uniform distribution, concluding with a general
proof.
2.3.1
2 Bidders
Consider a simple auction with 2 bidders valuations independently drawn from a Uniform
distribution on the interval [0, 100] with the following:
• Bidder 1
Valuation,
bid,
v1 ,
b1 .
• Bidder 2
Valuation
bid,
v2 ,
b2 .
Since the valuation of each bidder is independent, they form a joint distribution and this
property of independence allows us to multiply probabilities together.
fV1 V2 (v1 , v2 ) = fV 1 (v1 )fV 2 (v2 ),
Assume bidder 1 will bid a fraction
α
(only if v1 , v2 are independent).
of their true value, which is less than
b2 ,
(2)
such that:
αv1 < b2 ,
which can be rearranged to
v1 <
b2
.
α
b2 (provided b2 < v2 )
F (v1 ):
b2
b2
b2
F
= P rob X ≤
=
,
α
α
100α
The probability of winning the item with bid
is represented by the
cumulative distribution function (CDF)
where
X
is the random variable.
(If
b2 > v 2 ,
prot). Let us now calculate the expected prots
the bidder will be subject to a negative
kF (b2 )
that could be obtained using the
First Price mechanism. If bidder 2 wins the auction with a bid
kF (b2 ) = (v2 − b2 ) ∗ F
14
b2
α
,
b2 ,
their expected prot is:
= (v2 − b2 ) ∗
kF (b2 ) =
b2
100α
,
(3)
1
(b2 v2 − b22 ).
100α
(4)
In order to obtain the maximum return prot, we must rst nd the turning points of the
bidding function for bidder 2. This requires dierentiating equation (4) (with respect to
and equating this to zero to nd the desired parameter
b2 )
b2 :
d 1
d kF (b2 )
=
(b2 v2 − b22 ),
db2
db2 100α
1
=
(v2 − 2b2 ) = 0,
100α
⇒ v2 = 2b2 ,
b2 =
v2
.
2
(5)
Expression (5) tells us that in the case of two bidders, the best response to the rst bidder
1
using strategy b1 = α ∗ v1 is for bidder 2 to use the strategy b2 =
∗ v2 i.e. submit a
2
bid equivalent to half of their valuation. This is referred to as the Nash Equilibrium since
it occurs at stationary points. Using the second derivative test, we can verify that it is a
maximum point:
d
1
d2 kF (b2 )
=
(v2 − 2b2 ) ,
db22
db2 100
2
=−
,
100
1
= − < 0, ⇒ maximum point.
50
Note that in the First Price auction, a higher bid has a greater chance of winning the
b2
auction. As α increases, the probability of the bidder winning also increases. Hence, F
α
in equation (3) would allude towards an increasing function. Conversely, since expected
prot will naturally decrease if a bidder increases their bid towards their maximum valuation,
(v2 − b2 )
in equation (3) would allude towards a decreasing function, hence:
lim v2 − b2 = 0.
b2 →v2
We now calculate the optimum bidding function for
15
n bidders under the First Price auction.
2.3.2
n Bidders
This section calculates the optimum bidding function for
who are distributed along
n
bidders in a First Price auction
b
is the highest amongst all other
U ∼ [0, 100].
Suppose bidder 1 proposes a bid
b.
The probability that
bidders is equivalent to the probability that all
n−1
other bidders have a bid
bi
less than
b.
If each bidder bids a fraction α of their value, the probability that any one bid is less than
b
as in the previous case. As bidders are independent, values form a joint
b is F ( αb ) = 100α
distribution and they can be multiplied together (by expression (2)). The probability that
all
n−1
bidders have a bid less than
b
is therefore:
n−1
Y
n−1
Y b
b
Prob[max{b2 ...bi } < b] =
F
=
,
α
100α
i=1
i=1
n−1
b
=
.
100α
Using equation (3), the expected prot for bidder 1 is therefore:
kF (bn ) = (v − b)
=
b
100α
n−1
,
1
n−1
n
vb
−
b
.
(100α)n−1
(6)
Calculating the rst derivative of equation (6) now gives the maximum value for
b:
d
1
d
n−1
n
kF (bn ) =
vb
−
b
,
db
db (100α)n−1
1
n−2
n−1
(n
−
1)vb
−
nb
=
.
(100α)n−1
Hence, the best bidding function can be obtained by letting
d
k (b )
db F n
=0
and nding the
stationary points:
d
1
n−2
n−1
(n
−
1)vb
−
nb
= 0,
kF (bn ) =
db
(100α)n−1
n−1
nb
(n − 1)v
bn−2
− n−2 = 0,
b
(n − 1)v − nb = 0,
nb = (n − 1)v,
b=
(n − 1)v
.
n
16
(7)
This expression denes the optimal bid function for
n
risk neutral bidders with valuations
U ∼ [0, 100].
We can conclude that with n bidders, they will reduce their
n−1
from their valuation. This is also the gradient of their linear bid
n
function which has a constant (y -intercept) of zero, especially since bidders do not result in
distributed on
bid by a fraction of
a negative utility when losing the auction. To verify that this expression holds, it can be
tested with the case for 2 bidders:
(2 − 1)v
v
bn=2 =
= ,
2
2
which is exactly as established in expression (5) i.e. in the case of 2 bidders, they will each
tend to bid half of their valuation. It can also be proven that as the number of bidders rise,
each bidder bids closer to their respective valuations. Figure (5) helps to visualise this. By
taking limits of expression (7):
(n − 1)v
,
n→∞
n→∞
n
⇒ lim b = v.
lim b = lim
(8)
n→∞
b
v
b=
(n - 1) v
n
n
Figure 5: First Price bidding prole. As the number of bidders increase, bids are made closer to their
valuations such that n → ∞, b → v .
17
For example, this bidding prole can be veried with the case of 3 bidders:
2v
v
(3 − 1)v
=
> .
bn=3 =
3
3 |{z}
2
Optimum bid
f or 2 bidders.
2
v
Hence an auction with three bidders would result in each bidder submitting a bid of
3
1
which is closer to v than v .
2
This section proved the optimum bidding strategy to give the highest payos for the First
Price auction. Examples used the Uniform distribution to show the Nash Equilibrium on
an interval [0,100]. It was found that as the number of bidders increase, the tendency is for
bidders to bid closer to their valuation, to win the auction. The next section discusses the
bidders payos in the Second Price auction as well as the optimal strategy for bidding in
various scenarios.
2.4 Expected Payos in a Second Price Auction
In a Second Price auction, once a bidder has decided on their valuation
submit a bid
b.
How they bid relative to
v
v
of a product, they
will be explored in this section.
Section 2.1.2.2 discussed how the goods are allocated to the second highest bidder
bi+1
under
b < v . The idea would
b > v (as this results
auction at b < v , they earn
the Second Price mechanism. Consider the case where a bidder bids
be to win the item at the lowest possible cost, since there is no gain if
in a negative prot to the bidder). In fact, if a bidder wins an
v − b.
These cases assume that
v
does not change during the auction.
If the highest amongst all bidders (except bidder 1) submits
bidders valuation, i.e.
still pay
bi
bi < v ,
then provided
b > bi ,
bi
which is less than the rst
the bidder would win the auction and
due to the second price mechanism whilst earning
v − bi .
If
b < bi < v ,
then the
bidder loses the auction and does not pay anything. But note they could have potentially
won the auction, had they bid
b = v,
as
b > bi .
18
Figures (6a) and (6b) help to visualise this.
Bidder coincidently wins the auction
since b > bi and b < v meaning a prot of
v − bi .
(a)
Bidder loses the auction as b < bi , hence
there is no advantage to bidding less than v .
(b)
Figure 6: Bidders payos when underbidding, b < v .
If bidder 1 decides to overbid such that
(Figure (7)) and earn
v − bi
bi < v < b ,
they would still win the item and pay
as the highest bid is remains below
bi
v.
Figure 7: Overbidding when b > v . The bidder wins the auction only because bi < v and the Second Price
auction allows bidders to pay the second highest price, which in this case is bi .
If
bi > v
and the bidder bids their valuation i.e.
b=v
(Figure 8a), they lose the auction as
b > bi > v , the
bi − v , as they pay more than anticipated.
they are not the highest bidder. If they decide to submit a higher bid such that
bidder wins the auction but at a negative prot of
Figures (8a) and (8b) illustrate these scenarios. Therefore, overbidding the true valuation
leads the bidder to a worse position potentially having to pay more for the item than they
initially considered it worthy of, contrary to bidding truthfully and not winning the object.
19
(a)
Bidder loses the auction since bi > v and
Bidder wins the auction since b > bi but
as bi > v , they bidder makes a loss of bi − v .
(b)
b = v.
Figure 8: Bidders payo in the Second Price Auction, bi > v .
In the case of underbidding where
b < v < bi ,
the bidder loses the auction by default,
since their valuation has been superseded (Figure (9a)). Similarly, if
b < bi < v ,
they lose
the auction since they have been outbid, as illustrated in Figure (9b). Notice from Figure
(10b), if they had bid their valuation
making a return of
b = v,
they would have won the auction whilst also
v − bi .
Bidder bids b < v when bi > v and therefore loses the auction since both b and v have
been passed.
(a)
Bidder bids b < bi < v and loses the
auction.
(b)
Figure 9: Second Price payo scenarios when b < v .
From these cases, we can therefore conclude that the best strategy for the Second Price
auction is to indeed bid your true valuation i.e.
b = v [28] .
Bidding below will only result
in losing the item with regret, particularly if the item is sold for a price within a bidders
valuation. In contrast, bidding above
v
can be a costly decision as a bidder would have to
[8]
pay more than their initial valuation, resulting in a negative prot from the sale
20
.
If bi < b < v the rst bidder wins the
auction, but only by chance of other bids. If
another bidder increases their bid by such
that bi + > b, then bidder 1 would lose the
auction.
(a)
If the bidder bids b = v , the Second Price
mechanism allows them to win the auction at
bi making a prot of v − bi .
(b)
Figure 10: Second Price payo scenarios when b < v but with truthful bidding.
We have shown that it is a Nash Equilibrium to bid truthfully in the Second Price auction.
Bidders use a bid function bi
P (bi )
= β(vi ) = vi such that the expected payo is vi −bi ×P (bi ) where
bi ]. Note that this is a `strictly increasing function'
represents Prob[all other bids<
(since bids monotonically increase during an auction) reiterating the fact that the highest
value wins the auction i.e. it is a bidders optimal strategy to bid
v [26] .
This chapter discussed key concepts in game theory and the relevance and importance of
auction design as far as maximising a sellers revenue is concerned. We will see in the next
chapter that there is an interesting result from game theory called the Revenue Equivalence
Theorem, where examples are used to prove that the revenue from the First and Second Price
auctions are indeed equivalent. Scenarios are also discussed where the Revenue Equivalence
Theorem may not hold when certain conditions are violated.
21
3
Revenue Equivalence Theorem
This chapter discusses the Revenue Equivalence Theorem using the First and Second Price
auctions as examples, where it will be proven that the expected revenue from the auction is
identical. Cases are also discussed, where conditions for Revenue Equivalence can be violated
to show that the theorem would not always hold true.
A seller is unaware of the values of bidders, hence would have diculty deciding which
auction mechanism to select in order to achieve the highest revenue. The return from any
type of auction constitutes to a random variable
[7]
.
Hence, if the distribution of bidders'
values is known, the expected revenue can be calculated illustrating equivalence across any
mechanism. This is subject to the following restrictions:
•
Risk neutral bidders - bidders are interested in expected return and may or may not
take a slight risk to obtain their return.
•
Symmetric bidders - valuations must be distributed independently.
•
The bidder with the highest value wins the item.
•
The bidder with the lowest value leaves the auction with a prot of zero
[12]
.
Now we investigate the expected revenue to a seller if they were using the First Price or the
Second Price mechanism.
3.1 Expected Revenue
In the previous chapter, the best strategy for two dierent auctions were discussed; one
where bidders shave their bid (First Price) and the other where truthful bidding is the best
strategy (Second Price).
From a bidders perspective, what they contribute to the sales
revenue is dierent, as was proven. This section veries the expected revenue to the seller
and proves the Revenue Equivalence Theorem for both First Price and Second Price auctions.
Specic examples are used where bidders valuations are independently drawn from a Uniform
distribution on the interval [0,100].
3.1.1
First Price Auction
Examples start with a simple case of 2 bidders and are then generalised to case of
to promote understanding of more realistic environments.
22
n
bidders
3.1.1.1
2 Bidders
When two bidders are competing in an auction, there are two possibilities for the outcome
of
b:
v1 = x and bidder 2 has a valuation v2 ≤ x. The probability
f (x) × F (x) (f (x) and F (x) hold the same meanings as described
x
1
and F (x) =
⇒ f (x) × F (x) =
By U ∼ [0, 100], f (x) =
100
100
1. Bidder 1 has a valuation
of this outcome is
in section 2.1.1).
x 1
.
100
100
2. Bidder 2 has a valuation
v2 = x
and bidder 1 has a valuation
v1 ≤ x.
Again, the
v2 = x drawn from U ∼ [0, 100] equates for f (x) and thus the chances
of v1 ≤ x equates to F (x). Therefore the probability of this occurrence is f (x)×F (x) =
probability that
1
100
x
.
100
Let the probability of
by the equation
x
taking the highest bid from either of the two outcomes be governed
pF 2 :
pF 2 = Prob[x > max{v1 ∪ v2 }],
1 x 1 x +
,
=
100
100
100
100
2 x =
.
100
100
We can now calculate the expected revenue
value is
x,
rF 2 .
Since
pF 2
is the probability that the highest
the expected value of the highest bid is therefore:
Z
100
E[pF 2 ] =
x.pF 2 dx,
Z 100 2
x x.
dx,
=
100 0
100
Z 100
2
=
x2 dx,
(100)2 0
3 100
2
x
=
,
2
(100)
3 0
"
#
1
2
1003
=
,
2
(100)
3
0
=
200
.
3
Recall that in the First Price equilibrium phase for two bidders, they only bid half of their
23
valuation. The expected revenue for the seller is therefore half of the expected value of the
highest bidders value; i.e.:
rF 2 =
100
1 200
×
=
.
2
3
3
We will now develop on this method to prove the case for
(9)
n
bidders and show that we can
still arrive that this same result.
3.1.1.2
n Bidders
Let pF n represent the probability that out of n bidders, x is the highest bid, drawn from
U ∼ [0, 100] as previously. As there are n bidders, this could occur in n dierent ways such
th
that bidder 1 could have the highest bid up until the n
bidder. If we let any one bidder
take the highest bid x, this implies there are n − 1 bidders who now have a bid less than or
equal to x. Again, since each bid is independent, we can multiply probabilities. Hence:
1 x n−1
n−1
.
pF n = nf (x) (F (x))
=n
100
100
The expected value of the highest bid can now be calculated:
Z 100
n
x.pF n dx =
x.xn−1 dx,
E[pF n ] =
n−1
(100)(100)
0
0
#
"
n+1 100
n
x
n
100n
100n+1
=
.
=
=
n
(100)n n + 1 0
n+1
n+1
(100)
Z
100
n−1
of their valuation (expression
n
n−1
times the expected
(7)). Therefore, the expected revenue from the auction would be
n
value of the highest bid:
It was previously proven that bidders will bid a fraction
rF n
n−1
=
n
Z
100
x.pF n dx =
0
n−1
n
=
100n(n − 1)
,
n(n + 1)
=
100(n − 1)
.
(n + 1)
100n
n+1
,
This can be veried for the case of 2 bidders:
100(2 − 1)
100
rF n n=2 =
=
.
(2 + 1)
3
It is clear that this is equivalent to the revenue found in expression (9) for the specic case
of 2 bidders and we have now calculated the expected revenue for the case of
24
n
bidders also.
The above derivations also show that as the number of bidders increase, bidders are likely
to bid closer to their true value, hence the revenue for the seller is likely to increase. (Figure
(5)). We now prove this for a general case with any distribution.
3.1.1.3
General Case For First Price Revenue
Consider the bid function of
provided
b
b = β(v).
Recall that the payo for a winning bidder is
is the highest bid in the auction
b > bi .
(v − b)
We begin by dening a utility function
for bidders, that is the probability that a bidder wins multiplied by the bidders payo:
U (v, b) = (v − b) × Prob(v wins),
= (v − b) × Prob(b > bi ),
= (v − b) × G[v].
(10)
It was previously established that truthful bidding was not a Nash Equilibrium in the First
b 6= v . In fact, bidders shave their bid as a function of their valuation
implying b < v and b − v 6= 0. Also, since the bid was a function of v , the inverse can be
−1
found such that v = β
(b). Hence:
Price auction so
⇒ (v − b) × G[β −1 (b)].
Now, to nd the optimum bid, we dierentiate this function and equate to zero to nd the
turning points. This represents the Nash Equilibrium optimal strategy. Using the random
variable
x:
dG dz
du
= −G(x) + (v − b) . ,
db
dz db
= −G(x) + (v − b)g(x).
Since
x = β −1 (b) = v
and
β(x) = b ⇒
dβ
db
=1
dz
.
db
and by the chain rule
dx
1
dz
1
dz
1
= dβ =
= 0
⇒
= 0 −1
.
db
dβ
β (x)
dβ
β (β (b))
dx
25
(11)
dβ
db
=
dβ dx
. Therefore,
dx db
Substituting this into expression (11) gives:
U (v, b) = −G(x) + (v − b)g(x).
1
β 0 (β −1 (b))
= 0,
= −G(x)β 0 (β −1 (b)) + (v − b)g(x),
= G(x)β 0 (x) + β(x)g(x) = vg(x),
d
= [β(x)G(x)] = vg(x).
dz
This rst order ordinary dierential equation can now be integrated directly. Solving for the
bid function
β(v) :
Z
⇒ β(v)G(v) =
v
xg(x)dx,
0
Rv
xg(x)dx
β(v) = R0 v
≡ E[x|x < v].
g(v)dx
0
(12)
This implies that regardless of the distribution function attributed to the bidders, the optimum bidding function is the conditional average of the bidders valuations i.e. bidders would
have to pay
b×
Prob[winning the auction]. Hence the revenue for the First Price auction
can be calculated as:
MFAP = β(v) × Prob(v|wins),
= G(v) × E[x|x < v].
(13)
The next section will consider the revenue equivalence for the Second Price auction mechanism using examples with the Uniform distribution on [0,100] with 2 bidders and
n
bidders
followed by the general proof to demonstrate the revenue equivalence.
3.1.2
Second Price Auction
Recall that under the Second Price auction, the best strategy is for bidders to bid their true
valuation and the winning bidder pays the price of the second highest bid. This implies that
the sellers expected revenue is the expected value of the second highest bid. This section
considers the expected revenue for the case of 2 bidders and then
n
bidders. The expression
will also be used to verify that 2 bidders revenue case holds true, nally concluding with a
general proof to show the same revenue as that of the First Price design.
26
3.1.2.1
2 Bidders
Suppose 2 bidders with values independently drawn from a
the probability that the second highest bid is
x.
U ∼ [0, 100].
Let
pS 2
represent
In the case of 2 bidders, this can occur in
two possible ways:
P (X ≥ x) =
1 − P (X ≤ x) = 1 − F (x). Hence, the probability that bidder 1 has value x and bidder
2 has a value at least x is:
x 1 1−
.
f (x)(1 − F (x)) =
100
100
1. Bidder 1 has a valuation of
x (f (x))
and bidder 2 has a valuation of
x and bidder 1 has a valuation of at least x i.e. P (X ≥
x) = 1 − P (X ≤ x) = 1 − F (x). This possibility also happens with a probability
f (x)(1 − F (x)).
2. Bidder 2 has a valuation of
The probability that either of these two cases occur is therefore:
pS 2 = 2f (x)(1 − F (x)) =
The revenue
rS 2
is therefore the expected value of
Z
2
100
pS 2
is:
1−
x .
100
100
rS 2 = E[pS 2 ] =
x.pS 2 dx,
Z 100 2
x =
x. 1 −
dx,
100 0
100
2
Z 100
2
x
=
x−
dx,
(100) 0
100
2
100
2
x
x3
=
−
,
(100) 2
300 0
"
#
1
2
2
1002
1003
=
−
,
2
300
(100)
0
1002
,
150
200
= 100 −
,
3
= 100 −
rS 2 =
100
.
3
27
Note that this is the same expression we obtained for the case of 2 bidders when using the
First Price auction mechanism where bidders implement equilibrium strategies, i.e.
with
2 bidders, the expected value of the second highest bidders' value is equal to half of the
expected value of the highest bidders value. We will now prove the revenue equivalence for
the case of
3.1.2.2
n
bidders.
n Bidders
In order for bidder 1 to be the second highest bidder with bid
bidders must bid at least as much as
equal to
x.
x
and all other
n−2
x,
one of the remaining
n−1
bidders have a bid less than or
This occurrence is comprised of the following 3 probabilities:
•
Probability of bidder 1 having a value
•
Probability that one of the
•
Probability that the remaining n −
x n−2
.
P (X ≤ x) = (F (x))n−2 = 100
x ⇒ P (X = x) = f (x) =
n − 1 bidders have a bid
x
P (X ≥ x) = (n − 1)(1 − F (x)) = (n − 1) 1 − 100
.
2
1
.
100
greater than or equal to
x ⇒
bidders have a value less than or equal to
x⇒
Hence,
rS n = nf (x)(n − 1)(1 − F (x))(F (x))n−2 .
The expected value of
rS n
can now be found by integrating with respect to
done in previous cases:
Z
100
x.rS n dx,
E[rS n ] =
0
Z
100
x.nf (x)(n − 1)(1 − F (x))(F (x))n−2 dx,
Z 100 1 x x n−2
= n(n − 1)
x.
1−
dx,
100
100
100
0
Z
n(n − 1) 100
x2 x n−2
=
x−
dx,
100
100
100
0
Z 100 n(n − 1)
x2
=
x−
xn−2 dx,
(100)(100)n−2 0
100
Z 100 n(n − 1)
xn
n−1
=
x
−
dx,
(100)(100)n−2 0
100
n
100
n(n − 1)
x
xn+1
=
−
,
(100)(100)n−2 n
100(n + 1) 0
=
0
28
x
as we have
n(n − 1) 100n
100n+1
=
−
,
(100)n−1
n
100(n + 1)
n
100
n(n − 1)
100
1
=
− ,
n 100(n + 1)
(100)n−1
1
1
= n(n − 1)100
−
,
n (n + 1)
(n + 1)(n − 1)100 − n(n − 1)100
=
,
(n + 1)
=
n−1
n+1
100.
(14)
This expression is identical to that obtained for the First Price auctions with
n
bidders. We
can verify this expression with the case of 2 bidders as found in Section 2.2.2.1:
n−1
n+1
2
−
1
100
100 n=2 =
100 =
.
2+1
3
This is the same revenue as we achieved with the First Price mechanism. The next section
proves the general case for the Revenue Equivalence in Second Price auction.
3.1.2.3
General Case For Second Price Revenue
Dene a bidding function
b = β(v)
where variables hold the same meaning as dened pre-
viously. Using the utility function dened in expression (10), we know that a bidder wins
when their valuation is highest resulting in payment of the second highest bid,
revenue is therefore calculated as a product of the probability that a bidder with
bi+1 . The
v wins the
auction and the expected value of the second highest bid:
A
(v) = P rob(v wins) × E[bi+1 |v = highest],
MSP
= P rob(Y1 ≤ v) × E[Y1 |Y1 < v] ,
{z
}
|
{z
}
|
CDF
Conditional Expectation
Rv
Z v
yg(y)dv
=
f (v)dv × R0 v
,
f (v)dv
0
0
= G(v) × E[Y1 |Y1 < v],
where
G(v) represents the distribution of the bidders.
for First Price auction (expression (13)).
29
(15)
Notice that this is identical to revenue
The revenue equivalence therefore holds and we have now proven that whichever of the First
Price or Second Price auctions are used, the expected revenue is indeed identical if the bidders
are independently drawn from a Uniform distribution on [0,100] and also in the general case
with any distribution. Provided bidders valuations are only known to themselves and the
auction allocates zero payos to bidders with
v = 0, the Revenue Equivalence Theorem holds
regardless of the auction mechanism implemented.
The theorem extends from the Independent Private Values (IPV) model, which assumes that
a product can be sold through any auction design. However it can be argued that owers
for example were traditionally sold through Dutch auctions and as there is little evidence to
show that they have been sold through other auctions, the Revenue Equivalence cannot be
guaranteed. This implies that there could be other factors involved which are not considered
by the IPV model.
The next section considers scenarios where the Revenue Equivalence
[8]
theorem may not hold
.
This section proved the revenue equivalence theorem for
n bidders and it was veried for the
case of 2 bidders using both First Price and Second Price auction mechanisms. Examples in
this section were used on the Uniform with an interval [0,100], but a more general proof is
given in Appendix C. The next section proceeds to discuss circumstances where the Revenue
Equivalence Theorem may not hold due to violation of the principle conditions.
3.2 Violating Equivalence
There are circumstances where the Revenue Equivalence Theorem does not hold - particularly
when the conditions listed at the beginning of this chapter are no longer satised. Bidders
may not be risk neutral. They may be risk averse or risk takers. They may or may not take
a risk to optimise their returns. This section will discuss some of the properties that can
violate the RET resulting in a dierent expected return.
3.2.1
Risk Approach
A rst price auction with valuations independently and randomly distributed with a bidding
b = v leads to a zero payo to the bidder since v − b = 0. A bid lower than
valuation, b < v leads to an increase in expected payo since now v − b > 0. For a risk
strategy of
the
neutral bidder, the expected payo uctuates and results in an increase in variance. A risk
averse bidder would prefer a reduced expected return if it means a lower risk and hence bids
higher than the equivalent risk neutral bidder
[22]
.
[22]
Risk implies linearity
function say,
b =
. Figure (3) shows that a risk neutral bidder has a linear bidding
β(v). It is a simple process to nd the inverse such that v = β −1 (b).
But suppose that a bidder is risk averse or risk seeking. Their function is no longer linear
as illustrated by Figure (3).
It may be a function of order 2 or greater as suggested by
the curve, in which case it now becomes dicult to nd the inverse function. A quadratic
bidding function will have 2 roots and any higher ordered function could have many roots.
30
In the second price auction however, when valuations are drawn from a common probability
distribution, a bidders risk attitude is independent of the optimal bidding strategy. In fact
it is still the weakly dominant strategy to bid
b=v
whatever the risk attitude of bidders
[22]
.
When comparing the risk attitudes, it becomes evident that the revenue generated by the
First Price mechanism would be greater with risk averse bidders than with risk neutral
bidders (since they bid higher). It would also be greater than the revenue generated by the
Second Price mechanism (since risk has no eect on the outcome)
3.2.2
[19]
.
Auction Rules
One of the assumptions made in the Revenue Equivalence Theorem is that the rules of the
auctions are followed by the bidders.
However, some rules are not as easy to enforce as
others. For example collusion between bidders helps them to learn valuations of each other
and discuss information about the item for sale
[22]
. They can then come to an agreement on
bidding practices between them. This is not easy to police in an environment where there
are a large number of bidders. In the First or Second price auction, collusion would force
the lower value bidders out of the auction unless they were willing to bid higher than the
collusive bids
[24]
.
Bidders each tend to have individual intentions for the object which may or may not be
common knowledge to the rest of the bidders in the auction.
A bidder who knows this
information along with the probability distribution(s) of the other bidders' valuations, will
typically use this information when proposing their bid.
[22]
equivalence would no longer hold
This implies that the revenue
.
3.3 Example of Auctions Violating Equivalence
Consider an auction where bids are displayed on a screen during an auction. Other bidders
can see each others current bids, as well as a previous history of their bids.
also notice how many other bidders are participating in the auction.
Bidders can
Future bids would
now correlate to this information, inuencing bidder functions and behaviours. This means
bids are no longer independent and the condition of the Revenue Equivalence Theorem
that requires independence of bidders has now been violated.
The auction may now not
produce the same expected revenue compared to other mechanisms that could have been
implemented. eBay is an example of this. Bidders can view a history of bids for an item
including the increments between each bid - see Figure (11). This combined with the number
[9]
of participants in the auction can inuence a bidders behaviour
.
Where there is collusion, bidders can discuss their thoughts (giving each other information)
whilst they observe each others behaviour, meaning independence is no longer true.
One
bidder may decide to take more of a risk whilst another would take less risk after discussing
with other bidders. In fact, when bidders are risk averse, First Price auctions yield a higher
expected revenue than in a Second Price auction
31
[5]
.
Hello. Sign in Sell
My eBay
Search...
All Categories
Back to item description
Bid history
Bidders:
17
31
Bids:
Time left:
3 mins 46 secs
Duration:
3 days
Refresh
Item info
iPad Air 2 128gb Gold 4G+WIFI
*BRAND NEW SEALED ­
UNLOCKED*
Automatic bids may be placed days or hours before a listing ends. Learn more about bidding.
Hide automatic bids
Bidder
Bid Amount
t***a ( 4 )
i***. ( 81
Current bid:
23­Apr­15 10:11:15 BST
£500.10
22­Apr­15 23:02:18 BST
)
£496.25
23­Apr­15 02:34:49 BST
) £485.00
22­Apr­15 23:02:18 BST
o***0 ( 36
)
£475.00
20­Apr­15 22:14:54 BST
o***0 ( 36
) £461.00
20­Apr­15 22:14:54 BST
)
£451.00
20­Apr­15 21:53:53 BST
£450.00
20­Apr­15 22:14:43 BST
£330.00
20­Apr­15 21:53:53 BST
n***n ( 780
o***0 ( 36
)
n***n ( 780
) 3***1 ( 40
)
£320.00
20­Apr­15 21:27:45 BST
3***1 ( 40
) £310.00
20­Apr­15 21:27:45 BST
k***1 ( 71
)
£300.00
20­Apr­15 19:11:46 BST
3***1 ( 40
)
£300.00
20­Apr­15 21:27:39 BST
k***1 ( 71
) £295.00
20­Apr­15 19:11:46 BST
3***1 ( 40
)
£290.00
20­Apr­15 21:27:34 BST
k***1 ( 71
) £280.00
20­Apr­15 19:11:46 BST
3***1 ( 40
)
£275.00
20­Apr­15 21:27:27 BST
k***1 ( 71
) £260.99
20­Apr­15 19:11:46 BST
p***h ( 247
)
£255.99
20­Apr­15 17:20:38 BST
p***h ( 247
) £245.00
20­Apr­15 17:20:38 BST
£240.00
20­Apr­15 19:11:37 BST
k***1 ( 71
)
p***h ( 247
) £235.00
20­Apr­15 17:20:38 BST
u***_ ( 223
)
£230.00
20­Apr­15 14:50:27 BST
8***p ( 36
)
£225.00
20­Apr­15 16:00:01 BST
u***_ ( 223
) £216.00
20­Apr­15 14:50:27 BST
r***o ( 275
)
£211.00
20­Apr­15 14:31:33 BST
e***d ( 28
)
£209.00
20­Apr­15 12:43:26 BST
e***d ( 28
) £206.00
20­Apr­15 12:43:26 BST
t***i ( 91
)
£201.00
20­Apr­15 11:12:28 BST
t***i ( 91
) £200.00
20­Apr­15 11:12:28 BST
£195.00
20­Apr­15 12:43:20 BST
£185.00
20­Apr­15 11:12:28 BST
£180.00
20­Apr­15 12:43:14 BST
£171.00
20­Apr­15 11:12:28 BST
£166.00
20­Apr­15 12:43:05 BST
£160.55
20­Apr­15 11:12:28 BST
£155.55
20­Apr­15 11:27:28 BST
£127.10
20­Apr­15 11:12:28 BST
e***d ( 28
t***i ( 91
)
) e***d ( 28
t***i ( 91
)
) e***d ( 28
t***i ( 91
)
) n***n ( 780
t***i ( 91
£510.10
Postage: £8.75 ­­ Express Shipping (Royal
Mail Special Delivery (TM) 1:00 pm). )
a***s ( 90
i***. ( 81
Bid Time
£510.10
)
) Item number: 301601377066
Enter your maximum bid:
Place bid
(Enter £520.10 or more)
Figure 11: List of recent bids for an iPad advertised on eBay. As this information is publicly available
to all participants of the auction, factors such as interest, demand and bid functions can inuence another
bidders behaviour or willingness to participate in the auction[9] .
32
It is well known that the Revenue Equivalence can be violated and it can be shown for
example, bidders with a risk averse attitude produce a dierent revenue compared to risk
neutral bidders in a First Price auction. Che and Gale (2006) discuss some of these examples.
This chapter discussed the Revenue Equivalence Theorem and proved it for the First and
Second Price mechanisms using the Uniform distribution on [0,100] as an example. Principles
of violating the RET were also explored to demonstrate occasions where it may not always
hold, using eBay as an example to illustrate how this happens in real life.
The following
chapter concludes the nding of this research project, with an evaluation and reection.
33
4
Conclusion
This researched project found that advertising has revolutionised the online medium and
in particular, online ad auctions have increased massively over the last decade becoming
the predominant revenue model for search engines. This was used as motivation to discuss
auction design and implement a mathematical theorem known as Revenue Equivalence i.e.
irrespective of auction design, the expected return to the seller is unvaried.
General uses for auctions were discussed, particularly when sellers were curious to sell their
product at the market value or when they did not have a xed sale price.
Although this
research proved the revenue equivalence for generic auctions, four common auction designs
were discussed. These were English auction, Dutch auction, First Price and Second Price
with given real world examples where auction design has been largely protable (e.g. sales
of government assets, real estate and telecommunications 3G spectrum licenses).
Bidders payos for the First and Second price auctions were explored including von NeumannMorgenstern utility functions.
The relevance of game theory in auctions was emphasised
through Nash Equilibrium and bidders optimal strategies for the First and Second Price
mechanisms. The concepts of a bid shaving dominant strategy for the First Price and truthful bidding for the Second Price auctions were also explored. Explicit examples were derived
to prove that the revenue would indeed be identical to the seller if the conditions of the theorem were met and then this was proven for a generic case. (The general theorem is given
in Appendix C). All bidders were required to be risk neutral, valuations to be private and
independently drawn from a distribution, as well as the item being allocated to the bidder
with the highest bid, whilst the remaining bidders do not pay anything or make a prot from
the auction.
My research found auction designs to be unique to sellers preferences and therefore may
not always meet the conditions for revenue equivalence. Taking ad auctions as an example,
they are designed for more participants with the aim of obtaining the best revenue.
The
conditions for revenue equivalence are therefore not always met, suggesting that auctions
can be designed to obtain diering revenues. Cases were also discussed where bidders could
be risk averse or risk seeking, violating the conditions of the Revenue Equivalence Theorem.
Ongoing research and experiments into the validity and application of the theorem still
continue.
4.1 Evaluation
This research project has taught me new concepts about auctions, extending the knowledge
learnt from the AM30MR Mathematics Report. It is exciting to see how auctions can be
designed for specic purposes and customised to the needs of sellers (and buyers) in mind
and how no one size ts all
[19]
. Provided the aforementioned conditions are satised, it can
be mathematically proven that all these designs will result in the same revenue for the seller.
34
If more time was available I would research into the stability of an auction design. Suppose
there is a Nash Equilibrium in a highly unstable design, it would be interesting to learn
how random uctuations of bidders behaviour could disrupt the auction or potentially cause
it to collapse. Take for example the 3G Telecoms auction which was of a complex nature.
Supposedly, a more complex phenomenon would be more vulnerable to instability. The 3G
auction design may have been marginally close to collapsing and the backing out of bidders
or unexpected behaviour could have caused the auction to fail, potentially losing large sums
of money for the government.
I would also like to learn how xed prices (e.g.
`buy it now' on eBay) inuence bidders
behaviour in auctions and how the bidding functions adjust when the auction exceeds the
`buy it now' price, since they are aware of a vital piece of information which could alter
their valuation. I could use these concepts to explore how auctions such as all pay react to
violation of the RET conditions.
35
Appendices
1
Basic Statistical Theory
A random variable
X
is a function of possible outcomes of a sample space. This function
quanties these outcomes into a set of real numbers known as the range space.
the set of values which
X
x
denotes
can take. A discrete random variable (DRV) takes a nite and
countable number of values, whereas a continuous random variable (CRV) can take on any
value within a given range.
The probability distribution function (pdf )
fX (x) maps the
R → [a, b][7] .
probabilities. It is dened as a function from
P (X = x)
domain sample space to their
In the discrete case,
fX (x) ≡
and the following conditions must hold:
X
fX (x) ≥ 0 (always positive),
fX (x) = 1 (normalisation).
x
The probability that a DRV
X
lies within an open interval
X
P (a < X < b) =
(a, b)
is given as:
fX (x).
a<X<b
An example of a DRV would be the tossing of a fair coin, where we have assigned probabilities
fX (1) = P (X = 1) = P (H) and fX (0) = P (X = 0) = P (T ).
P (H) = P (T ) = n. Therefore by normalisation we can see:
of H and T respectively so that
Since this is a fair coin,
1
P (H) + P (T ) = n + n = 2n = 1 ⇒ n = .
2
H
The probability of either a
or a
T
occurring is therefore
1
.
2
The following criteria must be met in the continuous case:
Z
∞
fX (x), ≥ 0 (always positive),
fX (x)dx = 1 (normalisation).
−∞
The probability that a CRV
X
lies within a closed interval
Z
fX (x) = P r[a ≤ X ≤ b] =
[a, b]
is:
b
f (x)dx.
a
An example to illustrate a CRV is assuming every value is equally likely within the interval
[0, 1]
i.e.
fX (x ∈ [0, 1]) = n
or 0 otherwise. For normalisation,
Z
∞
Z
fX (x)dx =
−∞
1
n dx = n = 1.
0
36
An example of a continuous pdf is the Uniform distribution such that
fX (x) =
U ∼ [a, b]:
1
.
b−a
f (x)
1
(b - a)
a
b
x
Figure 12: Uniform Probability Distribution Function (f (x)).
Now, the cumulative distribution function (cdf ) is dened using the symbol
the subscript
X
FX (x),
where
represents the associated random variable:
x
Z
FX (x) ≡ P (X ≤ x) =
fX (t)dt,
(continuous RVs),
−∞
=
X
fX (x),
(discrete RVs).
∀x
X is less than or equal a given value x. CDF's
xi ≤ xj ⇒ F (xi ) ≤ F (xj ) for i < j [7] . For example,
This represents the probability that the RV
are monotonic and nondecreasing i.e.
we can take the cdf of the Uniform distribution mentioned above. Notice in Figure (13) how
the function is strictly increasing or constant and never decreases at any point. The sum of
all the intervals also total 1.
It is worth noting that the derivative of a CDF results in the associated PDF:
d
FX (x) = fX (x).
dx
37
(16)
F(x)
1.0
0.8
0.6
0.4
0.2
- 1.0
- 0.5
0.5
1.0
1.5
2.0
x
Figure 13: Uniform Cumulative Distribution Function (F (x)).
The examples used in Section 3 use a modied version of the Uniform distribution dened
on the interval
[0, 100]
(as opposed to the general form dened on
[a, b]).
U ∼ [0, 100],
1
1
=
,
b−a
100
Z x
Z x
1
x
FX (x) =
fX (t) dt =
dt =
.
100
−∞
−∞ 100
fX (x) =
38
2
Proof of Revenue Equivalence Theorem
This section explores the proof of the Revenue Equivalence theorem in a general case taken
from Krishna (2002). We rst introduce the following components:
•
standard auction,
•
symmetric equilibrium of the auction,
•
bidders value,
•
bidders bid,
•
equilibrium expected payment,
•
equilibrium payment function,
A,
x,
b,
M A (·) is independent
with b = x = 0 i.e.
Note that
bidder
β A,
M A (x),
M A (·).
of the auction form provided the expected payment of a
M A (0) = 0.
(17)
Now suppose values are independently and identically distributed, and all bidders are risk
neutral. Then any symmetric and increasing equilibrium of any standard auction such that
equation (17) hold true, yields the same expected revenue to the seller.
Consider bidder 1 in an auction, where all other bidders are following the equilibrium strategy,
β.
It is useful to abstract away from the details of the auction design and consider
expected payo for the bidder with value
equilibrium bid. Let
β(Y1 )
x
and when he bids
β(z)
denote the highest bid. Bidder 1 wins when his bid
the highest competing bid i.e
β(z) > β(Y1 ) ⇒ z > Y1 .
β(x), the
β(z) exceeds
instead of
Their expected payo can therefore
be represented as:
ΠA (z, x) = G(z)x − M A (z),
G(z) represents the cumulative distribution of Y1 for N − 1 bidders; G(z) ≡ F (z)N −1
A
and M (z) is dependent on other bidders strategies β and z but is independent of the true
value x. To nd the maximum payo for a bidder, the optimal point can be found at the
where
stationary point when the rst derivative equals zero. Using expression (16):
d
d A
Π (z, x) = g(z)x − M A (z) = 0.
dz
dz
39
At equilibrium, a bidder would bid their maximum valuation hence
z = x.
Letting
y represent
the equilibrium status:
d A
M (y) = g(y)y,
dy
Z x
A
A
yg(y)dy,
M (x) = M (0) +
⇒
∀y,
⇒
0
Z
=
x
yg(y)dy,
0
= G(x) × E[Y1 |Y1 < x],
E[Y1 |Y1 < x] is the conditional expectation for the random variable Y1
A
strictly less than x. Since it is assumed M (0) = 0 and the right hand side
of any particular auction form A, this proves that the same revenue would
where
(18)
given that it is
is independent
be achieved in
any design (provided certain conditions are met). An example using a Uniform distribution
is discussed in Section 3 of this report.
40
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