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 Online Advertising ...... 1 1.2 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 1.3 Revenue Equivalence Theorem ...... 7

2 Auctions 9 2.1 Common Auction Designs ...... 9 2.1.1 Open Bid Auctions ...... 9 2.1.2 Sealed Bid Auctions ...... 11 2.2 Importance of Auction Design ...... 13 2.3 Bidding Functions in First Price Auction ...... 14 2.3.1 2 Bidders ...... 14 2.3.2 n Bidders ...... 16 2.4 Expected Payos in a Second Price Auction ...... 18

3 Revenue Equivalence Theorem 22 3.1 Expected Revenue ...... 22 3.1.1 First Price Auction ...... 22 3.1.2 Second Price Auction ...... 26 3.2 Violating Equivalence ...... 30 3.2.1 Risk Approach ...... 30 3.2.2 Auction Rules ...... 31 3.3 Example of Auctions Violating Equivalence ...... 31

4 Conclusion 34 4.1 Evaluation ...... 34

Appendices 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. Exam- ples 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, predom- inantly 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 Univer- sity, 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 2 Global Internet Advertising Spend by Category ...... 3 3 Risk-Utility Indierence Curves ...... 6 4 Hierarchy of Auction Designs ...... 10 5 First Price Bidding Prole ...... 17 6 Bidding in the Second Price Auction b < v ...... 19 7 Overbidding in the Second Price Auction bi < v < b ...... 19 8 Second Price Payos bi > v ...... 20 9 Second Price Auction Payos b < v ...... 20 10 Second Price Auction Payos b < v ...... 21 11 Bid History in eBay Auctions ...... 32 12 Uniform Probability Distribution Function (f(x)) ...... 37 13 Uniform Cumulative Distribution Function (F (x)) ...... 38

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 Gener- alised 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 implement- ing 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

Online advertising has matured dramatically over the past decade[30], from selling advertisement spaces, to the introduction of clickability moving towards pop up adverts[16] and then the paid placement model (PPM)[4], an approach that allowed advertisers to purchase more desirable advertising slots on search engines[13]. Why online? User behaviour and surng habits can be measured more eectively online compared to o ine[27]. In 1998, a company called GoTo.com introduced the rst auction feature where online adverts were ranked subject to bids submitted by advertisers[13]. Known as ad auctions, they became a new economic revenue stream and by 1999, the online advertising medium had become a $1 billion industry[13]. Today this form of advertising is closing in to become the largest entertainment and media advertising segment, forecast to be worth $194.5 billion by 2018[30]. 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 to that of supermarket shelf stacking[40]. Product manufacturers pay for their items to be stacked on prime shelves i.e. at eye level or within arm's reach of customers[40]. 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 in order of prominence[13]. Each position has a desirability unique to advertisers[10]. A user is more likely to click a link at the top of their search rather than further down the page[10]. Hence, a website more likely to receive a click has a greater chance of improving business[4]

1 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]. When a user's search query matches the advertiser's keyword, an auction commences[38]. The auction concludes when the page, including the ads, has fully loaded[1]. 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 pays the search engine a fee. This approach is called pay per click (PPC)[38]. 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, and mesothelioma can reach an excess of $50 per click, for every user that visits the site[8].

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 to key words that are searched[27]. A report by PricewaterhouseCoopers LLP discusses how search will retain its dominant position in online advertising[30]. Figure 2 shows the global spending on various online advertising mechanisms.

2 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 websites 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- tiveness of these adverts[1]. 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[16] - all of which rely on mathematics to ensure an advert reaches its target audience. Auction theory and game theory play a signicant role when a company is trying to ensure they attract business customers over their competitors[16].

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 s. There may be multiple slots si = s1, s2, ..., sn 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 [3] of s1 > s2 > ... > sn. The auction can be represented as a game .

A game involves n 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 the winnings or losses for each player in every game[2]. 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 appropriate to their business[28]. There are three possibilities:

1. b > v,

2. b < v,

3. b = v.

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. (In the case of Google, a quality score is combined with b to decide on the ranking. This prevents

4 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 reach their optimum strategies and are unable to progress or improve any further[19].

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 do), optimal bidding behaviour is reached which is known as a Nash equilibrium[19].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 Γ = {N, Ω, Θ, {vi}}, where:

• N = 0, ..., n represents the list of participants in the game,

• Ω = {ω1, ...ωn} represents the strategy space of all game players,

• Θ = {θ1, ...θn} represents the payo functions for each player,

[28] •{ vi} is the valuation for each bidder .

The utility function for the players can be expressed as Ui(Ω) = U1(ω1, ..., ωn), ..., Un(ω1, ..., ωn). The strategy prole ∗ ∗ is a Nash Equilibrium if every agent maximises their utility {ω1, ..., ωn} given every other player sticks to their equilibrium strategy[28]:

∗ ∗ ∗ (1) Ui(ωi , ω−i) ≥ Ui(ωi, ω−i).

If a change is made to the equilibrium status, it could result in the player bearing responsi- bility for heavy costs as well as negative consequences[19]. 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- ter o, without taking a strategy that makes at least one of the other bidders worse o[35]. Section 2.1 discusses various auctions designs as well the Nash Equilibrium strategies for the First and Second Price auctions.

Game theoretic analysis of auctions assume bidders are rational[42]. 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 to provide them with an advantage[33].

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 averse Risk neutral

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 During an auction, bidders may choose to bid a fraction of their valuation for example v b = 2 or even b = v. Whichever tactic bidders employ during the bidding process is subject to the 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[19]. An auction in Germany for 3 spectrum

6 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 for 5 telecomms licenses, which led to a return of $34 billion ( 1 billion) for the UK ≡ £22 2 government[17], through utilising fundamental concepts of auction theory and game theory. The cumulative return of over $100 billion from worldwide 3G auctions has since provoked great interest and developments in these areas[19].

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 for a simple case using the Uniform distribution[39], 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. However, the theorem only holds provided the general conditions below have been satised[8].

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 to the seller (and each bidder making the same expected payment): [19]

∗ 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 F (v). ∗ The object is allocated to the player with the highest bid. ∗ A player with v = 0 does not make a prot from the auction[12].

7 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 particular auctions in practice[24]. 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 by the seller, creating a binding contract[11].

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 as a mechanism for price discovery[19]. 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, v, in mind i.e. they bid up to but not exceeding their valuation of the product being sold[8]. In a typical auction, this is rarely the case[6]. Bidders tend to adjust their idea of v whilst learning more about the product

9 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 auction once their intrinsic value has been reached[8].

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[15]. Often, bidders may hold crucial information which inates the value of the item[6]. As more information is revealed during the bidding process, further bids are provoked[15]. 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[6]. For example, for an item with v = £10, it would be bad practice for a bidder to exit an ascending auction well before v has been reached or even bid once v has exceeded[6]. This strategy encourages eciency as well as participation of more bidders[6].

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 successful, before placing their bid[19].

10 2.1.1.2 Descending Auction After the established tradition to use this mechanism for the sale of owers in the Nether- lands, descending auctions were branded Dutch auctions[8]. As the name suggests, the seller starts from a high price and decreases the bid sequentially. In the Dutch ower auctions, potential buyers are seated at desks in the same room with buzzers connected to a clock[19]. 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 Canada[19].

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 bid until someone's valuation is reached[19]. Bidders do not learn any new information about other bidders during the course of the auction, except that no one's v has been reached. Each bidder has their own bid bi 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 value of their bid[8].

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 `sealed' bids cannot be adjusted[8].

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 otherwise this would result in a zero or negative prot[19]. 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. Hence it is not a Nash Equilibrium or optimal strategy to bid truthfully (i.e. b = v) in the First Price auction[28]. These types of auctions are commonly used for submitting government contracts and selling real estate property[19].

11 One major disadvantage to the First Price auction is collusion between bidders. As bidders discuss their perceptive valuations of the product, bids can be rigged[29]. Bid rigging in procurement auctions for school milk has resulted in 45 corporations and 49 individuals being convicted with nes totalling $46 million[29].

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 auctions, the approach of bidders bidding up to v (their optimum value) remains the same. Hence it is a Nash Equilibrium to bid truthfully[28]. If a bidder quits the auction (perhaps because v 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 over- come 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 or lose, have to pay their bid[8]. 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 bids and signicantly large asymmetry between bidders[18].

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 would be rendered inecient[18]. It is in the interest of the seller to use a design which mitigates the challenge of ensuring competition between bidders and preventing collusion between bidders[18]. One means of enforcing this could be through punishments[18]. In fact, unreasonable reserve bids provoke predation and collusion between bidders[18].

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, auctions should be designed to be appealing and robust. Auctions may have entry fees[13]. 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 disadvantageous to the seller[29].

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. It can be argued that this is subject to the risk attitude of the bidder. 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, v1,

bid, b1. • Bidder 2

Valuation v2,

bid, 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.

(2) fV1V2 (v1, v2) = fV 1(v1)fV 2(v2), (only if v1, v2 are independent).

Assume bidder 1 will bid a fraction α of their true value, which is less than b2, such that:

αv1 < b2, which can be rearranged to b v < 2 . 1 α

The probability of winning the item with bid b2 (provided b2 < v2) is represented by the cumulative distribution function (CDF) F (v1): b b b F 2 = P rob X ≤ 2 = 2 , α α 100α

where X is the random variable. (If b2 > v2, the bidder will be subject to a negative prot). Let us now calculate the expected prots kF (b2) that could be obtained using the First Price mechanism. If bidder 2 wins the auction with a bid b2, their expected prot is: b k (b ) = (v − b ) ∗ F 2 , F 2 2 2 α

14 b = (v − b ) ∗ 2 , (3) 2 2 100α 1 k (b ) = (b v − b2). (4) F 2 100α 2 2 2

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 b2) and equating this to zero to nd the desired parameter b2:

d kF (b2) d 1 2 = (b2v2 − b2), db2 db2 100α 1 = (v − 2b ) = 0, 100α 2 2 ⇒ v2 = 2b2,

v b = 2 . (5) 2 2

Expression (5) tells us that in the case of two bidders, the best response to the rst bidder using strategy is for bidder 2 to use the strategy 1 i.e. submit a b1 = α ∗ v1 b2 = 2 ∗ v2 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:

2 d kF (b2) d 1 2 = (v2 − 2b2) , db2 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 auction. As increases, the probability of the bidder winning also increases. Hence, b2 α 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 n bidders under the First Price auction.

15 2.3.2 n Bidders This section calculates the optimum bidding function for n bidders in a First Price auction who are distributed along U ∼ [0, 100].

Suppose bidder 1 proposes a bid b. The probability that b is the highest amongst all other 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 is b b as in the previous case. As bidders are independent, values form a joint b F ( α ) = 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 n−1 Y b Y b Prob[max{b ...b } < b] = F = , 2 i α 100α i=1 i=1 b n−1 = . 100α

Using equation (3), the expected prot for bidder 1 is therefore:

b n−1 k (b ) = (v − b) , F n 100α 1 = vbn−1 − bn . (6) (100α)n−1 Calculating the rst derivative of equation (6) now gives the maximum value for b: d d 1 k (b ) = vbn−1 − bn , db F n db (100α)n−1 1 = (n − 1)vbn−2 − nbn−1 . (100α)n−1 Hence, the best bidding function can be obtained by letting d and nding the db kF (bn) = 0 stationary points: d 1 k (b ) = (n − 1)vbn−2 − nbn−1 = 0, db F n (100α)n−1 n−1 n¨−¨2 nb (n − 1)v¨b − = 0, bn−2 (n − 1)v − nb = 0, nb = (n − 1)v,

(n − 1)v b = . (7) n

16 This expression denes the optimal bid function for n risk neutral bidders with valuations distributed on U ∼ [0, 100]. We can conclude that with n bidders, they will reduce their bid by a fraction of 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 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 b n=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 lim b = lim , n→∞ n→∞ n ⇒ lim b = v. (8) n→∞

b v

(n-1)v b= 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:

(3 − 1)v 2v v b n=3 = = > . 3 3 2 |{z} Optimum bid for 2 bidders.

Hence an auction with three bidders would result in each bidder submitting a bid of 2 3 v which is closer to than 1 . v 2 v 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 v of a product, they submit a bid b. How they bid relative to v 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 the Second Price mechanism. Consider the case where a bidder bids b < v. The idea would be to win the item at the lowest possible cost, since there is no gain if b > v (as this results in a negative prot to the bidder). In fact, if a bidder wins an auction at b < v, they earn v − b. These cases assume that v does not change during the auction.

If the highest amongst all bidders (except bidder 1) submits bi which is less than the rst bidders valuation, i.e. bi < v, then provided b > bi, the bidder would win the auction and still pay bi 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. Figures (6a) and (6b) help to visualise this.

18 (a) Bidder coincidently wins the auction (b) Bidder loses the auction as b < b , hence since b > b and b < v meaning a prot of i i there is no advantage to bidding less than v. v − bi.

Figure 6: Bidders payos when underbidding, b < v.

If bidder 1 decides to overbid such that bi < v < b, they would still win the item and pay bi (Figure (7)) and earn v − bi as the highest bid is remains below 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 they are not the highest bidder. If they decide to submit a higher bid such that b > bi > v, the bidder wins the auction but at a negative prot of bi − v, as they pay more than anticipated. 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 (b) Bidder wins the auction since b > bi but b = v. as bi > v, they bidder makes a loss of bi − 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 b = v, they would have won the auction whilst also making a return of v − bi.

(a) Bidder bids when and there- b < v bi > v (b) Bidder bids and loses the fore loses the auction since both and have b < bi < v b v auction. been passed.

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 pay more than their initial valuation, resulting in a negative prot from the sale[8].

20 (a) If bi < b < v the rst bidder wins the auction, but only by chance of other bids. If (b) If the bidder bids b = v, the Second Price another bidder increases their bid by such mechanism allows them to win the auction at that bi + > b, then bidder 1 would lose the bi making a prot of v − bi. auction.

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 = β(vi) = vi such that the expected payo is vi−bi×P (bi) where P (bi) represents Prob[all other bids< bi]. Note that this is a `strictly increasing function' (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 n bidders to promote understanding of more realistic environments.

22 3.1.1.1 2 Bidders When two bidders are competing in an auction, there are two possibilities for the outcome of b:

1. Bidder 1 has a valuation v1 = x and bidder 2 has a valuation v2 ≤ x. The probability of this outcome is f(x) × F (x) (f(x) and F (x) hold the same meanings as described in section 2.1.1). By , 1 and x U ∼ [0, 100] f(x) = 100 F (x) = 100 ⇒ f(x) × F (x) = 1 x . 100 100

2. Bidder 2 has a valuation v2 = x and bidder 1 has a valuation v1 ≤ x. Again, the probability that 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) = 1 x . 100 100

Let the probability of x taking the highest bid from either of the two outcomes be governed by the equation 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 rF 2. Since pF 2 is the probability that the highest value is x, the expected value of the highest bid is therefore:

Z 100 E[pF 2] = x.pF 2 dx, 0 2 Z 100 x = x. dx, 100 0 100 Z 100 2 2 = 2 x dx, (100) 0 2 x3 100 = 2 , (100) 3 0 " 1 # 2 100¡3 = 2 , (100) 3 200 = . 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.:

1 200 100 r = × = . (9) F 2 2 3 3

We will now develop on this method to prove the case for 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 that bidder 1 could have the highest bid up until the nth 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 p = nf(x)(F (x))n−1 = n . F n 100 100

The expected value of the highest bid can now be calculated:

Z 100 Z 100 n n−1 E[pF n] = x.pF n dx = n−1 x.x dx, 0 (100)(100) 0 " # n xn+1 100 n 100n¡+1 100n = = = . n n (100) n + 1 0 (100) n + 1 n + 1

It was previously proven that bidders will bid a fraction n−1 of their valuation (expression n (7)). Therefore, the expected revenue from the auction would be n−1 times the expected n value of the highest bid:

n − 1 Z 100 n − 1 100n rF n = x.pF n dx = , n 0 n n + 1 100n(n − 1) = , n(n + 1) 100(n − 1) = . (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 n bidders also.

24 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 b = β(v). Recall that the payo for a winning bidder is (v − b) provided b is the highest bid in the auction b > bi. 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 Price auction so 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 found such that v = β−1(b). Hence:

⇒ (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: du dG dz = −G(x) + (v − b) . , db dz db dz = −G(x) + (v − b)g(x). . (11) db

Since −1 and dβ and by the chain rule dβ dβ dx . Therefore, x = β (b) = v β(x) = b ⇒ db = 1 db = dx db dx 1 dz 1 dz 1 = = = ⇒ = . db dβ dβ β0(x) dβ β0(β−1(b)) dx

25 Substituting this into expression (11) gives:

1 U(v, b) = −G(x) + (v − b)g(x). = 0, β0(β−1(b)) = −G(x)β0(β−1(b)) + (v − b)g(x), = G(x)β0(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 v ⇒ β(v)G(v) = xg(x)dx, 0

R v xg(x)dx 0 (12) β(v) = R v ≡ E[x|x < v]. 0 g(v)dx

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:

A MFP = 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 U ∼ [0, 100]. Let pS2 represent the probability that the second highest bid is x. In the case of 2 bidders, this can occur in two possible ways:

1. Bidder 1 has a valuation of x (f(x)) and bidder 2 has a valuation of 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: 1 x f(x)(1 − F (x)) = 1 − . 100 100

2. 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)).

The probability that either of these two cases occur is therefore:

2 x p = 2f(x)(1 − F (x)) = 1 − . S2 100 100

The revenue rS2 is therefore the expected value of pS2 is:

Z 100 rS2 = E[pS2] = x.pS2 dx, 0 2 Z 100 x = x. 1 − dx, 100 0 100 2 Z 100 x2 = x − dx, (100) 0 100 2 x2 x3 100 = − , (100) 2 300 0 " 1 2 # 2 100¡2 100¡3 = ¨ − , ¨(100)¨ 2 300 1002 = 100 − , 150 200 = 100 − , 3

100 r = . S2 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 n bidders.

3.1.2.2 n Bidders In order for bidder 1 to be the second highest bidder with bid x, one of the remaining n − 1 bidders must bid at least as much as x and all other n − 2 bidders have a bid less than or equal to x. This occurrence is comprised of the following 3 probabilities:

Probability of bidder 1 having a value 1 . • x ⇒ P (X = x) = f(x) = 100 • Probability that one of the n − 1 bidders have a bid greater than or equal to x ⇒ x . P (X ≥ x) = (n − 1)(1 − F (x)) = (n − 1) 1 − 100 • Probability that the remaining n − 2 bidders have a value less than or equal to x ⇒ n−2 x n−2. P (X ≤ x) = (F (x)) = 100

Hence, n−2 rSn = nf(x)(n − 1)(1 − F (x))(F (x)) .

The expected value of rSn can now be found by integrating with respect to x as we have done in previous cases:

Z 100 E[rSn] = x.rSn dx, 0 Z 100 = x.nf(x)(n − 1)(1 − F (x))(F (x))n−2 dx, 0 Z 100 1 x x n−2 = n(n − 1) x. 1 − dx, 0 100 100 100 n(n − 1) Z 100 x2 x n−2 = x − dx, 100 0 100 100 Z 100 2 n(n − 1) x n−2 = n−2 x − x dx, (100)(100) 0 100 Z 100 n n(n − 1) n−1 x = n−2 x − dx, (100)(100) 0 100 n(n − 1) xn xn+1 100 = n−2 − , (100)(100) n 100(n + 1) 0

28 n(n − 1) 100n 100n+1 = − , (100)n−1 n 100(n + 1) n(n − 1)100n 1 ¨100¨ = − , (100)n¡−1 n ¨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 = 100. (14) n + 1

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 2 − 1 100 100 n=2 = 100 = . n + 1 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 previously. 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, bi+1. The revenue is therefore calculated as a product of the probability that a bidder with v wins the auction and the expected value of the second highest bid:

A MSP (v) = P rob(v wins) × E[bi+1|v = highest],

= P rob(Y1 ≤ v) × E[Y1|Y1 < v] , | {z } | {z } CDF Conditional Expectation Z v R v 0 yg(y)dv = f(v)dv × R v , 0 0 f(v)dv

= G(v) × E[Y1|Y1 < v], (15) where G(v) represents the distribution of the bidders. Notice that this is identical to revenue for First Price auction (expression (13)).

29 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 theorem may not hold[8].

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 strategy of b = v leads to a zero payo to the bidder since v − b = 0. A bid lower than the valuation, b < v leads to an increase in expected payo since now v − b > 0. For a risk 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].

Risk implies linearity[22]. Figure (3) shows that a risk neutral bidder has a linear bidding function say, b = 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)[19].

3.2.2 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. This implies that the revenue equivalence would no longer hold[22].

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. Bidders can also notice how many other bidders are participating in the auction. 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 of participants in the auction can inuence a bidders behaviour[9].

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[5].

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Bidder Bid Amount Bid Time Current bid: £510.10 t***a ( 4 ) £510.10 23Apr15 10:11:15 BST Postage: £8.75 Express Shipping (Royal Mail Special Delivery (TM) 1:00 pm). i***. ( 81 ) £500.10 22Apr15 23:02:18 BST Item number: 301601377066 a***s ( 90 ) £496.25 23Apr15 02:34:49 BST

i***. ( 81 ) £485.00 22Apr15 23:02:18 BST Enter your maximum bid:

o***0 ( 36 ) £475.00 20Apr15 22:14:54 BST Place bid

o***0 ( 36 ) £461.00 20Apr15 22:14:54 BST (Enter £520.10 or more)

n***n ( 780 ) £451.00 20Apr15 21:53:53 BST

o***0 ( 36 ) £450.00 20Apr15 22:14:43 BST

n***n ( 780 ) £330.00 20Apr15 21:53:53 BST

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

k***1 ( 71 ) £240.00 20Apr15 19:11:37 BST

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

e***d ( 28 ) £195.00 20Apr15 12:43:20 BST

t***i ( 91 ) £185.00 20Apr15 11:12:28 BST

e***d ( 28 ) £180.00 20Apr15 12:43:14 BST

t***i ( 91 ) £171.00 20Apr15 11:12:28 BST

e***d ( 28 ) £166.00 20Apr15 12:43:05 BST

t***i ( 91 ) £160.55 20Apr15 11:12:28 BST

n***n ( 780 ) £155.55 20Apr15 11:27:28 BST

t***i ( 91 ) £127.10 20Apr15 11:12:28 BST

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 Neumann- Morgenstern 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. x denotes the set of values which X 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 domain sample space to their probabilities. It is dened as a function from [7]. In the discrete case, R → [a, b] fX (x) ≡ P (X = 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 (a, b) is given as: X P (a < X < b) = fX (x). a

The probability of either a or a occurring is therefore 1 . H T 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 [a, b] is: Z b fX (x) = P r[a ≤ X ≤ 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 1 fX (x)dx = n dx = n = 1. −∞ 0

36 An example of a continuous pdf is the Uniform distribution such that U ∼ [a, b]: 1 f (x) = . X b − a

f(x) 1 (b-a)

x a b Figure 12: Uniform Probability Distribution Function (f(x)).

Now, the cumulative distribution function (cdf) is dened using the symbol FX (x), where the subscript X represents the associated random variable:

Z x FX (x) ≡ P (X ≤ x) = fX (t)dt, (continuous RVs), −∞ X = fX (x), (discrete RVs). ∀x

This represents the probability that the RV X is less than or equal a given value x. CDF's [7] are monotonic and nondecreasing i.e. xi ≤ xj ⇒ F (xi) ≤ F (xj) for i < j . For example, 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 F (x) = f (x). (16) dx X X

37 F(x)

1.0

0.8

0.6

0.4

0.2

x -1.0 -0.5 0.5 1.0 1.5 2.0

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 f (x) = = , X b − a 100 Z x Z x 1 x FX (x) = fX (t) dt = dt = . −∞ −∞ 100 100

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, A,

• symmetric equilibrium of the auction, βA,

• bidders value, x,

• bidders bid, b,

• equilibrium expected payment, M A(x),

• equilibrium payment function, M A(·).

Note that M A(·) is independent of the auction form provided the expected payment of a bidder with b = x = 0 i.e.

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 x and when he bids β(z) instead of β(x), the equilibrium bid. Let β(Y1) denote the highest bid. Bidder 1 wins when his bid β(z) exceeds the highest competing bid i.e β(z) > β(Y1) ⇒ z > Y1. Their expected payo can therefore be represented as:

ΠA(z, x) = G(z)x − M A(z),

N−1 where G(z) represents the cumulative distribution of Y1 for N − 1 bidders; G(z) ≡ F (z) and M A(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 stationary point when the rst derivative equals zero. Using expression (16):

d d ΠA(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 ⇒ ∀y, M A(y) = g(y)y, dy Z x ⇒ M A(x) = M A(0) + yg(y)dy, 0 Z x = yg(y)dy, 0

= G(x) × E[Y1|Y1 < x], (18)

where E[Y1|Y1 < x] is the conditional expectation for the random variable Y1 given that it is strictly less than x. Since it is assumed M A(0) = 0 and the right hand side is independent of any particular auction form A, this proves that the same revenue would 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 References

[1] Angel, S. and Walsh, M. (2013) Veriable Auctions for Online Ad Exchanges. Hong Kong:University of Texas.

[2] Barron, N. (2013) Wiley Series in Operations Research and Management Science : Game Theory : An Introduction. 2nd Edition. New Jersey:John Wiley & Sons. [3] Binmore, K. (2007) Game Theory: A Very Short Introduction. UK:Oxford University Press.

[4] Chaey, D. et al. (2009) Internet Marketing: Strategy, Implementation and Practice. 4th Edition. Essex:Pearson. [5] Che, Y. and Gale, I. (2006) Revenue Comparisons For Auctions When Bidders Have Arbitrary Types. Theoretical Economics, 1(1), pp.95-118.

[6] Cramton, P. (1998) Ascending Auctions. European Economic Review, 42(3), pp.745- 756.

[7] Deep, R. (2006) Probability And Statistics. Amsterdam: Elsevier Academic Press.

[8] Easley, D. and Kleinberg, J. (2010) Networks, Crowds, and Markets: Reasoning About A Highly Connected World. New York:Cambridge University Press.

[9] eBay Inc. (2015) Auction Format. Available from: http://pages.ebay.co.uk/help/sell/ auction.html [Accessed on: 23/04/2015]

[10] Edelman, B., Ostrovsky, M. and Schwartz, M. (2007) Internet Advertising and The Generalized Second Price Auction. American Economic Review, 97(1), pp.242-259.

[11] Encyclopaedia Britannica (2006) Britannica Concise Encyclopedia. Chicago:Encyclopaedia Britannica Publishers.

[12] Fibich, G. et al. (2003) Revenue Equivalence in Asymmetric Auctions. Journal of Economic Theory, 115(2004), pp.309-321.

[13] Geddes, B. (2010) Advanced Google AdWords. Indianapolis:Wiley Publishing Inc.

[14] Goldfarb, A. (2014) What Is Dierent about Online Advertising?. Review of Industrial Organization, 44(2), pp.115-129.

[15] Haites, E. and Hornung, R. (1999) Analysis of Options For Distributing Allowances By Auction. Canada:Renouf Publishing Co. Ltd.

[16] Khan, S. (2015) The Mathematics Behind Online Ad Auctions. BSc. Birming- ham:Aston University.

41 [17] Klemperer, P. (2001) How (Not) To Run Auctions: The European 3G Telecom Auc- tions. Oxford:Oxford University Press.

[18] Klemperer, P. (2002) What Really Matters in Auction Design. The Journal of Eco- nomic Perspectives, 16(1), pp.169-189.

[19] Klemperer, P. (2004) Auctions: Theory and Practice. Oxford:Princeton University Press.

[20] Krishna, V. (2002) Auction Theory. San Diego:Academic Press.

[21] Leme, R. (2012) Design and Analysis of Sponsored Search Mechanisms. PhD. Cornell University.

[22] Miller, R. and Prasnikar, V. (2008) Strategic Play:Experimental Methods for Business and Economics. Stanford University Press.

[23] Milgrom, P. (2004) Putting Auction Theory to Work. Cambridge:Press Syndicate.

[24] Milgrom, P. (1989) Auctions and Bidding: A Primer. The Journal of Economic Per- spectives, 3(3), pp.3-22.

[25] Menezes, F. and Monteiro, P. (2005) An Introduction To Auction Theory. Oxford: Oxford University Press.

[26] Myerson, R. (1981) Optimal Auction Design. Mathematics of Operations Research. 6(1), pp.58-73.

[27] Parkes, D. and Seuken, S. (2012) Chapter 7 - Sponsored Search Auctions. (Pre- publication Copy). Harvard University.

[28] Parkes, D. and Seuken, S. (2013) Economics and Computation: A Design Approach. (Pre-publication Copy). Harvard University.

[29] Pesendorfer, M. (2000) A Study of Collusion in First-Price Auctions. The Review of Economic Studies Ltd., 67(3), pp.381 − 411. USA:Yale University. [30] PricewaterhouseCoopers LLP (2014) Global Entertainment and Media Outlook 2014- 2018. London: PwC-Informa Telecoms & Media.

[31] Quint, D. (2014) Optimal Auctions I. Madison:University of Wisconsin.

[32] Riley, J. and Samuelson, W. (1981) Optimal Auctions. American Economic Review, 71(3), pp.381-392.

[33] Richter, K. (2004) Revenue Equivalence Revisited: Bounded Rationality in Auctions. Germany:University of Kiel.

[34] Szentes, B. (2005) Equilibrium Transformations and The Revenue Equivalence Theo- rem. Journal of Economic Theory, 120(2), pp.175-205.

42 [35] Tenorio, R (1999) Multiple Unit Auctions with Strategic Price-quantity Decisions. Eco- nomic Theory, 13(1), pp.247-260.

[36] Ungern-Sternberg T. (1988) Cartel Stability in Sealed Bid Second Price Auctions. The Journal of Industrial Economics, 36(3), pp.351-358. Wiley Publishers.

[37] Varian, H. (2009) Online Ad Auctions. American Economic Review, 99(20), pp.30-34.

[38] Varian, H. and Harris, C. (2014) The VCG Auction in Theory and Practice. American Economic Review, 104(5), pp.442-445.

[39] Vickrey, W. (1961). Counterspeculation, Auctions and Competitive Sealed Tenders. The Journal of Finance, 16(1), pp.8-37.

[40] Vieria, V. and Camilo, L. (2014) An Analysis of the Eects of Promotion and Shelf Position on Store Revenue. Brazilian Business Review (English Edition), 11(4), pp.28- 52.

[41] Vijay, K. (2002) Auction Theory. USA:Academic Press.

[42] Waehrer, K., Harstad, R. and Rothkopf, M. (1998) Auction Form Preferences of Risk- averse Bid Takers. Journal of Economics, 29(1), pp.179-192.