June 20th 2014 Academic year 2013/2014
Dr. Audrey (X.) Hu Prof. Dr. J.H. Sonnemans
Master Thesis Semester 2, block 5 & 6
Faculty of Economics and Business
Unique Bid Auctions
A new means of improving the tender acquisition process
Adriaan Bron
A Master’s Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of: Master of Science in Economics
Acknowledgements I would like to extend my gratitude to Dr. Audrey Hu for her immense contribution, advice and assistance throughout the process and duration of writing this research report.
2 Table of content 1. Introduction 3 2. Auction theory 2.1 Auction theory 5 2.2 Valuations models 5 2.3 Standard auctions 6 2.4 Tullock contest 9
3. Unique Bid Auctions
3.1 Platforms of Unique bid Auctions 12
3.2 Review of the literature on the Unique bid auctions 14 3.3 Unique bid auctions in the spectrum of the other auction 18 3.4 Drawing an alternative unique bid auction (AUB auction) 19
3.5 Motivation of the made choices 20
4. The Model
4.1 The case with two potential bidders 22
4.2 The three potential bidder’s case: a generalized outcome 23 4.3 Using the general model for numerical examples 28 4.4 Usability of the proposed model for tenders 35
4.5 Comprising it with other auctions 38
5. Conclusion 41
6. References 44
Appendices 46
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1. Introduction
When it comes to procurement, public tenders play a vital role for governments1. In recent years the rise of the European Union has brought about a dramatic increase in public tenders across Europe. Research has shown that contracts given out by governments via public tenders count for over 19% of the European Union’s Gross Domestic Product. When it comes to the acquisition process of public tenders by governments it is clear that there is a general need to study this topic in greater depth. Over the past few years it has come to light that there has been a lack of emphasis in the past on important factors such as quality control in the tender acquisition process (Europe.eu, 2014).
There are several instances where the use of a public tender has not led to a socially optimal outcome. One of the more notable examples is the Dutch government’s recent acquisition of the production of high speed trains in the Netherlands. The new HSL line required the purchase of a set of high speed trains. A public tender for the production of these trains was announced in 2004. European law stipulated that the Dutch government would have no choice but to award the tender to the party who offered the lowest price in order to make the operation as cost effective as possible. As a result the tender was won by a party who had very little experience with high speed trains. The Dutch government was very aware of the risks involved in choosing this party but as result of the rules and laws of the tender they had no other option. This tender inevitably proved to be extremely unsuccessful leaving the Dutch government with nineteen expensive trains that will never be used (NOS, 2014).
The European Union has since realized the error of their ways and made it clear that they will try and prevent mistakes, such as the HSL line, from occurring in the future. In recent years the European Union has emphasized the need and importance of ensuring that quality plays a more significant role in the acquisition process of public tenders (Europe.eu, 2014)2.
1 When choosing a company to supply it with goods or services, European Union governments are expected, to
make use of specific tendering procedures. The rules of such a tender are determined beforehand and are mostly based on the factors price and quality and is open for all qualified bidders (Europa.EU, 2014) (businessdictionairy.com, 2014).
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On July 31st 2013, the European parliament voted in favor for changing the procedures of public tendering. These changes mostly focus that quality should play a more important role in the tendering processes. All the countries of the European Union have to include these new set of rules in their tendering procedures between now and two years (Europe.EU, 2014).
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The acquisition process of public tenders is of great importance to governments however it is a subject matter that has not been looked at in great detail. When it comes to studying public tenders the primary theory used by economist is that of the auction theory, there are two reasons for this: Firstly, there is a direct link between a tender and an auction, as for instance discussed in Luton & McAfee (1986). Secondly, the auction theory can help to formalize a tender in a more game theoretical way (as for instance done by Che & Gale, 1998).
There has been a significant amount of emphasis put on both standard and all-pay auctions and to date several in depth analysis’s and studies have been done on both these auction types in efforts to explain real-life economic events (see for instance Che & Gale (1998)). There is however another type of auction, one that has come about more recently and is commonly known as the ‘unique bid auction’. Very little is known about the unique bid auction and very little research has been done on this subject despite the fact that the use of unique bid auctions could vastly improve the tender acquisition process. This type of auction format gives one a better understanding of (public) tenders as well as other real-life economic happenings and could form as a strong foundation for improvement.
This research will show that unique bid auctions are an important tool for governments when it comes to formulating a tender which ensures that quality is a factor. Furthermore this research will show that unique bid auctions are a useful tool to analyze procurements from a more quality based perspective. The overall hypothesis of this research shows that the unique bid auction is a better way to draw and explain a tender in comparison to other auction methods previously used.
In the second chapter of this research an overview will be given of the importance, the implications and usability of the existing auction theory. In the third chapter a literature overview will be given of the unique bid auction and a comparison will be drawn between the unique bid auction and other types of auctions. The fourth chapter will focus on formalizing the unique bid auction and how it can be implemented when it comes to tenders. In the last section some concluding remarks will be made and advice for future research will be given.
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2. Auction theory
In this section an overview will be given on existing literature about the auction theory and its importance. The implications and differences between the formats common valuation with interdependent values and private independent values will be discussed. The outcomes of the auctions with respect to the other types of auctions will also be looked at in this section.
2.1 Auction theory
As argued by Klemperer (2002) auction theory plays an important role for theoretical, empirical and practical reasons. Game theorists have studied this phenomenon extensively in order to create theoretical frameworks. Klemperer (2002) presents three key arguments as to why auction theory is so important:
Firstly, he argues that auctions take place on a daily basis. One can simply think of the flowers that are sold in Aalsmeer every morning. The auction format that is chosen can have a huge impact on the outcome of prices as well as revenue for the seller. In the article written by McMillan (1994) he uses the ‘auction of spectrum rights’ as one of his primary examples. He argues that the amount of effort he spent designing the auction to sell these spectrum rights in the USA in 1993 was essentially because he argues that some auctions simply work better than others. So not only do auctions take place on a daily basis but the use and the design of these auctions can have a significant impact as well.
Klemperer (2002) further argues that auctions are well defined. This gives the ideal option to test the game theory empirically. Finally, Klemperer (2002) states that auctions play an important role in the testing and building of economic theoretical frameworks. He argues that auctions can also give good insights of price forming and believes that there is a close connection between how competitive prices and prices in auctions evolve.
2.2 Valuation models
There are several different models and formats that can be used to analyze auctions. A relatively simple format is that of independent private values. In this model every bidder has a certain private valuation of the auctioned object. Every bidder only knows its own value of
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the auctioned object but the values of others are unknown. Bidding strategies in this model are based on the distribution of these valuations.
Another format used that is of interest to auction theory is that of common valuations. This theory is contradictory to the private independent valuations theory because the value of the product for every bidder is the same. An important feature of common valuations is that every bidder has some private information about the auctioned object. This private information, known as signals, says something about the value of the auctioned object. This format is however problematic in that the valuations are also dependents on the signals of the others (Osborne, 2008). Osborne (2008) uses the auction of oil tact as a real life example. Every participant has some information about how much oil is in the tract but none of the bidders knows the exact amount. The exact amount is somehow dependent on the information all bidders have (Osborne, 2008).
As Klemperer (1999) argues, within a common valuation model it is of great importance that participants take into account that the signal of others influences the common value of the object as well. If this does not happen, there is a high chance that one may overbid. The person with the highest signal who does not take this into account will inevitably overprice the object. This phenomenon is better known in the literature as the ‘winner’s curse’ (Klemperer, 1999).
Krishna (2009) gives a specific example of a common value model, which he calls the ‘mineral right model’. Within this model, a clear outline is given demonstrating the distribution of signals and the value of the object. The use of this model makes it possible to test and compare different auctions. In this model the distribution of the common value ranges between zero and one. This value is indicated with a V. The signals of the bidders are uniformly and independently distributed over the interval zero to twice v. The letter v is assumed to be equal to capital V (Krishna, 2009).
2.3 Standard auctions
The four most commonly used auctions are that of the English, the Dutch, the first-price sealed and second-price sealed auction. These four auction types are commonly seen as the standard auctions. The four auctions have been discussed in detail within in the literature and the implications of each one are roughly the same (Milgrom & Weber, 1982).
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The first auction that will be discussed is the English auction of which is the oldest of the four auctions. Krishna (2009) believes that the English auction is perhaps the most prevalent auction form to date. In this auction the auctioneer calls a price and then raises this price as long as two bidders are still interested in buying the auctioned object for that price. The last player that is willing to pay the final price wins the object and pays the price equal to the amount where the second bidder stopped participating (Krishna, 2009).
The second auction, the Dutch auction, is well known because of its use in the flower trading markets in the Netherlands. The auction is seen as the descendant counterpart to the English auction. In this auction the auctioneer starts with a price and slowly lowers the price. At the moment a bidder is willing to pay the price the auction stops and that bidder will obtain the auctioned item for that price. Both the English as the Dutch auction can be seen as open format auctions (Krishna, 2009).
The second-price sealed bid auction is an auction where every player submits a bid at the same time. The idea (as the name already implies) is that bidders are unable to see the other bids being made by the other bidders. The winner of the auction is the bidder who submits the highest bid and pays the price of the second bid (Vicky, 1961). Finally we have the first-price sealed auction. This type of auction differs very slightly from the second-price sealed bid auction. Instead of the second bid the winner pays the price of its own bid. Both the first- and second-price auction can be seen as closed format auctions (Krisna, 2009).
In an independent private value model the Dutch and first price auction are strategically equivalent. The second price and the English auction are also equivalents but in a weaker sense than the other two auctions (Milgrom & Weber, 1982).
In a first price-sealed bid auction everyone submits a bid without any information of others and their bid is purely based on their own information. In a Dutch auction the same can be seen; participants only have their own information and only obtain ex post information. The only information bidders have is in a Dutch and first-price auction is the price that is paid for the auctioned object. Since this happens when the auction is over it does not influence the strategies of the bidders. As a result every single strategy in a first price auction can also be used in a Dutch auction. A main feature in these types of strategies is that it is never rational to bid higher than your own valuation (Krishna, 2009).
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The second-price auction and the English auction are equivalent but not similar to the first-price and Dutch auction. The optimal strategies are only similar when values are private and independent. In the case of independent private values bidders in a second-price auction submits a bid as high as its own valuation; this results in that the person who values the object the highest receives the object for the price equal to the second highest bidder’s valuation. In an English auction the same price occurs only. Here the highest bidder waits till it is the last one in the auction, which will be at the same moment when the second bidder reaches its valuation (Milgrom & Weber, 1982).
As in the independent private value model, the Dutch and the first-price sealed auction are strategically equivalent in a common value model where bidders’ types are affiliated. Contradictory to the independent value model, the English auction and the second-price auction are not equivalent to one another. In an English auction the expected price is assumed to be larger than is the case in a second-price auction (Milgrom & Weber, 1982).
The above statement that the equivalence between the second price and English auctions does not hold in an interdependent values model where bidders’ types are affiliated is further emphasized by McMillan (1994). McMillan and fellow colleagues constructed an auction model for the American government to sell spectrum rights. This auction is seen as an example of one that can be analyzed with the mineral right model since every bidder has some information about the value of the spectrum. In the auction of the spectrum rights it was chosen to make use of an open auction (such as an English auction) instead of a sealed bid auction. The argument is that bidders can update their information during the auction as they observe what other bidders do. In doing so, this leads to less uncertainty (a decrease in ‘winner’s curse) over the price of the auctioned product and possibly to a higher price (McMilan, 1994).
Lastly, Milgrom & Weber (1982) found that if the interdependent values are statically dependent and bidders’ types are affiliated (as in the mineral right model) the second price auction leads to a higher expected price/revenue than a first price auction. This in turn concludes that looking at the standard auction and the average generated price that it is evident that the English auction more often than not obtains the highest price. This is followed by the second-price auction. The first-price and Dutch auction generate the lowest price (Milgrom & Weber, 1982).
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An overview of the standard auctions is given in figure 4. In which it is clearly drawn which auctions are equivalent and in what extent.
Figure 1: Standard Auctions
Figure 1: An overview of the standard auctions and their equivalence. Source: Krishna (2009)
2.4 Tullock contests
A Tullock contest otherwise known as a Tullock auction is a model, originally created by Gordon Tullock (1980). This model is useful in that it can help model the rent seeking behavior of a bidder. The model has many applications and has been analyzed in depth over the years. Furthermore the model is helpful in creating understanding around certain related aspects such as lobbying (which can be related with a tender). The general formula is as following (Che & Gale, 1997);
( ) ∑
In this above equation is the made bid by player i. v is the value of the auctioned prize and ∑
the total amount of the made bids. So the chance of winning v depends on the other bids and how much you bid.
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The factor R can be interpreted in many ways. If the model is used for lobbying R may stand for the political culture (Che & Gale, 1997). Two values of R are studied in depth; R=1 and R=∞. Both can be seen as a special case of Tullock’s model.
All pay auction (R=∞)
The special case in Tullock’s model where R=∞ is called the all pay auction. An all pay auction in many ways looks like the standard auctions but has one major difference, instead that only the winner pays a price every bidder pays the amount of its made bid (Osborne, 2009).
This special auction is used to explain and model real life (economic) events. The most common relation with the all-pay auction is that of lobbying. Lobbying costs can be seen as a bid that is not refundable. Assumed that the bidder who invests the most wins the object, the other bidders will essentially lose their lobbying costs (Che & Gale, 1998).
Baye, Kovencock & de Vries (1996) give three other examples where an all-pay auction can be used. The three examples are (1) first the R&D costs; where every bidder has to pay the R&D cost but only one can win the market. Secondly (2) political contests in which parties all invest in promotion but only one can win the elections. Lastly (3) the search for job promotions where individuals invest time (opportunity cost) to make the promotion but only one person can obtain this (Che & Gale, 1996).
As in the standard auctions a difficulty of using the all-pay auction is the fact that capital constrains can play an important role. Contradictory to the standard auctions, although, is that one has to pay the bid up front. This means that no individual is able to invest more than their wealth, especially knowing that your investment can result in no return. A possible solution, when bidders’ values are commonly known, is an exogenously imposed bidding cap/constraint which stays uniform for all bidders. Che and Gale (1998) find that in some cases a limit on the value of the bids can lead to higher aggregate expenditures. They see two reasons for this: Firstly because firms and individuals with a lower valuation for the auctioned item will participate more aggressively and secondly there is no option that one firm or individual can outspend the rest of the potential participants (Che & Gale, 1998).
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Lottery
The other special Tullock case is where R=1. This gives the following function:
( )
∑
The above function can be seen as a lottery. Every bidder’s chance of winning is equal to the percentage of their bid and it is relative to the total made bids. This means, different from the all-pay auction, that a bidder with a lower bid can also be the winner (Fang, 2002).
Fang (2002) shows that, assuming that the valuation of the prize among the participants is high enough and values are common knowledge, a lottery generates more money for the seller than an all-pay auction would. It is however questionable as to whether this outcome is more socially efficient or not. Another thing that Fang (2002) finds is that the earlier discussed result of Che & Gale (1998), introducing a cap on bidding increases aggregate expenditures are not in fact needed in lotteries. Constraint participants can still hope for rent-seeking opportunities in a lottery although their chances may, depending on the invested amount by others, be smaller than that of a wealthier competitor (Che & Gale, 1997).
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3. Unique Bid Auctions
The use of unique bid auctions has grown dramatically in recent years. The basic rules for this type of auction are that either the highest or the lowest unique bid wins. Although seemingly simplistic, there are in fact many different features in this type of auction. As argued by Bruss et al. (2008), very little is known about the unique bid auction, whereas standard type auctions have been studied in great depth. In the following section a brief overview will be given showing the platforms in which unique bid auctions are used. A review will be conducted of the literature regarding the latter and an alternative unique bid auction will be drawn.
3.1 platforms of unique bid auctions
Media is one of the most common platforms that regularly use unique bid auctions. It is able to reach vast populations of people through the use of media such as radio, television and the internet (Rapoport et al., 2008). In these types of auctions the lowest unique bid usually wins and bidders have to pay a small price per bid. This can result in prizes worth up to $360.000 won by bids as small as $1000. The data provided by the website Auction air LTD gives one a better understanding of this concept i.e. This website, gave out prizes totaling $700.000, where total amounts of winning bids were as little as $12.000 (Eichenberger et al., 2008).
Some other examples of unique bid auctions (where in this case the lowest unique bid wins) are given in figure 2. Examples from different radio stations, TV stations and newspapers are given. Figure 2 shows that prizes are won with bids that are not even 1% of the auctioned object. In figure 3 other examples of lowest unique bid auctions from AuctionAir.com are given. A notable difference that can be seen in figure 2 is the way in which the auction ends. Where the example from figure 2 has a time limit to where participants can submit a bid, are the examples in figure 3 capped with a maximum amount of bid. This shows that in the framework of a unique bid auction small features can be different. One primary feature that these auctions all have in common is that the highest or lowest unique bid wins.
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Figure 2: Different lowest Unique bid auctions
Figure 2: Examples of lowest unique bid auctions. The table shows the medium it is used at, the prices that could be won, the cost of participating, the number of bidders, the total made bids, duration and the winning bid.
In the figure it is shown that prizes with a monetary value of a €1.000 can be won by bids as little as €0.60. This shows the rent seeking opportunity in this auction.
Source: Eichenberger & Vinogradov (2008)
Figure 3: Example of lowest bid auction on Auctionair.com
Figure 3: Some examples of lowest unique bid auction from AuctionAir. The difference of the auctions in figure 2 is that in the auction of auctionair instead of a duration the number of bids was determined to decide the end of the auction.
Also in this figure it is shown that high valued prizes (up to 16.900 pounds) are sold to bids as small as 6 pounds.
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It is not questionable as to why these kinds of auctions exist on the internet. As explained by Scarsini et al. (2010) the motivation of the auctioneer is clear because unique bid auctions generate more money than the value of the object3. Nevertheless, individuals do participate in this auction regardless. As argued by Raviv and Virag (2009), do they participate because they believe that there is a high value prize available for a low price (rent-seeking). This type of behavior is also seen in lotteries and gambling. Since the auctioneer makes more money than the auctioned object, the bidders’ expected payoff has to be negative. In this regard, the auction is usually only of interest to those who are risk loving individuals or individuals who are addicted to gambling (Rivav et al. (2009)4.
3.2 Review of the literature on Unique Bid Auctions
In the literature examples of research to find equilibriums for a unique bid auction can be found. Although the basic rules are simple there is not an easy ‘rule of thumb’ that shows optimum choices (Bruss et al, 2008). Houba et al (2008) agrees with the above mentioned fact stating that the auction may look simple but the mathematical analysis is rather difficult. Furthermore some small but significant differences can be used to formalize the auction which influences the outcomes of equilibriums. Some research has been done to find ‘Nash equilibriums’ and ‘optimal strategies’ within unique bid auctions. All of these studies differ in assumptions, rules of the auction and outcomes (e.g. Raviv and Virag (2007), Pigolotti et al. (2012), Eichberger and Vinogradov (2008)).
Raviv and Virag (2007) have done research on the ‘highest unique bid auction’. Their interest in this subject was influenced by the activities found on auctions4acause.com. This website makes use of the highest unique bid auction with a certain cap on the bids (mostly around the
3This can be simply shown by using figure 2 and figure 3. Looking at the first row of figure 2 a simple
calculation can be made. The prize the bidder can win is €10.000,-. So the cost for the auctioneer for the auctioned object is €10,000. The income it makes is much higher than the auctioned value. The total number of bids in this example is 47,872. For every bid a participant paid €0.49 and the winning bid was €14.55. A simple calculation shows that the income of the auction for the auctioneer was €23,471.83 (47.872*0.49+14.55). This is more than twice the amount of the auctioned price. This proves that these kinds of auctions, assuming you are able to reach a huge audience, can lead to high profits.
4This is especially interesting in the United States of America where some states do not allow people
to gamble. A way out for this group, and indirectly an interesting source of income for the auctioneers, are the unique bid auctions on the internet that are still available
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10% of the value of the auctioned object). In this auction there is a cost of participating and only the winner pays the bid. They assume that differences in the winning bid are so small that it is of interest to look at the probability of winning.
The theoretical findings of Raviv and Virag (2007) are that of different equilibriums over the number of participants. The main finding of their model is that the bids closest to the maximum bid are, in line with their hypothesis, the bids that attain the highest chosen probability. Their findings are tested and supported with the empirical data. Besides does this empirical data shows that the chance of winning the auction by submitting the highest possible bid decreases with an increase in the number of participants. In a unique bid auction with three people, submitting the highest bid ensures a much higher probability of winning than if there were a hundred people. This means that when more people attend the auction relatively lower bids are made (as seen in the data).
Pigolotti et al. (2012) look for symmetric Nash equilibriums in a lowest unique bid auction. Similar to Raviv and Virag (2007) do they also look at winning probabilities. Pigolotti et al. (2012) find a general solution based on a Poissionan distribution in respect to the number of participants.
Pigolotti et al. (2012) compare their findings with a data set. They find a remarkable change in the data when N increases. In what they call the lower participants segment (which holds less than 200 people) their theory adequately predicts the outcomes of the data. In a segment with more than two hundred people however their theory is unable to adequately predict the outcomes.
These finding are shown in figure 5. In the figure the black line follows from the theoretical predictions of Pigolotti et al (2012). The red boxes are the actual bid frequencies by participants drawn from their empirical data. The figure clearly shows that when the number of participants (N) is smaller than 200 the theory predicts the data quite accurate (graph A and B). While looking at the other graphs (C and D), where N is larger than 200, the empirical data does not follow the theory anymore. Another finding in the data is that individuals are more likely to submit odd and prime numbers. This is also shown in figure 5 where outliers can clearly be seen i.e. in graph A where bid frequencies at 17 and 23 are much higher than their surrounding numbers.
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Figure 5: Bidding frequencies
Figure 5: The theory compared to the empirical outcomes. L is the number of auctions held. N the number of players and the monetary value of the price is mentioned as first.
Source: Pigolotti et al. (2012)
Eichberger and Vinogradov (2008) find similar theoretical outcome to that of Pigolotti et al. (2012). Within their empirical data an interesting phenomenon can be found. In this data groups of players clearly show a different bidding distribution to that of others. These kinds of players known as ‘super-rational’ players do not choose to give positive bidding probabilities to the first 100 possibilities (since they believe that these numbers have no chance to win). The distribution difference of a ‘super rational’ player and the bidding behavior following the equilibriums drawn by Eichberger and Vinogradov (2008), Pigolotti et al. (2012) and Raviv and Verag (2009) can be seen in figure 6.
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Figure 6: Alternative distribution proposed
Rational, Theoretical, Distribution
Distribution Super Rational Bidders
P b P roba bi li ti es Bids 100
Figure 6: A different distribution that may explain spikes in the empirical results, theory based on the thought that there are super rational bidders.
Source: Eichberger and Vinogradov (2008), Pigolotti et al (2012), Raviv and Virag (2007)
There are other economists/academics that have done research to try and find equilibriums in unique bid auctions. Rapoport (2009) looks for a mixed-strategic equilibrium that can handle endogenous and exogenous entries using a fixed number of participants. Bruss et al (2007) found a mathematical solution for the optimal bid that assumes a certain distribution of bids. They have used a distribution on the set between 1 and V (where V is the common known, public, value). Lastly, Ostling et al. (2011) looks for equilibriums by using poison games. They argue that on a day to day basis the number of participants can change. Instead of using a lowest unique bid auction they use a lowest unique positive integer auction which differs from a unique bid auctions in that the winner of the auction does not have to pay its submitted bid. A lowest unique positive integer is closely related to a lowest unique bid auction because it still has the same incentives; choosing the lowest unique value.
The main findings in the literature can be summarized in the following way:
1. In a unique bid auction is it not possible to have a symmetric equilibrium in pure strategies. This is explained by the fact that the winning bid has to be unique.
2. Looking at asymmetric equilibrium is not of concern as it will always include a person that choses the lowest (or highest) possible bid and the rest stays out.
3. In a mixed equilibrium the maximum or minimum bid will always have a positive probability with a decrease in probabilities over the bids that follow.
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3.3 Unique bid auctions in the spectrum of the other auctions
In chapter two an overview of other auctions is given. It is of interest to compare these auctions to the unique bid auction. In doing so, a spectrum where auctions move can be drawn as demonstrated in figure 7. When drawing this spectrum two ‘extremes’ are used; on one end standard auctions are used and on the other a pure lottery is used. In the middle we find the unique bid auction and the ‘all pay auction’ that both share features of the lottery and that of standard auctions.
In the standard auction the highest bid wins and only the winner pays their bid. In the all pay auction the highest bid also wins but every bidder pays their submitted bid too. This can also be seen within in a lottery where all players also pay the full amount of all the lottery tickets they buy. Essentially an all pay auction looks more or less like a standard auction except for one feature which makes sure that everybody pays their own bid. It is for this reason that the auction can be placed somewhere in between a lottery and a standard auction.
Figure 7: The spectrum of auctions
Figure 7: A spectrum on at one side a lottery and the standard auction on the other side. The all-pay auction and unique bid auction can be found somewhere in the middle.
Standard
auctions
Lottery
Unique
bid
auction
All-Pay
auction
• Highest bid wins • Only the winner pays its bid • Winning based on chance • All bidderspay their bid • Highest bid
wins • All bidders
pay their bid
• Winning (partly) based on chance • Participation costs
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The unique bid auction has various features of both the lottery and that of the standard auction. In the unique bid auction the highest bid does not always win. This can be seen in a lowest unique bid auction but also in a highest unique bid auction where the winner is not always the one that submits the highest bid.
Raviv and Virag (2009) argue that a unique bid auction is not a lottery because winning the auction is influenced by your own choice. Therefor winning is only partly based on chance. Furthermore, bidders do not pay their full bid in a unique bid auction. In a lottery, bidders pay the full amount for the tickets he or she buys. In a unique bid auction every bidder pays a certain participation cost but not the submitted bid. The submitted bid is, like in a standard auction, only paid by the winner.
This results that in the spectrum both the unique bid and the all-pay auction can be placed somewhere in the middle. In some regards these auctions are similar in the way that all participants are required to pay a certain amount to the seller regardless of whether they win the auction or not. In the unique bid auctions everyone pays the participation costs where in the all-pay auction also losing bids are forfeit.
A unique bid auction lies slightly further away from a standard auction than that of the all-pay auction. The all-pay auction shares more similar aspects to that of a standard auction as the only difference is that every participant pays its submitted bid. The unique bid auction lies further from the standard auction since the chances of winning are partly based on luck. It is clear that both auction types are not seen as either a pure lottery or a standard auction.
3.4 Drawing an alternative unique bid auction (AUB auction)
When analyzing a unique bid auction it is important to determine the rules beforehand. In this section an alternative unique bid auction will be drawn (from now on called the AUB auction). This auction will then be used in further analysis and these features will be discussed in detail in the following chapter. The following set or rules are used when analyzing the AUB auction:
1. The highest ranked unique bid wins the auction 2. Participating in the auction has a fixed cost of C
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4. It is assumed that the only bids that can be made are x,y and z.
o It is important to note that the highest ranked unique bid wins from the second highest etc. From now on bid x>y>z. One should remember that it is assumed that the cost of bid x=y=z=B.
5. In the case of a draw the auction will be repeated 6. B and C are always higher than zero
The AUB auction also differs from standard auctions with respect to decision moments. Contradictory to the more standard auctions this auction has two moments where a decision is needed, namely:
- Whether to participate or not - What bid to make
When analyzing the AUB auction it is important to take both these decisive moments into account. As a result of participation costs bidders already have to make a decision whether they want to participate or not and a bidding choice is made afterwards.
3.5 Motivation of the made choices
Usually in a unique bid auction the price of the submitted bid determines the winner. In this article the bidding costs are assumed to be the same. A ranking between the qualities of the bids is made and will be used in the analysis. In other words this means that the auction is not about the price a bidder is willing to pay but about the quality they are willing to offer. Participants are fully aware of the ranking of the bids and as a result they are aware that the winning bid has to be unique.
In the setup only three bidding options will be considered. We know from the research done by Raviv and Virag (2007), Pigolotti er al. (2012) and Eichberger and Vinogradov (2008) that in a lowest unique bid auction with three players the probabilities that are given to the three bids closest to 0 almost add up to 1. Thus as long as only two or three potential participants are analyzed in the AUB auction more than three option can be neglected and the focus on only three bidding options is sufficient.
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It is evident from the literature that this auction also has a certain participation cost when a bidder wants to submit a bid. In this case the participation cost could be interpreted in the following two ways: Firstly, it is an auction where a seller sells a product and the cost of participating goes to the seller. Secondly, interpretation of this cost can be the cost of preparing a bid.
The choice to fix the bidding cost results from the fact that bids are close to one and other. As Bruss et al. (2009) also argue, due to these small differences in the values, the cost of a bid set the same. A person who is willing to bid 100.91 is also willing to bid 101, - if it his or her winning chances significantly increase. Because there are only three different bidding options, the difference in the price of bids would only be as small as two cents. Furthermore, the difference between x, y, z can also be seen as a different quality offer which all come with the same price B. The latter interpretation will be mainly used in this research.
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4. The model
This chapter will analyze and discuss the new auction conveyed in the previous chapter. It will then compare this auction type with the standard and the all-pay auctions. This chapter will also take a look at the ‘two player case’ and an equilibrium will be established. An analysis will further be conducted using an example of an auction with a third potential bidder. The introduction of the third bidder brings about an array of complex issues and therefor makes for an interesting analysis. A brief study will then be conducted to show how the alternative unique bid auction (AUB auction) can be a useful tool for the overall tendering process.
4.1 The case with two potential bidders
Houba et al (2008) argue that when there is the potential for two participants in a unique bid auction it can be viewed as a ‘Hawk-Dove Game’. The AUB auction differs from this idea since there is no payoff when both potential participants choose not to participate. Assuming that there are only two potential bidders the following analysis can be made: Firstly, the bidders must make the decision as to whether they will participate or not. If one participates they then have the liberty to decide what bid to submit. In this auction bidders can obtain a certain value (V) where the cost of winning the bid (B) is already subtracted. When two players participate in this auction the only rational bid is to bid the lowest (highest) in a lowest (highest) unique bid auction (bidding option x in the AUB auction). As a result if both players participate neither will win the auction and thus both players end up having to pay the participation costs (-C). If only one of the two bidders decides to participate in the auction, then that bidder will inevitably win the auction and further has to pay the participation and bidding costs. Secondly, there is the possible case where both players do not participate, in this instance the payoff will be zero for both players. This can be better demonstrated using a game theoretical picture, as seen in the symmetric game shown in figure 8.
Using figure 8 the expected payoffs can be made which helps to find symmetric equilibriums in this auction. The expected payoffs resulting from figure 8 can be drawn as following;
̅ ( ) ( )
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Figure 8: A two player unique bid auction
Figure 8: The AUB auction with two potential participants. When two bidders decide to participate it only makes sense to bid the highest possible bid. C is the participation cost and V the personal value of the auctioned item minus the bidding cost. This game differs from a regular hawk dove game since payoffs are zero when both do not participate.
The letter p, in the above expected payoff functions, stands for the possibility that a bidder participates. This, obviously, makes (1-p) the assigned possibility that a bidder does not participate in the auction. ̅ and ̅ respectively stands for the expected profit when a bidder participates and when it chooses not to participate. Using the above formulas the following equilibrium can be found:
̅ ̅ If p=
This auction has a strategic equilibrium for two bidders where every potential participant participates with a probability of p= and chooses the highest (or lowest) bid if it participates. In this equilibrium the expected payoffs are zero. In the following section the designed auction will be analyzed with the inclusion of a third potential participant.
4.2 The three potential bidders’ case; a generalized outcome
In this section the new auction type (AUB auction) discussed in chapter three when there are three potential bidders will be assessed. A more general outcome will be drawn based on
-C
V-C
0
0
Participate
Not Participate
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expected payoffs. A theorem showing expected payoff will also be given. Prove of theorem one will be given by showing three propositions. In the analysis it is assumed that all potential participants are able to off quality x,y and z.
Theorem 1
In the auction with three potential bidders the expected payoff when choosing to participate can be given in one general formula, namely:
̅ ( ) ( -B) - C
In this expected payoff function and holds the probability that respectively two, one or zero other players choose to participate the auction. The expected payoffs when one makes the decision not to participate in the auction are zero, which results in:
̅
Proposition 1
When all three players choose to participate in the first round, there is an equilibrium outcome which gives the same probability of winning to all participant; namely one third. Within the equilibrium participants will give the following probabilities to bidding options x,y and z:
=0.464, = =0.278
Proof proposition 1
In the AUB auction participants have three bidding options; x, y, z. The probability is the chance a bidder plays x. is the probability that it plays y and of playing z. It also known that x is the highest ranked bid followed by y and z (so x>y>z) and that only unique bids can win the auction. In this case C is the cost of participating and V is the value of the auctioned bid after subtracting B. To look for an equilibrium the probability of winning the auction by
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either playing x, y and z will be used. These winning probabilities will then be multiplied by V to give the average payoff playing a certain bid.
It is of interest to look at the probabilities of a win by the bidder if it plays x. stands for the winning probability when choosing bid x.
Simplifying the above function results in the following: = [ ]
The same kind of analysis can be done with bidding options y and z. The probability of winning when playing y is:
Simplifying the above formula gives: = [ ]
The probability of winning when playing z is:
Simplifying the above formula gives: = [ ]
To find an equilibrium probabilities given to the bidding options x,y and z should be in such a way that the expected chance of winning is equal. This means that should hold. This gives the following result;
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if =0.464, = =0.278
These calculations give an equilibrium where each participant has the same chance of winning. In this equilibrium the chance of winning the auction is for every player. To determine the expected payoff for every participant in a three player AUB auction one can use the change of winning the auction, namely one third. The expected payoff when three bidders participate is, in equilibrium, one third of the value of the auctioned object minus the bid. Furthermore C should also be subtracted as being the cost of participating. If V (the value of the auctioned object) is unknown, V can be substituted by the individual value ( ) or expected value (E ( )). This outcome proves that the first part of the expected payoff function in theorem one is correct. This all together results in the following expected payoff function:
̅ ( ( ) )-C
Proposition 2
When two potential bidders choose to participate in the auction the expected payoff, in equilibrium, is –C for both bidders.
Proof proposition 2
If only two bidders decide to participate the payoffs can be presented in a regular game form. In earlier analysis the dynamics of two potential bidders in the AUB auction were shown. The analysis given here differs from the latter analysis because participants have already paid C and decided to participate. It is known that the winner of the auction has to come with a unique bid that is ranked the highest among the bids. In figure 8 the symmetric game displays the payoffs.
In figure 9 only one example can be found of a pure symmetric Nash equilibrium that will be played by both participants. In this equilibrium both players will play x. Furthermore when the game is repeated, players will not deviate from playing x (in other words it can be
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seen as an evolutionary stable point where both bidders continuously make the choice to bid x). As a result of this stable equilibrium, the expected income when ending up with two bidders in the AUB auction is –C. The participation costs are already paid but there is no payoff in return. This results in the following expected payoff:
̅ C
Figure 9; The payoffs when two potential bidder choose to participate in the AUB auction
Figure 9: the choices and payoffs when two bidders decide to participate. This is transformed in a symmetric game which shows us that both player will choose x.
Proposition and proof of proposition 3
If a bidder is the only one who participates in the auction it will obtain the auctioned object after paying the participation and bid costs. This results in the following expected payoff:
̅ ( )-B)-C
-C
-C
x yV-B-C
-C
-C
V-B-C
-C
-C
-C
z28
Proof theorem 1
When proving theorem one; propositions one, two and three can be used. The proposition from above can be used to draw the following expected payoff when participating:
̅ ( ( ( )-B)-C) + ( C) + (( ( ))-b)-C)
In this expected payoff function the probabilities of the different situation are included. It should be noted that when an individual participates; =1. By using this equality the payoff function can be rewritten in the following way:
̅ ( ( )-b) + ( ( )-b) - C
This can be simplified as:
̅ ( ) ( ( ) B) – C
In the formulas and functions shown above it can be seen that the expected payoff when participating is based on the cost of participating, the expected value of the auctioned item, the pre-determined price of a bid one has to make and the probabilities of others participating in the auction.
4.3 Using the general model for numerical examples
In this section the ‘independent private value model’ and the ‘mineral right model’ will be used as tools to further analyze the AUB auction. The independent private value model offers various numerical examples of the general model. These numerical examples give one a better idea of the influence of chosen participation and bidding costs. A brief analysis will further be done showing the introduction of the mineral right model into the AUB auction.
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Independent private value model
There are various assumptions that can be made when analyzing the AUB auction type within the independent private value model. In this case the independent values of the auctioned object for every participant are distributed in a uniformly linear manner over {0,1}. This results in the following individual cumulative distribution function of F( )=V.
The above functions can be used to measure the probability of individuals participating in the new AUB auction. To calculate the probability that a potential bidder chooses to participate, an assumption that determines this has to be made. In the following analysis it is assumed that every bidder expects the other to participate when its value is higher than three times the cost of participating plus the bidding cost ( > 3C+B).
The expected probability that a potential bidder will participate in the auction can be shown by using the distribution of . This makes it possible to determine the chances of individuals having a private value higher than 3C plus B. The probability that a player receives a signal higher than 3C plus B is:
1-F( ) ( )
By squaring the above results one can measure the probability of both players participating: = ( ) = ( )
The probability that a bidder will be the only one participating in the auction is: =[F( )
This result in the following expected payoff function as a function of B, C and :
̅ ( )( ( ) ( ) ( )
When using the above model it is important to take into account that it is built on a strong assumption. The probabilities of participation is based on the fact that a bidder expects the
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other bidders to only participate when they have a private value that is three times higher than the participation cost plus the bidding cost ( > 3C+B).
It is of interest to find what the influences of the participation cost are. To draw an example of such it is assumed that a bidder receives a private value of 0.5. The bidding cost in this case is set as 0.1 (B=0.1). With C changing the expected payoff will change as well. In figure 10 these dynamics are shown. One can see that when C becomes 0.14 there is a notable decrease that drops below an expected payoff of 0. When C is above 0.17 there is once again positive expected payoff.
The drop below 0 occurs because in the range where C is between 0.14 and 0.17 the chance of ending up in a two player situation increases. As shown earlier, the expected payoff in a two player situation is minus C. The influence of this effect is of significance and is shown in figure 10. When the participation cost increases above the point of 0.17, the expected payoffs become positive again. At the moment the chances of being the only bidder increase and the expected payoff increase as well.
Figure 10; Expected payoffs keeping private value and cost of bids constant
Figure 10: The expected payoff of a bidder conditional on C. Assumed is a private value of 0.5. The cost of the bid (B) is 0.1.
-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 0 .0 9 0 .1 0 .1 1 0 .1 2 0 .1 3 0 .1 4 0 .1 5 0 .1 6 0 .1 7 0 .1 8 0 .1 9 0 .2 0 .2 1 0 .2 2 0 .2 3 0 .2 4 0 .2 5 E x pected pay off Cost of particpating Expected payoff
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It would be of interest to compare two private values and their different dynamics. Two different private values are compared in figure 11. The different values are respectively 0.5 and 0.7, and it is still assumed that B=0.1. In figure 11 these lines are drawn. The red line follows the assumption that the private value is 0.5, where the blue line follows a private value of 0.7. It is of interest to note that both lines show a drop at the beginning and start increasing again when the participation costs passes a certain point.
Figure 11;Expected payoffs keeping expected value and cost of bids constant
Figure 11: The expected payoff of a bidder conditional on C. The cost of a bid in this graph is 0.15
Furthermore it is of interest to see what the influence is of a difference in the bidding cost. In figure 12 the same figure is drawn as figure 11 only this time B is set as 0.2. Interesting to note is that there is only a small difference in the graphs compared to figure 11. This can be interpreted as that the set of B is not as influential as the choice of the participation cost.
In figure 13 these differences in expected payoffs are shown. It can be seen that changing the bidding costs from 0.1 to 0.2 show small changes in expected payoffs which never exceed a change in expected payoffs of 0.035. This shows that, in this case, the bidding cost have almost no influence on the dynamics of the participation cost.
5
A corresponding table can be found in appendix A
-0.05 0 0.05 0.1 0.15 0.2 E x p e cte d p a y o ff Cost of paticipating Expected payoff B = 0.1 V=0.5 V=0.5
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Figure 12;Expected payoffs keeping expected value and cost of bids constant
Figure 12: The expected payoff of a bidder conditional on C. The cost of a bid in this graph is 0.26
Figure 13;Difference in expected payoffs when changing B from 0.1 to 0.2
Figure 13: The difference in expected payoff of a bidder conditional on C when B changes by 0.17
6 A corresponding table can be found in the appendix B 7
A corresponding table can be found in the appendix C
-0.05 0 0.05 0.1 0.15 0.2 0.25 E x p e cte d p a y o ff Cost of paticipating Expected payoff B = 0.2 V=0.5 V=0.5 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 E x p e cte d p a y o ff Participations costs Difference in Expected payoffs B=0.1 to B=0.2
E(v)=0.5
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To gain an even better idea about the effect of different prices per bid it is of interest to draw a figure conditional on B. If the expected value of a bidder and the cost of participating are fixed the influence of B on the expected payoff can be drawn. This has been demonstrated in figure 14. The different lines stand for different costs of participating and private values. Within figure 12 it can be seen that the influence of B on the expected payoff differ over the different participation costs and private values. Where the movement of the lines over the different values is almost the same, the participation costs do change the influence B has on the expected payoff. Although the main thing that can be concluded, supporting the analysis above, is that the influence of B compared to C is rather small and can almost be neglected.
Figure 14:Expected payoffs keeping expected value and participation cost constant
Figure 14: The expected payoff of a bidder conditional on B. The cost of a participating and values differ over the different lines. Interesting is to see that the lines show different movement patterns when C changes8.
More analysis like this can be interesting, especially when one may be able to relax the made assumptions. Until this point assumptions have been made to come to numerical examples. Although, the general influence of the cost of a bid and the participation cost will be of the same kind when assumptions change. Especially the drop in expected payoffs due to an
8
A corresponding table can be found in the appendix D
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 E x p e cte d p a y o o ff s Cost of a Bid (B) Expected payoffs C=0.1 V=0.5 C=0.1 V=0.7 C=0.2 V=0.5 C=0.2 V=0.7
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increase of participation cost until a certain point and the influence of this participation cost in the dynamics in the bidding cost are of interest.
Mineral right model
When looking at the ‘mineral right model’ and the ‘AUB auction’ it is harder to determine the probabilities of someone participating. Strong assumptions should be made to come to numerical examples. This is because V is not known by any participant. A general outcome can be drawn which could be used when expected values, the bidding cost and participation costs are known.
In the mineral right model V is the value of the auctioned object and is uniformly distributed over {0,1}. In the model given by Krishna (2009) he assumes that V=v and that the distribution of for all individuals ranges between (0,2v). This signals ( ) are independently and uniformly distributed over this range. This leads to the following individual distribution function conditional on V=v; f( )=1/2v or a cumulative distribution function of F( )=x/2v
To determine the probability that someone participates an assumption can be made about both the v and the moment that participants participate (in the independent private value model this was fixed as V>3C+B). Using the same assumption as the one used in the independent private value model and substituting v as being the expected value after your signal ( | a general outcome can be drawn.
̅ ( | )[ ( ) ( | ] ( ) [( ) ( | ] ( | )
Determining the expected values for the participants would give the possibility to come with numerical examples. Furthermore the assumption that is made that participants are expected to participate when 3C plus B is higher than their expected value may also be relaxed. Further research on this topic is of interest when one may want to use this model into the AUB auction.
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4.4 Usability of the proposed model for tenders
The AUB auction type is of interest for two reasons: Firstly, it can be used as a mean to auction an object or to determine the appointment of a contract. Secondly, it can be used as a tool to analyze the procedures of an economic event. These examples are both of interest to economic theory as well as for decisions regarding formalizing a tender.
The introduction to this research gave an example that emphasized the need for the focus on the ‘quality’ of tenders. The tender in the example was based on the best offered price. With the benefits of hindsight it is now clear that costs that have been incurred are for trains that will never be used. If the government had only chosen a more quality oriented auction they may have achieved a more optimal outcome, that of possible high quality, trustable, trains.
The new AUB auction type, as drawn out in chapter three and analyzed in the last sections can be used for determining the appointment of a public contract to a producer or a service provider. As discussed in the introduction, studies that have looked into public tenders are of great importance as the current system in place does not always result in optimal outcomes. The unique bid auction that has been drawn out in this article can be used as an alternative method in the tender acquisition process. As it stands the tender acquisition process currently used by the government leans more towards a focus on cost rather than that of quality. The designed model given in this research could be of use in shifting the current focus from focusing on costs to rather focusing on the quality of public tenders. As discussed in the introduction, the European Union has been looking into alternative possibilities that make it possible for quality play a more important role in auctions.
The new AUB auction takes (unique) quality of the submitted bids into account. A seller (government) using this auction in a tender forces the bidders (the potential firms or individuals that can do the production or offer the service) to focus more on the quality they offer. This can be demonstrated using a simple example; Assume a public tender where the new AUB auction decides the winner. Participants can differentiate in the quality they offer but the price is already fixed up front. The government, in this case, is looking for a unique option with the highest quality for a certain value. The bidding options x,y and z can be seen as different quality functions that a bidder can offer the seller (the government). The only
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factor a bidder can outperform fellow bidders is in quality. It is important to note that in the example of a tender the bidding costs, B, stand for the price the government is willing to pay for the service or product. A higher B in this auction equals a lower price the government is paying for the service9.
Quality plays an important role in the new AUB auction type. All bidders have the choice to either bid x,y or z, where x>y>z. This results in quality being the determining factor when choosing the winner of the auction (and not the price). If the government has a certain income in mind it can influence this. By choosing the right amounts for B and C it can secure a certain income. With the income secured it can then base its decision on the quality offered.
More specific examples can also be used to analyze the usability of the AUB auction type. A certain government is searching for a product of quality x. There are three parties that can potentially produce this product but not all parties are able to offer the quality x. The government has a certain price in mind that it wants to pay (which can be partly compensated via the income of the participation cost). One of the parties, producer 1, is already familiar with producing this product and is able to bid x, y and z. Now there are two other potential parties that are also able to produce this product but can only offer quality y and z. If every producer participates, producer 1 will always win the tender if it bids x. Therefor it is essential for the government that at least one of the other producers participates in the auction so to ensure that it receives x. The other two producers are aware that there is a chance that there may be a third producer that can offer quality x. When determining the participation cost and price of the product the government should consider that keeping C low and compensating by choosing a higher B (so lower production price) gives more chance of more than one participant. By choosing C and B in such a way that B minus three times C results in the price the government aimed to pay it is able to accomplish the optimal result (quality x for the target price).
If the seller is not certain that all potential participants in the above example will participate in the auction it can also choose to ensure its income by making the participation cost in such a way that only one participant would be sufficient to reach the aimed price. A
9
This can be explained the following: A producer offers a product for a certain price. The value of the product minus the price can be seen as the value of the submitted bid. At the moment the price becomes lower the value of the submitted bid becomes higher. That is why it can be argued that a higher B equals a lower set price by a government in a tender.
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pitfall in doing this is, however, that it potentially leads to less competition. If this occurs and only one producer participates in the auction then there will be uncertainty as to whether it will receive quality x. This can result in a semi optimal outcome where the seller it is not assured it will receive quality x.
As well as the latter mentioned weakness, there are other pitfalls that should be taken into account by sellers (or the government) when choosing to use the AUB auction. Firstly, the choice of participation and bidding costs are of great importance and can influence the amount of potential bidders that participate in the auction. As shown in section 4.3 it is not easy to determine what an increase in B will lead to since it is dependent on the choice of C. In general we see a decrease when B increases although section 4.3 shows that this does not always have to be the case which only makes determining C harder. Furthermore one should be reminded that when choosing C an increase first leads to a decrease in expected payoffs and later results in a decrease of competition. Negative expected payoffs could essentially lead to potential participants choosing to stay out of the auction. Not only does this lead to less competition but it also results in a lower income for the seller (since there is a decrease in participants that pay C). Although the example in 4.3 is based on an assumption, the influence of C and B do not entirely depend on this assumption.
Secondly, it is important to note that sellers should always be aware that a situation involving two players in this auction type can often have unfavorable outcomes. In both the situation where there are only two potential bidders and where only two of the potential bidders end up in the bidding round difficulties can arise.
In the first case, with only two potential bidders, the best social outcome is that one of the two bidders participates. This results in a winner that will produce the product or offer the service. The highest chance of only one participant in a symmetric equilibrium is when the possibility of participating is half for both. The seller should choose C and B in such way that this is the case. As learned from section 4.1 this can be accomplished when 0.5 = ( )
For the second case, where only two bidders participate in the auction, it is harder to find a solution. In such a case both bidders (assuming that both are able to submit all possible bids) will bid x. This essentially leads to there being no winner in the auction. A sketch of the given problem is shown in figures 10 and 11 in section 4.3 where a clear drop in expected payoffs occur when the chances of ending up in a two bidders’ situation becomes high. A
