Global B2C and C2C Online Auction Models Timothy L. Y. Leunga, William J. Knottenbeltb

Abstract Online auctions have become an increasingly prevalent mechanism for the exchange of goods and services across the global e-marketplace both among consumers themselves, as well as between businesses and consumers. Such Internet auction mechanisms have the scope of incorporating procedures of much greater complexity and variety, and they exhibit characteristics and properties that are quite distinct from conventional auctions. In this paper, the authors provide an experimental study of the performance characteristics and operational behaviour of a number of online auction models, including the fixed time forward auctions, the Vickrey auctions, as well as models with soft close variable auction times. These online auction models are studied through systematic simulation experiments, based on a series of operational assumptions, which characterise the arrival rate of bids, as well as the distribution from which the private values of buyers are sampled. The behaviour of the average auction income and average auction duration are quantified and compared, and suggestions for efficient online auction design, and procedures for improving auction performance are given.

Keywords:

Online Auctions, Internet Auctions, Vickrey Auction, Auction Income, Auction Duration

                                                                                                                a

Doctoral Student, Department of Computing, Imperial College London, United Kingdom [[email protected]] b

Reader in Applied Performance Modelling, Department of Computing, Imperial College London, United Kingdom [[email protected]]

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Motivation and Related Work

In the early days of E-Commerce, the B2C (Business to Consumer) model was dominant, and the current trend is increasingly embracing the C2C (Consumer to Consumer) model. Thus, instead of companies selling items to the consumers, consumers are selling items to fellow consumers on a global-scale, and a common mechanism of achieving this is to use online auctions. Unlike conventional auctions, Internet auction mechanisms have the scope of incorporating procedures of greater complexity and can take on a wide variety of forms. Four types of auctions are commonly addressed in literature: •

The English auction involves public announcements of gradually increasing bids until a single bidder remains, who pays for the lot at the price of the last bid. This is also known as an open ascending price auction and the process is shown in Figure 1.

•

The Dutch auction is the reverse of this and a high asking price is gradually decreased until a single bidder agrees to pay for the lot at that price. This is also known as an open descending price auction.

•

In the reverse auction, public announcements of gradually decreasing bids are made until a single bidder remains, who agrees to the exchange of services or goods at the price of the last bid. This is also known as an open descending price auction and the process is shown in Figure 2.

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  •

In the sealed-bid first-price auction, all bids are submitted in private and the winner will pay the price that he had bid.

•

In the Vickrey auction, the winner will pay the price that the “runner-up” had bid, i.e. the next highest price. The process is shown in Figure 3.

Further to this, there are other independent properties that can be incorporated when designing an auction as described by Parsons (2009). These other properties are listed below and a taxonomy of auctions is shown in Figure 4. •

Combinatorial: auctions that are combinatorial see multiple heterogeneous goods auctioned together.

•

Dimensionality: in a singularly-dimensional auction, the bid is completely defined by the price of the lot, whereas in a many-dimensional auction, the bid may be a function of other attributes such as the timely delivery of the lot or the length and amount of the insurance contract taken out on that lot.

•

Sidedness: in a one-sided auction, bidders are either all sellers or all buyers. In a two-sided auction, both buyers and sellers submit bids and these are matched by the auctioneer.

In fact, in auctions without time restrictions, the English and the sealed-bid secondprice auctions are shown to be equivalent, while the Dutch and the sealed-bid firstprice auctions are also shown to be equivalent. Furthermore, this is generalised by the Revenue Equivalence Theorem which is shown by Myerson (1981) and displayed in Theorem 1, which states that the seller will obtain the same revenue for all auctions where: (i) the bidder with the highest bid always wins, (ii) the bidder with the lowest bid expects zero surplus, (iii) all bidders are risk neutral, and (iv) the private values of all bidders are drawn independently from the same distribution. By allowing sellers to

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equate the revenue generating power of different types of auctions that may otherwise seem incomparable, the Revenue Equivalence Theorem assists sellers in auction mechanism design. It is equally interesting to observe cases where the Revenue Equivalence Theorem does not hold, e.g. if the private values of bidders are shown to be related and not independently drawn from the same distribution; Milgrom (1982) shows that the English auction yields a higher level of revenue than the sealed-bid first-price auction. Theorem 1 from Myerson, 1981 The seller's expected utility from a feasible auction mechanism is completely determined by the probability function p and the numbers Ui(p, x, ai) for all i. That is, once we know who gets the object in each possible situation (as specified by p) and how much expected utility each bidder would get if his value estimate were at its lowest possible level ai, then the seller's expected utility from the auction does not depend on the payment function x. Thus, for example, the seller must get the same expected utility from any two auction mechanisms which have the properties that (1) the object always goes to the bidder with the highest value estimate above to and (2) every bidder would expect zero utility of his value estimate were at its lowest possible level. If the bidders are symmetric and all ei = 0 and ai = 0, then the Dutch auctions and progressive auctions studied in Vickrey (1961) both have these two properties, so Vickrey's equivalence results may be viewed as a corollary of our equation. However, we shall see that Vickrey's auctions are not in general optimal for the seller. Internet auctions have begun to pervade large sections of the Internet economy and there is an increasing amount of literature in this field. Internet auctions exhibit characteristics which are often not shared by conventional auctions. For example, Internet auctions are asynchronous and allow the bidders to be in different places in

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different times, and although the recent addition of telephone bidding in conventional auctions allows bidders to be in different places yet participating in the same auction, they still have to bid at the same time or in close succession. A few features of Internet auctions can be attributed to the sheer size and volume of the buyer and seller markets. This allows for a variety of different auctioning mechanisms, e.g. timeshift auctions and penny auctions, and yet still have enough liquidity and market activity to attract the interest of both buyers and sellers. In this sense, conventional auctions depend on getting the right people to the auctions in order to obtain an accurate valuation of the good and service sold – people that are interested and willing to buy. Internet auctions take the other side and depend on the large amount of traffic generated on the site in such a way that with the number of items sold and the amount of potential buyers browsing the site, they are bound to see something they are interested in. A final point to note is the full range of services offered by a conventional auction house as opposed to a online auction site such as eBay. In an auction house, items being sold are underwritten and buyers are often permitted to inspect the item beforehand, whereas with Internet auctions sites, this is often not possible. Studies of Internet auction bidding behaviour have been undertaken in Ockenfels and Roth (2002, 2006) and Wenyan and Bolivar (2008). In Wenyan and Bolivar (2008), different properties of online auctions such as consumer surplus, sniping, bidding strategy and their interactions are studied, where a significant correlation between sniping and surplus ratios is found. It also examines the efficiency of online auctions, where Pareto efficiency is used as the optimality criterion. In Ockenfels and Roth (2006), it is suggested that the strategic advantages of sniping are eliminated or severely eroded in auction mechanisms that apply an auction extension rule, and that

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there is noticeable difference between sniping on eBay and Amazon in proportion to user experience. Experimental studies of Internet auction behaviour have been undertaken in LuckingReiley (1999), Vragov (2009), and Katok and Kwasnica (2008). In Katok and Kwasnica (2008), it concentrates on the Dutch auction and first-price sealed bid auction formats, using laboratory experiments and human subjects, where values are drawn from the uniform distribution between 0 and 100, focusing primarily on the effect of clock speed on seller’s revenue. In Vragov (2009), laboratory experiments with human subjects are also conducted, and the operational efficiency of Internet auctions is studied. Collusion behaviour such as shilling, in which the seller plays a part in the bidding process, is studied in Kauffman and Wood (2005), where two types of shilling strategies are examined, which deploys competitive bidding and reserve price mechanism and each of these exhibits a characteristic pattern of behaviour. While an auction can be defined as a market institution whereby offers are made only by the buyers, i.e. bids, or only by the sellers, i.e. asks, a double auction is one where both buyers and sellers are able to make offers, as described in Friedman (1993). Viewing the interlinking relationship between bidders and sellers as networks is proposed in Dass and Reddy (2008), and the competitions in auctions is investigated in Haruvy et al. (2008). Price variation characteristics and consumer surplus are studied in Bapna et al. (2008) and Jank et al. (2006). The use of various types of curves for fitting price data for Internet auctions have been proposed in Hyde et al. (2007), in which monotone splines and beta functions are used. Empirical investigations of eBay auctions have also been undertaken in Lucking-Reiley et al. (2007) where the auction of coins is conducted. It makes use of regression models to estimate the price of items and examines the influence of seller ratings (which measures the reliability and services provided by the seller) on the final price. It has

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also found that the effect of positive and negative ratings is not symmetrical, with the latter having a much greater (adverse) influence on the price. It also suggests that longer auctions tend to have a beneficial effect in achieving a higher price. Moreover, in Dellarocas and Wood (2008), it has found that there is a reluctance on the part of users to give negative feedbacks compared with giving positive feedbacks. The Kmeans clustering algorithm has been employed in Bapna et al. (2004) which classifies bidders into five categories based on factors such as entry time, number of bids placed, and exit time. It also examines the use of automated agents in carrying out bidding as well as the different experience levels of bidders. The use of analogies from physics to study price movements have been applied in Hyde et al. (2007) and Jank et al. (2008) and Wang et al. (2008), which make use of the concepts of pricevelocity to characterise the dynamics of price changes and may subsequently be exploited to produce forecasts.

Experimental Evaluation using an Auction Process Simulator

To enable the comparison between observed and theoretical values and to validate the mathematical models, an auction process simulator that implements the pseudo-code for the different auction algorithms, has been constructed in C++. In order to sample values from the uniform and exponential distributions for the private value of bidders and the rate of bids respectively, the Boost C++ Library is used. In particular, we use the

variate_generator

with

the

uniform_01

and

exponential_distribution headers, which is implemented on top of the mersenne_twister psuedo-random number generator. The result is outputted as a space-delimited text string that states lambda, which is the incoming rate of bids and

 

 

usually the variable we change, the duration of that auction, and the revenue generated from that auction. Ten thousand trials are run for each arrival rate, which is sampled in .01 intervals in the units concerned over the desired interval. An Internet auction website contains a number of buyers and sellers and runs multiple auctions, each of which contains a subset of the auction website’s buyers and sellers exchanging money for lots of goods and services. Each auction has a set of parameters, which equate to the parameters of the various algorithms listed in this paper. Buyers have a utility function for each lot that is present at the auction website, and if the utility gained from buying a lot at a specific price is greater than that derived from holding the money, or using that lot to buy another lot, then the buyer will offer to bid for that lot in an auction. Both the utility of a win and a loss is required in determining how close a buyer should bid to his true valuation. If the loss of an auction is very costly, the buyer should bid the true valuation of the lot straight away, instead of trying to augment his utility by obtaining the lot at a lower price that he was willing to pay for it. The confidence that a buyer has in his valuation determines whether the valuation will be adjusted in the face of significantly higher or lower bids from other bidders present in the auction. A high confidence in the values means that the list of utility for each lot remains the same throughout an auction, while a lower confidence may mean that the auction itself is, to a certain extent, a price discovery mechanism for the bidder and his utility of obtaining the lot may vary in accordance with the current bid levels. In the same way, the seller also has lists of utilities and confidences, as well as a value for reputation, which may result in a price discount or premium in an auction depending on factors that may be external to the lot, e.g. timeliness of delivery.

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Performance Assessment of Internet Auctions and Non-Internet Auctions

Fixed Time Forward Auctions A forward auction is an electronic auction where buyers compete for items or services, with the price going up over time, and the items or services for sale are displayed and specified in a particular website (e.g. uBid.com). Fixed time auctions are adopted by website such as eBay and are susceptible to sniping, or bidding at the very last moment in order to prevent other users from submitted a counter-bid. Pseudo-code describing how the fixed time forward auction is implemented is presented as follows: begin L = 0; accept_id = null; while clock < T do begin for an arriving bid of magnitude R, if L < R, then do begin L = R; accept_id = bidder_id; end; end; return bid L offered by accept_id; end;

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Figure 5 shows the average auction income for different values of λ, or the income rate of bids in terms of bids per unit time, for L=100 (bids are uniformly distributed between 0 and L), T=10, which is the fixed duration of the auction expressed in time units. We see that the increase in bid rate up to λ=4 produces rather steep average auction income improvement. There seems to be a critical bid rate at around λ=6, above which the improvement in income becomes less pronounced.

Variable Time Forward Auctions with Fixed Inactivity Window Here, unlike the fixed time forward auction, sees the auction terminate when there is no bid arrival for a fixed window of length α. On termination, the largest bid received will be accepted. This type of auction is implemented by many penny auction websites such as swoopo.co.uk. Pseudo-code describing how the variable time forward auction with fixed inactivity window is implemented is presented as follows: begin L = 0; accept_id = null; counter = clock; while clock < counter + alpha do begin for an arriving bid of magnitude R, if L = 1/R, then do begin L = R; accept_id = bidder_id; counter = clock;

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  end; end; return bid L offered by accept_id; end;

The fixed inactivity window α may be adjusted, and for meaningful operation, it should not be significantly smaller than the bid inter-arrival time; a high value for α will produce a higher income but the auction will take longer. Unlike the fixed time forward auction, in which the auction duration is always bounded by T, here it is possible for the auction duration to go on for an indefinite period (unbounded) without a predictable end point. In particular, if the average bid inter-arrival time 1/λ is significantly less than α, i.e. α ≫ 1/λ, or λα ≫ 1, then there is little chance of having a clear interval of length α without any arrival. Figure 6 shows how the auction duration varies with λ for α = 1, 2, 3 and 4. We see that the duration increases gradually for small values of λ, but accelerates for large values. As the bid rate increases, there is reduced chance of a no-bid interval occurring, which lengthens the auction, and the difference between α = 1 and α = 4 also becomes more pronounced as λ→1. Figure 7 shows how the auction income varies with λ for α = 1, 2, 3 and 4. We see that the income increases as the bid rate increases, and it also tends to grow as α increases. However, the growth in income dampens somewhat for larger values of α.

Fixed Time Vickrey Auctions Various forms of Vickrey auctions, which are sometimes called second-price auctions, are common on the Internet. In eBay’s system of proxy bidding, for

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example, the auction winner does not pay the highest bid, but the one below it plus a bid increment. Variations exist for the increment charged by the auction mechanism which may be a fixed increment, or one that depends on the value of the relevant bid. An advantage of Vickrey auctions is that it gives bidders an incentive to bid for an item’s true value without needing to worry about overpaying. As in the first price forward auction presented above, here the auction time is fixed with duration T. For effective auction operation, one would need to have at least two bids, and thus T should be significantly greater than twice the mean inter-arrival time 1/λ. If N is the number of bid arrivals in the time interval (0,T), then at the close of the auction, the auctioning mechanism will pick the bidder of maximum bid to be the winner while the amount the winner pays will be the amount bid by the second highest bid. Pseudo-code describing how the fixed time Vickrey auction is implemented is presented as follows: begin L1 = 0; L2 = 0; accept_id1 = null; accept_id2 = null; increment = d; while clock < T do begin for an arriving bid of magnitude R, if L < R, then do begin L2 = L1; accept_id2 = accept_id1;

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  L1 = R; accept_id1 = bidder_id; end; end; L2 = L2 + increment; return payment L2 offered by accept_id1; end;

Figure 8 shows the average auction income for different values of λ, or the income rate of bids in terms of bids per unit time, for L=100 (bids are uniformly distributed between 0 and L), T=10, which is the fixed duration of the auction expressed in time units. This is compared with the first-price auction in Figure 11 where we can see a marked decrease in the auction income. Outside the simulation, however, bidders may hesitate to bid their true private value and may shade down their bids in case they overbid the second-bidder by a large margin, also resulting in a reduction of auction income for the seller. Thus, it is plausible for the seller to opt for the Vickrey auction option as opposed to the first price auction option, even though there is a theoretical reduction in auction income.

Variable Time Vickrey Auctions with Fixed Inactivity Window In the same way, a Vickrey auction can be applied to the second algorithm we have mentioned, i.e. the Variable Time Forward Auctions with Fixed Inactivity Window, and simply allowing the winner to pay the price of the second bid at the termination of the auction. Pseudo-code describing how the variable time Vickrey auction with fixed inactivity window is implemented is presented as follows:

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begin L1 = 0; L2 = 0; accept_id1 = null; accept_id2 = null; increment = d; counter = clock; while clock < counter + a do begin for an arriving bid of magnitude R, if L < R, then do begin L2 = L1; accept_id2 = accept_id1; L1 = R; accept_id1 = bidder_id; counter = clock; end; end; L2 = L2 + increment; return payment L2 offered by accept_id1; end;

Figure 9 shows how the auction duration varies with λ for α = 1, 2, 3 and 4. In a similar fashion to the corresponding graph for the first price auction in Figure 6, we see the duration increases gradually and then accelerating for increasing values of λ.

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Figure 10 highlights the increase in auction income for varying λ, which is again similar to the first price graph, as seen in the comparison shown in Figure 12. It is interesting to note that the difference in the income earned by an auction in the first price and Vickrey instances are not always equal, with the peak difference occurring around the .5 λ mark. This is because at smaller values of λ, the likelihood that there are two bids in an auction is not very high; in an auction with only a single bid, the income derived from a first price auction is the same as a Vickrey auction. For larger values of λ, since the private values are drawn from the same distribution, i.e. uniformly distributed between 0 and L = 100, with more bidders participating in an auction, the mean difference between bids also decreases.

Summary of Findings and Observations Thus, from these experimental results, a number of general observations can be made: 1. If auctions are run sufficiently long and there exists a sizeable pool of interested and competing bidders, then from the sellers point of view, running a Vickrey auction has two distinct advantages: i) there is virtually no difference in price, as evident in Figure 11 where the auction income tends towards the same level for a high rate of incoming bids, and ii) the psychological assurance from the point of buyer, that he/she is paying no more than other interested parties encourages participation and bidding at his or her private valuation of the lot. 2. The inactivity window, α, should not be too large, otherwise the auction may last for a long time especially if the arrival rate of bids is also high. Also, in all auction income graphs where α is compared, the differences where α = 1, 2

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  shows more drift than α = 3, 4. This signifies that the marginal benefit derived from increasing the inactivity window lessens, thus when designing auctions, the length of the inactivity window should not be unreasonably large. 3. Auction income increases steeply at first but slows down as it approaches the buyer’s valuation. This is because as more and more bidders join the auction, the price quickly closes in on the valuation held by the majority of bidders. 4. Compared with first-priced auctions, the increase in auction income for Vickrey auctions, tends to be relatively subdued and gradual. 5. Figures 13 and 14 show the corresponding fixed time and variable time auctions for different distributions of the bidder’s private value. Under the uniform distribution, the value of the bids are uniformly distributed between 0 and 100, while under the normal distribution, the value of the bids have normally distributed with the same mean and standard deviation as the uniform distribution, i.e. a mean of 50 and a standard deviation of mean 50/√3. For bid rates between 0 and 1, the two curves show remarkable similarity while at larger bid rates, the values slowly diverge. This is due to the uniform distribution having a cap of 100 on its values, which is approached as a large number of bids are received, while in the normal case, there is no such restriction. Thus, from the graphs, it can be seen that for low to moderate bid rates, the simulations seem to suggest that results are largely insensitive to the underlying distribution for the bidder’s private value.

Summary and Conclusions

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In this paper, we have studied four different types of online auctions: i) the Fixed Time Forward Auction, ii) the Variable Time Forward Auction with Fixed Inactivity Window, iii) the Fixed Time Vickrey Auction, and iv) the Variable Time Vickrey Auction with Fixed Inactivity Window. The performance evaluation of these auction types, has been based on a series of operation assumptions which characterise the arrival rate of bids, as well as the distribution from which the private values of buyers are sampled. Performance behaviour of the average auction income and average auction duration have been quantified, in particular comparing the auction income between the fixed time first price and Vickrey auctions, and between the variable time first price and Vickrey auctions with fixed inactivity windows. From this, several general observations have been made: i) if auctions are run for a long time and with many interested bidders, running a Vickrey auction creates psychological assurance for buyers in that he/she is paying no more than other interested parties, thus encouraging participation and bidding, while there being little difference in the auction income, ii) a large inactivity window extends the length of an auction unnecessarily while offering little increase in auction income for the seller, and iii) where auction income is concerned, this usually increases steeply at first and then levels out due to the price quickly closing in on the valuation held by the majority of bidders as more bidders join the auction. With the prevalence of Internet auctions in recent times, they have shown increasing commercial value. A model and analysis of these auctions is useful and may eventually shed light on the auction behaviour and dominant strategy of the various participants,

and

this

paper

is

a

useful

step

in

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References

Bapna, R., Goes, P., Gupta, A., and Jin, Y. (2004). User heterogeneity and its impact on electronic auction market design: An empirical exploration. MIS Quarterly, 28(1):21–43.

Bapna, R., Jank, W., Shmueli, G., and Hyderabad, G. (2008). Consumer surplus in online auctions. Information Systems Research, 19:400–416.

Dass, M. and Reddy, S. (2008). An analysis of price dynamics, bidder networks and market structure in online auctions. In W. Jank and G. Shmueli, eds. Statistical Methods in eCommerce Research.

Dellarocas, C. and Wood, C. (2008). The sound of silence in online feedback: Estimating trading risks in the presence of reporting bias. Management Science, 54(3):460–476.

Haruvy, E., Popkowski Leszczyc, P., Carare, O., Cox, J., Greenleaf, E., Jank, W., Jap, S., Park, Y., and Rothkopf, M. (2008). Competition between auctions. Marketing Letters, 19(3):431–448.

Hyde, V., Jank, W., and Shmueli, G. (2007). A family of growth models for represent- ing the price process in online auctions. In Proceedings of the ninth international conference on Electronic commerce, page 408. ACM.

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Jank, W., Shmueli, G., and Wang, S. (2006). Forecasting Online Auctions using Dynamic Models. In Workshop on Data Mining for Business Applications, page 1. Citeseer.

Jank, W., Shmueli, G., and Wang, S. (2008). Modeling price dynamics in online auctions via regression trees. In W. Jank and G. Shmueli, eds. Statistical Methods in eCommerce Research.

Katok, E. and Kwasnica, A. (2008). Time is Money: The effect of clock speed on sellers revenue in Dutch auctions. Experimental Economics, 11(4):344–357.

Lucking-Reiley, D. (1999). Using field experiments to test equivalence between auction formats: Magic on the Internet. American Economic Review, 89(5):1063– 1080.

Lucking-Reiley, D., Bryan, D., Prasad, N., and Reeves, D. (2007). Pennies from Ebay: The Determinants of Price in Online Auctions. The Journal of Industrial Economics, 55(2):223–233.

Milgrom, P. and Weber, R. (1982). A theory of auctions and competitive bidding. Econometrica: Journal of the Econometric Society, 50(5):1089–1122.

Myerson, R. B. (1981). Optimal Auction Design. Mathematics of Operations Research, 6(1):58.

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Ockenfels, A. and Roth, A. (2002). The timing of bids in internet auctions: Market design, bidder behavior, and artificial agents. AI Magazine, 23(3):79.

Ockenfels, A. and Roth, A. (2006). Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction. Games and Economic Behavior, 55(2):297–320.

Parsons, S., Rodriguez-Aguilar, J. A., and Klein, M. (2009). Auctions and bidding: A guide for computer scientists. ACM Computing Surveys.

Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16(1):8–37.

Vragov, R. (2010). Operational efficiency of decentralized Internet auction mechanisms. Electronic Commerce Research and Applications, 9:111–125.

Wang, S., Jank, W., and Shmueli, G. (2008). Explaining and forecasting online auction prices and their dynamics using functional data analysis. Journal of Business and Economic Statistics, 26(2):144–160.

Wenyan, H. and Bolivar, A. (2008). Online Auctions Efficiency: A Survey of eBay Auctions. In Proceedings of WWW 2008 Conference, Beijing, China, pages 925– 933.

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Figures

Figure 1: English auction process.

Figure 2: Reverse auction process.

Figure 3: Vickrey auction process.

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Figure 4: A taxonomy of auctions.  

Figure 5: Average auction income for fixed time forward auction.

Figure 6: Average auction duration for variable auction time with fixed inactivity window.

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Figure 7: Average auction income for variable auction time with fixed inactivity window.

Figure 8: Average auction income for fixed time Vickrey auction.

Figure 9: Average auction duration for variable time Vickrey auction with fixed inactivity window.

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Figure 10: Average auction income for variable time Vickrey auction with fixed inactivity window.  

Figure 11: Average auction income for fixed time first price and Vickrey auctions.

Figure 12: Average auction income for variable time first price and Vickrey auctions with fixed inactivity windows (α = 3).

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Figure 13: Average auction income for fixed time first price auctions with uniformly and normally distributed private values.

Figure 14: Average auction income for variable time first price auctions with fixed inactivity windows (α = 3) and uniformly and normally distributed private values.

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