Bid or Buy-it-Now: A Real Options Approach for Single and Sequential Auctions

Bid or Buy-it-Now: A Real Options Approach for Single and Sequential Auctions por Nuno Miguel de Vasconcelos Monteiro Dissertação de Mestrado em Economia Orientação por Professor Doutor Paulo Jorge Marques de Oliveira Ribeiro Pereira Faculdade de Economia da Universidade do Porto 2012 Nota Biográfica Nuno Miguel de Vasconcelos Monteiro nasceu no Porto em 1989. Em 2007 iniciou a Licenciatura em Economia na Faculdade de Economia da Universidade do Porto, que viria a terminar 3 anos depois. Em 2010 ingressou no Mestrado em Economia, com especialização em Modelação e Simulação Económica, na mesma faculdade. Ao nível de trabalhos académicos relevantes, em 2012 participou no 5ºEncontro de Investigação Jovem da Universidade do Porto com o trabalho “Efficiency Analysis of Portuguese Hospitals”, realizado em conjunto com Ana Silva e Liliana Fonseca e com orientação da Professora Ana Brochado. i Agradecimentos Em primeiro lugar, gostaria de agradecer ao meu orientador, o Professor Paulo Pereira, pelo incentivo que me deu para iniciar este desafio, por todo o apoio e pela extrema dedicação na orientação desta dissertação. Sem a sua colaboração este resultado não teria sido possível. Gostaria também de deixar um agradecimento aos meus colegas André Monteiro, Diogo Barbosa e João Guimarães por me terem acompanhado durante o desenvolvimento deste trabalho. Deixo também uma referência aos que fizeram parte do meu percurso pela Faculdade de Economia, todos contribuiram de alguma forma para a minha formação ao longo dos últimos cinco anos. Finalmente, um agradecimento especial aos meus pais e à minha irmã. Apesar de este ser um trabalho académico, é obrigatório o agradecimento à minha família por me terem dado todas as condições necessárias para a sua realização, por todo o apoio ao longo dos anos e por terem contribuido para o que sou hoje. ii Resumo Nesta dissertação, propõe-se um novo modelo capaz de tratar o problema de leilões com opções de compra. Este formato de leilão tem vindo a tornar-se cada vez mais popular em websites de leilões, mas toda a literatura sobre este tema é reduzida e bastante recente. Com recurso à teoria das opções reais, o problema das opções de compra permanentes é abordado e um modelo capaz de determinar o ponto de decisão óptimo é proposto. Os resultados do modelo permitem ao agente decidir se deve colocar mais licitações ou se deve optar por exercer a opção e comprar o bem imediatamente, utilizando apenas parâmetros de calibração de expectaticas e o preço de exercício da opção. Habitualmente, em websites de leilões, existem vários leilões disponíveis para bens semelhantes. Num segundo modelo (baseado no modelo inicialmente apresentado), aborda-se o problema considerando a existência de leilões sequenciais, determinando a estratégia óptima em cada um dos leilões considerados. Para ambos os modelos são apresentados exemplos numéricos, bem como uma representação dos efeitos de revisões de expectativas nos resultados obtidos. Palavras-Chave: Leilões, Opções Reais, Buy-it-Now Códigos de Classificação JEL: D11, D44, D81 iii Abstract This dissertation proposes a new model for auctions with buy-out options. This type of auction has become a common practice in major auction websites, but all the literature about this subject is scarce and fairly recent. By using real options theory, the problem of permanent buy-out options is addressed and a model capable of retrieving the optimal timing to buy-out is proposed. The model results allow the agent to decide whether to bid in the auction or to buy the item through the buy-out option using only expectation parameters and the buy-out price. Usually, in auction websites, there are multiple auctions for similar items. A second model (based on the first presented model) accounts for the sequential auctions scenario, granting the optimal strategies for each auction considered. For both models, a numerical example is presented, as well as the effects of expectation reviewing on the results. Keywords: Auctions, Real Options, Buy-it-Now JEL Classification Codes: D11, D44, D81 iv Contents I Introduction 1 1.1 Motivation................................. 1 1.2 Previous Research on Auctions with Buy-Out . . . . . . . . . . . . . 2 1.3 The Approach Proposed in this Dissertation . . . . . . . . . . . . . . 4 II The Models 6 1.1 TheModelSetup ............................. 6 The Model for Single Auctions 10 2.1 Bidding .................................. 11 2.2 Option Value and Optimal Behaviour . . . . . . . . . . . . . . . . . . 14 2.3 Analytical Comparative Statics . . . . . . . . . . . . . . . . . . . . . 18 2.4 Numerical Example and Results . . . . . . . . . . . . . . . . . . . . . 21 The Model for Sequential Auctions 25 3.1 Bidding .................................. 26 3.2 Option Value and Optimal Behaviour . . . . . . . . . . . . . . . . . . 27 3.3 Analytical Comparative Statics . . . . . . . . . . . . . . . . . . . . . 30 3.4 Numerical Example and Results . . . . . . . . . . . . . . . . . . . . . 33 III Conclusions 38 Suggestions for future research . . . . . . . . . . . . . . . . . . . . . . 39 PossibleExtensions............................ 39 Appendix 41 A The Model for Single Auctions with monitoring costs . . . . . . . . . 41 v B The Model for Sequential Auctions with large differences in Buy-Out Prices ................................... 43 References 47 vi List of Figures 2.1 Mean-Reversion Processes in variables Pand w............ 13 2.2 Option’s Value and Optimal Trigger for Single Auction . . . . . . . . 22 2.3 Option’s Value and Optimal Triggers for Single Auction - Comparison 24 3.1 Buy-out prices effects on the first auction’s threshold . . . . . . . . . 32 3.2 Option’s Value and Optimal Trigger for Sequential Auctions - First Auction .................................. 34 3.3 Option’s Value and Optimal Trigger for Sequential Auctions - Last Auction .................................. 34 3.4 Revieweffects............................... 36 3.5 With λ2unchanged............................ 36 3.6 With λ1unchanged............................ 36 A.1 Option’s Value and Optimal Trigger . . . . . . . . . . . . . . . . . . . 42 B.1 Single vs Sequential Auctions . . . . . . . . . . . . . . . . . . . . . . 46 List of Tables 2.1 Model Parameters - Single Auction Example . . . . . . . . . . . . . . 22 2.2 ThresholdResult ............................. 22 2.3 Model Parameters - Expectations Review . . . . . . . . . . . . . . . . 24 2.4 ThresholdChanges ............................ 24 3.1 Model Parameters - Sequential Auction Example . . . . . . . . . . . 33 3.2 ThresholdResults............................. 33 3.3 Model Parameters - Expectations Review . . . . . . . . . . . . . . . . 35 3.4 ThresholdChanges ............................ 35 A.1 Threshold Results with monitoring costs . . . . . . . . . . . . . . . . 42 B.1 ModelParameters............................. 45 B.2 Threshold Results - Sequential Auctions . . . . . . . . . . . . . . . . 45 B.3 Threshold Results - Single Auction . . . . . . . . . . . . . . . . . . . 45 vii Chapter I Introduction 1.1 Motivation Auctions have become a popular means for buying items in the past decades since the arrival of online auction platforms in the 90s. Auction theory itself has been around for much longer, but the current auction formats are a bit different from the traditional ones. In fact, since the appearance of auction sites like eBay or Yahoo, auctions were opened to wider ranges of bidders and adapted to fit the needs of such large demand. It’s a new reality, constantly growing and changing. It is also important to note that in the US, the online auction market changed recently since eBay practically has no competition. In 2007 Yahoo closed the auction website after giving up on other countries as well, now operating only in Japan, Taiwan and Hong-Kong. Today one can access eBay (or any similar website) and practically buy anything, and can do so without even leaving home. While the traditional idea of auctions associates with high value items or collection pieces and small audiences, today we see auctions for virtually every kind of item, available to any interested customer. With the changes in the auction setup it’s natural that some new auction concepts and rules arise, evolving from traditional auction formats. Longer durations (usually several days), proxy bidding systems1, unknown reserve prices2, and many other rules have been adopted by online auction websites. All of them are introducing new variables into the auction game, and changing the outcomes of their traditional versions. One particular example is the Buy-it-Now clause from eBay or from Yahoo, where the bidders have also the chance to buy the item at a fixed price if they don’t want to participate anymore in the auction. Since 2000, buy-out options started appearing in online auction sites and spread to sites like uBid, Bid or Buy, eBay 1A bidder tells the auction website the absolute maximum that he is ever willing to bid for an item. Then, website places a bid on his behalf and continues to bid on his behalf whenever he is outbid by another member’s bid, until the maximum is exceeded or the auction is won (www.ebay.about.com). 2A reserve price is an optional feature used by sellers to allow them to void the results of an auction if bidding does not reach their desired price. Buyers can, however, identify reserve price auctions by the presence of “Reserve not met” (or “Reserve met”) in the auction information, indicating that a reserve price has been set and it has not (or has) been exceeded by bidding (www.ebay.about.com). 1 expectations of auction evolution will determine who keeps the good. “How much are my opponents willing to bid?” and “Is the current price high enough to assume that someone will buy-out the item soon?” are some questions that can describe the kind of expectations that a bidder must have, in order to set a threshold for the price evolution that will trigger his buy-out of the item through the option. This threshold will be compared to bidding evolution and will help the buyer decide when it is time to buy-out the item. It’s important to say that this trigger works for both single bidding or incremental bidding strategies. For single bidding strategy consider setting one single bid (equal to the maximum value that you are willing to bid before buying out the item) 5right at start and follow the auction evolution. Incremental bidding is studied in Roth and Ockenfels (2002); Ockenfels and Roth (2006); Peters and Severinov (2006). It’s basically a strategy defined by not bidding your maximum at start, and raising your bid cap as you get outbid. For single bidding strategies, the optimal value to bid will be the threshold that comes from our model, and buying out the item will only be optimal if the agent gets outbid and, in that case, should be done immediately. In case of incremental strategies, the difference is that the agent will place several lower bids before setting the threshold as his maximum value. If he doesn’t get outbid for bids lower than the threshold, then he wins the auction. If he is forced to raise his bids and place one with the threshold value, then he will not be willing to raise the bid any higher and should exercise the option to buy-out the item if (and when) a competitor manages to outbid him. Related to the buy-out option, consider the following assumption: Assumption 1. Assume that when the agent wants to exercise the option, only he can get the item, meaning that a competitor who also wants to buy-out at the same time as our agent will lose it. This implies that, when exercising the option, the item is won by the agent with probability 1, not having to account for the possibility of another player also exercising his option at the same time. For modeling we use an increasing bids approach, replicating the implicit iterative process of the proxy bidding system. This means that, in our model, bids will be observed as a continuous increasing process instead of an increase with random jumps. In the end, the trigger itself will provide a threshold strategy. By setting the thresh5Reservation Value/Price is the highest price the buyer is willing to pay for the item. Equals the agent’s valuation. 8 old as the maximum bid in the proxy bidding program, being outbid provides the optimal timing to exercise the option. If expectations are reviewed during the auction due to some new information or even auction evolution itself, the agent updates the threshold and the strategy, transforming the optimal decision into a dynamic process. Some of the parameters that we must address in order to define our problem are “Reservation Price”, “Buy-out Price” and “Expected Price”. These three will be focused during the analysis and some assumptions must be made. Let’s define these parameters: 1. “Reservation Price” (PR) - the valuation that the agent gives to the item, the highest value he’s willing to pay for it. 2. “Buy-out Price” (PB) - the price that one must pay to end the auction and buy-out the item. 3. “Expected Price” ( ¯ P) - represents the expectation of the ending price of the auction without exercising the buy-out option. It’s treated as an exogenous constant in our model. Later we will transform these variables in their wcounterparts, being each wa valuation-to-price ratio. In order to study the dynamics of choice between bidding and buy-out, we must ensure that the buy-out price is a possible price to pay. This means that the reservation price must be higher than the buy-out price. If, for a given agent, the buy-out price is higher than the reservation price, there is no option. If the potential buyer cannot pay the buy-out price, then the only way he can get the item is by bidding. In this case, the option doesn’t have any value for the agent. At the same time, the expected price must be lower than the buy-out price. Expecting that the bids may be higher than the buy-out price is unreasonable. If this is not the case, when a bidder is deciding whether to bid the amount of the buy-out price (not knowing for sure if he will win) or to exercise the option (paying the same buy-out price and getting the item surely and immediately), he will always buy-out the item and end the auction. When the bids reach the buy-out price, exercising the option is the rational decision, so we can assume that no one will ever bid the buy-out price or any other above it. 9 Assumption 2. The expected auction final price must be lower than the buy-out price, which cannot be higher than the reservation price of the agent. This implies that PR≥PB>¯ P. The Model for Single Auctions There is only one auction available for the selected item and each potential buyer values the item differently. Every one of them has a reservation price (PR), independent from its competitors’ valuations. Additionally, monitoring the auction has no costs 6. Assume that none of the bidders knows neither how many competitors there are, nor their reservation prices. All the information available to an agent is his valuation (PR), the buy-out price (PB) and any kind of expectation he has about the evolution and outcome of the auction. So he acts, and must decide, with important information constraints. Let’s define was the ratio PR P. The higher w, the higher the amount of satisfaction for agent. Higher possible gains in the auction, paying a lower Pfor a given PR, are translated into achieving a higher w. This valuation-to-price ratio (w) reflects the consumer surplus from the transaction (in a ratio). Assumption 3. Assume that the agent’s welfare (w)is simply given by the ratio between his reservation price and the price paid for the item, i.e. w=PR P. So, for each Pwe will have a correspondent w. Since the buyer has a reservation price PR, defined as the maximum value he is willing to pay for the item, we can surely assume that his bids cannot surpass PR, therefore, w⩾1. Note that wR=PR PR= 1. This implies that the buyer will only consider exercising the option if the buy-out price fulfills this condition, PB≤PRand wB=PR PB≥1. And, as the buy-out price is the safe outcome of the auction and is lower than the reservation price of the buyers, we can also say that no one will ever bid higher than PB(which implies w > wB). Assumption 2 can be transformed into: Assumption 4. PR≥PB>¯ P⇒wR= 1 ≤wB<¯w=PR ¯ P. 6An extension to this model, adding the effects of monitoring costs, is presented in Appendix A. 10 2.1 Bidding Each potential buyer only has access to information about the current price, the buy-out price and his own expectations of the future evolution of the auction. He must decide whether to bid until the end and hope to win the auction by normal bidding process, or to buy-out the item for a previously determined buy-out price. Since the goal of this model is to determine the optimal trigger for exercising of the buy-out option, the time component of the auction - its maturity and the behaviour of bids in each phase (beginning, mid and ending) of the auction - will be set aside and the focus shall be on the other variables that define the trigger. Let’s assume that before any bid is made, the potential buyer has an expectation of the bidding behaviour throughout the auction. He believes that the bids will follow a mean-reverting process towards that expectation value ( ¯w=PR ¯ P). The results of the model will, of course, be linked to this expectation so there should be a reviewing process of these expectations as the auction evolves. For modeling purposes, and because bids are placed through a proxy bidding system, we assume that there is an infinitely small minimum bid increment. Assumption 5. Assume that bid increments are infinitesimal. Consider F(w)to be the value function of the option. For convenience, it is assumed that there is no pre-determined time limit for the auction (remember that the placement of a bid increases the auction’s duration), and so, the passage of time, by itself, does not change the value of the option, nor does it influence the optimal timing to exercise the option. The decision between buying-out right now or waiting to buy-out later is only influenced by the evolution of bids. But if no new bids are placed, then the value function doesn’t change. Assumption 6. The value of the option to buy-out is time-independent: ∂F (w, t) ∂t = 0 All eBay auctions behave like English auctions with hard closing time, but we will not consider this duration constraint in order to exclude strategies like sniping (Roth and Ockenfels, 2002) from our model. But not all auction websites follow the same rules, in fact Yahoo auctions use an extending clause where any bid made in the last 11 minutes adds a few more minutes to the auction, virtually extending any auction for long periods of time. We assume that every auction is long enough to enable this kind of behaviour, as if it were an auction with no hard closing time, where the deadline extends every time a bid is made. So, all bids converge to a given wfinal independently of the duration of the auction. Some auctions will reach it sooner, others later, but the the auction ends at a given at w=wfinal. When no additional bids are made, the auction ends and the winner pays the current bid. This final bid is, of course, unknown during the auction, but the agent establishes some expectations about its value: Ewfinal= ¯w. Each individual agent has his own ¯w, meaning that in an auction with Nparticipants each bidder ihas a given ¯wi,i∈ {1, N}. Since agents don’t know each other’s valuations or expectations, our model is set in an asymmetric information environment. The individual agent will make his decision based solely on the information that he has available, so he won’t consider the competitor’s ¯wi. In the model, only the individual agent’s ¯wis important, and that’s the one that will be used. ¯wwill be the expectation formulated by the decision-maker (the protagonist in our model) for the value of wfinal. This expectation will be used to model the expected bidding behaviour next. But how should bids evolve? Let’s keep it simple and make a deterministic expectations function. We know that each competitor has a different valuation (PR) for the item, and we also know that some of the participants in the auction don’t value the item high enough to buy it out. There is a scale of reservation values in the group of competitors so, when higher value bids are placed, some of the players step out because they don’t value the item high enough to beat those bids. As this happens, less and less competitors stay in the auction until only one remains and wins the item. There is also a larger number of players with lower valuations so, bids should raise very quickly in the beginning of the auction. As a consequence, in the last stages of the auction, bidding evolution is slower as a result of the smaller number of remaining bidders. Since we determined that bids could not surpass the buy-out-price, we can state that ¯w > wB. So, our bidding process will have to account for the reducing number of bidders as prices rise and also a cap at wB. Notice that since wis an inverse function of P, while Pincreases, wdecreases. As the auction develops, wwill show a decreasing behaviour. It’s expected that bids will converge to a final bid, so Pwill increase towards ¯ P 12 (the expected Ptrue for our agent), meaning that wwill decrease towards w(the expected wfinal for the agent). Assumption 7. Bidding behaviour (in terms of w) is defined by the following meanreversion process: dw =η( ¯w−w)dt (2.1) Where ¯wis the expected wfinal and ηis the speed of adjustment (higher ηmeans that the auction will have a faster bidding evolution). It will be a parameter to calibrate the expected amount of time needed to achieve ¯w. The next figure shows a mean reversion process representation for PR= 125e,¯ P= 100e(which is the same as having ¯w= 1.25) and η= 0.3, with P= 25e(meaning w= 5) when t= 0. This would be the behaviour of the processes with continuous time increments: 5 10 15 20 25 30 t 20 40 60 80 100 PH€L 5 10 15 20 25 30 t 0.5 1.0 1.5 2.0 2.5 3.0 w Figure 2.1: Mean-Reversion Processes in variables Pand w Even though the bids follow a deterministic process, the model and the final outcome are not deterministic since there is expected competition through buy-out. Notice how during the entire auction the option is present. It is possible that any other buyer decides to exercise it, ending the auction by doing so. This event will be treated as a catastrophic event, following a Poisson process with occurring intensity λ. The auction that we’re considering will be treated as the last chance to get the item. If someone wins by buy-out, all the other competitors lose the chance to get the item. If that happens, then the option ceases to exist and so, it stops having value for the agent. 13 2.2 Option Value and Optimal Behaviour To address the problem we follow the real options approach (Dixit and Pindyck, 1994). Let F(w)be the value function for the option to buy-out the item at strike price PB. F(w)will represent some sort of net welfare from exercising the option to buy-out, considering the surplus wBand the loss of the possibility to win the auction at a price lower than PB(and therefore, the possibility to obtain a higher surplus). When a competitor buys the item using the option, our agent loses his option and it’s value disappears. This catastrophic event will be traduced in the value function as a jump (a sudden change of value). With occurring intensity λ, the value function will jump from F(w)to 0. In equation (2.3), this event will appear as: λ[0 −F(w)] (2.2) Considering the mean reverting process in equation (2.1) and catastrophic event from (2.2), by using Ito’s lemma we achieve the following differential equation: dF(w)dt =1 2 ∂2F(w) ∂w2(dw)2+∂F (w) ∂w dw +λ[0 −F(w)]dt (2.3) From equation (2.1), (dw)2is given by: (dw)2=η2( ¯w−w)2dt2 Since dt2tends to zero faster than dt, by the standard argument it can be ignored, so: dt2= 0 ⇒(dw)2= 0 (2.4) Combining equations (2.1), (2.3), and (2.4), and after dividing by dt we get: dF(w) = ∂F (w) ∂w η( ¯w−w) + λ[0 −F(w)] dF(w) = ∂F (w) ∂w η( ¯w−w)−λF (w)(2.5) The buy-out option available in the auction is free and, at the same time, automat14 ically given to anyone who follows the auction. Accordingly, the buy-out option is not tradable in the market and the agent doesn’t earn any returns by holding it: dF(w)=0 (2.6) The value function that we are trying to determine will be divided in two branches. The first branch will represent the intrinsic and the remaining time value of the option, whereas the other one will represent the payoff of the option when all the time value has been depleted. The trigger w∗will provide the optimal timing to exercise the option, the moment where the option loses all its time value and so, provide the threshold between the two branches. Let’s start by the continuation region, where w > w∗and it is not yet optimal to exercise the option. Combining equations (2.5) and (2.6): ∂F (w) ∂w η( ¯w−w)−λF (w) = 0 (2.7) The differential equation (2.7) has the following general solution: F(w) = C1(wη −¯wη)−λ η where C1is an arbitrary constant that needs to be determined. As for the second region (where w≤w∗), the payoff of the option is simply given by: F(w) = wB−φw This is the intrinsic value of the option, the agent receives wBwhen exercising it but gives up on φw. The parameter φrepresents the probability7of not being placed more bids and not being exercised the option by any competitor (for that level of w) or, in other words, the probability of the auction ending without the option being exercised and the agent getting the item with the current surplus (w). So, −φw traduces an opportunity cost (in terms of welfare) for exercising the buy-out option. 7Since φis a constant expectation parameter it should be revised frequently. Additionally, by considering φto be weighted by the risk aversion coefficient, we can introduce risk aversion into the model. In this dissertation we don’t explore this possibility and treat the model as if risk neutrality was assumed. 15 Our goal is to determine the the trigger (w∗) and the value function. But to achieve the value function we need to determine the trigger (w∗) , which will mark the optimal threshold (i.e. the value of wfor which is optimal to exercise the option to buy-out), and also the constant C1for the first branch. To determine two unknown variables we need two boundary conditions: F(w∗) = wB−φw∗(2.8) ∂F (w) ∂w w=w∗ =−φ(2.9) These represent the so-called “value-matching” (2.8) and the “smooth-pasting” (2.9) conditions. Equation (2.8) is usually called the “value-matching condition” because it matches the values of the function F(w)to those of the payoff function. When exercising the option, the buyer earns wBbut loses the opportunity to win the auction at the current bid and receive w∗(that would occur with probability φ). As PB> P∗, the buy-out option eliminates any uncertainty about the outcome of the deal, but does so by making the buyer pay a premium over the current bid (P∗) and settle for an inferior level of welfare (wB< w∗). To achieve certainty about the terms of the transaction, the buyer loses the chance to get higher gains through the auction. Notice that the value matching condition brings another condition, we need the option value to be positive in w∗in order to ensure economic rationality to the problem. The option can not have a negative value when the agent is supposed to exercise it. So, we get wB−φw∗>0. Recalling Assumption 4 and considering this condition, we can say that wB≥1∧ wB> φw∗. This means that, to be acceptable, the buy-out decision must give the agent a “value-for-money” greater (or equal) than 1 and must also compensate for the loss of welfare related to the chance (not certain) of acquiring the item at a lower price. This condition will be of use later on. As for equation (2.9), it ensures that the value function has smooth pasting when switching from one branch to the other. Proposition 1. Solving these equations, one can achieve the value function and trigger value. 16 F(w) =            1 λ[(w−¯w)η]−λ ηφhηλ(wB−¯wφ) (η+λ)φiη+λ η, w > w∗ wB−φw , w ≤w∗ (2.10) where: w∗=wBλ+ ¯wηφ (η+λ)φ(2.11) Further analysis of this trigger w∗can show a concealed and interesting finding. Proposition 2. By transforming w∗in the respective P∗, we achieve: P∗=(η+λ)φ¯ PPB ¯ Pλ +ηφP B(2.12) Proof. Recall that wB=PR PBand ¯w=PR ¯ Pand note that w∗can be written as w∗=PR P∗. Rearranging equation (2.11): w∗=wBλ+ ¯wηφ (η+λ)φ ⇔PR P∗= PR PBλ+PR ¯ Pηφ (η+λ)φ ⇔P∗=(η+λ)φ¯ PP B ¯ Pλ +ηφP B Notice the variables that change P∗: The model parameters η,λ, and φ, as well as the buy-out and the expected auction prices (PBand ¯ Prespectively). However, and contrary to some previous work on the matter (Hidvégi et al., 2006; Reynolds and Wooders, 2009; Chen et al., 2011), we find that the threshold in not dependent on PR. This is an interesting finding, which contradicts previous approaches to the problem. Focus on the choice at hand, either the agent chooses a certain deal paying PBor 17 Parameter Value Description wB1.10 Buy Out Valuation-to-Price Ratio PR PB ¯w1.25 Expected Final Valuation-to-Bid Ratio PR ¯ P ¯w01.15 Reviewed ¯w wValuation-to-Bid Ratio η0.30 Adjustment velocity in the mean-reverting process λ0.35 Probability of losing the auction to a competitor’s Buy-Out λ00.40 Reviewed λ φ0.40 Probability of winning through bidding at the threshold w∗ φ00.30 Reviewed φ Table 2.3: Model Parameters - Expectations Review Output Value w∗2.05769 w0∗ 2.58810 Table 2.4: Threshold Changes F(w) F '(w) w'* w* 2 3 4 5 w 0.2 0.4 0.6 0.8 FHwL (wdecreases as the auction progresses) Figure 2.3: Option’s Value and Optimal Triggers for Single Auction - Comparison As a consequence of his review of expectations, the agent now determines a harsher threshold. He recognizes a larger threat in his competitors and is willing to buy-out the item sooner (w0∗ > w∗). It is important to refer that this review can happen at any time during the auction, both to increase and to decrease expectations about competitors’ valuations, and it can happen as many times as the agent changes his mind. The results will change as demonstrated in this example and the optimal behaviour for the agent will adapt according to his new expectations. What this represents is a dynamic result, subject to the agents thoughts at every moment. In truth, to use this method in a real world scenario would require a con24 stant adjustment of the parameters to accommodate the dynamics of the strategy. If the new threshold hasn’t been reached yet, then the behaviour should be the same as before, just with a new trigger value. But if the review process results in a threshold that has already been surpassed then the option should be exercised immediately. The Model for Sequential Auctions Let’s consider a new auction for an equivalent item that starts after the first auction. The second (and last9) auction starts only after the first one ended, and is similar to the previous one. This will give the participants in the first auction a second chance to buy the item, decreasing the intensity of competition in their first interaction. Like in the single auction case scenario, none of the bidders knows neither how many competitors there are, nor their reservation prices. This time all the information available to an agent is his valuation (PR), his expectations and the buy-out prices of all the auctions. Every participant knows that there are sequential auctions and also knows the buy-out prices of each of these auctions. The decision for one auction takes into account the conditions of that auction but also considers the available information about the future auctions. All the assumptions made in the single auction model will be kept for the sequential auctions model, and some new assumptions will be added. Let w(P, N)≡wN(P)be the valuation-to-price ratio when Nauctions (including the current one) are available. Accordingly, considering only two sequential auctions (N∈ {1,2}), w2will represent the usual w=PR Pduring the first auction and w1 will refer to the last auction. All the potential buyers know that after the first auction ends, another one will start. And all of them, except for the one who bought the item in the first auction, will try to win the last auction as well. Most of the agents remain the same, but after the end of the first auction there is one less chance to get the item, so the remaining players will now change their behaviour and decide to buy-out the item sooner than they would in the first auction. Assumption 8. To traduce the increment in the level of competition between auctions we define λ1> λ2, meaning that the probability of a competitor exercising the option before w∗is reached is higher in the last auction than in the previous one. 9The model is defined for a 2 auctions environment but could be extended for N > 2auctions. 25 Additionally, the first auction’s buy-out price (represented by PB 2) must be lower than the second auction’s price (PB 1). If this is not the case, then there is no reason to buy-out in the first auction. By not buying the item and just keep bidding, a bidder ensures that he will bid with no cost because even if he looses the first item to a competitor, he will always have a chance to buy it out immediately after for an equal or lower price. So, it will never be optimal to buy-out at the first auction as the second buy-out is just as profitable and there is no cost for losing the first auction’s option. It’s as if the option was postponed to the second auction without any penalty, and according to the real options perspective (Dixit and Pindyck, 1994) the agent always chooses to defer the option to exercise if there is no associated cost. Assumption 9. In sequential auctions we assume that wB 1< wB 2, traducing PB 1> PB 2in terms of buy-out prices. 3.1 Bidding The bidding process is the same as in the previous case. The main difference is noticed in the event of losing the auction by buy-out from another participant. In the single auction case-scenario, if any competitor buys the item by buy-out, all the chances for ever getting the item are lost. With a second auction on the table, if the first auction is lost in those circumstances, there is still the opportunity to enter another one and buy the item there. This could be seen as a “pre-auction” interaction, that takes place before the “main” auction. The main implication from this is reflected in the differential equation, adding a positive component to the catastrophic event’s outcome. Instead of losing −λ2F2(w), the expected loss is given by −λ2[F1(w)−F2(w)], reflecting the loss of F2(w)and the gain of the last auction’s option value F1(w). F1(w)is the value function of the last auction at the point when it’s starting. Since the auctions are sequential, and as the buyers are mostly the same, it’s safe to assume that when the first auction ends, all buyers switch to the second auction and immediately set the bid they had in the previous one. So, the second auction starts exactly at the same point where the first ended, there is no break in the continuity of bids between both auctions. Therefore, the remaining value when the catastrophic event occurs, is F1(w). As the option value changes accordingly to the current w, when the first auction is lost due to buy-out, the remaining value is that of the last auction’s option valued at current bid. 26 3.2 Option Value and Optimal Behaviour Consider F2(w)and F1(w)the value functions for the first and last auctions, respectively. F2(w)is the function for the auction taking place first, when there are 2 auctions available (the current and the last one). F1(w)refers to the last auction. Let’s assume that η,φand ¯wwon’t change between the two auctions. The items must be similar and the competitors are most likely the same, so these expectations don’t change. The only reason to change them would be due to a review of expectations. Notice that wBwill change between auctions. In this case, the second buy-out price must be higher than the first as stated in Assumption 9 wB 2> wB 1. The parameter λwill differ between both auctions because the last auction is similar to the first in its structure and rules but the competition is higher now since the item for sale is the last one available. It is expected that during the transition to the second auction the competition through BiN becomes fiercer . We will solve this problem using a backwards approach, solving the last auction with the same method as before, and then using the results in the first auction. The last auction is treated as if it was a single auction, since there is no auction afterwards. The results are the same as the ones for the single auction environment. The problem is the same, with the same differential equation: dF1(w) = ∂F1(w) ∂w η( ¯w−w)−λ1F1(w) = 0 (3.1) Subject to the following boundary conditions 10: F1(w∗ 1) = wB 1−φw∗ 1(3.2) ∂F1(w1) ∂w1w1=w∗ 1 =−φ(3.3) Proposition 3. Equation (3.1) is the equivalent of equation (2.7) in the single auction scenario. And equations (3.2) and (3.3) are the same boundary conditions as before. As a result, the procedure is the same and the corresponding trigger and value function are as follow. 10The value function has to be positive to be rational for the agent to exercise the option, so F1(w∗ 1) = wB 1−φw∗ 1>0. 27 F1(w) =            1 λ1[(w−¯w)η]−λ1 ηhwB 1λ1+ ¯wηφ (η+λ1)φ−¯wηiη+λ1 ηφ, w > w∗ 1 wB 1−φw , w ≤w∗ 1 (3.4) where: w∗ 1=wB 1λ1+ ¯wηφ (η+λ1)φ(3.5) So far, we have simply set the single auction as the last auction in the sequential auction problem. Now, let’s use the value function of the last auction in the preceding one. Consider now the first auction of the two. Since the value doesn’t fully disappear when another participant buys the item through buy-out, the differential equation for the first auction suffers a small modification: dF2(w) = ∂F2(w) ∂w η( ¯w−w) + λ2[F1(w)−F2(w)] = 0 (3.6) − ∂F2(w) ∂w η( ¯w−w)−λ2F2(w) λ2 =F1(w)(3.7) Equation (3.7) can be rearranged, using the value function for the last auction (3.4), into: − ∂F2(w) ∂w η( ¯w−w)−λ2F2(w) λ2 =              [(w−¯w)η] − λ1 η" wB 1λ1+ ¯wηφ (η+λ1)φ−¯w!η#η+λ1 ηφ λ1, w > w∗ 1 wB 1−φw , w ≤w∗ 1 The boundary conditions 11 are the same, with the respective parameters: F2(w∗ 2) = wB 2−φw∗ 2 ∂F2(w2) ∂w2w2=w∗ 2 =−φ 11See footnote 10, F2(w∗ 2) = wB 2−φw∗ 2>0. 28 Proposition 4. So for the auction taking place first we have the following value function and trigger: F2(w) =                                [(w−¯w)η]−λ2 η       − [(w−¯w)η] −λ1+λ2 ηλ2φηλ1(wB 1−¯wφ) (η+λ1)φη+λ1 η λ1(λ1−λ2)+A       , w > w∗ 1 −wB 1+wB 2η[(w−¯w)η]−λ2 η(−wB 1+wB 2)λ2 φλ2 η η+λ2 +B , w∗ 2< w ≤w∗ 1 wB 2−φw , w ≤w∗ 2 (3.8) where: A= (wB 1−wB 2)η(λ1−λ2)(−wB 1+wB 2)λ2 φλ2 η−(η+λ1)φηλ1(wB 1−¯wφ) (η+λ1)φη+λ2 η (η+λ2)(−λ1+λ2) B=wB 1−¯wη +wλ2φ η+λ2 and w∗ 2= ¯w+(wB 2−wB 1)λ2 ηφ (3.9) The reason for F2(w)to have 3 branches lies with the possible scenarios when the transition between auctions happens. The last auction has a more strict threshold (w∗ 2< w∗ 1). As it is the last chance to buy the item, buyers are not willing to risk as much as before and decide to exercise the option sooner than they would in the first auction 12. This creates a gap between [w∗ 2, w∗ 1], where the behaviour is not simply to chose either to bid or to buy. With a multiple auctions environment, besides deciding whether to bid or not to bid, the player must also decide in which auction he should bid or buy. Imagine an agent going through the decision process: •For w≤w∗ 2< w∗ 1, it is optimal to exercise the option right away on the first auction. •For w∗ 2< w ≤w∗ 1, the agent’s trigger to exercise the option has not yet been reached in the first auction so the agent should wait. But in case some other 12This should happen for the majority of the cases, but in an exceptional case-scenario this may not be true due to high differences in buy-out prices. See Appendix B for further explanation. 29 buyer buys-out the item before his threshold w∗ 2is reached, the last auction will start and the agent will buy-out the item immediately. •For w∗ 2< w∗ 1< w, whatever happens, the agent will wait. He will wait for w∗ 2 in the first auction and keep bidding, and if someone ends the auction before that barrier is reached he will enter the last auction and bid normally until w∗ 1 is reached. Proposition 5. Using the same approach as before, when turning equation (2.11) into equation (2.12), we can rearrange equation (3.9) and achieve the optimal price threshold for the first auction: P∗ 2=PB 1PB 2ηφ ¯ P PB 1PB 2ηφ +¯ P(PB 1−PB 2)λ2 Proof. w∗ 2= ¯w+(wB 2−wB 1)λ2 ηφ ⇔PR P∗ 2 =PR ¯ P+PR PB 2−PR PB 1λ2 ηφ ⇔P∗ 2=PB 1PB 2ηφ ¯ P PB 1PB 2ηφ +¯ P(PB 1−PB 2)λ2 Once again, the price threshold is independent from PR. This means that, besides expectation related parameters, the trigger is only affected by changes in the buy-out prices. 3.3 Analytical Comparative Statics Each parameter influences w∗ 2according to these derivatives: 30 ∂w∗ 2 ∂¯w>0 ∂w∗ 2 ∂λ2 >0 ∂w∗ 2 ∂λ1 = 0 ∂w∗ 2<0 ∂w∗ 2<0 These parameters keep the same effects as in the previous scenario. The expected auction price ( ¯w) keeps affecting the trigger in a positive way, if the agent expects the bidding to stop at higher ¯w(lower price ¯ P), then he will also set a higher trigger w∗, exercising the option sooner. The effects of λs are divided in two since now we’re considering two different auctions with different λ(λ1and λ2): •Notice how ∂w∗ 2 ∂λ1= 0, this means that the threshold of the first auction is not affected by the last auction’s probability of losing the item for a competitor’s buy-out. Competition through buy-out in the last auction does not influence the decision in the first one because we are assuming that the agent wins the item when he reaches the trigger (Assumption 1), the option’s payout is independent from any λin our model. •∂w∗ 2 ∂λ2>0simply traduces the increased competition effect in the current auction. As before, higher competition will make the agent exercise the option sooner. As for φ, with higher probability of winning the item through auction at the threshold price the agent has a higher cost when exercising the option, so the the optimal timing is delayed (lower w∗) with an increase of φ. The effects of ηare the same as before, lower ηwill tighten the threshold as it represents a slower achievement of the final expected result and, therefore, a wider window of opportunity for competitors to buy-out the item. 31 Since now the model also has two different buy-out prices, let’s focus on their effects: ∂w∗ 2 ∂wB 2 =λ2 ηφ >0 ∂w∗ 2 ∂wB 1 =−λ2 ηφ <0 1.1 1.2 1.3 1.4 1.5 w1 B 1.1 1.2 1.3 1.4 1.5 w2 B 1.5 2.0 w2* Figure 3.1: Buy-out prices effects on the first auction’s threshold The analysis of the effects of wB 1and wB 2are shown in Figure (3.1). Imposing that wB 1< wB 2we ignore the right half the plane. As wB 2increases, the first auction’s threshold gets more restrictive to reflect the better result of the buy-out and the higher difference facing the second auction’s buy-out price. On the other hand, when the last auction’s buy-out valuation-to-price ratio increases (higher wB 1), the first auction’s trigger w∗ 2decreases to account for the better conditions in the last auction’s buy-out option. Additionally, the following proposition can be set: Proposition 6. As the buy-out prices become closer wB 1→wB 2, the first auction’s threshold approaches the expected final bid (w∗ 2→¯w). lim wB 1→wB 2 (w∗ 2) = ¯w 32 Proof. lim wB 1→wB 2 (w∗ 2) = lim wB 1→wB 2¯w+(−wB 1+wB 2)λ2 ηφ = ¯w When wB 1→wB 2the agent starts to have very few reasons to chose the first buy-out over the second and just bids to try to win through auction. He will only buy-out the item when wreaches the point where he thinks that bidding will stop. If his expectations are reviewed, he will keep waiting for the “new” final expected bid to buy-out the item, which means that the agent will never exercise the option to buy-out in the first auction. 3.4 Numerical Example and Results Like in the previous case, consider this example: Parameter Value Description wB 21.20 Buy Out Valuation-to-Price Ratio in the first auction PR PB 2 wB 11.10 Buy Out Valuation-to-Price Ratio in the last auction PR PB 1 ¯w1.30 Expected Final Valuation-to-Bid Ratio wValuation-to-Bid Ratio η0.30 Adjustment velocity in the mean-reverting process λ20.35 Probability of a competitor’s Buy-Out in the first auction λ10.45 Probability of a competitor’s Buy-Out in the last auction φ0.40 Probability of winning through bidding at the threshold w∗ Table 3.1: Model Parameters - Sequential Auction Example Applying equations (3.4), (3.5), (3.8), and (3.9), the results are as follow: Output Value w∗ 21.59167 w∗ 12.17 Table 3.2: Threshold Results 33 best choice by paying that price. Sequential auctions interaction could also be addressed. •Changing the type of option. As stated before, there isn’t only one type of option in online auctions today. In this work, we approached the problem of the permanent buy-out option, but the model could be extended for temporary options. There two main types of temporary options, some that disappear once the first bid is made, others after an unknown price has been reached. While the first can be simply traduced in the choice between a normal purchased at a fixed price, the second one resembles barrier options. This could be an interesting path for future investigation on the matter. 40 Appendix A The Model for Single Auctions with monitoring costs In this extension to the single auction model we will address the problem of existing costs for the agent (in terms of welfare) for monitoring the evolution of the auction. Taking equation 2.7 and adding an instantaneous loss of welfare (θ) we get: ∂Fθ(w) ∂w η( ¯w−w)−λFθ(w)−θ= 0 The general solution is to this differential equation is: Fθ(w) = −θ λ+ ((w−¯w)η)−λ ηCθ(A.1) where Cθis, once again, an arbitrary constant that needs to be determined. The value function will have two branches, just like in the original model. The continuation region is given by equation A.1 and, since the payoff of the option is the same, the second region is given by: Fθ(w) = wB−φw The boundary conditions are the same as before (equations 2.8 and 2.9), so the value function can be determined as: Fθ(w) =            1 λ[(w−¯w)η]−λ ηφhη[θ+λ(wB−¯wφ)] (η+λ)φiη+λ η−θ, w > w∗ θ wB−φw , w ≤w∗ θ where: w∗ θ=θ+wBλ+ ¯wηφ (η+λ)φ Let’s produce an example to show the changes. 41 Consider the parameters in Table 2.1 and θ= 0.1. The results are as follow: Output Value w∗2.05769 w∗ θ2.44231 Table A.1: Threshold Results with monitoring costs F(w) F Θ (w) w* wΘ* 1 2 3 4 5 w -0.2 0.2 0.4 0.6 0.8 1.0 FHwL (wdecreases as the auction progresses) Figure A.1: Option’s Value and Optimal Trigger Notice that we can rearrange the threshold w∗ θin the following manner: w∗ θ=θ+wBλ+ ¯wηφ (η+λ)φ =wBλ+ ¯wηφ (η+λ)φ+θ (η+λ)φ =w∗+θ (η+λ)φ Comparing the thresholds from the model with monitoring costs and the original model we can conclude that the threshold for w∗ θis always higher than the one for w∗, meaning that monitoring costs reduce option value and force the agent to exercise the option sooner, which is also visible in the numerical example. 42 B The Model for Sequential Auctions with large differences in Buy-Out Prices The goal of the sequential auctions model was to address the problem of similar auctions happening one after the other, creating compound options. The value function that we proposed was defined for the case of w∗ 2< w∗ 1, which should happen for “similar” auctions. The best suited situation for applying the sequential auctions model is one where the auctions have close buy-out prices for equal items (while maintaining Assumption 9 , wB 1< wB 2). But what if this doesn’t happen? Auctions happening one after another, for equal items, may have significantly different buy-out prices, so how should the agent decide in those situations? First we need to define which auctions should be approached with our original sequential auctions model and which auctions need a different approach. Since the model for sequential auctions should only be used in cases where w∗ 2< w∗ 1, we will make a division in two different groups: one where w∗ 2< w∗ 1and another where w∗ 2≥w∗ 1. The condition w∗ 2< w∗ 1can be turned into: ¯w+(wB 2−wB 1)λ2 ηφ <wB 1λ1+ ¯wηφ (η+λ1)φ wB 2−wB 1<ηλ1wB 1−¯wφ (η+λ1)λ2 So, for two auctions where wB 2−wB 1<ηλ1(wB 1−¯wφ) (η+λ1)λ2we should use the original model in this dissertation. For every other case, the approach must be slightly different, being the option value deduction based on w∗ 2≥w∗ 1. For the last auction, everything remains the same, changes will only occur in the previous auction. The differential equation is exactly the same as equation 3.6: dF2(w) = ∂F2(w) ∂w η( ¯w−w) + λ2[F1(w)−F2(w)] = 0 But when exchanging F1(w)for its expression we don’t consider two branches. As w∗ 2≥w∗ 1, the agent will want to exercise the option in the first auction for a higher level of wthan he would in the last auction. So, either it’s not yet optimal to exercise the option in the first auction and the agent waits, or it’s optimal to exercise and 43 the agent exercises the option immediately. But, if it’s not yet optimal to exercise the option in the first auction (w > w∗ 2), then if the option is lost (a competitor exercises the option before wreaches w∗ 2), the last auction begins and the agent will not exercise the option in the last auction until wreaches w∗ 1, since w∗ 2≥w∗ 1. So, F1(w)in the differential equation must always be a value in the continuation region (w > w∗ 1) because if whas not yet reached w∗ 2then it means that it hasn’t reached w∗ 1either. When considering w > w∗ 2≥w∗ 1the value function is obtained in the following manner: − ∂F2(w) ∂w η( ¯w−w)−λ2F2(w) λ2 =F1(w) − ∂F2(w) ∂w η( ¯w−w)−λ2F2(w) λ2 = [(w−¯w)η]−λ1 ηhwB 1λ1+ ¯wηφ (η+λ1)φ−¯wηiη+λ1 ηφ λ1 For which the general solution is: F2(w) = [(w−¯w)η]−λ2 η    − [(w−¯w)η] −λ1+λ2 ηλ2φhηλ1(wB 1−¯wφ) (η+λ1)φiη+λ1 η λ1(λ1−λ2)+C2    where C2is a constant to be determined. From here on the deduction is simple, the boundary conditions are the same: F2(w∗ 2) = wB 2−φw∗ 2 ∂F2(w2) ∂w2w2=w∗ 2 =−φ And the value function is given by: F2(w) =              (w−¯w)ηφ λ2+ ((w−¯w)η)−λ1 ηφηλ1(wB 1−¯wφ) (η+λ1)φη+λ1 η λ1, w > w∗ 2 wB 2−φw , w ≤w∗ 2 44 Where w∗ 2cannot be determined analytically, but can be achieved by using numerical methods. Consider the following numerical example: Parameter Value Description wB 23.0 Buy Out Valuation-to-Price Ratio in the first auction PR PB 2 wB 12.0 Buy Out Valuation-to-Price Ratio in the last auction PR PB 1 ¯w4.0 Expected Final Valuation-to-Bid Ratio wValuation-to-Bid Ratio η0.30 Adjustment velocity in the mean-reverting process λ20.35 Probability of a competitor’s Buy-Out in the first auction λ10.45 Probability of a competitor’s Buy-Out in the last auction φ0.40 Probability of winning through bidding at the threshold w∗ Table B.1: Model Parameters And the results are: Output Value w∗ 25.84466 w∗ 14.6 Table B.2: Threshold Results - Sequential Auctions As expected, w∗ 2> w∗ 1which means that the agent prefers the first auction and will fight for it even while knowing that there is another auction available for the same item later on. The difference is that, this time, the second auction is very little attractive because of the buy-out price difference. Let’s just compare this first auction with an identical one but in the single auction case-scenario. We’ll try to see if there is any effect in having a last auction available, even if it’s so much less interesting than the first. When considering a single auction with the same parameters the threshold is: Output Value w∗ single 5.88462 Table B.3: Threshold Results - Single Auction As one can see, w∗ single is slightly higher than w∗ 2. There is some effect in the existence of a second auction, it’s very small but there’s still postponing of the optimal moment to exercise the option in the sequential auction case-scenario when compared to the single auction result. 45 The value function also has a small difference, being the sequential auction option slightly more valuable than the single auction one (it’s barely visible, but the single auction’s option value is the darker line): w*single w2* 5.5 6.0 6.5 w 0.5 0.6 0.7 0.8 0.9 1.0 F2HwLvs FsingleHwL (wdecreases as the auction progresses) Figure B.1: Single vs Sequential Auctions As expected, even in this extreme case-scenario, a free option still adds some value. 46 References Ausubel, L. (2003). Auction theory for the new economy. New Economy Handbook, pages 124–162. Budish, E. and Takeyama, L. (2001). Buy prices in online auctions: irrationality on the internet? Economics Letters, 72(3):325–333. Chen, J., Chen, K., Chou, C., and Huang, C. (2011). A dynamic model of auctions with buy-it-now: Theory and evidence. Mpra paper, University Library of Munich, Germany. Dixit, A. and Pindyck, R. (1994). Investment under uncertainty. Princeton University Press. Friedman, L. (1956). A competitive-bidding strategy. Operations research, 4(1):104– 112. Gallien, J. and Gupta, S. (2007). Temporary and permanent buyout prices in online auctions. Management Science, 53(5):814–833. Hasker, K. and Sickles, R. (2010). ebay in the economic literature: Analysis of an auction marketplace. Review of Industrial Organization, 37(1):3–42. Hidvégi, Z., Wang, W., and Whinston, A. (2006). Buy-price english auction. Journal of Economic Theory, 129(1):31–56. Juda, A. and Parkes, D. (2009). An options-based solution to the sequential auction problem. Artificial Intelligence, 173(7-8):876–899. Mathews, T. (2003). A risk averse seller in a continuous time auction with a buyout option. Brazilian Electronic Journal of Economics, 5(2). 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. Peters, M. and Severinov, S. (2006). Internet auctions with many traders. Journal of Economic Theory, 130(1):220–245. 47 Reynolds, S. and Wooders, J. (2009). Auctions with a buy price. Economic Theory, 38(1):9–39. Roth, A. and Ockenfels, A. (2002). Last minute bidding and the rules for ending second price auctions: evidence from ebay and amazon auctions on the internet. American Economic Review, 92(4):1093–1103. Said, M. (2011). Sequential auctions with randomly arriving buyers. Games and Economic Behavior. Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. The Journal of finance, 16(1):8–37. Zeithammer, R. (2004). Forward-looking bidding in online auctions. working papers series, University of California, Los Angeles. Zeithammer, R. (2006). Sequential auctions with information about future goods. working papers series, University of California, Los Angeles. http://ebay.about.com/od/glossaryofebayterms/g/gl_proxy.htm , accessed on August 6th 2012 (Proxy Bidding) http://ebay.about.com/od/glossaryofebayterms/g/gl_reserve.htm , accessed on August 6th 2012 (Reserve Price) 48