Common Value Auctions with Buy Prices

Transcription

1 Common Value Auctions with Buy Prices Quazi Shahriar 3 June, 008 Abstract Risk aversion and impatience of either the bidders or the seller have been utilized to explain the popularity of buy prices in private value auctions. This paper, using a pure common value framework, models auctions with temporary buy prices. We characterize equilibrium bidding strategies in a general setup and then analyze a seller s incentive to post a buy price when there are two bidders. We find that, when bidders are either risk neutral or risk averse, a risk neutral seller has no incentive to post a buy price. But when the seller is risk averse, a suitably chosen buy price can raise his expected payoff when the bidders are either risk neutral or risk averse. This provides an explanation for the popularity of buy prices in online common value auctions. JEL Codes: C7, D44, D80, L86. Key Words: Auction, Buy Price, Buy-It-Now, Common Value. Department of Economics, San Diego State University, 5500 Campanile Drive, San Diego, CA qshahria@mail.sdsu.edu. Tel: (69) Fax: (69) I am thankful to Stan Reynolds and John Wooders for their guidance and helpful comments. 0

2 Introduction: In an auction with a buy price the seller sets a fixed price and a bidder can accept that price to win the item. A buy price thus allows a bidder to win and end an auction early, sometimes even without going to the auction at all. The buy-price auction format has become very popular in online auctions. In 999 Yahoo! introduced buy-price auctions, known as Buy Now auctions. Then, in 000 ebay introduced Buy It Now auctions, its version of such hybrid auctions. A buy price can be either temporary or permanent. A temporary buy price is available only at the beginning of an auction when bidding has not yet started. The buy price disappears as soon as someone places a bid. But a permanent buy priceisavailable during theentiretimeanauction takesplace. ebay s auctions with temporary buy prices have become hugely popular over the last few years. This paper, using a common value framework, characterizes symmetric equilibrium bidding strategies in a buy-price auction with a temporary buy price and then analyzes the effects of the buy price on seller payoff. An auction without a buy price is modeled as a second price sealed bid auction. We show that, for both risk neutral and risk averse bidders, it is not possible for the seller to raise his expected revenue by offering a buy price. This is different from the private value model results in which an appropriately chosen buy price can always raise seller revenue when bidders are risk averse. However, we find that, when the seller is risk averse, an appropriately chosen buy price can raise seller payoff with either risk neutral or risk averse bidders. There are several motivations for us to consider a common value model in a buyprice auction and to look at ebay-type temporary buy prices. The implication of the type of information available to the bidders regarding their valuation of the item has never been investigated in the context of buy-price auctions. All the existing theoretical investigations assume that each bidder has his own private valuation for the item (we discuss some of these studies below). But sometimes a bidder may only have some partial information (a signal) regarding his value for the item and other Buy prices on ebay, LabX and Mackley and Company are temporary while those on Yahoo!, ubid, Bid or Buy, MSN and Amazon are permanent. (See Mathews (006).) ebay s value of sales in the fixed price platform (Half.com and buy-price auctions together, but primarily from the latter one) totaled approximately $6.8 billion during the fourth quarter and $3.9 billion during the whole year of 007 which is about 40% of gross merchandise sales of $59.4 billion in 007. (See Quarterly Financial Results press releases available at

3 bidders possess information that, if known, would affect his assessment of valuation. This paper assumes a pure common value structure in which the ex post value is common to all bidders but unknown at the time they participate in an auction. Each bidder s ex ante information regarding the value consists only of a private signal and the common value is the average of all the signals. Within this structure, this paper analyzes second price sealed bid auctions with temporary buy prices. The focus of the paper is on the implication of risk preferences (for both bidders and the seller) on seller payoff in such buy-price auctions. Bajari and Hortaçsu (003) use data from ebay s coin auctions to explore the determinants of bidder and seller behavior. They argue that a common value model is the appropriate one to use for coin auctions as they found an inverse relation between the sale price and number of bidders. 3 They found that, for an average auction, the bidders lowered their bids by 3.% per additional competitor. This study, along with the popularity of ebay s buy-price auctions, motivates us to consider common value auctions with temporary buy prices. There are also some theoretical interests in analyzing a seller s incentive to post a buy price in a common value auction. When values are common, there is no known optimal auction format and prior work shows that bidders bid less aggressively as they become more risk averse. (See Milgrom and Weber (98).) A buy price, on the other hand, eliminates the uncertainty of payment but it does not eliminate the uncertainty of the winner s payoff. The unknown valuation of the item creates this uncertainty. As a result, it is not obvious how attractive a buy price can be to risk averse bidders and whether this attraction will be sufficient to raise seller revenue. We find that a buy price actually reduces expected seller revenue when bidders are either risk neutral or risk averse. So, there is no incentive for a risk neutral seller to post a buy price. But a risk averse seller may have another motivation to use a buy price as it reduces the variability in trading prices. We show that an appropriately chosen buy price gives a risk averse seller a higher expected payoff. This paper, therefore, establishes one explanation for the popularity of buy prices in common value auctions. Related Literature Budish and Takeyama (00) were the first to analyze buy-price auctions. They considered an independent private values (IPV) model with only two possible valua 3 The presence of the winner s curse in a common value auction requires bidders to shade their expected values and bids in an equilibrium, and the bidders shade their bids more as the number of bidders increases.

4 tions. In their model, two risk averse bidders participate in an ascending bid auction with a permanent buy price. They showed that the buy price raises expected seller revenue. Reynolds and Wooders (003) used a symmetric IPV framework with a continuous distribution of values for two or more bidders. They showed that both temporary and permanent buy prices can raise seller revenue compared to a standard ascending bid auction when the bidders have constant absolute risk aversion (CARA). A buy price eliminates the uncertainty of payment and reduces the uncertainty of winning in an auction. As a result, risk averse bidders agree to pay a risk premium by accepting a high buy price. In fact, this study shows that the more risk averse the bidders are the easier it is for the seller to generate higher revenues using buy prices. The reason for this, as they note, is that it s easier to make a high buy price attractive to the bidders when they are more risk averse higher risk aversion translates into higher risk premium. Mathews and Katzman (006) developed an IPV model which for the first time allowed a risk averse seller. They found that a temporary buy price added to a second price sealed bid auction with risk neutral bidders gives the seller a higher expected payoff. Since a buy price reduces the variance of seller revenue, expected seller payoff can increase even when his monetary payoffs are lower. Hidvégi, Wang and Whinston (006) examine an IPV model for a modified English auction in which a bidder does not observe rival dropouts. They show that a properly set permanent buy price increases social welfare and the utility of each agent when either the bidders or the seller is risk averse. 4 This paper contributes to the growing literature on buy-price auctions. It differs from the previous studies in two aspects: it considers a common value model and it allows risk aversion for bidders and seller at the same time. Our model combines the ones considered in Reynolds and Wooders (003) and Mathews and Katzman (006), and then extends it to the case when the value is common to the bidders. The private and the common values frameworks present different decision making scenarios to the bidders. As a result, the implications of a buy price are different in private and common values auctions. A common value auction involves an additional source of uncertainty arising from the unknown valuation of the item which a buy price cannot eliminate. To summarize, we find that a buy price that is accepted by bidders with a positive probability reduces expected seller revenue but an appropriately chosen buy 4 There are some other explanations for the use buy prices (e.g. bidder impatience in Mathews (004)). Since they are not directly related to the current study, we skip discussing them. 3

5 price can raise a risk averse seller s expected payoff. Intuitively, a buy price can be attractive to a risk averse seller since it reduces the uncertainty of revenue. The rest of this paper is organized as follows. Section describes a common value model of auctions with and without a temporary buy price. Section 3 characterizes equilibrium bidding strategies (for both risk neutral and risk averse bidders) in a general setup and applies them in a simple case of two bidders with signals that are identically and independently distributed as U[0, ]. In the two-bidder case, we also show that the equilibrium is unique among the class of equilibrium considered in the paper. Section 4 analyzes the seller s incentive to post a buy price when the seller is either risk neutral or risk averse and there are two bidders, both risk neutral or risk averse. The signals are assumed to be distributed identically and independently as U[0, ]. Section 6 makes some concluding remarks. The Model: Consider a seller selling a single item through an auction. There are n bidders in the market who have identical risk aversion. Each bidder i receives a private signal x i which is identically and independently distributed according to a cumulative distribution function F over thesupportof [x, x]. Let f(x) = F 0 (x) be the probability density function. We consider a pure common value model in which the ex post valuation of the item is the same for each bidder and equals the average of all the signals. 5 So, the value of the item to bidder i is x + x x n v i = v =. n In this formulation, each bidder s valuation is a symmetric function of other bidders signals. The common value of the item v is unknown to a bidder until after he buys the item. If bidder i buys the item and pays a price p, he receives a payoff of u(v p), otherwise he receives a zero payoff. We assume a Constant Absolute Risk Averse (v p) (CARA) utility function, u(v p) = e,where is the index of risk aversion. 5 This common value formulation has been used in Bikhchandani and Riley (99), Albers and Harstad (99), Klemperer (998), Bulow, Huang and Klemperer (999), Goeree and Offerman (00) and Goeree and Offerman (003). 4

6 Notice that lim u(v p) = v p. As a result, =0 corresponds to risk neutral 0 bidders. If the seller earns a revenue s, he receives a payoff of ue(s). Assume that ue(s) is continuous, ue(0) = 0, ue 0 (s) > 0 and ue 00 (s) 0. When the seller is risk neutral, ue(s) =s and ue 00 ( s) =0, but when he is risk averse, ue 00 ( s) < 0. We consider two auction formats. The firstformatisanauction withouta buy price and the auction proceeds as a second price sealed bid (SPSB, hereafter) auction. 6 Bidders simultaneously and independently submit their private bids and the bidder with the highest bid wins the item paying a price equal to the second highest bid. The second format is a buy-price auction which consists of two stages. In the first stage, the seller posts a buy price B and the bidders simultaneously and independently make their decisions to accept or reject B. 7 The auction ends immediately if one or more bidders accept B. If only one bidder accepts B, then he wins the item. If multiple bidders accept B, then the winner is chosen randomly from the bidders who accepted B. If none of the bidders accepts B, then the option to buy at B disappears and the auction proceeds to the second stage. In the second stage, the item is sold via a SPSB auction. 3 Equilibrium Bidding Strategies 3. An Auction without a Buy Price: In a SPSB auction, a strategy for a bidder is a function mapping his signal x into his bid b. Since this paper concentrates on symmetric auctions, throughout this paper we focus only on bidder s decisions. Bidder s unique symmetric equilibrium bid function b(x) when his signal is x = x is known from Milgrom and Weber (98). The bid function b(x) solves E[u(v b(x)) x = x, z = x] =0, () where z =max{x,..., x n } and z is distributed according to the cumulative distrin 8 bution function H(z) =F (z). If bidders are CARA and is their risk aversion index, then from () we get 6 Bajari and Hortaçsu (003) consider a symmetric common value model of ebay auctions with two stages. The first stage is conducted as an open-exit ascending auction and the second stage is as a SPSB auction. They show that, in this model, an ebay auction is equivalent to a SPSB auction. 7 A rational seller posts a buy price B [x, x]. 8 The density function of z is, therefore, given by h(z) =(n )F (z) n f(z). We can assume, 5

7 b (x) = ln. () E [e v x = x, z = x] If bidders are risk neutral (i.e., = 0), then () reduces to b 0 (x) =E [v x = x, z = x]. (3) From () it is easy to find that b(x) =x. Since b(x) is an increasing function of x, the bidder with the highest signal draw wins the auction. 9 Notice that b(x) does not depend on x, the upper bound of the support, since the maximum value that any of the random variables may take in () is x. Another important thing to notice is that the concavity of the utility function in () implies that b (x) <b 0 (x). A risk averse bidder, therefore, bids less than a risk neutral bidder. 3. A Buy-Price Auction: In a buy-price auction, a strategy for a bidder in the first stage is a function mapping his signal x into his decision to accept or reject the buy price B. This paper focuses on symmetric cutoff strategies. A cutoff strategy for bidder i is defined by a constant c [x, x] such that bidder i accepts B if x i >c and he rejects B if x i <c. Suppose that all the rivals employ the same cutoff c. Then, given a signal x = x, the expected payoff to bidder from accepting B is "Ã! # X n µ n U A (x, c) = F (c) n l ( F (c)) l u l (x) (4) l l + l=0 where u l (x) is the expected utility to bidder when his own signal is x, he wins the item at price B, l rival bidders have signals above c and n l rivals have signals 0 below c. without loss of generality, that x = z. Now, () can be written as Z Z µ P x x (n ) x + x + l> x l... u b(x) df (x n )...df (x 3 )=0. F (x) n n x x We can use the above equation to solve for b(x) under alternative functional specifications for F (.) and u(.). 9 For an example, if bidder has the highest signal, he wins the item and pays b(z). 0 When c (x, x), Z Z Z Z µ P x x c c x + j> x j f(x ) f(x n l ) f(x n l+ ) f(x n ) u l (x) = u B dx... dx n l dx n l+... dxn. n F (c) F (c) F (c) F (c) c c x x 6

8 When bidder rejects B he wins only when all his rivals also reject B and he has the highest signal draw. In the second stage (i.e., a SPSB auction) that follows after all the bidders reject B, bidder knows that his rivals signals are less than c. But this observation is uninformative in () as we condition on z = x and x<c (since bidder rejects B). As a result, b(x) is also the equilibrium bid function in the second stage of a buy-price auction. So, if bidder has the highest signal and z < min{x, c}, then by rejecting B bidder wins and pays b(z). We can assume, without loss of generality, that x = z. Now, given a signal x = x, the expected payoff to bidder from rejecting B is Z min{x,c} Z x Z µ P x x + l> x l f(x n ) f(x 3 ) U R (x, c) =... u b(x ) dx n... dx 3 h(x )dx x x x n F (x ) F (x ) Z min{x,c} Z x Z µ P x x + l> x l = (n )... u b(x ) df (x n )...df (x ), (5) n x x x where the second line uses h(x )=(n )F (x ) n f(x ). Acutoff c is a symmetric Bayes Nash equilibrium if a bidder receives a higher expected payoff by accepting B (than rejecting B) when x > c, and he receives a higher expected payoff by rejecting B (than accepting B) when x<c. That is, U A (x, c ) > U R (x, c ) when x >c and U A (x, c ) < U R (x, c ) when x <c. Now, before we propose an equilibrium we define two quantities, denoted by γ and γ, which are useful in characterizing the candidate equilibrium. () Let γ be theprice that makesbidder indifferent between buying the item at γ and participating in the SPSB auction when his signal is x = x. So, γ is implicitly defined by E[u(v γ) x = x] = E[u(v b(z)) x = x]. (6) The right side of (6) is the bidder s expected payoff in the SPSB auction (he is the winning bidder as he has the highest signal), while the left side is his expected payoff when he buys the item at a price of γ. Now, solving for γ from (6) gives for CARA bidders When c = x, all the rival bidders reject B. So, U A (x, x) =E[u(v B) x = x]. But when c = x, all the rival bidders accept B. So, U A (x, x) = E [u (v B) x = x]. n When c = x, all the rival bidders reject B. So, U R (x, x) =E[u(v b(z)) x = x]. But when c = x, all the rival bidders accept B. So, U R (x, x) =0. 7

9 γ and for risk neutral bidders = ln E[e (v b(z)) x = x] E[e v x = x] (7) γ 0 = E[b(z) x = x]. (8) () Let γ be bidder s maximum willingness to pay when his signal is x = x. So, γ is implicitly defined by Now, solving for γ from (9) gives for CARA bidders and for risk neutral bidders E[u(v γ) x = x] = 0. (9) γ = ln E[e v x = x] (0) γ 0 = E[v x = x]. () Now, Proposition below characterizes an equilibrium cutoff c in a buy-price auction. Proposition : In a common value buy-price auction with a buy price B, n CARA bidders, bidder i s signal x i distributed according to a CDF over the support of [x, x] and the x +x +...+x common value of the item v = n, n (i) if B (γ, γ), then there exists a symmetric equilibrium cutoff c (x, x) implicitly defined by U A (c, c ) = U R (c, c ), () (ii) if B γ, then c = x is a symmetric equilibrium cutoff and (iii) if B γ, then c = x is a symmetric equilibrium cutoff. Proof: Appendix-A. The equilibrium probability that the buy price is accepted is given by Pr[max{x,z} > c ]. Proposition shows that it is possible for the seller to select a buy price that the 8

10 bidders accept with a positive probability in an equilibrium. But if the buy price is too high (low) it is never (always) accepted in the equilibrium. Next we consider a case of two bidders where we establish the uniqueness of the equilibrium cutoff c given by Proposition. 3.3 A Case of Two Bidders: Let n = and the signals be identically and independently distributed as an uniform distribution over the support [0, ]. The bid function in the SPSB auction in () and (3) reduces to b(x) =x. In a buy-price auction, we get from (4) and (5) that Z c µ µ Z µ x + x x + x U A (x, c) = u B dx + u B dx, 0 c and Z min{x,c} µ x + x U R (x, c) = u x dx. 0 We find from h (7) and i (8) that γ = γ 0 =. Similarly, we find from (0) and () e that γ = ln and γ According to Proposition, if B (, =. ), then e an interior equilibrium cutoff c 0 (0, ) exists for risk neutral bidders which solves U A (c,c )=U R (c,c ),thatis, 3c Bc c B c + + =. (3) If B then c =0and if B then c =. 4 0 ³ h i 0 e In a similar way, if B ln,, then an interior equilibrium cutoff The value of γ inequality e > e depends on, but it s easy to show that γ e4 holds. So we get 4 <γ =. Since > 0, the 0 e ³ e e < <e 4 e e4 e e e ln < ln <. e 4 e 4 h i = e 6 For example, when =,weget γ ln = which is smaller than.so, γ <γ. e

11 c (0, ) exists for risk averse bidders which solves U A (c,c )=U R (c,c ),thatis, µ c (c ( c B) ( c + B) B) + + e e + e µ c = c + e. (4) h i e If B ln then c =0and if B then c =. Now, Proposition e establishes that the equilibrium cutoffsdefinedimplicitlyby(3) and(4)are unique. Proposition : In a common value buy-price auction with a buy price B (γ, γ), n = bidders (either risk neutral or risk averse) and bidder i s signal x i U[0, ], the equilibrium cutoff c (0, ) defined in Proposition is unique. Proof: Appendix-A. 4 A Seller s Payoff: In this paper we do not analyze a seller s optimal choice of a buy price. The comparison is between the seller s payoff in auctions with and without a buy price. The seller can set the buy price high enough so that it is never accepted by the bidders and in that case the expected seller payoff will be the same under both auction formats. We are definitely not interested in such buy prices. We look at the possibilities that one format may give the seller a higher payoff than the other one when bidders accept the buy price with a positive probability. Let n = and bidder i s signal x i U[0, ]. As we already know, with these assumptions, we get b(x) = x. We denote by x () and x () the highest and the second highest signals and f e be their joint density function. Then we get f(q, e r) = f(q)f(r) =where q and r aresomerealizationsof x () and x (), respectively. Now, the expected seller payoff in an auction without a buy price is Z Z r eu NBP = ue (q) dqdr. (5) 0 0 In a buy-price auction, let β(.) be the function that maps an equilibrium cutoff c to a buy price B so that β(c )= B. Itiseasytoshow using theindifference equation in Proposition that this function exists. 3 So, the expected seller payoff in the buy-price auction as a function of the equilibrium cutoff c is 3 We derive this function in the proof of Proposition. 0

12 Z c Z r Z Z r eu BP (c )= ue (q) dqdr + ue (β(c )) dqdr. (6) 0 0 c The firstparton the rightsideof (6) corresponds to a situation when a buy-price auction proceeds to the SPSB phase and the second part corresponds to a situation when at least one bidder accepts B. Notice that when B γ = so that c = x, we get Ue BP (x) = U e NBP. So, the seller can be indifferent between the two auction formatswhenhechooses ahigh B. But if he sets B < so that it is accepted with a positiveprobability therevenuesfrom the two auction formatswillhavedifferent variances and may also have different means. The seller payoff is thus affected by his attitude towards risk A Risk Neutral Seller: A risk neutral seller seeks to maximize his expected revenue which, in a common value auction (with or without a buy price), depends on bidders risk preferences. This happens because, as we mentioned in Section 3., bidding behaviors are affected by these preferences. In our discussion below, we consider risk neutral and risk averse bidders separately as we compare the expected seller revenues in auctions with and without buy prices. When the seller is risk neutral, we find from (5) that Ue NBP = and U e 3 BP = c ( c )+( c )B. When bidders are risk neutral, Proposition says that 3 any buy price B<γ = 0 has a positive probability of being accepted. For such buy prices, Proposition 3 below compares eu BP to Ue NBP = when the seller and the 3 bidders are risk neutral. Proposition 3: In a common value setting with n = risk neutral bidders (i.e. x +x = 0), bidder i s signal x i U[0, ], and the common value of the item v =,a buy-price auction with B<γ 0 = generates a smaller expected seller revenue than an auction without a buy price, that is, for a risk neutral seller, Ue BP < U e NBP =. 3 Proof: Appendix-A. We can numerically verify Proposition 3 by calculating Ue BP for all possible values of B <γ 0 = when the seller and both the bidders are risk neutral. If B γ = 0 4 then the equilibrium cutoff c 0 =0, B is accepted by both bidders and seller revenue is

13 B <.But if B (, 4 3 ) then (3) solves for c 4 0 (0, ) and we can calculate the corresponding U e BP using(6). Now, Figure shows that U e BP < for all B (, ). 3 4 So, a risk neutral seller earns a lower revenue by posting a buy price when the bidders are risk neutral. (Figure goes here.) In order to analyze the revenue implications of a buy price when bidders are risk averse, we assume that =. We get from (0) and (7) that γ = 6 and 5 γ =. So, according to Proposition, any buy price B<γ = has a positive 6 probability of being accepted. If B γ = 5 then the equilibrium cutoff c =0, B 6 is accepted by both bidders and seller revenue is B <.If B ( 6, ) then (4) solves for c e (0, ) and we can calculate the corresponding U BP using (6). Now, Figure shows that U e BP < 6 for all B (, ). So, a risk neutral seller earns a 3 5 lower revenue by posting a buy price when the bidders have CARA and =. (Figure goes here.) To see how we get the result in Figure, we may divide the set of buy prices 6 6, into two subsets: (i) B, and (ii) B,. For a B in the first subset, clearly eu BP = c ( c )+( c )B < because B < and c 3 <. For the 3 3 second subset, let s fix a buy price let B =.Thenweget c =0.6 from (4) and 5 eu BP = 0.35 < from (6). 3 A buy price helps a risk averse bidder to avoid some of the uncertainties regarding winning and payment upon winning in an auction. This may give the seller a chance to extractsomeriskpremium in theform ofahigher buyprice (as ithappens in the case of a private value model in Reynolds and Wooders (003)). But in a common value auction there is an additional source of uncertainty due to the unobservable common value which enters a bidder s expected payoff when he accepts a buy price. Risk averse bidders dislike this uncertainty. Intuitively, this feature of the common value model makes a buy price unattractive and the seller gets a lower revenue. 4 4 The revenue result and our intuition hold in examples involving more than two bidders with different values of and different values of upper and lower bounds of the uniform distribution of signals.

14 4. A Risk Averse Seller: A risk averse seller cares for not only the monetary payoff but also the level of uncertainties involved in a transaction. Although a buy-price auction does not offer a revenue advantage it might still be attractive to a risk averse seller as it involves less uncertainties in the revenue he obtains. In particular, a buy-price auction in which the buy price is accepted with a positive probability involves a lower variance of revenue than an auction without a buy price. Now, Proposition 4 states that a risk averse seller can actually prefer a buy-price auction to an auction without a buy price. Proposition 4: In a common value setting with n = bidders (either risk neutral or risk averse), bidder i s signal x i U[0, ], and the common value of the item x +x v =, there exist some buy prices B<γ = for which a buy-price auction gives a risk averse seller a higher expected payoff than an auction without a buy price, that is, for a risk averse seller, Ue BP > Ue NBP. Proof: Appendix-A. When B = γ 0 =,we find c = and the seller gets U e BP () = U e NBP. The proofofproposition 4 relieson the existenceofsome B<γ 0 = that generates an equilibrium cutoff c < for which Ue ) > e e BP (c U BP () = U NBP. In order to show du e BP (c ) the existence it is sufficient to establish that dc < 0. Intuitively, the lower c = variance of revenue in a buy-price auction (with a buy price which is high enough but is accepted with a positive probability) offsets the effect of lower expected revenue compared to that in an auction without any buy price. 5 Conclusion: In this paper, we modeled common value auctions with and without temporary buy prices. We used a pure common value framework where each bidder receives a signal and the unknown common value of the item is the average of all the signals. The paper characterizes a symmetric equilibrium in the general case of n CARA bidders. The effects of a buy price on seller payoff is then analyzed for n = and signals distributed as U[0, ]. We find that, regardless of bidder risk aversion, a buy price reduces seller revenue. So, there is no incentive for risk neutral sellers to post buy prices in common value auctions. But Reynolds and Wooders (003) show that a temporary buy price 3

15 can generate a higher seller revenue in an IPV auction with CARA bidders. The unknown valuation of the item in a common value setting makes a buy price less attractive which leads to this difference between private and common values results. We also show that a risk averse seller, however, can raise his payoff by using an appropriately chosen buy price. When a seller is risk averse he cares for both revenue and the uncertainty of his revenue. A buy price reduces the variance of revenue and thus the revenue uncertainties. This is similar to the results in Mathews and Katzman (006) for an IPV auction. This paper makes some contributions to the growing literature on buy-price auctions. First of all, it considers a common value framework where the implications of a buy price are quite different than those when the values are private. This paper also differentiates itself from prior works as it considers risk aversion for both bidders and the seller at the same time. Finally, it provides an explanation for the popularity of online buy-price auctions where bidder valuations can be common. So, we establish the possibility that seller risk aversion, regardless of the type of information bidders may possess, can explain the popularity of temporary buy prices. The revenue results, however, are limited by some simplifying assumptions we made. Future extensions of this study can focus on relaxing these assumptions. 4

16 References [] Albers, W. and Harstad, Ronald M. (99): Common-Value Auctions with Independent Information: A Framing Effect Observed in a Market Game, Game Equilibrium Models. R.Selten, ed., Vol. II, Springer, Berlin. [] Bajari, P. and Hortaçsu, A. (003): The Winner s Curse, Reserve Prices, and Endogenous Entry: Empirical Insights from ebay Auctions, RAND Journal of Economics, Vol.34, Issue, [3] Bikhchandani, S. and Riley, John G. (99): Equilibria in Open Common Value Auctions, Journal of Economic Theory, Vol. 53, [4] Budish, E. and Takeyama L. (00): Buy prices in online auctions: irrationality on the internet? Economics Letters, Vol. 7, [5] Bulow, Jeremy I., Huang, M. and Klemperer, Paul D. (999): Toeholds and Takeovers, Journal of Political Economy, Vol. 07(3), [6] Goeree, Jacob K. and Offerman, T. (003): Competitive Bidding in Auctions with Private and Common Values, Economic Journal, Vol.3,Issue 489, [7] Goeree, Jacob K. and Offerman, T. (00): Efficiency in Auctions with Private and Common Values: An Experimental Study, American Economic Review, Vol. 9(3), [8] Hidvégi, Z., Wang, W. and Whinston, A. B. (006): Buy-price English Auction, Journal of Economic Theory, Vol.9,Issue, [9] Klemperer, Paul D. (998): Auctions With Almost Common Values, European Economic Review, Vol. 4, [0] Mathews, T. (004) : The Impact of Discounting on an Auction with a Buyout Option: a Theoretical Analysis Motivated by ebays Buy-It-Now Feature, Journal of Economics, Vol. 8, No., 5-5. [] Mathews, T. and Katzman, B. (006): "The Role of Varying Risk Attitudes in an Auction with a Buyout Option, Economic Theory, Vol.7, No.3,

17 [] Milgrom, R. and Weber, P. (98): A Theory of Auctions and Competitive Bidding, Econometrica, Vol. 50, No. 5, [3] Reynolds, S., and Wooders, J. (003): Auctions with a Buy Price, forthcoming in Economic Theory. 6

18 Appendix A Proposition : Proof: (i) First we show the existence of some c (x, x) that satisfies U A (c, c) = U R (c, c) when B (γ, γ), and then we show that this c actually is a symmetric equilibrium cutoff. Now, let s define Û A (c), Û R (c) and Q(F (c)) as follows: Ã! X n µ n Û A (c) = U A (c, c) = F (c) n l ( F (c)) l u l (c), l l + l=0 Û R (c) = U R (c, c) Z c Z x Z x µ P c + l> x l = (n )... u b(x ) df (x n )...df (x ), x x x n Ã! X n µ n Q(F (c)) = F (c) n l ( F (c)) l l l + l=0 F (c) n = n( F (c)) where Q(F (c)) is defined for F (c) [0, ). Notice Q(0) = n and, for any c <x, Q(F (c)) > F (c) n. that Q() = lim Q(m) =, m (i)-: For the existence of a cutoff c it is sufficient to show that the two curves Û A (c) and Û R (c) intersect at some c (x, x). When c = x, from footnote 0 and, we get Û A (x) = E [u (v B) x = x], and n Û R (x) = 0. From (9) it is easy to see that Û A (x) = Û R (x) when B = γ. Since Û A (x) is a decreasing function of B, for any B >γ it must be that Û A (x) < Û R (x). Again, when c = x, from footnote 0 and, we get Û A (x) = E [u (v B) x = x], and Û R (x) = E [u (v b(z)) x = x]. Now, from (6) we find that Û A (x) = Û R (x) when B = γ. Since Û A (x) is a decreasing function of B, for any B <γ it must be that Û A (x) > Û R (x). 7

19 The above arguments verify that, for a buy price B (γ, γ), Û A (x) < Û R (x) and Û A (x) > Û R (x). Since Û A (c) and Û R (c) are continuous in c, therefore, there exists a c (x, x) for which the two curves Û A (c) and Û R (c) would intersect. Now, we show that this c is in fact a symmetric equilibrium cutoff. (i)-: Suppose that B (γ, γ) so that there exists a c that satisfies U A (c, c) = U R (c, c). In order to show that this c is an equilibrium cutoff, it is sufficient to show that U A (x, c) lies above U R (x, c) for any x (c, x], and U A (x, c) lies below U R (x, c) for any x [x,c). du l (x) For the CARA utility function, we find that u 0 (.) = u(.). So, dx = u n n l(x). Now, we get Ã! n µ U A (x, c) X n du l (x) = F (c) n l ( F (c)) l x l l + dx l=0 = Q(F (c)) U A (x, c). (7) n n Now, when x c, Z Z Z µ P U R c x (x, c) x x + l> x l = (n )... u b(x ) df (x n )...df (x ) x x x x n n Z (n ) c = F (x ) n df (x ) U R (x, c) n n that x = F (c) n U R (x, c). (8) n n Since, by definition of c, U A (c, c) = U R (c, c), and Q(F (c)) > F (c) n it must be U A (x, c) = Q(F (c)) U A (c, c) > x x=c n n U R (x, c) F (c) n U R (c, c) =. n n x x=c Now, when x <c, applying Leibniz rule gives Z Z Z µ P x x x U R (x, c) x + l> x l = (n )... u b(x ) df (x n )...df (x ) x x x x n n Z Z µ P x x x + x + l> x l +(n )... u b(x) df (x n )...df (x 3 ) Z x x x n (n ) = F (x ) n df (x ) U R (x, c)+ E[u(v b(x)) x = x, z = x]f (x) n n n x = F (x) n U R (x, c), (9) n n 8

20 where the third and the fourth lines use footnote 8 and (). Now, if U A (x, c) and U R (x, c) intersect at some x 0 (c, x], we get U A (x 0,c)= U R (x 0,c). Since Q(F (c)) > F(c) n,we find from (7) and (8) that U A (x, c) U R (x, c) > x 0 x 0 which contradicts with x=x x=x U A (x, c) U R (x, c) >. x x x=c Similarly, if U A (x, c) and U R (x, c) intersect at some x 00 [x,c), we get U A (x 00,c)= U R (x 00,c). Since Q(F (c)) > F(c) n >F(x) n for any x <c,we find from (7) and (9) that which contradicts with x=c U A (x, c) U R (x, c) > x 00 x 00 x=x x=x U A (x, c) U R (x, c) >. x x x=c As a result, U A (x, c) and U R (x, c) intersect only at x = c, and U A (x, c) is steeper than U R (x, c) at this intersection. So, U A (x, c) lies above U R (x, c) for any x (c, x], and U A (x, c) lies below U R (x, c) for any x [x,c). (ii) Firstweshow that, when B = γ and c = x, U A (x, c) lies above U R (x, c) for x (x, x]. Then we show that, when B<γ and c = x, once again, U A (x, c) lies above U R (x, c) for x (x, x]. This completes the proof that, when B γ, c = x is a symmetric equilibrium cutoff. Fix c = x. Suppose B = γ. Then we get U A (x, x) = U R (x, x). Since Q(F (x)) > F (x) n we find from (7) and (8) that, U A (x, x) x x=x > UR (x, x) x x=c x=x. (0) If U A (x, x) and U R (x, x) intersect at some x 0 (x, x],we get U A (x 0, x) = U R (x 0, x). Since Q(F (x)) > F(x) n we find from (7) and (8) that U A (x, x) U R (x, x) > () x 0 x 0 x=x which contradicts (0). So, if B = γ and c = x, then U A (x, c) lies above U R (x, c) for x (x, x]. 9 x=x

21 Now, suppose that B<γ. For c = x, we find U A (x, x) > U R (x, x) as U A (x, x) is a decreasing function of B. If U A (x, x) and U R (x, x) intersect at some x 0 (x, x], it implies () which contradicts U A (x, x) > U R (x, x). So, when B<γ and c = x, U A (x, c) lies above U R (x, c) for x (x, x]. (iii) First we show that, when B = γ and c = x, U A (x, c) lies below U R (x, c) for x [x, x). Then we show that, when B >γ and c = x, once again, U A (x, c) lies below U R (x, c) for x [x, x). This completes the proof that, when B γ, c = x is a symmetric equilibrium cutoff. Fix c = x. Suppose B = γ. Then we get U A (x, x) = U R (x, x). Since Q(F (x)) = F (x) n we find from (7) and (8) that U A (x, x) x x=x = UR (x, x) x x=x. () For a sufficiently small ε> 0, U A (x, x) = U R (x, x), together with (), implies that U A (x ε, x) = U R (x ε, x). Since Q(F (x)) > F (x ε) n we find from (7) and (9) that U A (x, x) U R (x, x) >. x x x=x ε So, there exist some x 0 < x ε for which x=x ε U A (x 0, x) < U R (x 0, x). (3) If U A (x, x) and U R (x, x) intersect at some x [x, x), we get U A (x, x) = 00 U R (x, x). Since Q(F (x)) > F (x 00 ) n we find from (7) and (9) that U A (x, x) > UR (x, x) x 00 x 00 x=x x=x (4) which contradicts (3). So, when B = γ and c = x, U A (x, c) lies below U R (x, c). Now, suppose that B>γ. For c = x, we find U A (x, x) < U R (x, x) as U A (x, x) 00 is adecreasingfunction of B. If U A (x, x) and U R (x, x) intersect at some x [x, x), it implies (4) which contradicts U A (x, x) < U R (x, x). So, when B>γ and c = x, U A (x, c) lies below U R (x, c). 0

22 Proposition : Proof: Suppose that the bidders are risk averse. With n = bidders and signal x i U[0, ], Proposition says that, when B (γ, γ 0 ) where γ = and γ 0 =, there exist some symmetric equilibrium cutoff c (0, ) that satisfies U A (c,c )= U R (c,c ). We show that this c is unique. c 0. We find from (3) that 3c Bc c B c + + = which gives c 0 =(4B ). Clearly, for any B (, ) there is one and only one 4 Now, suppose that the bidders are risk averse and that B (γ, γ ) where γ = h i ln e and γ =. According to Proposition there exist some symmetric e equilibrium cutoff c (0, ) that satisfies U A (c,c )= U R (c,c ). Wewantto show that thesolutiontothisequationisunique. We find from (4) that µ c (c B) ( c B) ( c + B) + + e e + e µ c = c + e. Solving for B from the above equation gives (ignoring the asterisk) " # c c c [ + ]e e B = ln c. ( c ) e e Let sdenotethe rightsideofthe aboveequationby β(c). Now, to show that, for any B (γ, γ ), there is a unique c (0, ) that satisfies the equation B = β(c), we provethatthere exists an inverseto the function β(c) such that c = β (B) (0, ) for any B (γ, γ ).For this, we need to show that β(c) is well-defined and monotone increasing or decreasing in c (0, ). We proceed in three steps as follows. (i) Let s define λ(c) as follows: c c c [ + ]e e λ(c) = c. (5) ( c e e )

23 Differentiating λ(c) with respect to c gives: 3c 3c c λ 0 [(4 + c)e +(c )e +(c 6)e c +4e ] (c) = c ) 4[e e ] ( c. Now, to show that λ 0 (c) > 0, we need to show that both the numerator and the denominator of the above expression have the same sign. Given > 0, the denominator is positive when c 0. Let s denote the numerator by η(c), so that 3c 3c c η(c) = (4+ c)e +(c )e +(c 6)e c +4e. Now, differentiating η(c) with respect to c gives η 0 (c) = 3c 3c 3c 3c 3 3c 3 3c 4e +3 e 3 ce e + ce e c 4e c + ce e c +e c and differentiating η 0 (c) with respect to c gives 3c 9 3c 9 3c 3 3c 3c 3c 9 9 η 00 (c) = 3 e + 3 e 3 ce e + 3 ce 3 e 4 4 c c e c + 3 ce 3 e c + e and differentiating η 00 (c) with respect to c gives 7 7 η 000 (c) = 4 e ( c)+ 4 e (c ) + 4 e c (c ) + 3 e c 3c 8 c = 4 ( c)(e e e c )+ 3 e c )+ 3c 3c 3c 3 c = 4 ( c)( e e + e e e 4 7 3c 3c c 7 3c 3c 3c c c = 4 ( c) (e e )+ (7e 6e ) + 3 e c 3c Now, given > 0 and c 0, ithas to be thecasethat e >e and 7e 3c > 6e c. So, η 000 (c) > 0, which implies that η 00 (c) is a strictly increasing function of c 0. Now evaluating η 00 (c) at c = 0 gives µ µ η 00 (0) = 3 3 e + e = 3 e + e Since > 0 implies 0 <e <, itmustbethat η 00 (0) > 0. Now, as η 00 (c) is an increasing function of c 0, we have η 00 (c) > 0 for c (0, ). Then, this implies that η 0 (c) is a strictly increasing function of c 0. We find that η 0 (0) = ( e )+ ( 3 e ). Since dη0 (0) =( e )+( e )+ 3 e 0 for d 4

24 0, and η 0 (0) =0 =0, itmustbethat η 0 (0) > 0 for >0. As η 0 (c) is an increasing function of c 0, we can say that η 0 (c) > 0 for c 0 and >0. Then, this implies that η(c) is an increasing function of c 0. We find that η(0) = ( + )e. dη(0) d Since = e 0 for 0, and η(0) =0 =0, itmustbethat η(0) > 0 for >0. As η(c ) is an increasing function of c, we can say that η(c) > 0 or η(c) > 0 for c 0 and >0. So both the numerator and denominator of λ 0 (c) are positive. So, λ 0 (c) > 0 for c 0 and >0, which implies that, when >0, λ(c) is a strictly increasing function of c 0. (ii) In this step we show that β(c) is a strictly increasing function of c (0, ). Since β(c) = ln[λ(c)] we find that β (c) = λ (c). We 0 0 λ(c) find from (5) that λ(0) = (/)/( e ). Clearly, for >0, λ(0) > 0. Since, given >0, λ(c) is an increasing function of c 0, we can say that λ(c) > 0 for c (0, ) and >0. Asa result, β 0 (c) > 0 for c (0, ) and >0. (iii) Nowwe showthat β(c) = ln[λ(c)] is well-defined, which boils down to showing that λ(c) is positive for c (0, ). We have already shown this in (ii). So, based on the above three steps we can say that β(c h ) is i well-defined and e monotone increasing in c (0, ). Moreover, β(0) = ln and β() =. e So, there ³ exists h an inverse i to the function β(c ) such that c = β (B) (0, ) e for B ln,, and this inverse function will be monotone increasing e in B. ³ So, there h is i only one e a B ln., e c (0, ) that satisfies U A (c,c ) = U R (c,c ) for 3

25 Proposition 3: Proof: Let s divide the set of buy prices B < into two subsets: (i) B and 4 (ii) B (, ). We want to show that Ue BP < U e NBP = for a B in any of these two 4 3 subsets. (i) When B 4, both the bidders accept B and the seller revenue is B. So, e = B e U BP 4 < 3 = U NBP. (ii) When B (, ), according to Proposition, there exists a symmetric 4 equilibrium cutoff c 0 (0, ) which solves (3). We find from (3) that c 0 =(4B ). We denote by x () and x () the highest and the second highest signals. If both the bidders reject B, the two auction formats generate the same revenue b(x () )= x (). The difference occurs when B is accepted. This happens when at least one of the bidders signal is higher than c 0, that is, x () >c 0. So, the revenue difference between a buy-price auction and an auction without a buy price is 5 4Π = Z Z r (B q)f e (q, r)dqdr c 0 0 = B Bc 0 + c = B B(4B ) + (4B ) It is easy to check that setting 4Π = 0 implies B =,and d(4π) =4(B ) > 0 db d(4π) for B = 6 and =4(B ) =0 for B =. Then, clearly, 4Π < 0 for db B (, ). That is, for any B (, ), we find that Ue < e =. 4 4 BP U NBP We denote by f(q, e r) the joint density function of x () and x ().So, f(q, e r) =. 4

26 Proposition 4: Proof: For n = risk neutral or risk averse bidders and signal x i U[0, ], Proposition states that c = when B and c < when B <. We want to show that U e BP > U e NBP for a risk averse seller for some B <.Since U e BP () = U e NBP du e BP (c ) it will be sufficient to show that dc < 0 to complete the proof. c = We find from (6) that Z c Z r Z Z r eu BP (c ) = ue (q) dqdr + ue (β(c )) dqdr 0 0 c 0 Z Z Z c c Z r = ue (q) dqdr + ue (β(c )) dqdr Now, using Leibniz rule, we get q 0 c 0 Z c = ue (q)(c q)dq + ue (β(c )) ( c ). 0 Z c d Ue BP (c ) = ue (q) dq + ue 0 (β(c )) β 0 (c )( c ) ue (β(c )) c. dc 0 So, Z d Ue ) BP (c = ue (q) dq ue (β()) dc 0 c = µ = E[ue (x i )] ue = [E[ue (x i )] ue (E[x i ])]. The second line utilizes that β() = and the last line utilizes that E[x i]=. Since ue(.) is strictly concave, following Jensen s inequality, E[ue (x i )] < ue (E[x i ]). So, due BP (c ) dc < 0. c =x 5

27 Figure : Expected Seller Revenue when n =, x ~U [0,] and = Expected Revenue Auction Without B Buy-Price Auction Buy Price

28 Figure : Expected Seller Revenue when n =, x ~U [0,] and = Auction Without B Expected Revenue Buy-Price Auction Buy Price