\documentclass[thmsa,12pt]{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=Latex.dll} %TCIDATA{Created=Mon Jun 16 14:43:25 1997} %TCIDATA{LastRevised=Tue Nov 03 14:33:35 1998} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{CSTFile=article.cst} \input tcilatex \topmargin -30pt \textheight 23cm \begin{document} \baselineskip=20pt \begin{center} \quad %TCIMACRO{ %\TeXButton{Large}{\Large% %}}% %BeginExpansion \Large% % %EndExpansion \textbf{Prices and the Winner's Curse } %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.4in}% %}}% %BeginExpansion \vspace{0.4in}% % %EndExpansion %TCIMACRO{ %\TeXButton{normal}{\normalsize% %}}% %BeginExpansion \normalsize% % %EndExpansion Jeremy Bulow% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.1in}% %}}% %BeginExpansion \vspace{0.1in}% % %EndExpansion Graduate School of Business, Stanford University, Stanford, USA% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.1in}% %}}% %BeginExpansion \vspace{0.1in}% % %EndExpansion Tel: \ 650 723 2160 Fax: 650 725 0468% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.1in}% %}}% %BeginExpansion \vspace{0.1in}% % %EndExpansion email: jeremy\_bulow@gsb.stanford.edu %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.3in}% %}}% %BeginExpansion \vspace{0.3in}% % %EndExpansion and% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.3in}% %}}% %BeginExpansion \vspace{0.3in}% % %EndExpansion Paul Klemperer% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.1in}% %}}% %BeginExpansion \vspace{0.1in}% % %EndExpansion Nuffield College, Oxford University, UK% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.1in}% %}}% %BeginExpansion \vspace{0.1in}% % %EndExpansion Int Tel: +44 1865 278588 Int Fax: +44 1865 278557% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.1in}% %}}% %BeginExpansion \vspace{0.1in}% % %EndExpansion email: paul.klemperer@economics.ox.ac.uk %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.3in}% %}}% %BeginExpansion \vspace{0.3in}% % %EndExpansion May 1998% %TCIMACRO{ %\TeXButton{vspace}{\vspace{0.3in}% %}}% %BeginExpansion \vspace{0.3in}% % %EndExpansion \textbf{Abstract} \end{center} We usually assume increases in supply, allocation by rationing, and exclusion of potential buyers will never raise prices. But all of these activities raise the expected price in an important set of cases when common-value assets are sold. Furthermore, when we make the assumptions needed to rule out these ``anomalies'' when buyers are symmetric, small asymmetries among the buyers necessarily cause the anomalies to reappear.% %TCIMACRO{ %\TeXButton{newline}{\newline% %}}% %BeginExpansion \newline% % %EndExpansion %TCIMACRO{ %\TeXButton{noindent}{\noindent% %}}% %BeginExpansion \noindent% % %EndExpansion \textbf{Keywords}: Auction theory, common value, winner's curse, PCS auction, spectrum auction, airwaves auction, initial public offerings, IPO.% %TCIMACRO{ %\TeXButton{newline}{\newline% %}}% %BeginExpansion \newline% % %EndExpansion %TCIMACRO{ %\TeXButton{noindent}{\noindent% %}}% %BeginExpansion \noindent% % %EndExpansion \textbf{JEL No}: D44 (Auctions), L96 (Telecommunications), G30 (Corporate Finance), G24 (Investment Banking)% %TCIMACRO{ %\TeXButton{newpage}{\newpage% %}}% %BeginExpansion \newpage% % %EndExpansion \section{Introduction} \baselineskip=23pt Increases in supply lower prices. It is never profitable to commit to rationing at a price at which there is surely excess demand. Excluding potential buyers cannot raise prices. These statements evoke almost universal agreement in our profession. Yet economists from Veblen (1899) to Becker (1991) have sought to explain examples of pricing that appear to contradict these truths. In fact, it is perfectly reasonable for these statements to be false. This paper shows why, and when this is most likely to happen. To understand our results, it is important to understand how a bidder determines the maximum he will be willing to pay for an asset. If a buyer's estimate of an asset's value is affected only by his own perceptions and not by the perceptions of others, he should be willing to pay up to his valuation. This is the Adam Smith world, where a buyer can easily maximize his utility given any set of prices, and a firm can easily maximize its profits. In this sort of ``private value'' model, the statements in the first paragraph are true. But in many important markets others' perceptions are informative. The extreme cases are ``common-value'' assets, or assets all buyers would value equally if they shared the same information. Financial assets held by non-control investors may be the best example; oil fields are commonly cited. Most assets have both a private and common value element, particularly if imperfect substitutes exist. For example, a house's value will have both common and idiosyncratic (private) elements. With common values, buyers may find it prudent to exit an ascending price auction at more or less than their pre-auction estimate of the value, so the statements in the first paragraph are often false in common-value auctions. The reason is the ``winner's curse''. Buyers must bid more conservatively the more bidders there are, because winning implies a greater winner's curse. This effect can more than compensate for the increase in competition caused by more bidders, so more bidders can \textit{lower} expected prices.% \footnote{% Steven Matthews (1984) has already provided an example with symmetric bidders and affiliated common values in which additional bidders reduce expected revenue in a first-price auction. Our paper provides insight into why results like ours and Matthews' can arise, and shows they are surprisingly likely.} Conversely, adding more supply, and/or rationing, creates more winners, so reduces the bad news learned by winning, and so may raise bids enough to \textit{increase} expected prices. This paper shows when this happens and why it is surprisingly often. A good example is provided by the market for Initial Public Offerings (IPOs). Rather than being priced to clear the market, many IPOs are made at prices that guarantee excess demand. By pricing low enough so that everyone will want to buy, potential shareowners are absolved of the winner's curse of only being buyers when they are among the most optimistic investors. This allows the pooling price to be quite high and, under quite reasonable conditions, as high or higher than the expected price in a standard auction.% \footnote{% A related example is when the value of an asset is not allowed to rise or fall more than a fixed amount in a day. The South Korean won was limited to a 10 percent daily decline through early December of 1997. The limitation prevented the market from fully aggregating bidders' information and the price fell by the maximum on several days. When the limitation was removed and the market was allowed to clear, the price actually rose: it is entirely possible that if a price is artificially fixed only slightly in excess of the expected market clearing price there will be an enormous excess supply, but if the market is allowed to clear the price will rise above the fixed rate. (Obviously, the Korean situation was very complex and relaxing the limit on the amount the won could fall may not have been important for the increase in the market price.)} Our results are especially likely in asymmetric ``almost common value'' markets in which some competitors have a small advantage, because the other bidder(s) then face an exacerbated winner's curse. This was illustrated in the A and B band spectrum auction held in 1994-95 by the Federal Communications Commission. Pacific Telesis was the natural buyer of the single Los Angeles license available for sale,\footnote{% AT\&T was ineligible to bid, and PacTel had the benefit of its name recognition and experience in California, as well as its familiarity with the California wireless market in which it was a duopolist prior to its spinoff of its cellular subsidiary, Airtouch. No one knows what PCS licenses are really worth, but it is fair to say that the LA license was worth more to PacTel than anyone else.} and was able to acquire it very cheaply.% \footnote{% While the FCC's mandate was for economic efficiency rather than revenue, and awarding the license to PacTel was almost certainly efficient, if PacTel had paid more there would have been an efficiency gain to the economy from being able to reduce the deadweight loss from taxation.} Markets where two licenses were sold generally yielded more competitive prices relative to their demographic characteristics.\footnote{% The most obvious example is Chicago, where the prices were about \$31 per head of population for each of the two licenses, compared with less than \$26 per head of population for Los Angeles' single license, in spite of Chicago's inferior demographic characteristics. (The famous long commutes of Angelenos and the population density in the area makes it a particularly desirable place to own a wireless telephone franchise.) The single New York license yielded only \$17 per head of population.} Even where one bidder had an advantage, the prices of both licenses were determined by aggressive competition for the second license. So prices were better, even though the third-highest bid set the price in these markets while the second-highest bid set the price in Los Angeles.\footnote{% Our model does not involve bidding costs, but these would tend to reinforce our explanation and result. See Avery (1998), Daniel and Hirshleifer (1995), and Hirshleifer (1995) for models including bidding costs.} In section 2 we set up a simple model of a standard ascending auction% \footnote{% The spectrum auction was an ascending auction, but also included a number of special features designed to allow licenses for different regions to be sold simultaneously (see, for example, McAfee and McMillan (1996)). However, we do not believe these additional features affect our basic argument.} among bidders with ``almost''common values. Section 3 shows when \textit{higher} prices are associated with selling \textit{more} units in the symmetric case. Section 4 shows that the results are dramatically different when bidders are asymmetric: greater supply \textit{raises} price precisely when it does not with symmetric bidders! Section 5 shows when rationing, as in Initial Public Offerings, is optimal. It also shows when restricting participation in an auction can raise expected revenues, and considers first-price auctions.\footnote{% The first (1997) draft of this paper shows how our model can be used to develop one possible explanation of the ``Declining Price Anomoly''. (See Ashenfelter (1989), Ayres and Cramton (1996), Beggs and Graddy (1997), Black and deMeza (1992), Levin (1997), McAfee and Vincent (1993), Pitchik (1995), Pitchik and Schotter (1988) and von der Fehr (1994), among others for further discussion of the ``anomoly'' and other explanations of it.) We plan to pursue this further in subsequent work.} Section 6 concludes.\footnote{% Other recent papers that use similar models to ours are Avery and Kagel's (1997), de Frutos and Rosenthal's (1997), and Krishna and Morgan's (1997) Working Papers. Krishna and Morgan develop important insights about the effects of collusion and joint-bidding in common-value auctions. Independently from the first (1997) draft of our paper, they also obtain results that are equivalent to the symmetric case of our section 5.2 about restricting participation. They do not tackle the asymmetric case because their model, unlike ours, is of pure common values, so has a vast multiplicity of equilibria, even when bidders are asymmetric (see note 14). (Nor, since their main focus is different, do they analyse the effects of increasing supply, or of rationing, which are the main focuses of our paper.) Avery and Kagel and de Frutos and Rosenthal address different concerns from ours; Avery and Kagel discuss experimental results in a two-bidder one-prize model, while de Frutos and Rosenthal obtain interesting results about sequential auctions. See also Bikhchandani and Riley (1991).} \section{The Model} We use the simplest possible model to make our points: each of 3 risk-neutral potential bidders observes a private signal $t_{i}$ independently and identically distributed according to the distribution $% F(t_{i}),i=1,2,3.$ We assume $F(\cdot )$ has a strictly positive continuous finite derivative $f(\cdot )$ everywhere on its range, and the lowest possible signal is $\underline{t}>0,$ so $F(\underline{t})=0.$ Conditional on all the signals, the expected value, $v_{i},$ of a unit to $i$ is \[ \begin{array}{lll} v_{1} & = & (1+\alpha _{1})t_{1}+t_{2}+t_{3}% %TCIMACRO{ %\TeXButton{medskip}{\medskip% %} }% %BeginExpansion \medskip% % %EndExpansion \\ v_{2} & = & t_{1}+(1+\alpha _{2})t_{2}+t_{3}% %TCIMACRO{ %\TeXButton{medskip}{\medskip% %} }% %BeginExpansion \medskip% % %EndExpansion \\ v_{3} & = & t_{1}+t_{2}+(1+\alpha _{3})t_{3}% %TCIMACRO{ %\TeXButton{medskip}{\medskip% %}}% %BeginExpansion \medskip% % %EndExpansion \end{array} \] That is, each unit has a \textit{common value}, $\sum_{i=1}^{3}t_{i},$to all the bidders, plus a \textit{private value,} $\alpha _{i}t_{i},$ to each bidder $i$. We will focus on two cases, ``the symmetric case'' in which $% \alpha _{1}=\alpha _{2}=\alpha _{3}=\alpha >0$ and ``the asymmetric case'' in which $\alpha _{1}>\alpha _{2}=\alpha _{3}=\alpha >0.$ In the latter case we will refer to bidder 1 as the ``advantaged'' bidder, and bidders 2 and 3 as ``disadvantaged'' bidders. We are interested in the case in which the private-value components, that is, the $\alpha _{i}$'s, are all small and so the sizes of bidders' advantages and disadvantages are also small. To make our points most starkly and straightforwardly, we consider an asymmetric case in which $\alpha /\alpha _{1}$ is also small, so we state our results throughout for the limits in which $\alpha _{i}\rightarrow 0,\forall _{i},$ and, for the asymmetric case, $\alpha /\alpha _{1}\rightarrow 0.$\footnote{% All we actually need is that the $\alpha _{i}$'s are small relative to the rates of change of bidders' inverse hazard rates, $\frac{1-F(t_{i})}{f(t_{i})% }$. So the order in which the limits is taken is unimportant.} No bidder wants more than one unit. We consider two cases: the auctioneer has one unit to sell, and the auctioneer has two units to sell. (The number of units is common knowledge.) We assume a conventional ascending bid ``English'' auction\footnote{% More formally, we area assuming what auction theorists call a ``Japanese auction''. Bikhchandani and Riley (1993) describe this as follows (for the single unit case): The auctioneer starts with a very low price and raises it continuously. Bidders indicate, by depressing a button, whether they are interested in buying the object at the current price. Once a bidder withdraws, he cannot reenter the auction. At each price level, the identities of all bidders active at that price are common knowledge. Whenever one or more bidders withdraw at a price, the auctioneer stops raising the price and asks the remaining bidders if they wish to withdraw. If additional bidder(s) withdraw, this is announced by the auctioneer and the remaining bidders are again asked if they wish to withdraw. This process continues until no additional bidders quit. When no additional bidders withdraw, and at least two bidders remain, the auctioneer starts raising the price continuously from the current level. The auction can end in one of two possible ways. If at any price there is only one active bidder, then this bidder is declared the winner and the auction ends. Else, if at any price all the remaining active bidders withdraw (either simultaneously or during the sequential quitting process described above) the auction ends and one of the last active bidders is randomly chosen as the winner. The winner gets the object and pays the current price. The other bidders pay nothing.} in which the price, $p,$ starts at zero and rises continuously until the number of bidders who are still willing to pay the current price equals the number of units the auctioneer has for sale. Each bidder observes the price at which any other bidder drops out. Each player's pure strategy specifies the price level up to which he will remain in the bidding, as a function of his private signal and of the price (if any) at which any other player quit previously. We assume symmetric bidders follow symmetric strategies, and restrict attention to the (Perfect Bayesian) equilibrium in which each bidder stays in the bidding just so long as he would be happy to find himself a winner, and stops bidding at that price at which he would be just indifferent were he to find himself a winner on the assumptions that any opponent(s) who drop out to make him a winner are of their lowest possible types assuming they have followed the equilibrium strategies prior to the current price.\footnote{% That is, strategies specified in this way yield a Perfect Bayesian equilibrium in the space of all strategies; in this equilibrium a player cannot do better by following any other strategy.},\footnote{% Restricting attention to equilibrium of this form both avoids trivialities (although there are other equilibria, they do not seem very natural) and greatly reduces the technical burden: See Bikhchandani and Riley (1993) for an exposition of how cumbersome and lengthy is a fully general analysis of even the completely symmetric version of our model, although they too make assumptions to obtain a unique equilibrium (the same equilibrium as ours, though their model is a special case of ours). See the Appendix for further discussion.} The Appendix shows that this yields a unique (Perfect Bayesian) equilibrium.\footnote{% By contrast, in a pure common values model with $\alpha _{1}=\alpha _{2}=\alpha _{3}=0$ this construction does not define a unique equilibrium. (For example, with just two bidders and $v_{1}=v_{2}=t_{1}+kt_{2},$ where $k$ is a positive constant, it is an equilibrium for 1 to quit at $\beta t_{1}$ and 2 to quit at $\left( \frac{\beta k}{\beta -1}\right) t_{2}$ for any $% \beta >1.$) Hence the need to include the $\alpha _{i}$ in the model, and to analyze a pure common value model as the limit of almost common value models; focusing on a particular equilibrium of the pure common value model can be misleading.} We write the actual $i^{th}$ highest signal as $t_{(i)},$ write $E(t)$ for the expectation of $t_{i},$ and write $E(t\mid t\geq t^{\prime })$ for the expectation of $t_{i}$ conditional on it exceeding $t^{\prime },$ etc. It will be useful to define bidder $i$'s \textit{marginal revenue}\footnote{% In analyzing our auctions using marginal revenues, we are following Bulow and Roberts (1989) who first showed how to interpret independent private-value auctions in terms of marginal revenues, and Bulow and Klemperer (1996) who extended their interpretation to more general settings such as this one. The marginal revenue of bidder $i$ with signal $t_{i}$ is exactly the marginal revenue extracted from the customer who is the same fraction of the way down the distribution of potential buyers of a monopolist whose demand is such that it has $q=1-F(t_{i})$ customers who have values $\geq p=v_{i}(t_{i})$ (i.e. there are $F(t_{i})$\ customers with values less than $v_{i}(t_{i})).$ This allows the direct translation of results from monopoly theory into auction theory, and so facilitates the analysis of auctions and the development of intuition about them.} as \[ MR_{i}=v_{i}-\frac{1-F(t_{i})}{f(t_{i})}\frac{\partial v_{i}}{dt_{i}}. \] Note that since (we assumed) the $\alpha _{i}$ are all small, $MR_{i}\approx v_{i}-h_{i}$ in which $h_{i}(t_{i})\equiv \frac{1-F(t_{i})}{f(t_{i})}$is the reciprocal of $i$'s hazard rate. \section{\textbf{The Symmetric Case}} We begin with the symmetric case in which $\alpha _{1}=\alpha _{2}=\alpha _{3}=\alpha >0$ (but $\alpha \approx 0)$. When three bidders compete for a single object, the lowest bidder quits first in symmetric equilibrium, and the other bidders can then infer (assuming equilibrium behaviour) his actual signal, $t_{(3)}.\footnote{% In fact, the lowest bidder quits at $(3+\alpha )t_{(3)},$ since if he stays in until a slightly higher price he will win only if both other signals are $% t_{(3)},$ but this fact is not necessary to our argument.}$ The next-lowest bidder then quits when the price reaches the point at which he would just be indifferent about winning were he the marginal winner, that is, were he tied for the highest signal, so he quits at $p=t_{(3)}+(2+\alpha )t_{(2)}.% \footnote{% It is easy to check that if he were to find himself a winner at any higher price he would lose money, since at price $p^{\prime }=t_{(3)}+(2+\alpha )t^{\prime }$ with $t^{\prime }>t_{(2)},$ the inferred value of the unit equals $t_{(3)}+(1+\alpha )t_{(2)}+t^{\prime }$ conditional on winning at price $p^{\prime },$ and conversely he would make money at any lower price, so should not quit before $p.$}$ We therefore have (since $\alpha \approx 0)$% :% %TCIMACRO{ %\TeXButton{newline}{\newline% %}}% %BeginExpansion \newline% % %EndExpansion \textbf{Lemma 1}\textit{: When 3 symmetric bidders compete for 1 object, the bidder with the highest signal wins and the price }$\approx t_{(3)}+2t_{(2)}. $ \textbf{Proof: }See appendix.\ $\Box .$% %TCIMACRO{ %\TeXButton{newline}{\newline% %}}% %BeginExpansion \newline% % %EndExpansion If instead, three bidders compete for two objects, the lowest quits in symmetric equilibrium at the price at which he would just be indifferent about winning were he the marginal winner, that is, were he tied with the second-highest signal. So the actual lowest-signal bidder with signal $% t_{(3)}$ quits at the value to him if the second-highest-signal bidder has the same signal, $\left( t_{(3)}\right) ,$ and the remaining signal equals its expected value given the two lowest signals are $t_{(3)},$ that is, $% E\left( \left. t\right| t\geq t_{(3)}\right) .$ So the lowest-signal bidder quits at $p=(1+\alpha )t_{(3)}+t_{(3)}+E(t\mid t\geq t_{(3)}).\footnote{% Again it is easy to check that if either of the other bidders were to quit and leave him as a winner at any higher price, $p^{\prime }=(2+\alpha )t^{\prime }+E(t\mid t\geq t^{\prime })$ with $t^{\prime }>t_{(3)},$ he would expect to lose money since he would then infer a unit's value to be $% (1+\alpha )t_{(3)}+t^{\prime }+E(t\mid t\geq t^{\prime })$ $
MR_{j}\Leftrightarrow t_{i}-h_{i}>t_{j}-h_{j}.$
In the common value case, since $v_{i}=v_{j},$ we have $MR_{i}>MR_{j}%
\Leftrightarrow -h_{i}>-h_{j}.$ So in the private-value case the result that
greater supply lowers (expected) price requires that $%
t_{i}-h_{i}>t_{j}-h_{j}\Leftrightarrow t_{i}>t_{j},$ which condition is
satisfied by many standard distributions $F(\cdot )$, and is often assumed
in the literature without comment. However in the common-value case the
result that more supply lowers price requires $-h_{i}>-h_{j}\Leftrightarrow
t_{i}>t_{j}$ which is a much more stringent condition on $F(\cdot )$.
In simple terms, the difference is that with private values when a bidder
has a higher signal it affects only his own value and marginal revenue. But
with common values when a bidder has a higher signal it also raises the
other bidders' values and so raises the others' marginal revenues. So it
takes a much stronger distributional condition to ensure that bidders with
higher signals have higher marginal revenues.
The condition in the private-value case is just that the bidder's marginal
revenue is downward sloping, that is, that a monopoly firm with demand $%
q=1-F(p)$\ has marginal revenue downward sloping in its\textbf{\ }\textit{own%
}\textbf{\ }output.\footnote{%
The demand curve $q=1-F(p)$\ is just the conventional demand curve that
would be created by a very large number of buyers with values $v_{i}(t_{i})$%
\ when the $t_{i}$\ are drawn independently from the distribution $F(t_{i}).$%
\ (We hold $t_{j}$\ and $t_{k}$\ fixed; buyers are atomistic with total mass
1.) For more discussion of the analogy between a bidder with signal
distributed as $F(t_{i})$\ and a market with demand curve $1-F(p)$\ see
Bulow and Klemperer (1996) and the first (1997) draft of our Working Paper.}%
\ The condition in the common-value case is that the same firm's marginal
revenue is steeper than its demand curve,\footnote{%
Since $-h_{i}>-h_{j}\Leftrightarrow t_{i}>t_{j}$\ implies $%
((v_{i}-h_{i})-v_{i})>((v_{j}-h_{j})-v_{j})\Leftrightarrow t_{i}>t_{j},$\
implies $(v_{i}-h_{i})-v_{i}$\ increasing in $t_{i},$\ hence decreasing in $%
q $, letting $q\equiv 1-F(t)$\ and $p\equiv v_{i}(t_{i}).$}\ or equivalently
that the firm's marginal revenue is downward sloping in a sufficiently small
\textit{opponent's}\textbf{\ }output;\footnote{%
Assuming the opponent is producing a homogeneous product, see Bulow,
Geanakoplos and Klemperer (1985a).}\ this is exactly the condition required
to guarantee strategic substitutes in quantity competition in
oligopoly---see Bulow, Geanakoplos, and Klemperer (1985a). And the
assumption of strategic substitutes, while commonly made, and \textit{perhaps%
} more plausible than the converse assumption of strategic complements, is
not a reasonable general assumption.\footnote{%
See Bulow, Geanakopolos and Klemperer (1985a) for more discussion, and also
Bulow, Genakopolos and Klemperer (1985b) for an example in which a
monopolist facing a new entrant views products as strategic complements.}
Indeed, among the most commonly used demand curves, linear demand $%
(p=A-Bq\Leftrightarrow q=\frac{A-p}{B})$ yields strategic substitutes,
constant elasticity demand $(p=Aq^{\frac{1}{\eta }}\Leftrightarrow q=\left(
\frac{P}{A}\right) ^{\eta },\eta <-1)$ yields strategic complements, and
logarithmic demand $(p=A-\frac{1}{\lambda }\log q\Leftrightarrow
q=e^{-\lambda (p-A)},$ i.e. quantity is exponential in price) yields
strategic independence (neither strategic substitutes nor strategic
complements) for a monopolist facing a small new entrant.
Corresponding exactly to the oligopoly cases we have:%
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\textbf{Example 1: }\textit{With uniformly distributed signals, }$%
F(t)=\left( \frac{t-\underline{t}}{\overline{t}-\underline{t}}\right) $%
\textit{, expected price is decreasing in supply. With constant-elasticity
distributed signals, }$F(t)=1-\mathbf{(}\frac{t}{\underline{t}}\mathbf{)}%
^{\eta },$\textit{\ expected price is increasing in supply. With
exponentially distributed signals, }$F(t)=1-e^{-\lambda (t-\underline{t})},$%
\textit{\ expected price is constant in supply.}\footnote{%
For example, if $F(t)=1-t^{-2}$ for $t\geq 1$ (which corresponds
probabilistically to a demand curve $q\equiv 1-F(p)=p^{-2}$, that is,
constant elasticity of -2) the expected values of the three signals would be
$1.2$, $1.6,$ and $3.2$. So the expected price in a 3 for 1 auction would be
$1.2+1.6+1.6=4.4,$ and the expected price with 3 for 2 would be $1.2+1.2+%
\frac{1.6+3.2}{2}=4.8$.
\par
If $F(t)=\frac{t}{4}$ for $4\geq $ $t\geq 0$ (which corresponds
probabilistically to a linear demand curve) the expected values of the three
signals would be 1,2, and 3. The expected price in a 3 for 1 auction would
be $1+2+2=5,$ and the expected price in a 3 for 2 auction would be $1+1+%
\frac{2+3}{2}=4.5.$}%
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(With a uniform distribution, twice the expected distance between the lowest
signal and the lower of two higher signals exceeds the expected distance
between the lowest signal and a single higher signal; with the
constant-elasticity distribution this fails; and the exponential
distribution is the intermediate case in which the ratio of the expected
distances is exactly one:two.\footnote{%
With a uniform distribution, $2t_{(3)}+E\left( t\mid t\geq t_{(3)}\right) $=
$3t_{(3)}+\frac{1}{2}(\overline{t}-t_{(3)}),$\ while $t_{(3)}+2E(t_{(2)}\mid
t_{(3)})=$\ $3t_{(3)}+\frac{2}{3}(\overline{t}-t_{(3)}).$\ With constant
elasticity distributed signals, $2t_{(3)}+E\left( t\mid t\geq t_{(3)}\right)
=$ $3t_{(3)}-\frac{1}{1+\eta }t_{(3)},$\ while $t_{(3)}+2E(t_{(2)}\mid
t_{(3)})=3t_{(3)}-\frac{2}{1+2\eta }t_{(3)}$. For exponentially distributed
signals, $2t_{(3)}+E\left( t\mid t\geq t_{(3)}\right)
=t_{(3)}+2E(t_{(2)}\mid t_{(3)})=3t_{(3)}+\frac{1}{\lambda }.$%
\par
The calculations are straightforward in the first and third cases. In the
constant elasticity case, $E\left( t_{(1)}\mid t_{(1)}\geq t_{(2)}\right) =%
\frac{\eta }{\eta +1}t_{(2)}$ and $(E\left( \frac{t_{(1)}+t_{(2)}}{2}\right)
\mid t_{(2)}\geq t_{(3)})=E\left( t\mid t\geq t_{(3)}\right) =\frac{\eta }{%
\eta +1}t_{(3)}.$ Combining the last two equations yields $\frac{2\eta +1}{%
1+\eta }E(t_{(2)})=\frac{2\eta }{1+\eta }t_{(3)}$ so $E(t_{(2)})=\frac{2\eta
}{1+2\eta }t_{(3)}.$ By substituting $E\left( t\mid t\geq t_{(3)}\right) =%
\frac{\eta }{\eta +1}t_{(3)}=t_{(3)}-\frac{1}{1+\eta }t_{(3)\text{ }}$we can
derive the constant elasticity revenue for when there are two winners, and
by substituting $E(t_{(2)}\mid t_{(3)})=\frac{2\eta }{1+2\eta }t_{(3)}$ we
can derive the expected revenue when there is one winner. (The formulae $%
E\left( t\mid t\geq t_{(3)}\right) =\frac{\eta }{\eta +1}t_{(3)}$ and $%
E\left( t_{(1)}\mid t_{(1)}\geq t_{(2)}\right) =\frac{\eta }{\eta +1}%
E(t_{(2)})$ are mathematically identical to the statement that, given
constant elasticity demand and a price $p,$ the average buyer has a value of
$\frac{p\eta }{\eta +1}.$ This must be true since\textbf{\ }$\frac{\eta }{%
\eta +1}$ is just the ratio of price to marginal revenue at each point along
a constant elasticity curve and therefore the ratio of average value to
average revenue (equals price). Here the calculations are the same, except
we use $t_{(2)}$ and $t_{(3)}$ instead of $p.)$})
So, just as in oligopoly it is an empirical matter whether firms' outputs
are strategic substitutes or strategic complements, so in symmetric pure
common-value auctions it must be an empirical matter whether price is
increasing or decreasing in supply.
The next section, however, will show that even (arbitrarily) small
asymmetries can make the relationship between supply and price even less
predictable.
\section{\textbf{\ The Asymmetric Case}}
This section will show that when the result that greater supply lowers
expected price holds for the perfectly symmetric case, it can fail when
there are even arbitrarily small asymmetries between the bidders. In
particular it fails if the item(s) for sale are almost pure common-values
but one bidder, say bidder 1, almost certainly has an arbitrarily small
private-value advantage. We assume $\alpha _{1}>\alpha _{2}=\alpha
_{3}=\alpha >0,$ but $\alpha _{1}\approx 0$ and $(\alpha /\alpha
_{1})\approx 0.$
We begin by analyzing bidding behaviour in more detail:%
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\textbf{Lemma 3}: \textit{When 3 bidders compete for 1 object in the
asymmetric case, the advantaged} \textit{bidder (almost always) wins and the
price} $\approx \underline{t}+t_{2}+t_{3}$ \textit{(in which} $t_{2}$
\textit{and }$t_{3}$\textit{\ are the actual signals of the disadvantaged\
bidders).}
\textbf{Proof: }See appendix.\quad $\Box .$%
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The logic is straightforward. Bidder $i$ quits where he would be just
indifferent about finding himself a winner, so his marginal type $\underline{%
t}_{i}$ quits at price $p=(1+\alpha _{i})\underline{t}_{i}+\underline{t}_{j}+%
\underline{t}_{k}$, where $\underline{t}_{j}$ and $\underline{t}_{k}$ are
his expectations of $j$'s and $k$'s signals conditional on his winning at
this price. That is, $\underline{t}_{j}$ is the marginal type of bidder $j$
who is just quitting if any type of $j$ is currently quitting,\footnote{%
If $j$ has already quit $\underline{t}_{j}$'s is $j$'s inferred signal, and
if $j$ has not quit but no type of $j$ is quitting then $\underline{t}_{j}$
is $j$'s lowest possible signal consistent with equilibrium.} and similarly
for $\underline{t}_{k}.$ Likewise, type $\underline{t}_{j}$ of $j$ is in
fact just quitting iff $p=\underline{t}_{i}+(1+\alpha _{j})\underline{t}_{j}+%
\underline{t}_{k}.$ So types $\underline{t}_{i}$ and $\underline{t}_{j}$
quit simultaneously iff $(1+\alpha _{i})\underline{t}_{i}+\underline{t}_{j}+%
\underline{t}_{k}=\underline{t}_{i}+(1+\alpha _{j})\underline{t}_{j}+%
\underline{t}_{k}\Leftrightarrow $ $\alpha _{i}\underline{t}_{i}=\alpha _{j}%
\underline{t}_{j},$ and $\underline{t}_{i}$ quits before (after) $\underline{%
t}_{j}$ iff $\alpha _{i}\underline{t}_{i}<(>)\alpha _{j}\underline{t}_{j}.$
So since $\alpha _{1}\underline{t}>\alpha t_{2}$ and $\alpha _{1}\underline{t%
}>\alpha t_{3}$ for almost all $t_{2}$ and $t_{3}$ for sufficiently large $%
\alpha _{1}/\alpha ,$ bidder 1 is almost always the winner. If, for example,
in fact $\alpha _{1}\underline{t}>\alpha t_{2}>\alpha t_{3},$ then bidder 3
quits first at $(1+\alpha )t_{3}+t_{3}+\underline{t}$ (since at this price
he knows $t_{2}\geq t_{3}$ so the current lowest types of bidders 2 and 1
that could remain are $\underline{t}_{2}=t_{3}$ and $\underline{t}_{1}=%
\underline{t}$), and bidder 2 quits next at $(1+\alpha )t_{2}+t_{3}+%
\underline{t}\approx t_{2}+t_{3}+\underline{t}.$
The intuition is that because bidder 1 (almost always) values the asset a
little more than bidders 2 and 3, there cannot be any equilibrium where
bidder 2 or 3 is willing to pay $p$ and bidder 1 is not willing to pay a
little more unless $t_{1}$ is almost zero. So bidders 2 and 3 face an
enormous winner's curse if bidder 1 ever exits, and they must therefore
assume $t_{1}\approx \underline{t}$ whenever he bids. So they quit at $%
\approx t_{2}+t_{3}+\underline{t}$, and bidder 1 almost always wins.
However, with three bidders competing for two units and increasing hazard
rates, bidder 1's advantage is almost eliminated and he wins only when he
has one of the two highest signals:%
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\textbf{Lemma 4}: \textit{When 3 bidders compete for 2 objects, in the
asymmetric case (i) if hazard-rates }$\left( \frac{1}{h_{i}}\right) $\textit{%
\ are increasing in signals, the\ bidders with the highest signals (almost
always) win and the price }$\approx 2t_{(3)}+E\left( \left. t\right| t\geq
t_{(3)}\right) ,$ (ii) \textit{if hazard-rates are decreasing, the
advantaged bidder and the disadvantaged bidder with the higher signal win
and the price }$\approx E(t)+2\min (t_{2},t_{3})$\textit{\ (in which }$t_{2}$%
\textit{\ and }$t_{3}$\textit{\ are the actual signals of the disadvantaged
bidders).}
\textbf{Proof: }See appendix.\ $\Box .$%
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To understand Lemma 4, again begin by observing that bidder $i$ quits where
he would be just indifferent about finding himself a winner. If $\underline{t%
}_{i},\underline{t}_{j},$ and $\underline{t}_{k}$ are the lowest possible
signals of bidders $i,$ $j$ and $k$ assuming equilibrium behaviour up to the
current price, type $\underline{t}_{i}$ of bidder $i$ has expected value $%
(1+\alpha _{i})\underline{t}_{i}+\underline{t}_{j}+E(t_{k}\mid t_{k}\geq
\underline{t}_{k})$ if $j$ quits now, and expected value $(1+\alpha _{i})%
\underline{t}_{i}+E(t_{j}\mid t_{j}\geq \underline{t}_{j})+\underline{t}_{k}$
if $k$ quits now. So type $\underline{t}_{i}$ quits at $p=(1+\alpha _{i})%
\underline{t}_{i}+\underline{t}_{j}+\underline{t}_{k}+x_{j}Prob(k$\textit{\ }%
quits now$\mid j$\textit{\ }or\textit{\ }$k$\textit{\ }quits now\textit{)}$%
+x_{k}Prob(j$\textit{\ }quits now$\mid j$\textit{\ }or\textit{\ }$k$\textit{%
\ }quits now\textit{)} in which $x_{j}\equiv E(t_{j}-\underline{t}_{j}\mid
t_{j}\geq \underline{t}_{j})$ and $x_{i}$ and $x_{k}$ are defined similarly.
Since $\alpha _{2}=\alpha _{3}=\alpha <\alpha _{1},$ some types of bidders 2
and 3 quit (symmetrically) before any types of bidder 1 quits. Now note that
for small enough $\alpha $ and $\alpha _{1}$ the differences between $\alpha
_{1}\underline{t}_{1}$ and $\alpha \underline{t}_{2}(=\alpha \underline{t}%
_{3})$ are very small relative to differences between $x_{1}$ and $%
x_{2}(=x_{3}).\footnote{%
The exception is when hazard rates are constant, $h_{1}=h_{2}=h_{3}=h,$ in
which case $x_{k}=h,\forall \underline{t}_{k},\forall _{k}.$}$ So if hazard
rates are increasing, so $x_{i}$ is decreasing in $\underline{t}_{i},$ then
if $\underline{t}_{1}$ were to fall much behind $\underline{t}_{2}(=%
\underline{t}_{3})$ then $x_{2}$ would become small relative to $x_{1}$ and $%
\underline{t}_{1}$ would wish to quit at a lower price than $\underline{t}%
_{2}$. So types of bidder 1 would have to quit until $\underline{t}_{1}$
roughly caught up to the value of $\underline{t}_{2}(=$ $\underline{t}_{3}).$
Therefore increasing hazard rates require $\underline{t}_{1}\approx
\underline{t}_{2}=\underline{t}_{3}.$ So bidder $i$ quits at (approximately)
$(1+\alpha _{i})t_{i}+t_{i}+E\left( \left. t_{k}\right| t_{k}\geq
t_{j}=t_{i}\right) ,$ just as in symmetric equilibrium with symmetric
bidders, and the bidder with the lowest signal, $t_{(3)},$ (approximately)
quits first, and we have part (i) of the Lemma.
The intuition is that even if bidder 1 had a large advantage, bidders 2 and
3 would compete against each other, symmetrically, for the second unit, and
in that competition they would not face an abnormally large winner's curse.
Because the prices of both units will be the same, the more aggressive
bidding by bidders 2 and 3 will force bidder 1 to pay more, and may cause
bidder 1 to exit if his signal is low enough (which further reduces the
other bidders' winner's curse). When bidder 1's advantage is small, it
becomes irrelevant with increasing hazard rates.\footnote{%
A numerical example may help some readers: assume, counterfactually, that
bidder 1 almost always wins when there are two units and $t$ is distributed
uniformly on $\left[ 0,10\right] $ . Then bidders 2 and 3 will not learn
anything about bidder 1's signal through the auction and will assume that $%
t_{1}=5$ (its average value)$.$ So bidder 2 will bid up to $\approx 2t_{2}+5$
and bidder 3 will bid to $\approx 2t_{3}+5.$ If, for example, $t_{2}=2 x_{1}$ so when $%
\alpha _{1}$ is small we require $\lambda $ small, that is, $\frac{H_{1}}{%
H_{2}}$ large$,$ hence $\frac{d\underline{t}_{1}}{d\underline{t}_{2}}$ is
large. So $\alpha _{1}\underline{t}_{1}-\alpha \underline{t}_{2}\rightarrow
0 $ and $\alpha _{1}\underline{t}_{1}=\alpha \underline{t}_{2}$ is achieved
for finite $\underline{t}_{2}.$ (Until this point $\underline{t}_{1}$ and $%
\underline{t}_{2}$ must just be following the differential equation
determined by $p=(1+\alpha _{1})\underline{t}_{1}+2\underline{t}_{2}+x_{2}=%
\underline{t}_{1}+(2+\alpha )\underline{t}_{2}+\left( \frac{H_{2}}{%
H_{1}+H_{2}}\right) x_{1}+\left( \frac{H_{1}}{H_{1}+H_{2}}\right) x_{2},$
that is, the same differential equation as in the increasing hazard-rate
case.) But at $\alpha _{1}\underline{t}_{1}=\alpha \underline{t}_{2},$ and
hence $\left( \frac{H_{2}}{H_{1}+H_{2}}\right) =0,$ we have the same
contradiction that we had with $\alpha _{1}\underline{t}=\alpha \underline{t}%
_{2}$ and $\lambda =0.$ (Any finite rate of quitting of player 2 would imply
all types close to $\underline{t}_{2}$ strictly wished to quit which is a
contradiction, but if no types of player 2 quit as the price, and hence $%
\underline{t}_{1}$, rises, then we will have $p=(1+\alpha _{1})\underline{t}%
_{1}+2\underline{t}_{2}+x_{2}>\underline{t}_{1}+(2+\alpha )\underline{t}%
_{2}+x_{2}$ which is also a contradiction.)
So the equilibrium we found, in which player 1 never quits while players 2
and 3 quit symmetrically according to $\underline{t}_{2}=\underline{t}_{3}$
and $p=\underline{t}+(2+\alpha )\underline{t}_{2}+\underline{x}$, is the
unique (Perfect Bayesian) Nash equilibrium satisfying our assumptions, and
the final price is $\underline{t}+\underline{x}+(2+\alpha )\min
(t_{2},t_{3})=E(t)+(2+\alpha )\min (t_{2},t_{3})\approx E(t)+2\min
(t_{2},t_{3})$ in which $t_{2}$ and $t_{3}$ are the actual signals of
bidders 2 and 3.%
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\textit{The Symmetric Case}
When $\alpha _{1}=$ $\alpha _{2}=$ $\alpha _{3}=\alpha $ it is
straightforward that it is a (Perfect Bayesian) equilibrium for bidders to
quit according to $\underline{t}_{1}=\underline{t}_{2}=\underline{t}_{3}$
and $p=(3+\alpha )\underline{t}_{1}+x_{1}$, and that this is the unique
equilibrium satisfying our assumptions. In this case the final price is $%
(3+\alpha )t_{(3)}+x_{(3)}\approx 2t_{(3)}+E(t\mid t\geq t_{(3)}).$%
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Thus we have proved Lemmas 2 and 4.%
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\textbf{C. Comparison of Sealed-Bid and Ascending Auctions}%
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In the asymmetric case, when 1 unit is sold, the ascending auction yields $%
\approx \underline{t}+2E(t)=\underline{t}+\frac{2}{3}E\left(
t_{(1)}+t_{(2)}+t_{(3)}\right) $ in expectation (Lemma 3). The sealed-bid
auction yields $\approx E\left( t_{(3)}+2t_{(2)}\right) $ in expectation ---
we assume the conjecture in Section 5.3 that the expected revenue from the
sealed-bid auction is almost unaffected by the small asymmetries between the
bidders,\footnote{%
See note 43 of the text.} so is almost Revenue Equivalent to the situation
in Lemma 1. Furthermore, $E\left( t_{(3)}+2t_{(2)}\right) >\frac{2}{3}%
E\left( t_{(1)}+t_{(2)}+t_{(3)}\right) +\underline{t}\Leftrightarrow E\left(
9t_{(3)}+6\left( t_{(2)}-t_{(3)}\right) \right) >$ $E\left( 6t_{(3)}+4\left(
t_{(2)}-t_{(3)}\right. \right) +2\left( \left. t_{(1)}-t_{(2)}\right)
\right) +3\underline{t}\Leftrightarrow \,$%
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$E\left( 3(t_{(3)}-\underline{t})+2\left( t_{(2)}-t_{(3)}\right) \right) >$ $%
E\left( 2\left( t_{(1)}-t_{(2)}\right) \right) $which is always true (false)
if hazard-rates are increasing (decreasing).\footnote{%
Thinking about marginal revenues is the quickest way to see the result for
the 1 unit case, since the sealed bid and ascending auctions yield the
expected marginal revenue of the highest-signal bidder and the average
bidder, respectively. For the 2 unit, decreasing hazard-rate case, however,
the calculations are trickier since this is the special case in which the
ascending auction gives positive expected surplus to the lowest type of
bidder 1 (see notes 19 and 31 of the text), so the expected revenue from the
ascending auction is the sum of the expected marginal revenues of the
winning bidders minus this expected surplus.}
When 2 units are sold the sealed-bid auction yields $\approx E\left(
2t_{(3)}+E\left( \left. t\right| t\geq t_{(3)}\right) \right) $ in
expectation, assuming approximate Revenue Equivalence to the situation in
Lemma 2. The ascending auction yields the same in expectation if
hazard-rates are increasing, but $\approx E(E(t)+2\min (t_{2},t_{3}))$ in
expectation if hazard-rates are decreasing (Lemma 4). But $E\left(
2t_{(3)}+E\left( \left. t\right| t\geq t_{(3)}\right) \right) =E(2t_{(3)}+%
\frac{1}{2}t_{(1)}+\frac{1}{2}t_{(2)})$ and $E(E(t)+2\min (t_{2},t_{3}))=E(%
\frac{1}{3}t_{(1)}+\frac{3}{3}t_{(2)},+\frac{5}{3}t_{(3)})$ (see the proof
of Proposition 2) and $E(2t_{(3)}+\frac{1}{2}t_{(1)}+\frac{1}{2}t_{(2)})>E(%
\frac{1}{3}t_{(1)}+\frac{3}{3}t_{(2)}+\frac{5}{3}t_{(3)})\Leftrightarrow
E(t_{(1)}-t_{(2)})>E(2(t_{(2)}-t_{(3)}))$ which is always true with
decreasing hazard rates.%
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\textbf{References}%
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Ashenfelter, Orley, ``How Auctions Work for Wine and Art'', \textit{Journal
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Avery, Christopher, ``Strategic Jump Bidding in English Auctions'', mimeo,
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Avery, Christopher, and Kagel, John H., ``Second-Price Auctions with
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Ayres, Ian and Cramton, Peter, ``Deficit Reduction Through Diversity: How
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Bikhchandani, Sushil and Riley, John G., ``Equilibrium in Open Auctions'',
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Bulow, Jeremy, Geanakoplos, John, and Klemperer, Paul, ``Multimarket
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Bulow, Jeremy, Geanakoplos, John, and Klemperer, Paul ``Holding Idle
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Bulow, Jeremy, Huang, Ming, and Klemperer, Paul, ``Toeholds and Takeovers'',
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DeGraba, Patrick, ``Buying Frenzies and Seller-Induced Excess Demand'',
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Denicolo, Vincenzo and Paulo Garella, ``Rationing in a Durable Goods
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de Frutos, Maria Angeles, and Rosenthal, Robert W., ``On Some Myths about
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