# Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

# Auctions

• Martin Pesendorfer
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_669-1

## Glossary

All-pay first-price auction

Bidders submit sealed bids. The high bidder wins the item. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. All bidders (including losing bidders) pay their bid.

Bayesian Nash equilibrium

A Bayesian Nash equilibrium is a collection of bidding strategies so that (i) no bidder has an incentive to deviate and (ii) beliefs are consistent with the underlying informational assumptions.

Bidding strategy

A bidding strategy for a buyer is a mapping from the buyer’s signal into bid prices.

Dutch auction

Price falls until one bidder presses her button. That bidder gets the object at the current price. Losers pay nothing.

English auction

Bidders call out successively higher prices until one bidder remains. The item is allocated to the last remaining bidder at the price at which the second last bidder dropped out.

First-price auction

Bidders submit sealed bids. The high bidder wins the item and pays her bid. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. Losers pay nothing.

Second-price auction

Bidders submit sealed bids. The high bidder wins the item and pays second highest bid. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. Losers pay nothing.

## Introduction

Auctions have been a common selling form throughout history; see Cassidy (1967) for an account. Roman legions sold their plunder at auction. Slave auctions were held throughout medieval times. Art auctions have been taking place for the last 300 years with Christie’s and Sotheby’s being two well-known auction houses. Real estate, treasury bills, flowers, livestock, even large corporations are sold at auction. Government procurement follows specific regulations and rules which give rise to an auction rule. The sale of mineral extraction rights and spectrum licenses are good sources for governmental revenues. eBay has become a successful marketplace with the arrival of the Internet.

This entry surveys contributions of the auction literature. It is a selected account from an economist’s perspective; see McAfee and McMillan (1987), Klemperer (1999), Krishna (2002), Hong and Paarsch (2006), and Hortacsu and McAdams (2016) for related surveys. Auctions are encountered in many settings, but specific rules and procedures may differ. Broadly speaking, we can distinguish single-item versus multi-item, sealed-bid versus open-outcry, and single-round versus multi-round auctions. The nature of the rules and format may depend on the items at hand but will also affect the behavior of bidders and what revenues the seller may get. Popular single-item auction rules include:
• English open-outcry auction in which bidders call out successively higher prices until one bidder remains. The item is allocated to the last remaining bidder at the price at which the second last bidder dropped out. Sometimes these are referred to as “hammer auctions” which are commonly used by Sotheby’s and Christy’s.

• Second-price sealed-bid auction in which bidders submit sealed bids. The high bidder wins the item and pays the second highest bid. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. Losers pay nothing.

• Dutch or descending-price auction is the opposite of English auction. Price falls until one bidder presses her button. That bidder gets the object at the current price. Losers pay nothing. This auction format is used to sell flowers in Holland.

• First-price sealed-bid auction has bidders submitting sealed bids. The high bidder wins the item and pays her bid. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. Losers pay nothing. The first-price auction format is commonly used for governmental procurement.

• All-pay first-price sealed-bid auction has bidders submitting sealed bids. The high bidder wins the item. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. All bidders (including losing bidders) pay their bid.

The seller of the item selects the auction format before bidding starts. The seller may additionally decide how much information to reveal about the item and may announce a reserve price, which is a minimum price at which the seller is willing to sell. If the seller knows what buyers are willing to pay, then the seller can post a selling price equal to the high willingness to pay. The resulting allocation extracts all the rent and gives the item to the buyer that values it most. When the seller does not know the willingness to pay, then this posting price scheme may perform poorly. An auction seems then a good choice as the seller may achieve higher revenues when using an auction than any posting price scheme, as was shown by Myerson (1981). An auction enables the seller to learn about buyers’ willingness to pay and to extract as much rent as possible.

The informational environment, that is, how much buyers know about their own willingness to pay, can vary. Environments range from buyers knowing precisely their value of the object or having a rough idea only. Let’s describe some informational structures. Suppose there are N buyers. Let θ i ∈ [0, A] denote buyer i’s signal drawn identically and independently from a cumulative distribution function F with probability density function f. Formally, the joint probability density function of signals equals the product of the marginals, $$f\left({\theta}_1,\dots, {\theta}_N\right)=\prod \limits_if\left({\theta}_i\right)$$ and this is a common knowledge. The independence assumption is for illustration purposes only. It can be relaxed leading to a correlated information environment; see Milgrom and Weber (1982) for affiliation which is a particular correlation structure among signals. Heterogeneity between buyers can be incorporated by allowing signal distributions to differ across bidders, F 1, …, F N . Following standard incomplete information terminology, we assess information at the interim stage in which buyer i has learned her own signal θ i but has not learned the value of competitors’ signals θ j , ji (In information economics, the term interim is used to distinguished from ex ante, in which types are not known yet, and ex post, in which everyone knows everyone else’s type.).

Private values arise when the signal equals the value of the object to buyer i, v i = θ i . Private values refer to a situation in which buyers know precisely their own value. An example may be a construction contract in which firms know their own opportunity costs of undertaking the project. However, a firm may be only vaguely informed about competing firms’ costs.

(Pure) common values arise when the value of the object is determined by the average signal, $$v=\frac{1}{N}\sum \limits_j{\theta}_j$$. This environment differs from private values in two respects: first, as buyer i only observes one signal θ i from a total of N signals, she knows only a little bit about the true value and, second, all bidders assign the same value v. An example may be an oil field auction. Each bidder conducts their own study of how much oil there is and comes up with an estimate θ i . The true value of the oil field will be some average across the bidders’ noisy signals and is the same to all bidders. A second example is the wallet game, in which two bidders compete for the joint value of their wallets. In that case the total value equals the sum of signals, v = θ 1 + θ 2, or twice the average signal.

Interdependent values are a mixture between private and common values. With interdependent values, the value of the object to buyer i is given by $${v}_i={\alpha \theta}_i+\left(1-\alpha \right)\frac{1}{N-1}{\sum}_{j\ne i}{\theta}_j$$ with 0 < α < 1. When the parameter α = 1, this formula reduces to v i = θ i , the case of private values. On the other hand, when the parameter $$\alpha =\frac{1}{N}$$, then this formula becomes the pure common value case.

Most of our following analysis will focus on the case of (independent) private values.

### Assumption

Buyers know their values privately, v i = θ i for all i.

The reader interested in the more advanced topic of interdependent valuations is referred to Krishna (2002) for a nice introductory exposition. We shall illustrate on occasions what may happen under alternative informational assumptions.

The seller may have information that is useful to bidders. For example, on eBay, the seller may decide how accurately to describe the object. Or a used car owner may know very well the pros and cons of the car. What should the seller do? Reveal the information prior to bidding, or conceal it. The seller may also impose a reserve price. Should the seller impose a reserve price? At what level? We shall return to these questions in the section entitled “Comparing Auction Outcomes” after having studied buyer’s behavior in standard auction formats.

The sections entitled “Second-Price Auction” and “English Auction” examine optimal bidding strategies and Bayesian Nash equilibria for standard auction formats. The payoff bidder i receives will depend on the auction rule, the equilibrium played, and attitudes toward risk. Attitudes toward risk matter as bidders face lotteries, winning the auction or not. We shall assume risk neutrality in which bidder i’s payoff equals the expected value of the lottery. We shall comment later on the extension to bidder risk aversion is introduced.

The strategy space and equilibrium concept is shared across the following sections.

A bidding strategy for buyer i is a mapping from signals into bid prices, b i : [0, A] → ℝ.

A Bayesian Nash equilibrium is a collection of bidding strategies, $${\left({b}_i\right)}_{i=1}^N$$, so that (i) no bidder has an incentive to deviate and (ii) beliefs are consistent with the underlying informational assumptions.

With a Bayesian Nash equilibrium, we mean a stable resting point in which every bidder adopts a strategy that maximizes her payoff.

The entry is organized as follows: We shall begin by studying bidder behavior at specific auction rules, including second-price and first-price auction. We then compare bidders’ payoffs and auctioneer’s revenues across distinct auction formats. We describe key issues for empirical work on auctions, the winner’s curse phenomenon, and issues concerning collusive behavior at auction.

## Second-Price Auction

The second-price auction format has been advocated in Vickrey (1961). It has the rule that the high bidder wins the item and pays the second highest bid. The payoff for a winning buyer i is v i b (2) where b (2) is the second highest bid. The payoff to a losing bidder is zero.

What is an optimal bidding strategy in a second-price auction? Let’s consider an example. Suppose your value is 60; what should you bid? You could bid your value. Is it optimal to bid your value? The answer is yes. Suppose another bidder bids 70. Do you regret? No as you would lose money if you outbid the other bidder. Suppose other bidders’ bid is 40. Do you regret? No. So, bidding your value is indeed optimal. It is a Nash equilibrium.

This example generalizes and leads us to the following result.

### Theorem 1

Bidding the true value, b i (v i ) = v i , is a Bayesian Nash equilibrium in weakly dominant strategies.

### Proof

Let the seller’s reserve price be denoted by R. Suppose buyer i’s valuation is below the reserve price, v i < R: It is (weakly) optimal for buyer i to bid v i . The only way that buyer i can win the item is if buyer i bids more than i, but in case of winning, she pays at least R and makes a loss, v i – R < 0. Next, suppose buyer i’s valuation is above the reserve price, v i R: Following the strategy implies that if buyer i wins, she pays the second highest bid b(2) < v i and makes a profit of v i − b(2). Consider a deviation: (a) Suppose buyer i bids more than her valuation, b i > v i . For b (2) v i , she pays b (2) and she gets the same payoff, but for b i > b (2) > v i , she makes a loss. (b) Suppose buyer i bids less than her valuation, b i v i . For b (2) < b i , she wins, pays b (2) , and gets the same payoff, but for b i b (2) < v i , she does not win the auction, gets a payoff that is zero, and makes a loss.

The theorem characterizes an equilibrium which has the feature that all bidders bid their true value, bidders bid “sincerely.” An interesting feature of the second-price auction is that sincere bidding is optimal irrespective what bidding strategies other bidders adopt.

Vickrey (1961) is the classic auction paper that has emphasized these features of second-price auctions and compares the second-price auction to a first-price sealed-bid auction. Vickrey has shown that the above equilibrium is in fact efficient. With efficiency we mean that the bidder that values the item the most gets it.

We can also assess the revenues to the seller. The expected revenues (for the seller) of the Vickrey auction equal the expected second highest valuation.

Are there other equilibria in second-price auctions? The answer is yes. Suppose there is no reserve price. Consider the following “pooling equilibrium” in which bidder 1 bids b 1 = A and all other bidder bid b i = 0. This is in fact a Bayesian Nash equilibrium. Nobody can benefit from deviating. Notice though that this is not a dominant strategy equilibrium. Moreover, the outcome is not efficient. We shall focus on the dominant strategy equilibrium when comparing auction outcomes across auction formats and return to the pooling equilibrium in the section describing empirical work.

Next, we shall illustrate that the equilibrium in the English auction shares features with the above equilibria.

### English Auction

We consider a continuous-price version of the English auction in which the price increases continuously, without any bidding jumps, and does so until only one bidder remains. As the price increases, bidders drop out irrevocably. When only one bidder remains, the item is allocated to the last remaining bidder at the price at which the second last bidder dropped out.

Consider a bidding strategy in the English auction in which bidder i stays in until the price reaches her value v i . If everyone adopts this strategy, does this constitute an equilibrium? Yes, for the same reason as given in the above proof.

In fact there is a strategic equivalence between an English auction and a second-price auction with independent private information. Think of an agent that bids on behalf of bidder i. The agent would receive a number to submit in a second-price auction and a dropout value in the English auction. With strategic equivalence, it is meant that the number should be identical in both auction formats.

Notice though that the strategic equivalence breaks down with interdependent values. In that case, we need to take into account that bidders form their valuation estimate based on all available information. In a second-price sealed-bid auction, the only available information is a bidder’s private signal. In an English auction, bidders learn something as the price increases. For instance, when an opponent drops out of the auction, something can be inferred about that bidder’s private signal which may influence the valuation estimate. Thus, as price increases, bidders will update their bidding strategy to take the additional information into account.

Next, we shall consider bidding behavior at a first-price auction.

## First-Price Sealed-Bid Auction

In a first-price auction, bidders submit sealed bids. The high bidder wins the item and pays her bid. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. Losers pay nothing.

We maintain our assumption of bidder risk neutrality. If bidder i wins the item, her payoff is v i b i , while she makes a payoff of zero if she loses. Ignoring the issue of ties for the moment, let Pr(p i > b j , for all j) denote the winning probability. It is the probability that bidder i submits the high bid.

Bidder i’s (interim) expected payoff U i will depend on the valuation v i and the bid b i submitted. For a first-price auction, the (interim) expected payoff equals
$${U}_i\left({v}_i,{b}_i\right)=\left[{v}_i-{b}_i\right]\cdot \Pr \left({b}_i>{b}_j,\mathrm{for}0.5em \mathrm{all}0.5em j\right)$$
(1)

Recall that a strategy is a mapping: b i : [0, A] → ℝ. We wish to find an equilibrium. One way to proceed is to invoke calculus and work through the first-order conditions. Another way to proceed is to impose assumptions on the equilibrium and use those in the derivation. At the end, we then have to verify that indeed it is an equilibrium. This is the approach we shall adopt.

We restrict attention to bidding strategies which are differentiable, strict monotone increasing, and symmetric. Thus, there exists a differentiable strict monotone increasing function b : [0, A] → ℝ so that
$${b}_i\left({v}_i\right)=b\left({v}_i\right)0.5em \mathrm{for}0.5em \mathrm{all}0.5em i0.5em \mathrm{and}0.5em {v}_i\in \left[0,A\right]$$
(2)
The restriction allows us to simplify the problem. The winning probability when other bidders use the strategy b(v j ) is given by Pr(b i > b j , for all ji) = F(b −1(b i )) N − 1. To see this, consider the following equivalent expressions:
$${\displaystyle \begin{array}{l}\Pr \left({b}_i>{b}_j,\mathrm{for}0.5em \mathrm{all}0.5em j\ne i\right)\\ {}=\Pr \left({b}_i>b\left({v}_1\right),\dots, {b}_i>b\left({v}_N\right)\right)\\ {}=\Pr \left({b}^{-1}\left({b}_i\right)>{v}_1\right)\cdots \Pr \left({b}^{-1}\left({b}_i\right)>{v}_N\right)\\ {}=F{\left({b}^{-1}\left({b}_i\right)\right)}^{N-1}\end{array}}$$

The first line writes out what bidder i’s winning probability is when other bidders follow the bidding strategy b(.). The second equation uses the independence assumption and the strict monotonicity property, which implies that the inverse function b −1 exists and is strict monotone. The final equation uses the definition of the cumulative distribution function F, which evaluated at the number b −1(b i ) gives the probability that a buyer’s valuation is less than equal to that number. In total this probability arises (N − 1) times as there are (N − 1) opponents.

The (interim) expected payoff becomes thus
$${U}_i\left({v}_i,{b}_i\right)=\left[{v}_i-{b}_i\right]\cdot F{\left({b}^{-1}\left({b}_i\right)\right)}^{N-1}$$
(3)

In a Bayesian Nash equilibrium, it must be that the strategy b(v i ) is optimal. Observe that b(0) = 0 since the valuation is zero. Observe also that any bid greater than b(A) will win for sure. Thus, possible deviation bids must be contained in the range [0, b(A)]. Since the bid strategy b(.) is strict monotone, we can express this range with b(w) and w ∈ [0, A].

If bidder i with valuation x = v i bids b(w) rather than b(x), her payoff is
$${U}_i\left(x,b(w)\right)=\left[x-b(w)\right]F{(w)}^{N-1}$$
Taking the derivative with respect to w yields
$$\frac{\partial {U}_i\left(x,b(w)\right)}{\partial w}=\left[x-b(w)\right]\frac{\partial \left(F{(w)}^{N-1}\right)}{\partial w}-{b}^{\prime }(w)F{(w)}^{N-1}$$
For bidding b(x) to be optimal, this derivative must be zero when evaluated at w = x
$$x\left[\frac{\partial \left(F{(x)}^{N-1}\right)}{\partial x}\right]=b(x)\left[\frac{\partial \left(F{(x)}^{N-1}\right)}{\partial x}\right]+{b}^{\prime }(x)F{(x)}^{N-1}$$
Integrating both sides with respect to x
$${\int}_0^{v_i}x\left[\frac{\partial \left(F{(x)}^{N-1}\right)}{\partial x}\right] dx=b\left({v}_i\right)F{\left({v}_i\right)}^{N-1}-b(0)F{(0)}^{N-1}$$
Since b(0) = 0 and F(0) = 0, we have an explicit solution for the bid function
$$b\left({v}_i\right)=\frac{\int_0^{v_i}x\left[\frac{\partial \left(F{(x)}^{N-1}\right)}{\partial x}\right] dx}{F{\left({v}_i\right)}^{N-1}}$$

Observe that this strategy is indeed differentiable and strict monotone. It is thus an equilibrium. The right-hand side is the conditional expected value of the random variable v (2) with probability density function $$\frac{\partial \left(F{(x)}^{N-1}\right)}{\partial x}$$ conditional on v (2) being less than the valuation v i . This leads us to the following result for first-price auctions.

### Theorem 2 (First-Price Auction Equilibrium)

The Bayesian Nash equilibrium bid function in the first-price auction is
$$b\left({v}_i\right)=E\left[{v}_{(2)}|{v}_{(2)}<{v}_i\right]$$

The equilibrium bid function has the feature that bidder i marks her valuation v i down. Bidder i bids the expected value of the high-valuation competitor conditional on her competitors’ valuations being less than her own.

### Example

Let’s consider a simple two bidder case with F the uniform distribution on [0,1]. The equilibrium bid function becomes
$${\displaystyle \begin{array}{c}b\left({v}_i\right)=\frac{\int_0^{v_i} xdx}{v_i}\\ {}=\frac{v_i}{2}\end{array}}$$

What is $$\frac{v_i}{2}$$? It equals the expected valuation of your competitor conditional on your competitor’s valuation being less than your own, E [v (2)|v (2) < v i ].

Observe that the first-price auction is efficient. The bidder with the high valuation wins the item. The reason is that the equilibrium bid function is identical and strict monotone increasing.

The equilibrium derivation was based on symmetric, strict monotone bid functions. The questions arise whether there are other equilibria. The answer is no. Maskin and Riley (2003) have shown that the above bidding strategy is the only equilibrium in first-price auctions.

Richer informational environments have been studied by a number of authors. Milgrom and Weber (1982) is the classic reference for equilibria in standard auction formats with symmetric bidders with interdependent values and affiliated signals encompassing both the common value and private value models. Bergemann et al. (2017) examine bidding implications in first-price auctions when the information structures specifying bidders’ information about their own and others’ values are not restricted.

We next illustrate the Bayesian Nash equilibrium in two variants of the first-price auction: (i) in which all bidders pay their bid and (ii) in the Dutch auction.

### All-Pay First-Price Auction

In an all-pay first-price sealed-bid auction bidders submit sealed bids. The high bidder wins the item. In case of a tie, the auctioneer randomly assigns the item to one of the high bidders with equal probability. All bidders (including losing bidders) pay their bid.

Bidder i’s (interim) expected payoff U i will depend on her valuation v i and her submitted bid v i . For an all-pay first-price auction, the (interim) expected payoff equals
$${U}_i\left({v}_i,{b}_i\right)={v}_i\cdot \Pr \left({b}_i>{b}_j,\mathrm{for}0.24em \mathrm{all}0.24em j\right)-{b}_i$$
(4)
Recall that a strategy is a mapping b i : [0, A] → ℝ. We wish to find an equilibrium. We follow a similar approach as in the first-price auction which leads to a differential equation. Suppose bidders use a strictly increasing bid function b(.). As before, this implies that Pr (b i > b j , for all j) = F(b 1(b i )) N − 1. If bidder i with valuation x = v i uses b(w) rather than b(x), her payoff is
$${U}_i\left(x,b(w)\right)= xF{(w)}^{N-1}-b(w)$$
Taking the derivative with respect to w yields
$$\frac{\partial {U}_i\left(x,b(w)\right)}{\partial w}=x\frac{\partial \left(F{(w)}^{N-1}\right)}{\partial w}-{b}^{\prime }(w)$$
For bidding b(x) to be optimal, this derivative must be zero when evaluated at w = x
$$x{\left[\frac{\partial \left(F{(w)}^{N-1}\right)}{\partial w}\right]}_{w=x}={b}^{\prime }(x)$$
Integrating both sides with respect to x
$${\int}_0^{v_i}x{\left[\frac{\partial \left(F{(w)}^{N-1}\right)}{\partial w}\right]}_{w=x} dx={\int}_0^{v_i}{b}^{\prime }(x) dx$$
The left-hand side can be integrated by parts, which yields
$${\left[ xF{(x)}^{N-1}\right]}_0^{v_i}-{\int}_0^{v_i}F{(x)}^{N-1} dx=b\left({v}_i\right)-b(0)$$
Since b(0) = 0, we have an explicit solution for the bid function
$$b\left({v}_i\right)={v}_iF{\left({v}_i\right)}^{N-1}-{\int}_0^{v_i}F{(x)}^{N-1} dx$$

This leads us to the following result for all-pay first-price auctions.

### Theorem 3 (All-Pay First-Price Auction Equilibrium)

The Bayesian Nash equilibrium bid function in the all-pay first-price auction is $$b\left({v}_i\right)={v}_iF{\left({v}_i\right)}^{N-1}-{\int}_0^{v_i}F{(x)}^{N-1} dx$$.

Observe that the all-pay first-price auction is efficient. The bidder with the high valuation wins the item. The reason is that the equilibrium bid function is identical and strict monotone increasing.

### Dutch Auction

The Dutch auction has price falling until one bidder jumps in. That bidder gets the object at the current price. Losers pay nothing.

A Dutch auction is strategically equivalent to the first-price auction. Think of an agent bidding on behalf of a bidder. In both formats, the bidder would instruct the agent with a number to bid. The number is the same in both formats.

We have now considered a number of auction rules and formats. Next, we shall compare the outcomes under those auction formats.

## Comparing Auction Outcomes

From an economic perspective, there are three key dimensions in which auction formats can be compared, based on (i) revenues to the seller, (ii) rents to buyers, and (iii) efficiency. We begin by comparing the first-price and second-price auction outcome under the independent private value framework with risk-neutral buyers. Then we explore how the auction comparison looks like when we depart from this set of assumptions.

A central result in the auction literature is the equivalence theorem expressed in terms of expected revenues, expected utilities, and efficiency; see Vickrey (1961), Riley and Samuelson (1981), and Myerson (1981) in increasing generality. We shall consider the comparison of first-price and second-price auctions only. The result has been extended to wider class of auctions. The following theorem is based on the dominant strategy equilibrium in the second-price auction characterized in Theorem 1. We shall return to the pooling equilibrium in second-price auctions later on in the section on empirics of auctions.

### Theorem 4 (Equivalence)

Consider the symmetric independent private value framework with risk-neutral buyers. The following properties hold:

(a) The expected revenue to the seller is the same in the first-price and second-price auction.

(b) The item goes to the buyer who values it the most in the first-price and second-price auction.

(c) The expected utility to the buyer is the same in the first-price and second-price auction.

### Proof

(a) The proof is an immediate consequence of our earlier theorems. Theorem 2 shows that in a first-price auction, buyers bid the expected second highest valuation conditional on being the high valuation. Furthermore, the high-valuation bidder wins. The expected price paid equals the expected second highest valuation. Consider next the sincere bidding equilibrium in the second-price auction described in Theorem 1. The high-valuation bidder wins at a price equal to the second highest valuation. The price paid equals the realization of the second highest valuation. In expectations, the realization will equal the expected value.

(b) In the standard auction formats, the high-valuation bidder wins the item. The reason is that the bid functions are identical across bidders and strict monotone.

(c) From part (b), the allocation is the same in both a second-price auction and a first-price auction. In both cases, the high-valuation buyer wins. Part (a) shows that the expected revenues are the same. Therefore, expected utility must be the same.

Notice that parts (a) and (c) are in terms of expectations. Revenue, or utility, realizations need not be the same as the price paid may differ across auction formats.

How robust is the above theorem to departures from the assumptions? We shall see that the result is very fragile. If any of the assumptions is modified, the equivalence result breaks down. We shall discuss some contributions in this literature.

Suppose buyers are risk averse, instead of risk neutral, while maintaining all other assumptions. In a second-price auction, the (weakly dominant strategy) equilibrium is not affected. It remains an equilibrium that bidders bid their value. The equilibrium construction and proof of Theorem 1 remain valid in this case. Next consider a first-price auction. Maskin and Riley (1984) show that the equilibrium changes. Bidders bid more aggressively. The intuition is that bidders face the following trade-off. Bidding higher increases the chances of winning but comes at a utility loss of the higher bid. The first effect is not affected by attitudes toward risk, but the second is valued less when bidders are risk averse instead of risk neutral. In terms of the revenue ranking, this means a seller is better off with a first-price auction, while buyers prefer the second-price auction; see Matthews (1987).

Suppose bidders are asymmetry, that is, bidders draw their valuation from distinct probability distribution functions, while maintaining all other assumptions. Maskin and Riley (2000) have shown that revenue (and utility) ranking can go either way.

There are parametric examples of valuation distributions in which the first-price auction does better for the seller and examples in which it does worse. In terms of efficiency, the second-price auction equilibrium remains efficient, while the first-price auction equilibrium is no longer efficient.

Third, consider interdependent valuations. The classic paper analyzing bidding equilibria in this case, also permitting that signals are correlated, is Milgrom and Weber (1982). They show that with affiliated interdependent valuations, the English auction performs best in terms of revenues followed by the second-price and then the first-price auction.

Myerson (1981) characterizes the revenue-maximizing auction. This pioneering paper develops a new approach, in which the auction rule is the choice variable. Myerson’s studies auction rules from a mechanism design perspective in which buyers announce their valuations and the mechanism determines the allocation and transfer payments. Myerson shows that with independent private values, both first- and second-price auctions are optimal, but not if bidders private values are asymmetric or correlated.

So far, we have considered the choice of auction format. Within an auction format, the seller can fine-tune the auction outcome. The seller may decide a minimum bid level below which bids are rejected, whether to charge bidder participation fees, or how much information about the object to make available to bidders. The minimum bid level or reserve price is easily understood with a single bidder. In the absence of a reserve price, the bidder would acquire the item at a price of zero. With a reserve price, the seller can force the bidder to pay a positive price at the cost of sometimes not selling the item. The optimal reserve is achieved at the point where the marginal benefit of increasing the reserve price becomes zero. Myerson (1981) gives an intuitive interpretation for the optimal reserve price formula. It is the valuation where information revelation costs (or incentive costs) equal the benefits from information.

The precision of information available to bidders can be influenced, by either the auctioneer or the buyers. For example, in a used car auction, the seller can decide whether bidders may inspect or even test-drive the car prior to the auction. Similarly, on eBay, the seller can decide on the informativeness of the item description. In oil auctions bidders may decide how much money to spend on geological studies. Persico (2000) considers an affiliated-values environment and shows that bidder incentives to acquire information differ across auctions with the marginal benefit of additional information being higher in a first-price than in a second-price auction which may overturn Milgrom and Weber (1982) revenue ranking result. Bergemann and Pesendorfer (2007) study the joint decision problem when the seller controls the information and the auction rule in a private value setting. They show that increased information has the benefit of enhancing efficiency but comes at an information revelation cost. The optimal seller’s policy has to balance these two elements. With few bidders, providing little information is optimal, while when the number of bidders increases, the efficiency motives dominate the information revelation cost element.

This section has examined some optimal auction design questions. Next, we shall consider empirics of auctions.

## Empirics of Auctions

Bid data are available for many auctions and allow researchers to study bidders’ behavior. The empirical literature has focused on two central questions: first, how to measure or quantify the underlying informational distribution from bid data and, second, how to design and assess the optimal auction for a market at hand. The first question in terms of econometrics is about the identification and inference of parameters determining the distribution of information. The second question is motivated by the fragility of the revenue equivalence theorem . Which elements, bidder asymmetry, risk aversion, and common versus private values, are key drivers? An answer to the second will tell us what auction rule is best used in practice, a market design question.

This section illustrates some empirical issues based on hypothetical and stylized data set. We shall ignore bidder heterogeneity, auction heterogeneity, and covariates. We shall comment on these extensions later on. We shall start with first-price auctions and then consider second-price (or English) auction data.

Before proceeding, let us raise one central issue for empirical work which concerns the well-known problem of selection bias . The theoretical analysis and equilibrium characterization for specific auction rules assumes a known number of “potential” bidders. In fact, not all of these potential bidders may submit a bid. For example, a reserve price or bidder participation cost may reduce bidder participation. Empirically, this poses a problem as we only observe the “actual” bidders and need to infer the set of “potential” bidders. Put differently, the number of observed bids may not be an accurate picture of the degree of competition. One way to proceed is to estimate the potential number of bidders by using the maximum number of observed bids across auctions or a subset of auctions. Such an extremum estimator has nice asymptotic properties. Another way is to model this selection explicitly; see Li and Zheng (2009) and Athey et al. (2011). We shall ignore this issue here in this exposition and assume that the actual number of bidders equals the potential number of bidders.

### Empirics of First-Price Auctions

Consider the following assumptions about the data-generating process consisting of a cross section of first-price sealed-bid auctions.

### Assumption

The data-generating process for the bid data $${\left({b}_i^t\right)}_{i=1}^N$$ is equilibrium bidding for independent private-value first-price auctions in which each auction t, t = 1, … T had (i) an identical object and (ii) a fixed (and known) number of identical bidders N.

Equilibrium bidding means that bidders follow the unique bidding strategy characterized in Theorem 2. Independent private values mean that a bidder i’s valuation equals her signal, v i = θ i , and is drawn identically and independently from a distribution F. The fixed number of bidders means that we do not have to worry about the distinction between “actual” and “potential” bidders.

The empirical question is how to estimate the cdf F from bid data. Distinct estimation methods exist. Our description shall focus on a popular and commonly used inference approach . This approach looks at the optimal bid choice vis-à-vis the empirical distribution of opponent bids.

Following Guerre et al. (2000), the problem can be formulated based on H(b) the probability distribution function of bids b. Bidder i’s problem of finding a bid that maximizes the expected payoff in a first-price auction under risk-neutrality, private values, and when bids are independently distributed, can be written as:
$$\underset{b_i}{\max}\left[{v}_i-{b}_i\right]H{\left({b}_i\right)}^{N-1}.$$
The first-order condition is
$$-H{\left({b}_i\right)}^{N-1}+\left[{v}_i-{b}_i\right]\left(N-1\right)H{\left({b}_i\right)}^{N-2}{H}^{\prime}\left({b}_i\right)=0,$$
which can be rewritten to obtain an explicit expression for the valuation v i
$${v}_i={b}_i+\frac{H\left({b}_i\right)}{\left(N-1\right){H}^{\prime}\left({b}_i\right)}.$$
(5)

Equation 5 is the inverse of the theoretical bid function characterized in the section entitled “First-Price Sealed-Bid Auction”. It tells us which valuation rationalizes the observed bid. The right-hand side elements are the bid b i , and the number of bidders N, the cdf H, and the pdf H′. The pdf and cdf can be estimated from the bid data $${\left({b}_i^t\right)}_{i,t}$$ by using a suitable estimator. For example, the cdf H can be consistently estimated with the empirical cdf $$\widehat{H}(x)=\frac{1}{TN}{\sum}_{i,t}1\left({b}_i^t\le x\right)$$ where 1(.) is the indicator function. The indicator function equals one, if the argument is nonnegative, and zero, otherwise.

Guerre et al. (2000) advocate a nonparametric estimator in which first the cdf and pdf of bids are estimated by using kernel estimators and then in the second step, (pseudo) valuations and their distribution are inferred. In practice this approach has gained a lot of popularity. One reason is its simplicity. A second reason is that it extends readily in various directions including bidder heterogeneity, different informational distributions, different auction rules, multi-unit auctions, and even sequential auctions. In practice, researchers tend to use parametric approaches to estimate the distributions to allow for covariates to enter; see Hong and Paarsch (2006).

### Empirics of Second-Price Auctions

Suppose the data-generating process is a second-price auction instead of a first-price auction.

### Assumption

The data-generating process for the bid data $${\left({b}_i^t\right)}_{i=1}^N$$ is equilibrium bidding for independent private-value second-price auctions in which each auction t, t = 1, … T had (i) an identical object and (ii) a fixed number of identical bidders N.

We have seen earlier that there can be multiple equilibria in second-price auctions. One equilibrium, in which one bidder bids high and all other bidders bid low, which is called a “pooling” equilibrium, has the feature that multiple valuations rationalize a bid. If this equilibrium arises in the data, then it may not be possible to infer the details of the distribution of valuations. However, bounds on the support of valuations could be inferred.

On the other hand, if the dominant strategy equilibrium is played, in which a bid equals the value, then inference of the valuation distribution is straightforward. In this case, the distribution of valuations can be estimated by using the empirical cdf of bids
$$\widehat{F}(x)=\frac{1}{TN}\sum \limits_{i,t}1\left({b}_i^t\le x\right).$$

The dominant strategy equilibrium allows the econometrician to readily infer the underlying valuations and distribution of valuations from the observed bids.

In practice the researcher may not know the type of equilibrium played. Furthermore, different equilibria may be played in the cross section of auctions. How to deal with inference in this case has been an ongoing research area not only in the empirical auction literature but in economics in general; see Tamer (2003).

## Winner’s Curse

In addition to using field data, economists also use laboratory experiments to study market outcomes. Bazerman and Samuelson (1983) conducted a first-price sealed-bid auction experiment with MBA students at Boston University. The number of student bidders varied between 4 and 26 across classes, and in each class, a jar of 800 pennies was offered for sale. The value of the jar was unknown to students. Students were asked to provide their bid and best estimate for the jar value. The average value estimate equaled $5.13 which is$2.87 below the true value. The average winning bid was $10.01 which amounts to an average loss of$2.01 with losses occurring in over half of all the auctions. The evidence suggests a “winner’s curse.”

The curse can be explained by using a (pure) common value environment in which bidder i’s signal θ i is bidder i’s unbiased estimate of the value v, E[θ i | v] = v (A framework that satisfies this assumption is when signals are noisy estimates of the true jar value, θ i = v + ε i with i = 0 iid.). If bidders use a monotone increasing bidding strategy, then the auction winner will be the bidder with the high signal, max i {θ i }. The curse arises as the high-signal bidder in fact overestimated the true value, E[max i {θ i }| v] > max i E[θ i | v] = v. The result follows since max is a convex function and from Jensen’s inequality, which says that for a convex function f, it must be that Ef (θ) > f(). Thus, winning confers bad news in the sense that the bidder learns ex post that she was the high-signal bidder.

How can we interpret the winner’s curse in terms of equilibrium bidding? The section entitled “First-Price Sealed-Bid Auction” has shown that bidders will shade their bid down in a private value setting. Now, with common values, bidders will shade their bid down even further in anticipation of the bad news effect. In particular, bidders will calculate the value of winning based on having the high signal for the object, E[v| θ i , θ i θ j  for all j].

Evidence from field data about a winner’s curse is mixed. Porter (1995) summarizes his joint work with Hendricks on field data for common value auctions. They analyze the US offshore mineral rights program in which the right to extract oil of the US coast has been awarded in form of a first-price sealed-bid auction since the 1950s.

Table 1 reports selected summary statistics for newly explored areas, Wildcat tracts, see Porter (1995, Table II). All dollar figures are in million of 1972 dollars. Standard deviations are in parenthesis. To the extent that the value of the oil field is the same for all bidders, say because of competitive raw oil prices, the informational environment can be viewed as common values.
Table 1

Summary statistics for Wildcat tracts

No of bidders

1

2

3

4

5–6

7–9

10–18

Total

b (1)

1.50

2.76

4.17

5.62

7.9

14.2

21.8

6.07

(0.1)

(0.2)

(0.4)

(0.5)

(0.6)

(1.2)

(1.4)

(0.2)

(b (1)b (2))/b (2)

0.55

0.49

0.46

0.39

0.34

0.30

0.44

Fraction drilled

0.61

0.74

0.86

0.85

0.91

0.90

0.99

0.78

Fraction productive

0.41

0.47

0.47

0.51

0.49

0.63

0.69

0.50

Discounted revenues

13.5

15.5

19.5

25.1

26.2

28.8

33.4

22.5

(2.0)

(2.1)

(2.5)

(4.1)

(3.9)

(3.3)

(5.1)

(1.3)

Interestingly there is substantial uncertainty about the price of these oil fields. The amount overpaid by the winning bidder, (b (1)b (2))/b (2), or “money left on the table ,” averages to 44% or about 2.67 million of the average price of 6.07 million dollars. The uncertainty does remain substantial even when the number of bidders equals ten or more and amounts to 30% of the final price paid.

A winner’s curse implies a positive correlation between the price paid and the number of bids. Indeed, the data exhibit a positive correlation, as is evident in row 1. With one bidder, the average price paid, b (1), equals 1.50 million dollars, while with ten or more bidders, the price paid increases to 21.8 million. The price paid increases as the number of bidders increases. This positive correlation can alternatively be attributable to endogenous bidder participation decisions. For example, high-valued tracts may attract more bidders. To consider this alternative hypothesis, Table 1 reports what fraction of tracts were drilled and also productive (oil was found). Conditional on being drilled and oil is found, the last row reports the discounted revenues (ignoring drilling costs). Table 1 shows that tracts of higher value attract more bidders.

Table 2 reports a crude assessment of net returns calculated from Table 1. Net returns are defined as discounted revenues times the probability of drilling times the probability of being productive minus the winning bid. This calculation ignores drilling costs. Table 2 shows that net returns are positive throughout. There is no evidence that winning bidders make a loss in these oil field auctions. The evidence appears to reject the winner’s curse.
Table 2

Net returns

No of bidders

1

2

3

4

5–6

7–9

10–18

Total

Net returns

1.9

2.6

3.7

5.3

3.8

2.1

1.0

2.7

Investigating bidding strategies further, Porter reports an interesting (ex post) best response test which is a precursor to the (interim) best response formula developed by Guerre et al. (2000). Consider the set of tracts on which a given firm submits bids. Assume that the bids of rival firms and ex post returns are held fixed. Suppose the vector of bids submitted by the firm in question is varied proportionally. If all of the firm’s bids are increased, it will win more tracts but earn less per tract. In that way the optimal bid proportion that maximizes ex post returns can be calculated. Porter reports that few firms did not behave optimally and overbid. The calculation is ex post and does not take into account the uncertainty bidders were facing at the time of bid submission.

So far our analysis assumed that bidders behave competitively. Next, we shall describe some issues when bidders collude.

## Collusive Bidding

Collusive bidding is illegal in many settings, but the temptation exists which has led to many bid rigging cases pursued by antitrust authorities. Exceptions include joint bidding in OCS auctions, which is legal at least among some bidders as described in Porter (1995), and subcontracting, which is legal in some procurement auctions. We shall discuss some issues relating to collusion among bidders. Collusion may also arise between the auctioneer and one or more bidders, but we shall leave that aside.

A bidding ring has to designate a cartel bidder. Suppose all bidders collude, so that the bidding ring is all-inclusive. Ideally, the ring would like to send only one bidder to the seller’s auction and ask all other ring members to refrain from bidding or to submit phony bids. How can the ring determine their designated cartel bidder? Graham and Marshall (1987) point out that a pre-auction knockout can achieve this. Suppose the colluding ring holds an English auction prior to the seller’s auction and shares the proceeds periodically in equal shares. The winner of the pre-auction knockout gets the right to be the only (serious) bidder at the seller’s auction and pays the difference between the second highest bid and the seller’s reserve price, b 2R. Theorem 1 shows that there exists an equilibrium in which bidders bid their value. Thus, the high-value bidder wins the knock-out auction and pays the second highest bid minus the seller’s reserve price (provided the value is above the reserve price).

Observe that the collusive outcome achieves an efficient allocation. The high-value bidder wins. The rent distribution is now shifted relative to the competitive outcome, with a larger share of the rent going to bidders instead of the seller. What could the auctioneer do in response? Well, anticipating that a ring is in operation, the seller can increase the reserve price. The optimal level would be based on the expectation that the seller faces a single cartel bidder with valuation being the high valuation from all ring.

The pre-auction knockout requires periodic division of spoils. Such side payments may leave a paper trail which increases the risk of detection by antitrust authorities. To minimize the risk of detection, the ring may refrain from using side payments all together. Instead the ring may use some other scheme or mechanism. Let $${q}_i\left(\widehat{v}\right)$$ denote the probability that bidder i is the designated cartel bidder when ring members announce a valuation profile $$\widehat{v}=\left({\widehat{v}}_1,\dots, {\widehat{v}}_N\right)$$ to the ring mechanism. For bidders to tell the truth, it has to be that the expected probability of being the designated cartel bidder is independent of the own announcement $${\widehat{v}}_i$$. Otherwise, bidder i would announce the valuation that achieves the high probability. In turn this implies that the ring mechanism must assign items irrespective of the valuation. One scheme that achieves this is the “phases of the moon” allocation scheme, as used by companies involved in the great electrical conspiracy during the 1960s. Notice though that such a scheme is not efficient as not necessarily the bidder with the high valuation is selected. Thus, the collusive spoils will not be as high as when side payments are available.

The empirical literature on collusion in auctions is small. Two questions have been the focus: first, how can collusive behavior be detected based on bid data and, second, how cartels behave in practice. Porter and Zona (1993) propose a statistical test procedure to determine whether a subset of bidders colluded or not. The test is applied to highway procurement auctions. Pesendorfer (2000) shows that cartels may adopt distinct collusive schemes in practice. Pesendorfer studies school milk auctions and finds that in one regional market, Florida, cartel firms appear to use side payments, while in another regional market, Texas, cartel firms refrained from using side payments.

## Concluding Remarks

Since the seminal papers by Vickrey (1961) and Milgrom and Weber (1982), research on auctions has created a large body of literature. This entry examined well-established results including competitive bidding behavior in standard auction formats, revenue and utility comparison across auction formats, empirics of auctions, winner’s curse, and collusion. The setup involved a one-shot single-unit auction using a standard auction format.

Richer settings involving more goods, sold sequentially or simultaneously, and/or more involved market rules have already received attention in the economic literature and will attract more interest in the future. Market and mechanism design has become a successful area in economics from a theoretical, empirical, and practical perspective. It is to be expected that this will continue to be a fruitful research area in the future.

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