Lecture Notes 18: Auctions
Basic Characteristics
An auction consists of many bidders. The value for each bidder is the true worth that the bidder attaches to an item. In other words, a bidder’s value is his true maximum willingness to pay.
The critical feature of an auction is that a bidder’s value is private information. The bidder keeps it hidden not only from other bidders but from the seller as well.
If all values were public, auctions would be efficient. The buyer with the highest valuation would obtain the item, with the surplus split between the buyer and the seller. But bidders might have an incentive not to reveal their true valuations, so that they can obtain the item at the lowest possible cost. In other words, bidders and sellers are both trying to maximize their own surplus.
Auctions are a special case of a whole subdiscipline called mechanism design, which studies how a principal (e.g. a seller or a firm owner) can design rules to elicit certain actions or information from agents (e.g. bidders or employees).
Auction Classifications
Auctions are classified according to their rules and according to the environment in which they are operated (i.e. the characteristics of bidder valuations).
There are four basic auction rules. Two of them involve open auctions, where bids are public. Two of them involve closed auctions, where bids are private.
• Open Auctions
• English Auction / Ascending-Price Auction – The seller starts at a low price and bidders bid the price up until nobody offers a higher bid.
• Dutch Auction / Descending-Price Auction – The seller starts at a very high price and gradually reduces it. The winner is the first bidder to agree to the stated price.1
• Closed Auctions
• First-Price Sealed Bid Auction – Bidders submit private bids. The winner is the bidder with the highest bid, and he pays his bid.
• Second-Price Sealed Bid Auction / Vickrey Auction – Bidders submit private bids. The winner is the bidder with the highest bid, but he pays the amount of the second-highest bid.
Now, auctions are also distinguished from each other in terms of their environments.
• Private Value Auction – Each individual has his own private valuation of the object. An example might be a baseball card or stamps, which have a different value to different bidders.
• Common Value Auction – The value of the item is the same to all bidders, but bidders do not know precisely what it is. An example is leasing rights to extract minerals from land. All the bidders agree that the value of the lease is the market value of the minerals on the land, but the best anybody can do is to estimate it.
Buyer Strategy
Let’s work out how bidders should behave for each of the four major auction rules in a private value setting. To do this, we will suppose that there are 𝑛𝑛 bidders and that their true valuations are
𝑉𝑉1 > 𝑉𝑉2 > ⋯ > 𝑉𝑉𝑛𝑛. Also, for simplicity, we will assume that all bidding is in increments of $1
English Auction
If your value is 𝑉𝑉, you should continue to add $1 to the previous bid as long as it’s under 𝑉𝑉. Thus, the winner is 𝑉𝑉1 – the bidder with the highest valuation. He ends up paying 𝑉𝑉2+ $1 for the item. If the bidding increment is small, then we say simply that the winner is 𝑉𝑉1 and he pays 𝑉𝑉2.
First-Price Sealed Bid Auction
If your value is 𝑉𝑉, you will not submit a bid of exactly 𝑉𝑉. You want to win the item, but you want to win it at as low of a price as possible. If you bid exactly 𝑉𝑉, then you exactly break even and get no surplus. In this case, we say that bidders shade their bids, bidding less than their true valuations.
Ultimately, how much to shade depends on the number of bidders with whom you are competing. If there are a large number of bidders, you should shade less because there is a higher chance of losing to one of the other bidders.2
Combining, the winner is 𝑉𝑉1 who pays an amount somewhat less than 𝑉𝑉1. The extent of the shading depends on the total number of bidders.
Dutch Auction
A Dutch Auction creates exactly the same incentives as a first-price sealed bid auction. You don’t want to stop and pay the asking price too soon, because you’d rather win at a lower price. But you don’t want to wait too long, because another bidder might jump in and win. The incentives are exactly the same as shading in a first-price sealed bid auction.
The winner is 𝑉𝑉1 who pays an amount somewhat less than 𝑉𝑉1. The extent of the shading depends on the total number of bidders.
Vickrey Auction
In a Vickrey auction, bidding honestly is a weakly dominant strategy. To demonstrate this, suppose that your true value is 𝑉𝑉 = 100, and you are considering whether to shade. Say you are considering whether to bid $75 instead. (The same argument will work for any shaded bid less than $100).
We will demonstrate that, in a Vickrey auction, it is a weakly dominant strategy to bid your true valuation. In other words, no matter what your opponent bids, you are either indifferent or better off if you bid your true valuation ($100) rather than shading ($75). Let your opponent’s bid be 𝑋𝑋.
Opponent bids
𝑿𝑿 < 𝟕𝟕𝟕𝟕 Opponent bids 𝑿𝑿 > 𝟏𝟏𝟏𝟏𝟏𝟏 Opponent bids 𝟕𝟕𝟕𝟕 < 𝑿𝑿 < 𝟏𝟏𝟏𝟏𝟏𝟏
You bid $75 You win, pay $𝑋𝑋 You lose You lose
You bid $100 You win, pay $𝑋𝑋 You lose You win, pay $𝑋𝑋
• If your opponent bids less than $75, then you will win the auction whether you bid $75 or whether you bid $100. Either way, you win and pay the second-highest bid $𝑋𝑋.
2 In the special case where there are 𝑛𝑛 bidders whose values are uniformly distributed over the interval [0,1], then
your optimal bid is �𝑛𝑛−1
• If your opponent bids more than $100, then you lose whether you bid $75 or $100.
• What if your opponent bids something between $75 and $100? If you bid $75, you lose. But if you bid $100, then you win and you pay whatever your opponent bid. Since your opponent’s bid is less than $100, you prefer to win. If an object is worth $100 to you, it’s better to win it and pay a price less than $100 than it is not to win it at all.
The incentives in a Vickrey auction are different than in a price sealed bid auction. For a first-price sealed bid auction, it was good to bid low because you paid less if you won. But in a Vickrey auction, what you pay is actually unrelated to what you bid – If you win, you pay the second-highest bid. Thus, there is no reason to shade. Bidding honestly is a weakly dominant strategy. Regardless of your opponent’s bid, you are either the same or better off by bidding honestly.
Summarizing, all bidders bid honestly in a Vickrey auction. So the winner is 𝑉𝑉1 and he pays the amount of the second-highest bid 𝑉𝑉2.
Comparing Auction Outcomes and Revenue Equivalence
The Dutch auction and the first-price sealed bid auction lead to the same outcome. The winner is
𝑉𝑉1 who pays an amount somewhat less than 𝑉𝑉1, with the extent of the shading depending on the
total number of bidders.
The English auction and the Vickrey auction lead to the same outcome (plus or minus $1). The winner of the English auction is 𝑉𝑉1, who pays 𝑉𝑉2+ 1. The winner of the Vickrey auction is 𝑉𝑉1 who pays 𝑉𝑉2. As the bidding increment gets smaller, the two are basically equivalent.
The Vickrey auction has a special place in mechanism design theory because it is the only auction of the four that elicits the truth from all the bidders about their real valuations. This is called a
truth-telling mechanism. Of course, the truth comes at a price. The seller only gets the second-highest bid rather than getting the second-highest bid.
What about the seller’s incentives?
• The English and Vickrey auctions end up selling the item for the second-highest valuation. If the second-highest valuation is very close to the highest valuation, this will generate more revenue for the seller.
Can we say that one is better than the other on average? Actually, no. The following result is fundamental in auction theory.
• Revenue Equivalence Theorem: For any given distribution of valuations, all four types of auctions will lead to the same revenue on average over many auctions.
Revenue equivalence does not say that revenue is the same for every particular auction. But the cases where the English auction is better (when the second-highest valuation is very close to the highest valuation) will be “cancelled out” by the cases where the Dutch auction is better (when the second-highest valuation is lower). On average, the two generate the same revenue for the seller.
What about experimental evidence? Dutch auction prices tend to be lower than first-price sealed bid auction prices even though the two are theoretically equivalent. Maybe in a Dutch auction there’s some positive utility from the suspense of watching the price go down lower and lower. As for Vickrey auctions and English auctions, there is some limited evidence of overbidding in Vickrey auctions, although studies from Internet auctions show near equivalence between the two, as theory would predict.
Seller Strategy
In the simple setup we have been dealing with, revenue equivalence implies that, over many auctions, the seller makes the same revenue for any of the four major auction types. But there are two complications that might change our answer.
Risk-Averse Bidders
We have been assuming risk-neutral bidders who only care about their expected payoffs. But a risk-averse bidder doesn’t like the uncertainty associated with potentially losing. She would rather win the item for sure at a somewhat higher price rather than risk losing by shading a lot.
This suggests that, with risk-averse bidders, Dutch auctions and first-price sealed bid auctions will generate more revenue for the seller because bidders will not shade as much. For risk-neutral bidders, revenue equivalence implies that all the auctions are the same. But since risk-averse bidders shade less, Dutch auctions and first-price sealed bid auctions (where shading is relevant) will generate more revenue than English and Vickrey auctions when bidders are risk averse.
Correlated Estimates
Conventional wisdom is that an English auction is better in this case. Bidders in an English auction can get into a “bidding frenzy”, where all of them are influenced in real-time by rising bids from other bidders. This is only possible in an English auction.
Common-Value Auctions – The Winner’s Curse
In a common-value auction, an item has the same value 𝑉𝑉 to everyone, but 𝑉𝑉 is unknown. For a simple example, consider auctioning off a jar full of quarters. We all agree that its true value is whatever the quarters are worth, but the best we can do is to guess what that value is. Similarly, for mineral rights to land, all the bidders agree that the land is worth whatever is the market value of the land or the oil contained on the land, but the best they can do is guess.
Consider the jar of quarters. Some people will underestimate its value and bid too low; some people will overestimate its value and bid too high. The winner of the auction will be the person with the most inaccurate bid on the high end! This is called the winner’s curse. If you won this auction, it’s probably because you guessed too high and you bid more than what the quarters are actually worth. After all, you won by guessing higher than what anyone else guessed!
Here is a simple example. Suppose that the true value of an oil field is $100 million. Ten companies estimate the value correctly on average, but with a potential error of $10 million on either end. If companies bid their honest estimates of the true value, then the largest bid will be $108 million, on average. In other words, the winner will typically be paying more than the true value of the oil field. This is the winner’s curse.
Of course, bidders will figure this out and will adjust their bids to account for the winner’s curse. This is a complicated problem, though, because the bidders have to account for the fact that other bidders also make the same adjustment. Basically, bidders should not ask themselves “Am I willing to pay $108 million?” Rather, they should ask “Am I willing to pay $108 million knowing that I will win only when nobody else was willing to bid this much?”
Here’s an even simpler example of this general principle. Suppose that a seller is selling mineral rights to his land. Whatever the land is worth to the seller, it’s worth 1.5 times that to you. You bid
𝑏𝑏 for the land. The seller will only accept your bid when the value of the land to him is in the interval [0, 𝑏𝑏]. The average value to the seller of an accepted bid is 0.5𝑏𝑏. Thus, the average value of an accepted bid to you is 1.5 ⋅ 0.5𝑏𝑏 = 0.75𝑏𝑏. In other words, any bid 𝑏𝑏 is a money-loser for you since the average value to you of what you will receive is only 0.75𝑏𝑏. You should bid 0.
Evidence on the winner’s curse suggests that the problem can be very serious.
• There is usually significant overbidding for jars of coins, and in fact the problem gets worse with repeated play.
• Companies typically make large losses on oil and gas drilling rights when the government auctions off these rights.
• New sports players who are auctioned off to teams are severely overpaid relative to their productivity to the team. This is the winner’s curse. The teams all try to estimate what the player will be worth, and the team that wins the player is the team that overestimates the player’s value by the largest margin.
Escalation Auctions
In an escalation auction (sometimes called an all-pay auction), both winners and losers have to pay their bids.
For a simple example, imagine that a seller auctions off a $1 bill to two people using a standard English auction. The winner has to pay and the loser has to pay his last bid. Suppose that the bidding has already gotten up to $1.00. Should you go for $1.01? Sure – at least winning will give you $1.00 to offset your losses. People end up bidding way more than the item is actually worth just so that they can win and recover some of their losses.
Evidence on escalation auctions suggests that the problem can be very serious.
• Dollar-bill type auctions generally end up selling the bill for around 6 times its actual value before one party or the other gives up.
• Election contributions – Everyone donates money to election campaigns, but only the people who support the winner get any return on their investments.
• Olympic games – Everyone bears heavy costs in training and attending the games, but only the winners get any return.
• Wars – People often wonder why parties engage in extremely costly wars, and continue to escalate them past the point where a victory would justify the costs already invested in fighting it. Escalation auctions provide an interesting game theoretic interpretation. Even if you have already invested so much that a victory in the war wouldn’t be worth the costs you have incurred in fighting it, those are sunk costs. It might be worth holding out just a little bit longer so that at least you can eke out a victory and recover some of your losses. So hold out just a little bit longer, and a little bit longer, and a little bit longer…
Multi-Unit Auctions
A seller is auctioning off a group of items that have some value individually, but the items have a higher total value if they are owned together instead of separately. A simple example is a collection of dishes. Each dish is worth something on its own, but the value of the dishes is higher as a collection than the sum total of the value if the collection is pieced out. Another example might be ten one-acre plots of land. Each acre could contain a house, but if a single person owned all 10 acres then he could build a shopping mall on the property, which is more valuable than 10 houses.
A seller has 10 acres of land. In Town A, the seller auctions all 10 acres together. In Town B, the seller auctions off each of the ten acres separately. How can we compare these two auction rules?
• Buyers prefer Town A. In Town B, bidders will have to bid very aggressively to win all 10 acres, but it’s risky because they don’t want to go over the total value at the end.
• The seller will probably make more money in Town B, especially if there are some buyers who are interested in single units and will compete with a buyer interested in all 10 acres. But one caveat is that buyers may not show up to the auction at all.
Collusion and Shilling
For a motivating example, consider a group of 5 buyers who always come together and bid competitively against each other. Each ends up winning 1/5 of the time, but ends up paying a high price because of competition from the other bidders. The bidders might consider getting together and colluding, agreeing in advance who will win each auction and not to bid the price up too high. Each bidder would win just as often on average, but would pay a lower price. Like all collusion, this would only work in the context of a repeated game since each bidder would have an incentive to renege on the agreement and bid for a good deal on an item otherwise.3
3 This is illegal, just like collusion among sellers. A group of builders who regularly bid on surplus government
All types of auctions are potentially subject to collusion. For example, in a Vickrey auction, bidders might agree on the highest bid and then lowball all the other bids. However, it may be easier to collude in open auctions than in closed auctions because cheating is easier to detect and punish.
While collusion is dishonest behavior by buyers, shilling is dishonest behavior by sellers. Shilling
occurs when a seller plants false bids at his own auction in order to force higher bids by the bidders.
Shilling is common in English auctions, especially on the Internet. Similarly, in Vickrey auctions, the seller can plant a high second-highest bid in order to force the winner to pay more.
Online Auctions
Online auctions are interesting and they provide a wealth of data for economists who study auction theory.
eBay and Amazon use proxy bidding, where you enter the highest amount that you are willing to pay. When there are new bids, the computer automatically bids up for you in increments of $0.01 until the current bid goes over your maximum. This is essentially an English auction, where the highest bidder wins and ends up paying slightly over the second-highest bid. Dutch auctions are difficult to conduct online because they depend on live, simultaneous action.
The biggest difference between live auctions and online auctions is that online auctions have to have some kind of ending rule since not everyone is at their computers simultaneously. eBay auctions end at a specified time, but Amazon auctions extend automatically if there is a lot of action close to the closing time.
The setup on eBay opens the door to sniping. It’s better to wait and issue your bid late because there is less opportunity for others to jump in and beat you. It also means less opportunity for a bidding war if there is another bidder interested in the item.
Evidence suggests that sniping is a good strategy on eBay. It avoids bidding wars and it also protects private information you might have on what an item is worth. For example, if you are knowledgeable about coins, you might not want to jump in right away with a high bid for a rare coin, because it could give other buyers a signal that the coin is valuable and that they should jump in too. Not surprisingly, there is more early bidding on Amazon and more late bidding on eBay.
Problems
1. A house painter has a regular contract to work for a builder. On these jobs, his cost estimates are generally right: sometimes a little high, sometimes a little low, but right on average. But the painter also bids competitively for other jobs. “Those are different”, he says. “They almost always end up costing more than I estimate.” If we assume that his estimating skills are equally good for any type of job, what might explain this difference?
2. Three bidders are interested in purchasing a baseball card. Bidders 1, 2 and 3 have valuations of $12, $14 and $16 respectively. Bids are made in $1 increments.
a. Which bidder wins in an English auction? What price does he pay?
b. Which bidder wins in a second-price sealed bid auction? What price does he pay? c. What causes the difference in the seller’s revenue between parts (a) and (b)? d. If, in a first-price sealed-bid auction, all bidders bid at least $2 less than their true
valuations, how does the likely outcome compare to the outcome from parts (a) and (b) from the perspective of the seller?
e. If the bidders in (d) were extremely risk averse and did not shade their bids at all, what would be the winning price and profit for the seller?
3. (The no-trade theorem) Brenna owns a painting. She knows whether it is “good” (worth 1 to everyone) or “bad” (worth 0 to everyone). Buyers do not know whether the painting is good or bad, only that it is good and bad with equal probability. A buyer has an opportunity to make a take-it-or-leave-it offer to Brenna for the painting. What bid maximizes expected profit for the buyer? What is this trading opportunity worth to the buyer on average?
