CHAPTER 17. Auctions and Competitive Bidding

Transcription

1 CHAPTER 17 Auctions and Competitive Bidding Two rules for succeeding in business: (1) Never underestimate the importance of money, and (2) everything is up for bid. ANONYMOUS Years before each Olympic Games, a competition takes place that is just as intense as the Olympiad itself namely, the high-stakes bidding competition by the U.S. networks to televise the games. The following table shows the spectacular growth in the price paid for these television rights. Part of the revenue growth is attributable to price inflation over the past 25 years. Part, too, is due to the growth in U.S. television audiences. (Larger audiences allow networks to charge higher rates to advertisers.) However, the greatest part of the revenue increase is owing to the skill with which Olympic organizers have arranged the bidding competition. Since 1976, when ABC was awarded the games without any real competition (before the other networks had a chance to bid), the organizers have implemented a number of bidding innovations. 1 For instance, in 1980, the Soviets organized a ruthless bidding competition involving multiple rounds. At each stage, the current leader was announced, and losers were required to up the bidding by at least 5 percent. For the 1988 Seoul summer Olympics, the organizers limited the competition to two rounds of sealed bids. NBC s winning bid included a novel revenue-sharing agreement: a $300 million guaranteed payment plus two-thirds of any gross advertising revenues in excess of $600 million (up to a maximum $500 million total payment). Bidding to Televise the Olympics 1 John McMillan provides an incisive analysis of this competition. See Chapter 12 of J. McMillan, Games, Strategies, and Managers (New York: Oxford University Press, 1992). 505

2 506 Chapter 17 Auctions and Competitive Bidding Winning Bids for Televising the Olympics Summer Games Winter Games 1976 Montreal (ABC) $25 million 1976 Innsbruck (ABC) $10 million 1980 Moscow (NBC) $ Lake Placid (ABC) $ Los Angeles (ABC) $ Sarajevo (ABC) $ Seoul (NBC) $ Calgary (ABC) $ Barcelona (NBC) $ Albertville (CBS) $ Atlanta (NBC) $ Lillehammer (CBS) $ Sydney (NBC) $ Nagano (CBS) $ Athens (NBC) $ Salt Lake City (NBC) $ Beijing (NBC) $ Turin (NBC) $ London (NBC) $1, Vancouver (NBC) $820 For the 1992 winter games, the Olympic Committee specified single sealed bids, before returning to multiple bidding rounds (adding a minimum required bid of $360 million) for the 1992 summer games. In 1995, a surprising turn occurred in the competition for the rights to the games from 2000 to NBC made early (high) preemptive offers that the organizers mulled and accepted, thereby short circuiting the bidding competition altogether. However, in 2003, the committee returned to rounds of competitive bidding for the 2010 and 2012 games. The history of the Olympics bidding raises a number of questions. As a network representative, how should you determine the size of your bid? What difference do the bidding rules make? How should the organizing committee marshal the bidding competition to maximize its revenue? This chapter studies an important application of decision making under uncertainty: the use of auctions and competitive bidding. Indeed, auctions are among the oldest forms of economic exchange. One of the earliest references was given by Herodotus, who noted a peculiar auction used by the ancient Babylonians as a way of distributing wives. In modern economies, a common means of selecting a best alternative is to solicit competitive bids. In the simplest (and also most common) bidding settings, the objective is to get the best price. This is the case in the networks bidding for the Olympic Games television rights. Here, a single seller faces a number of competing buyers, and best price means highest price. The two most frequently used methods are the English and sealed-bid auctions. In the familiar English ascending auction, the auctioneer calls for higher and higher bids, and the last and highest bid claims the item. The English auction enjoys a secure place in a bewildering variety of settings the sale of art and antiques, rare gems, tobacco and fish, real estate and automobiles, and liquidation sales of all kinds. By contrast, in a sealed-bid auction, buyers submit private bids, bids are opened, and the highest bidder claims the item and pays its bid. The sale of public

3 The Advantages of Auctions 507 and private companies has been accomplished by sealed-bid auction as have the sales of real estate, best-seller paperback rights, theater bookings of films, U.S. Treasury securities, and offshore oil leases, to name a few. A third kind of auction method is the Dutch auction, used in the sale of a variety of goods but especially in the sale of flowers in Holland. The auctioneer s initial price is set very high, and then the price is lowered at intervals. The first buyer who announces a bid obtains the item at the current price. Auctions are also a common means for conducting competitive procurements. Here, a single buyer solicits bids from a number of competing suppliers with the objective of obtaining the lowest possible price. In complex procurements, the ultimate objective is best thought of as source selection. The buyer seeks to select the best supplier, measured not only in terms of price but also product quality, management capability, service performance, and the like. The most common institution for complex procurements is the submission of sealed bids (possibly in multiple rounds) before the buyer makes a final selection. Whatever the particular institution or setting, auctions share the common feature that competition exists on only one side of the market. The auctioning party occupies a monopoly position and faces competing buyers or sellers. The auction determines at once with whom a transaction will take place and at what terms. Thus, the study of auctions raises two main questions: As a competitor, how should a firm bid to maximize its profit? In turn, how can the auctioning party design competitive bidding institutions for maximum advantage? This chapter considers each of these issues in turn. THE ADVANTAGES OF AUCTIONS Competitive bidding institutions are widespread because of the advantages they bring in obtaining the best price. As a means of effecting transactions, auctions take a place alongside competitive markets, posted prices, and negotiated transactions. Auctions are viable when a well-functioning, competitive market fails to exist. In other words, a producer of a standardized good that is bought and sold in a competitive market at predictable prices hardly can expect to have much success holding its own auction; any would-be buyers already can obtain the good from the market at the best available price. Thus, a prerequisite for an auction sale is that the good be differentiated from others. Indeed, auctions are a ready means of sale for unique items: artwork, antiques and other rare objects, paperback rights, oil and mineral leases, and the like. Roughly speaking, auctions occupy a middle ground between posted pricing and negotiated prices. A common means of sale (and the universal means for U.S. retail sales) is for sellers to post prices, leaving buyers the choice to purchase at that price or not at all. Ideally, posted prices should be set in line with supply and demand. But given the difficulty in judging these forces (or changes in them), this is not always the case. At the other extreme are negotiated prices, which are freely flexible. Although it has obvious advantages, pricing flexibility also has its costs. Negotiations can be time-consuming

4 508 Chapter 17 Auctions and Competitive Bidding and expensive. Moreover, in the bargaining process, both buyer and seller have a significant influence on the final price. If everything is negotiable, a seller surrenders much of its monopoly power over price. Auctions can be viewed as combining the best of the posted and negotiated pricing worlds. An auction ensures that competition among buyers sets the final price the highest price the market will bear. In effect, the auction allows the seller to compare all buyer offers simultaneously and choose the best one. The auction is less time-consuming than rounds of one-on-one negotiations, and it preserves the seller s monopoly position. Auctions are more flexible than posted pricing. The current state of market demand determines the good s price, not the seller s best guess as to demand. The following examples illustrate these points. A STOCK REPURCHASE A company is considering buying back a portion of its common stock. Top management believes the value of its company s stock to be about $80 per share. (The current market price of $67 is indicative of the market s undervaluing the firm s shares.) Management is considering offers to buy back shares at one of three possible prices: $70, $72, and $74. However, there is a great deal of uncertainty about how many shares might be tendered at these prices. The following table lists the number of shares tendered (in millions) at the different prices for three kinds of shareholder response: strong, medium, and weak. Management does not know which of the cases will hold; its best prediction is that all three are equally likely. The company reckons its profit from any repurchase at ($80 P)Q, where P is the price it pays and Q is the quantity of shares it succeeds in buying. What price offer should the firm make to maximize its expected profit? Shareholder Response Price Strong Medium Weak $ A simple expected-value analysis identifies the optimal offer as $70. With this offer, the firm earns a profit of $130 million, $90 million, or $60 million, depending on the market response. For instance, under strong demand, the firm s profit is (80 70)(13) = $130 million. Since each market response is equally likely, the firm s expected profit is $93.3 million. It is easy to check that both the $72 and $74 offers deliver lower expected profits. Now suppose that instead of setting a buyback price, the firm uses a particular kind of auction to repurchase shares. In this system, each shareholder tenders any number of shares at a price he or she names. After all tenders are collected, the firm buys shares

5 The Advantages of Auctions 509 (as many or as few as it wishes) at a single common price consistent with the tenders submitted. 2 For example, if the response proves to be strong, the firm has a number of options. By choosing a common price of $70, it could buy back 13 million shares (from all shareholders who named prices of $70 or below). It could name $72 as the common price and repurchase 14 million shares, or it could name $74 and repurchase 18 million shares. By waiting for the shareholders response, as revealed by the auction, the firm can select the best repurchase price, given current market conditions. It is easy to check that the firm s most profitable offer is contingent on demand. If demand proves strong, the firm s best price is $70 (yielding $130 million in profit). If demand is medium, its best price increases to $72. Here, the firm s resulting profit is (80 72)(12) = $96 million. If demand is weak, its best offer is $74 (with $72 million in profit). Using the auction method, what is the firm s expected profit? Since each response is equally likely, the firm s expected profit is simply (1/3)(130) + (1/3)(96) + (1/3)(72) = $99.3 million. The firm s expected profit has increased by $6 million relative to the profit under its best posted price, $70, which is fixed regardless of the market response. Using the auction method, the firm effectively has acquired perfect information about demand. As noted in Chapter 13, the difference in expected profit in this case $6 million measures the expected value of perfect information. To sum up, in an uncertain environment, this auction method has a clear advantage over posted pricing. BIDDING VERSUS BARGAINING In Chapter 15, we considered the classic example of bilateral monopoly, where a single seller faced a single buyer with the aim of negotiating a mutually beneficial price. The following example illustrates the potential benefit of competitive bidding versus bargaining in securing a better price. 3 With the aid of its investment banker, a firm is seeking to sell its division at the highest possible price. The division is worth $40 million under the company s own management. The investment banker is hopeful that it can find as many as four to six potential buyers for the division. The banker believes the range of buyer values to be between $40 million and $64 million with all values in the range equally likely. In addition, the banker believes that buyers values are independent of one another. Thus, if one buyer is willing to pay $44 million, the next buyer s independent value might be $55 million (or any other equally likely value in the $40 $64 million range). What price could the firm expect to obtain in negotiations with a single buyer? There is certainly room for a mutually beneficial agreement. For instance, if the buyer s actual value were $52 million, a negotiated price of $46 million (halfway between the parties values) would generate a profit of $6 million for each side. If the bargainers were equally matched, one would expect the final price to be close to this split-the-difference prediction. Moreover, since $52 million is the single buyer s expected value for the transaction, 2 The financial community commonly refers to this procedure as a Dutch auction. However, this label is misleading because the method is very different from the usual Dutch auction described later in this chapter. 3 You encountered an earlier version of this example in Chapter 13 s discussion of optimal search.

6 510 Chapter 17 Auctions and Competitive Bidding one-on-one bargaining by equally matched parties should result in a price of $46 million on average. The firm can obtain a much higher sale price on average by putting the division up for competitive bid and enlisting as many potential buyers as possible. Suppose the firm solicits sealed price bids from the buyers. In placing its bid, each buyer will assess the (independent) monetary value it places on the division and will submit a sealed bid below this value, aiming to win the division at a profit. As we show later in the chapter, with seven buyers, the price paid by the highest bidder will be $58 million on average. The firm obtains a much better price for its division by soliciting bids from multiple competitors than from a single one-on-one negotiation. The sources of the advantage are twofold. First, as the number of potential buyers increases, it is more likely that one will hold a high value (in the upper part of the $40 $64 million range) and make a high bid. Second, the increase in the number of competitors forces each bidder (including the high-value buyer) to place a bid near to its true value. This implies lower profit for the bidder and a better price for the seller. In an auction, the bidder must compete against other would-be buyers, instead of against the seller alone, as in a one-on-one negotiation. In sum, competitive bidding serves to marshal the competition among a number of buyers to deliver the best price to the seller. CHECK STATION 1 In attempting to sell an item, firm S has approached buyer A, whose last best price offer is $24. It now plans to approach firm B but is uncertain of the price it might get. Its best assessment is that B s final price offer lies in the range $20 to $28, with all (continuous) values in between equally likely. Show that firm S can improve its payoff (to $25 on average) by selling to the firm offering the better price. BIDDER STRATEGIES A firm s optimal bid in a given situation depends on many factors: its value for the good at auction, its assessment of the extent of bidding competition, and most important, the type of auction in which it competes. To model bidding behavior, we begin by considering the so-called independent private value setting. Here, each bidder assesses an individual value (or reservation price) for the item up for bid. We denote buyer i s value by v i, for i = 1... n, where n denotes the number of bidders. (In the converse case of a competitive procurement, each firm would hold a private cost estimate c i.) Each bidder s value is independent of the others and is private: that is, it is known only to itself. Although values are private, all bidders are aware of the common probability distribution from which buyer values are independently drawn. (After the fact, buyers hold different values. But before the fact, one buyer is no more likely to have a high value than any other.) If winning bidder i obtains the item at price P, its profit is simply v i P. We now consider three common types of auction.

7 Bidder Strategies 511 English and Dutch Auctions In the oral, ascending English auction, bids continually increase until the last and highest bidder wins the item at his or her bid price. Optimal bidder strategies in the English auction are remarkably simple: When buyers hold independent private values, each buyer s dominant strategy in an English auction is to bid for the good up to the buyer s reservation price if necessary. In the English auction, a buyer never should place a bid above his or her true value; this would imply a loss if the bid were to win. Nor should the buyer stop short of his or her value; this needlessly precludes earning a profit should a slightly higher bid win. Bidding up to full value (if necessary) is optimal regardless of the competitors values or the bid strategies they might use; that is, this strategy is dominant. Notice that the bidding stops when the price barely rises above the next-to-last bidder s value. Thus, the auction delivers the good to the buyer holding the highest value among the bidders. The price in the English auction stops at a level approximately equal to the secondhighest reservation price: P E = v 2nd. The seller can achieve exactly the same price result using a closely related auction method: the so-called Vickrey auction or second-price auction. In the Vickrey auction, bidders submit sealed bids, and the highest bidder claims the item but pays a price equal to the second-highest bid (instead of paying his own bid). It is easy to check (see Problem 1) that in the second-price auction, each buyer s dominant strategy is to bid its true reservation price. For instance, if you believe a Persian rug up for bid is worth $800, this is exactly what you should bid. If yours is the highest bid and $620 is the next highest bid, you win the rug at a price of $620. It doesn t pay to shade your bid below your true value because doing so does not affect the price you pay when you win. (In fact, if you were persuaded to bid $720, you would be fortunate to still win, but the price of $620 wouldn t change. However, foolishly lowering your bid would be a disaster if the next-highest bid were $750, causing you to lose the rug altogether.) Although they appear to be quite different, the English and Vickrey auctions are strategically equivalent when bidders hold private values. To see this, imagine that every buyer in an English auction were to authorize a personal representative to bid on his behalf. (For instance, this is routinely done if the buyer cannot be present at the auction.) Each buyer privately submits its reservation price to this representative, instructing him to bid up to this value if necessary. (In fact, each buyer should always submit his true reservation price.) Once all bids are reported, the auction house could hold a virtual English auction. It simply identifies the two highest submissions, bids them against one another, until the price just matches the second-highest submission (whereupon this bidder is dropped from the auction), and thereby awards the item to the high bidder at the second-highest bid price (and this corresponds to the second highest reservation price). Thus, the final prices in the Vickrey and English auctions are identical: P 2nd = P E = v 2nd.

8 512 Chapter 17 Auctions and Competitive Bidding Auctioning Google s Online Ads Google dominates the online search market, from which it generates more than $30 billion in annual revenues in the form of online advertising. When a user searches for a keyword (such as Hawaii ), relevant links appear as well as up to 11 sponsored links, that is, paid advertisements, listed from top to bottom in a column on the righthand side of the Web page. An Internet seller has paid Google for a particular sponsored link attached to a specific keyword. How was the sponsored link sold? Via an automated auction indeed, new bids are submitted to Google continually for thousands of keywords, minute by minute. 4 Google s online ads were not always sold this way. Before 2002, its top-of-web page ads were sold the old-fashioned way by a human sales force pursuing clients and negotiating prices. Clients were charged whenever a user called up the Web page showing the ad, whether or not the user actually clicked on it. But top management soon realized that selling online ads this way was too limited, too labor intensive and slow, and (most important) too uncertain as to what the right price should be. Google took what was a huge risk at the time; it phased out its ad sales force that had been pushing campaigns worth hundreds of thousands of dollars in fees in favor of automated ad auctions paying Google mere pennies per clicks on its sponsored links. Top management did the math; millions of clicks per hour (across thousands of keywords) added up to billions of dollars per year. Google s automated auction method is called a Generalized Second Price auction or GSP and works like this. Suppose a number of online advertisers are vying for three sponsored links positions. The top slot is the most valuable; its prominence means it gets the most clicks and ultimately the greatest expected sales revenue. (The second position is next most valuable, and so on.) The advertisers submit price bids via the automated system, and the highest bidder claims the top slot, the second-highest bidder gets the next slot, and the third-highest gets the last slot. The twist is that the high bidder pays Google a price equal to the second-highest bid; the second bidder pays the third-highest bid, and the third bidder pays the fourth-highest price. As in the Vickrey or second-price auction discussed earlier, winning bidders benefit via this automatic price break. For instance, if the top four bids are 30 per click, 25, 22, and 19, the three slots would sell for 25, 22, and 19. For Google, the method is economical and scalable. A single auction (rather than three separate auctions) suffices to allocate all the slots at once. It s also dynamic. New online advertisers can submit bids at any time. If the newcomer outbids a current slot holder, that advertiser and each slot holder below are bumped down a slot. Finally, the GSP method is known to promote stable bids. By their bids, advertisers find their spots in the pecking order. An advertiser in the third slot is there because it has discovered that it is too costly (and less profitable) to beat the going bid price for the second slot (even accounting for the increased click-through rate). Google s CEO Eric Schmidt sees automated auctions fueling all sorts of transactions. In 2004, Google surprised the investment banking world by using a variation on a Dutch auction to sell shares in its Initial Public Offering. Rather than hiring an investment 4 This account is based on B. Edelman, M. Ostrovsky, and M. Schwarz, Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords, American Economic Review (March 2007): ; and S Levy. Secret of Googlenomics: Data-Fueled Recipe Brews Profitability, Wired Magazine, May 22, 2009.

9 Bidder Strategies 513 bank to set the offer price and control the sale of shares, Google hoped to secure the highest price for its shares by soliciting bids from all investors, large and small. DUTCH AUCTIONS Dutch auctions are used to sell many commodities worldwide, including produce, fish, and most notably flowers in Holland. In a Dutch auction, the auctioneer starts the sale by calling out a high price and then lowers the price by small increments until a bid is made. The first bidder obtains the item at the current price. Optimal bidding strategies are significantly different in the Dutch and English auctions. The Dutch bidder faces a decision under uncertainty. Once the price descends below the bidder s reservation price, the bidder must decide how long to wait that is, until how low a price before placing a bid. A buyer with value 10 might choose not to bid when the price has fallen to 9, hoping to win the bid when the price drops to 8.5. The risk, of course, is that another buyer will be first to bid and win the item at a price of 8.7. Thus, the decision when to bid depends on one s value and the assessed strength of the competition (as embodied in the number of rival buyers and their likely values for the item). By contrast, the English bidder faces no such risk because there is always the opportunity to better the current price. It is also interesting to compare the Dutch auction and the sealed-bid auction. In the latter, buyers submit sealed price bids, and the highest bidder wins the item at his or her bid price. The two auctions appear to be very different in form, but as some simple reasoning can readily show, the two methods are strategically identical for bidders. In each auction, a buyer must choose how high to bid (or how low to let the price drop), trading off the probability and profitability of winning. Holding a given reservation value and facing the same set of rivals, any buyer should make the same bid in either auction. After all, a Dutch bidder could just as well write down its bid-in price beforehand. If all bidders did this, the Dutch auction could be run as a sealed-bid auction: Prices are opened and the highest-price buyer obtains the item at its bid-in price. In short, the Dutch and sealed-bid auctions are expected to induce identical bidding behavior and, therefore, to generate identical expected sale prices. Flower sales in Amsterdam occupy the pinnacle of the Dutch auction. Auctions take place in a building roughly the size of 10 football fields. Each day thousands of lots of flowers are brought for sale before hundreds of bidders occupying steeply tiered seats in separate auction halls. To expedite sales, prices are displayed on the hand of a computerized clock. The price descends with the downward counterclockwise sweep of the clock hand, and buyers bid by pushing a button that stops the hand and automatically records the sale price. Sales are completed at the rate of 600 transactions an hour (one every six seconds). Flowers for sale come not only from Holland but also by air from Europe, Israel, Africa, and parts of Asia. Upon sale, the flowers are shipped to Canada, the United States, and scores of other developed countries. Sealed-Bid Auctions Sealed competitive bidding is frequently used to sell unique items: certain antiques, real estate, oil leases, or timber and mineral rights by the U.S. government. Submitting a bid in a sealed-bid auction is a classic example of decision making under uncertainty. Each bidder

10 514 Chapter 17 Auctions and Competitive Bidding faces a fundamental trade-off between the probability and profitability of winning. In raising its bid to purchase an item, the company increases its chances of winning but lowers its profit from winning. (Similarly, in a competitive procurement, a lower price increases a supplier s chance of being selected but reduces its potential profit.) Given this trade-off, it is natural to ask, What bid will maximize the bidder s expected profit? STRATEGY AGAINST A BID DISTRIBUTION The key to formulating a profitmaximizing bidding strategy is to anticipate the distribution of competing bids. Obviously, to win the auction, the firm must beat the best competing bid. If the firm could predict this bid perfectly, its most profitable bid would be the one that wins by the smallest margin. But a perfect prediction clearly is impossible; at best, the firm possesses a probability assessment of competing bids. Based on this assessment, the firm determines its optimal (i.e., profit-maximizing) bid. As an example, consider a typical sealed-bid auction say, for a small suburban office building expected to receive bids from three firms. Each bidding firm plans to occupy the building (if it wins the auction) and places a greater or lower value on the building, depending on its main features: location, office space, amenities, and so on. Needless to say, an additional desired feature is a low purchase price at auction. Let s consider the bidding problem faced by a typical firm, say firm 1. Its bidding strategy begins with an assessment of its reservation price, that is, its monetary value for the building. For concreteness, suppose its reservation price is v 1 = $342 thousand. This value can be thought of as a break-even price. The firm is just indifferent to the alternatives of acquiring the building at this price or not acquiring it at all. It never would pay more and would be happy to pay less. 5 Firm 1 s profit from winning the auction at bid b is $342 thousand b, the difference between its value and its bid. If it does not win the auction, its profit is, of course, zero. It follows that the firm s expected profit is E(π) = [342 b][pr(b wins)], where the second term denotes the probability that bid b wins (i.e., is the highest bid). The key to determining a profit-maximizing bid is to assess accurately the way the firm s winning chances depend on its bid. Recognizing that it faces two other bidders for the building, the firm has thought carefully about its winning chances and has made the probability assessments listed in the third column of Table As we would expect, the firm s winning chances increase steadily with its bid. Looking at the last column of the table, we see that the firm s expected profit is minimal (1) for very low bids, which have little chance of winning, and (2) for bids near its reservation price, since these generate little profit. Expected profit is maximized at a bid of $328 thousand, which the firm predicts has a 49 percent winning chance. This is firm 1 s optimal bid. The firm s probability assessment of its winning chances usually is based on its past bidding experience: how often its bids have won auctions against varying numbers 5 In fact, the $342,000 estimate probably represents an expected value; that is, the company recognizes that the value of the building is more or less uncertain. Given this uncertainty, a risk-neutral buyer values the building at its expected value.

11 Bidder Strategies 515 TABLE 17.1 Winning Probability Expected Bid Profit of Winning Profit $300 $42.00 $ Finding a Profit- Maximizing Bid (Thousands of Dollars) Raising a sealed bid increases the probability, but lowers the profitability, of winning. A bid of $328,000 maximizes the buyer s expected profit. of competitors in the past. A useful way to think about the firm s winning chances is in terms of the distribution of the best (i.e., highest) competing bid. The probability of the firm s winning is simply the probability that the best competing bid (BCB) is smaller than the firm s own bid. Figure 17.1 shows the graph of the cumulative distribution of BCB (labeled H). The curve s height indicates the probability that the best competing bid is smaller than the value shown on the horizontal axis. For instance, at $320 thousand the height of the curve is.25, meaning that there is a.25 chance that the highest competing bid will be lower than $320 thousand (and, of course, a.75 chance that it will be higher than this value). The median of the BCB distribution is about $328 thousand (actually, very slightly higher). There is a 50 percent probability of BCB being lower than the median value. The BCB distribution curve is important because it precisely measures the firm s winning chances for its various bids. Thus, a bid at the distribution median has a.5 chance of winning the auction because half the time the best competing bid will be below this value. According to the BCB curve, a $320 thousand bid has a.25 chance of winning, and so on. Using the BCB curve, there is a simple geometric means of describing the firm s optimal bid. In Figure 17.1, a vertical line has been drawn at the firm s reservation price, $342 thousand. Suppose the firm chooses bid b. Then the firm s profit, if the bid wins, is $342 thousand b. This profit is measured by the horizontal distance between $342 thousand and b. The probability that the bid wins the auction is given by the height of the curve H(b). It follows that the firm s expected profit, (342 b)h(b), is measured by the area of the rectangle inscribed under the BCB curve. Maximizing the firm s expected profit is equivalent to choosing a bid that maximizes this rectangle s area. 6 The figure shows the firm s optimal bid, $328 thousand, and 6 The interested reader may wish to apply calculus to find the optimal bid. A bidder with value v that submits bid b earns an expected profit of E(π) = (v b)h(b). Marginal profit is Mπ = de(π)/db = (v b)h(b) H(b), where h(b) = dh/db is the density function of the BCB distribution. Setting Mπ equal to zero, we find b = v H(b)/h(b). The optimal bid is below the firm s value, v, and the size of this discount is given by H(b)/h(b).

12 516 Chapter 17 Auctions and Competitive Bidding FIGURE 17.1 Probability Distribution of the Best Competing Bid The curve s height shows the probability that the best competing bid is smaller than the value shown on the horizontal axis. The buyer s optimal sealed bid is $328,000. Probability H curve Probability b = 328 wins Firm 1 s profit v b Firm 1 s optimal bid Range of Competing Bids (Thousands of Dollars) Firm 1 s reservation price the associated inscribed rectangle. This rectangle has a larger area than any other possibility. In general, the same geometric procedure can be used to determine an optimal bid for any reservation price the firm might hold. The right side of the expected-profit rectangle simply is set at the reservation price. (For instance, if the firm s value is $330 thousand, we can find the best bid to be $320 thousand after some experimentation.) A useful way to arrive at the BCB distribution is to begin by assessing the distribution for a typical competing bid. For instance, let G(b) denote the cumulative distribution function for the bid of a single competitor. In other words, if the firm submits bid b, the chance that it will better the bid of this single competitor is simply G(b). What if the firm faces two competitors? For the firm to win, its bid must be greater than both competing bids. Under the assumption that the competing bids are independent of one another, this occurs with probability [G(b)] 2. More generally, suppose the firm faces (n 1) competitors

13 Bidder Strategies 517 whose bids are independent and come from the common distribution G(b). Then the probability that bid b is greater than all the others is H(b) = [G(b)] n 1. To illustrate this result, consider the sealed-bid competition for the office building, again taking firm 1 s point of view. Suppose the firm s best assessment is that each competitor s bid will be somewhere between the limits $300 thousand and $340 thousand, with all values in between equally likely. This assessment implies 7 G(b) = (b 300)/40. Therefore, it follows that H(b) = [(b 300)/40] 2, because the firm faces two competitors. In fact, this is exactly the probability function depicted in Figure 17.1 and listed in Table As we saw earlier, the firm s optimal bid against this distribution is $328 thousand. To sum up, based on an assessment of possible bids by a typical competitor, the firm should compute the distribution of BCB and then fashion an optimal bid against this distribution. Now consider how the BCB distribution is affected if the number of bidders increases, say, to five firms. Facing four competitors, firm 1 would assess H(b) = [(b 300)/40] 4. Notice that the firm s win probability for any given bid goes down drastically with the number of bidders. For example, a bid of $330 thousand has a.75 chance of beating any single bidder, a.56 chance of beating two other bids, but only a.32 chance of beating four others. Not surprisingly, the increase in the number of competitors causes firm 1 to increase its optimal bid. In fact, there is a second effect reinforcing firm 1 s raise in bid: With the increase in competition, firm 1 would expect other firms to raise their bids as well. Thus, the bid distribution of the typical firm, G(b), will be shifted toward higher bids. In short, firm 1 faces an increase not only in the number of competing bids but also in their levels. For both reasons, its optimal bid increases. Your firm is competing in a sealed-bid auction for a piece of computer equipment valued (by the firm) at $30,000. You are contemplating one of four bids: $18,000, $20,000, $24,000, and $27,000. Given the bid distribution of a typical buyer, these bids would win against a single competitor with respective probabilities.4,.6,.8, and.9. What is the firm s optimal bid against one competitor? Against two competitors? Against three competitors? CHECK STATION 2 EQUILIBRIUM BIDDING STRATEGIES 8 Thus far, we have taken the point of view of a single firm whose task is to formulate an optimal bid, given a prediction about the distribution of competing bids. Although this approach has certain advantages, it also suffers from two problems. First, firms often are severely limited in their information about their competitors bidding behavior; that is, there may be little empirical basis (i.e., a history of past bidding tendencies) for assessing H(b) or G(b). An optimal bid that assumes the wrong competing bid distribution will show a poor profit performance. Second, a purely empirical approach involves only one-sided optimization. It ignores the important fact that competitors are profit maximizers that they themselves are attempting to set optimal bids. (Rather, it simply takes the bid distribution as given.) 7 To check this formula, note that G is.5 at the distribution median, $320,000. At a bid of $330,000, G is.75 as expected. For a uniformly distributed value (as here), G = (b L)/(U L), where L and U denote the lower and upper limits of the distribution, respectively. 8 The material in this section is advanced and can be skipped without loss of continuity.

14 518 Chapter 17 Auctions and Competitive Bidding A key point is the significant element of interdependence among bidding strategies. As we saw earlier, one firm s optimal bid depends on the number of competitors and how those competitors are expected to bid. For instance, higher bids from competitors may call for higher bids from the firm itself. Recall that in Chapters 9 and 10 we introduced the concept of equilibrium strategies in the context of oligopoly competitive interdependence. Equilibrium analysis is equally applicable to sealed competitive bidding. Firms strategies constitute an equilibrium if each firm is profit maximizing against the behavior of the others that is, if there is no opportunity for any firm to make a profitable unilateral deviation from its current bidding strategy. The simplest example of equilibrium bidding occurs when buyers compete for a good with a known, common value. For example, suppose all bidders have the same reservation price for the office building, say, $348 thousand (and all recognize this as the common value). The unique equilibrium has each bidder submitting a sealed bid exactly equal to this common value, so this value becomes the final price. The seller obtains full value for the item. Any set of bids with the high bid below $348 thousand is not in equilibrium because any one of the losing bidders can increase its profit by slightly topping the current high bid. 9 Profit-increasing deviations are exhausted when bids match the item s full value. With this simple observation in hand, let s examine equilibrium bidding in the case of differing private values. Again, there are n bidders, with bidder i holding value v i and placing sealed bid b i. Buyer values are drawn independently from a common distribution; that is, each buyer s value comes from the common, cumulative probability distribution F(v). To illustrate, consider the office building example and assume for the moment that there are only two bidders. Suppose each buyer s value is uniformly distributed between $300 thousand to $360 thousand. (This means that all values in this range are possible and equally likely.) Furthermore, bidder values are independent of one another. Knowing its own value, but not knowing its opponent s, how should each buyer determine its optimal bid? The answer is provided by the equilibrium bidding strategy b i = (.5)(300) +.5v i, where values and bids are measured in thousands of dollars. Using this strategy, a buyer with value $300 thousand bids $300 thousand; with value $340 thousand it bids $320 thousand; and with value $360 thousand (the maximum value) it submits a maximum bid of $330 thousand. In short, the buyer bids a price midway between its true value and the lowest possible value (here, $300 thousand). Let s check that this strategy constitutes an equilibrium. Consider a typical firm (say, firm 1) whose expected profit is E(π) = [v 1 b 1 ][Pr(b 1 wins)]. [17.1] 9 In the real world, prices below full value would be temporary at best. In repeated auctions, losing bidders would certainly raise their bids, seeking to claim any positive profit. These upward bid adjustments cease when there is no longer any bid profit available, that is, when all buyers are bidding full value.

15 Bidder Strategies 519 The competing bids of its rival, firm 2, are distributed uniformly between $300 thousand and $330 thousand (since firm 2 is presumed to employ the equilibrium bidding strategy just shown). Thus, bid b 1 wins with probability (b 1 300)/30. For instance, a bid of $300 thousand never wins, a bid of $315 thousand wins with probability.5, and a bid of $330 thousand wins with certainty. Substituting the probability expression into Equation 17.1 yields E(π) = (v 1 b 1 )a b b. [17.2] 30 The bidder s marginal profit is Mπ = de(π)/db 1 = [v 1 2b ]/30. Setting this equal to zero implies b 1 = (.5)(300) +.5v 1. [17.3] Note that the firm s expected profit is zero for the extreme bids, $300 thousand (which never wins), and v 1 (for which the firm makes no profit). The firm s optimal bid is halfway between these extremes. Thus, we have confirmed that the suggested equilibrium strategy is indeed optimal for the typical firm and for any value the firm might hold. Two firms compete in a sealed-bid auction, with each firm s private value uniformly distributed between $0 and $50. (i) Suppose firm 2 is expected to use the (nonequilibrium) bidding strategy, b 2 =.6v 2. Confirm that firm 1 s optimal response is b 1 =.5v 1. (In other words, firm 1 still does best by bidding one-half its value as in Equation 17.3.) (ii) Against b 2 =.4v 2, show that firm 1 s optimal sealed-bid strategy remains b 1 =.5v 1. CHECK STATION 3 How are equilibrium bidding strategies affected by changing the number of competing firms? Let buyer values be uniformly and independently distributed between lower and upper bounds, denoted by L and U, respectively. (In the preceding example, L is $300 thousand and U is $360 thousand.) Then the common equilibrium strategy when n firms compete is simply b i = a 1 1 bl + an n n bv i. [17.4] The firm s equilibrium bid is a weighted average of the firm s actual value and the lowest possible bidder value. Note that, as the number of bidders increases, the equilibrium bid rises and comes closer and closer to the firm s value. Holding v i = $348 thousand, the firm bids $324 thousand when it is one of two bidders, but $342 thousand when it is one of eight bidders Here is how to check Equation 17.4 s equilibrium strategy. With this strategy, each buyer s bids are in the range L to (L/n) + [(n 1)/n]U. Therefore, bid b i wins with probability [(b i L)/ ((n 1)(U L)/n))] n 1. Thus, the firm s expected profit can be written in the form of k(v i b i ) (b i L) n 1, after collecting miscellaneous constant terms into the coefficient k. Therefore, marginal profit is k[(n 1)(v i b i )(b i L) n 2 (b i L) n 1 ]. Setting this equal to zero and canceling out the common factor, (b i L) n 2 yields Equation Thus, we have confirmed that using the proposed strategy is an equilibrium.

16 520 Chapter 17 Auctions and Competitive Bidding In fact, there is a very simple rule describing equilibrium bidding strategies with any number of bidders and any common probability distribution: In a sealed-bid auction, the equilibrium bidding strategy of the typical risk-neutral buyer is to submit a bid, b i, equal to the expected value of the highest of the n 1 other buyer values, conditional on these values being lower than v i. Formally stated, b i = E(v ƒ v v i ), where v denotes the largest of the other bidders personal values. Since this bidding rule is something of a mouthful, a concrete example is useful. Consider, once again, the office building auction this time with two buyers. Suppose firm 1 s value is v 1. Since firm 1 knows its opponent s value is uniformly distributed between $300 thousand and $360 thousand, the distribution of this value, v, conditional on v being smaller than v 1, is uniform between $300 thousand and v 1. Therefore, the conditional expected value of v is simply (.5)(300) +.5v 1 the average of $300 thousand and v 1. This is the firm s best bid. But this is exactly the equilibrium bidding strategy depicted in Equation More generally, if there are n 1 other bidders whose personal values are distributed uniformly between 300 and v 1, the expectation of the greatest of these values is 300/n + [(n 1)/n]v 1. This confirms Equation 17.4 s equilibrium strategy. 11 Although there is no easy intuitive argument to explain the preceding bidding rule, one comment is in order. The key to crafting an optimal bid is to assume for a moment that yours is the highest value. (If this is not the case, the buyer will be outbid by a rival anyway.) Accordingly, the buyer bases its bid on the expected value of v, conditional on v being smaller than v 1. To sum up, we have examined two approaches to finding the firm s optimal bidding strategy. The first approach takes the distribution of opposing bids as given and asks what is the firm s profit-maximizing bid strategy in response. The equilibrium approach starts from a prediction concerning the underlying values of the firms and identifies bid strategies such that each buyer is profit maximizing against the bidding behavior of its competitors. By the way, the BCB distribution in Figure 17.1 is derived from equilibrium bidding behavior among three bidders, each with a value distributed uniformly between $300 thousand and $360 thousand. Here, the common equilibrium bidding strategy is b i = (1/3)(300) + (2/3)v i. In turn, competing bids range from $300 thousand to $340 thousand. Therefore, the BCB distribution is given by: H(b) = [(b 300)/40] 2 ; this is graphed as the H curve in Figure Common Values and the Winner s Curse Frequently, bidders are uncertain about the true value of an item put up for competitive bid. For instance, the United States periodically sells offshore oil tracts via sealed-bid auctions. The value of any tract is highly uncertain, depending on whether oil is found, at what depth and at what cost, and future oil prices. Except for differences in costs, the 11 For uniformly distributed values, the equilibrium bidding strategy can be expressed in a neat formula. This is not the case for many other distributions, such as the normal distribution. However, tables of conditional expected values for many distributions are readily available.

17 Bidder Strategies 521 FIGURE 17.2 The Winner s Curse Bids (B) Estimates (E) When the value of the item is highly uncertain, a winning bid drawn from the right tail of the bid distribution may exceed the true value of the item. B V profit from the tract is likely to be similar across firms. Thus, to a greater or lesser degree, the tract has a common value for all bidders. The difficulty is that this value is unknown. In making its bid, typically the individual firm first will form an estimate of the tract s potential value. Obviously, this estimate will be subject to error, and firms may hold very different estimates of value. How should a profit-maximizing firm bid in this situation? Figure 17.2 depicts bidding behavior when the item for bid has a common unknown value. The true (but unknown) value is labeled V. Centered at V is the distribution of possible bidder estimates (curve E). The figure depicts a normal distribution of estimates. On average, estimates reflect the true value, V, but there is considerable dispersion. Some buyers underestimate and others overestimate the true value. The figure also depicts a typical distribution of bids (curve B). Obviously, each bidder, seeking a profit, submits a bid well below its estimate of value. This explains why the bid distribution is centered well to the left of (i.e., below) the estimate distribution. Here is an important observation to draw from Figure A winning bid drawn from the right tail of the bid distribution may exceed the true value of the item. The shaded tail shows the portion of the distribution of bids that exceeds the true value. For a bid in this region, the buyer is said to fall prey to the winner s curse. After the fact, the auction winner finds that the good obtained is worth less than the price paid for it. The source of the winner s curse lies in the fact that the winning bidder has been too optimistic and has grossly overestimated the good s value. When the firm s (upward) estimation error exceeds its bid discount, it buys the good at a price greater than its value. For instance, suppose the true value of the oil tract is $2 million, but the most optimistic bidder believes it s worth $3 million. Hoping to win the lease at a profit, this firm bids $2.3 million (discounts its estimate by $.7 million) but still ends up overpaying and experiencing a $.3 million loss on the tract.