Favoritism in asymmetric procurement auctions

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International Journal of Industrial Organization 26 (2008) 1407 – 1424 www.elsevier.com/locate/ijio

Favoritism in asymmetric procurement auctions ☆ Joon-Suk Lee ⁎ Department of Economics, Bowdoin College, 9700 College Station, Brunswick, ME 04011, USA Received 24 October 2006; received in revised form 29 January 2008; accepted 10 February 2008 Available online 18 April 2008

Abstract I examine the costs and benefits of granting a right of first refusal (ROFR) to one bidder in a first-price procurement auction with two bidders. This right permits the favored bidder to win a contract by matching the bid of the competing bidder and is often observed in procurement auctions. I show that the auctioneer prefers to grant the ROFR to the ex-ante weak bidder and that granting this right can increase the auctioneer's expected payoff. The results continue to hold even when the auctioneer can set an optimal reserve price. Both the reserve price and the right of first refusal serve to elicit more aggressive bids and hence, to a certain degree, are substitute tools. © 2008 Elsevier B.V. All rights reserved. JEL classification: D44; D82; C72 Keywords: Auctions; Favoritism; Right of first refusal

1. Introduction The mention of “favoritism” generally evokes negative images of corruption and cronyism. As common thinking goes, only the favored party gains while everybody else loses. I show in this paper that this is not always the case: There are cases in which not only the favored party wins but the auctioneer wins as well.

☆

For very helpful comments and discussions, I am much indebted to the Editor, an anonymous referee, Gary Biglaiser, Claudio Mezzetti, James Friedman, Sergio Parreiras, Sandra Campo, as well as Andreas Dische and Wolfram Schlenker. All remaining errors are my own. ⁎ Tel.: +1 207 725 3732. E-mail address: [email protected]. 0167-7187/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2008.02.002

While favoritism takes on multiple forms, such as outright preference, subsidies, credits or special quotas, I focus on a mechanism known as the “right of first refusal” (ROFR). The “right of first refusal” allows the favored bidder to win the contract by matching the best bid of the competing bidders. The ROFR is interesting because it is a mechanism that is simple, both to implement and to observe, and hence is widely used as a form of favoritism. From the point of the view of the bidders the ROFR is a transparent policy (no subjective adjustment of bids) and easy to take into consideration when bidding. From the point of the view of the auctioneer the ROFR is an attractive mechanism because, its application requires a minimum of information about the bidders' characteristics and involves the setting of a single binary parameter: whom to favor.

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The practice of granting a ROFR is often found in real estate transactions,1 entertainment contracts,2 business partnerships3 and sports contracts. The ROFR in these cases gives one or each contracting party the right to purchase the property or business interest before it can be offered to anyone else. For instance in the National Football League (NFL) a player becomes a “restricted free agent” upon expiration of his initial contract. The restricted free agent can negotiate a new contract with another team, but the player's original team has the right to match the other team's offers to retain the player.4 While in the above mentioned cases the ROFR is an explicitly stated right, quite often it is extended implicitly as well. For instance, in 2003 Pratt & Whitney, a leading manufacturer of jet turbines for airplanes, cried foul when it lost out in a bid to supply Airbus Industries with engines for the production of the Airbus military transportation plane A400M. 5 The company that ultimately was awarded the contract was EuroProp International, a European consortium of engine makers. For much of the bidding process Pratt & Whitney was confident to win the contract, as all sources indicated that its bid was by far the most competitive. However, last minute interventions by European governments led to Airbus' concession that EuroProp was allowed to revise its bid. This revised bid reportedly matched Pratt & Whitney's offer and ultimately won the contract. One may interpret the fact that Airbus allowed EuroProp to resubmit its bid while no such option was given to Pratt & Whitney as implicitly granting a ROFR to a favored (domestic) bidder. Given the politically sensitive nature of contract, it is fair to assume that from the outset both bidders were aware of the possibility that the European government would intervene on behalf of “their” firms. Similarly, when the bankrupt South Korean brewery Jinro was put up for sale in 1999, the bidding process was also marred by last minute interventions.6 Oriental 1

Either as a contractual clause in residential and commercial property contracts as described in Walker (1999) part I.B. or by legal statute such as the Tenant Opportunity to Purchase Act (TOPA) in the District of Columbia. TOPA requires the owner of rental property to grant a ROFR to the current tenant, if the owner wants to sell the property. 2 Grosskopf and Roth (in press) describe the renegotiations of the contract between NBC and Paramount Studios for the broadcasting rights of the TV show “Frasier”. The contract included a ROFR provision. 3 As a common provision for the eventual dissolution of a business partnership (see Brooks and Spiers, 2004). 4 Smith, Timothy, “N.F.L. Is Dusting Off Its Shield For Another Free Agency Battle”, New York Times, June 14, 1992. 5 Lunsford, J. Lynn, “Airbus Bypasses Pratt & Whitney, Keeps Engine Contract in Europe”, Wall Street Journal, NY, May 7, 2003. 6 Modern Brewery Age, “Coors in battle for bankrupt Jinro Ltd.”, July 12, 1999, vol. 50, no. 28, pp. 1–2.

Brewery, a domestic bidder apparently favored by the seller, learned the terms of the bid made by Coors and submitted a second bid, which was accepted even though the original deadline for bids had already expired. Or, consider Carnival Corporation, a shipping cruise firm, soliciting bids for the contract to build the Queen Mary II ocean liner in 2000.7 Having chosen two finalists, it allowed one of them, Harland & Wolff, a troubled but tradition-rich British shipyard, to revise its bid. Harland & Wolff in fact had built the predecessor ship Queen Mary some 70 years ago and Carnival had stated publicly that for sentimental reasons it was predisposed to choose Harland & Wolff as long as they offered a competitive bid. Harland & Wolff failed to meet the bid put forward by its rival, Chantiers de L'Atlantique, and ultimately failed to win the contract. This led to suggestions that Carnival may have used the yard's bid merely to put pressure on Chantiers de L'Atlantique.8 In all these cases some of the ostensible motivation for granting a ROFR is political: to satisfy some notion of fairness (if say a long-term tenant or a small disadvantaged business is involved) or to simply reward longterm business partners or political allies. However, as the example of Harland & Wolff hypothesizes, granting a ROFR may also have economic motivations. This may be particularly true, if the competing bidders are unequal in their bidding strengths and the ROFR is given to the weaker bidder. In the Airbus example, the favored EuroProp consortium lagged Pratt & Whitney substantially in market share, while in the Queen Mary II case the competition between Harland & Wolff and Chantiers de L'Atlantique pitted a financially near-bankrupt shipyard with no recent experience in building ocean liners against a proven maker of cruise ships.9 The literature on optimal auctions, such as Myerson (1981), supports the idea that the ROFR can serve as a tool to extract higher payoffs by levelling the playing field between competitors. It states that the auctioneer does best when subsidizing bidders who are more likely to draw unfavorable private values. The intuition here is that by levelling the playing field, the auctioneer is able to elicit more aggressive bids from the otherwise stronger bidder. While the ROFR proves to be a rather crude instrument, that lacks the ability to fine-tune the The Irish Times, “Cunard gives Harland & Wolff extra time to revise bid for ‘Queen Mary’ project”, March 7, 2000, p.17. 8 The Guardian, “Harland & Wolff locks horns with DTI”, March 11, 2000. 9 Based on sales data from Vlachos-Dengler, Katia, “From National Champions to European Heavyweights: The Development of European Defense Industrial Capabilities Across Market Segments”, RAND Corporation, 2002, p. 111 and The Irish Times, ibid. 7

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degree of subsidization, it is nevertheless capable to imperfectly implement the idea of handicapping the stronger bidder. The simplicity of the mechanism, both in terms of information requirements and implementation, makes it an attractive candidate for handicapping. I capture this intuition by formulating a model with a single buyer and two asymmetric sellers, who compete against each other in a first-price auction with independent private values. The buyer has the option of granting a ROFR to one of the sellers beforehand. My model confirms the levelling of the playing field notion on two levels. When the buyer is required to grant a ROFR, he will always prefer to favor the weak bidder. In fact, I show that for lower levels of asymmetry between the bidders granting a ROFR to the weak bidder induces the strong bidder to behave as if he was facing an equally strong bidder. When the buyer has complete discretion over whether or not to offer a ROFR, he will only do so if the asymmetry is sufficiently large. The ROFR proves to be too powerful a tool when the two bidders do not differ much in the cost distribution. In such a situation the ROFR overshoots in tilting the competition in favor of the weak right-holder who extracts rents that would otherwise have gone to the auctioneer. While the strong (non-favored) bidder, as mentioned before, acts as if facing a symmetric opponent and therefore bids more aggressively than otherwise, the favored weak bidder will take advantage of his right and pursue a “wait and see” strategy. He will initially only place a token bid and later decide whether or not to match his opponent's bid. As the degree of asymmetry increases, the gains that the buyer makes from neutralizing the strong bidder's original advantage becomes larger and larger. At the same time, the weak bidder will become less capable of actually exercising his right, because increasingly he would lose money if he matched his strong opponent's bid. Eventually, granting a ROFR does manage to provide a balanced adjustment of the bidding strengths of the two bidders. This rationale mostly carries over to the case in which the auctioneer additionally has the freedom to set the reserve price. The ROFR still proves to be too heavy of an instrument when differences between the bidders are small. However, in this setting the reserve price can serve as a tool to not only to increase expected surplus but also to counterbalance the effect of a ROFR. In fact, I show that the reserve price and ROFR exhibit somewhat substitutable characteristics with regard to influencing expected surplus. For high degrees of asymmetry, the reserve price completely displaces the ROFR as the tool of choice for the auctioneer as the ROFR by itself proves

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to be too weak to offset the asymmetry. For low degrees of asymmetry, as mentioned, the ROFR proves to be actually harmful, so the auctioneer prefers to rely on setting the reserve price only. Only for intermediate degrees of asymmetry can the auctioneer combine the two policy tools in a non-conflicting way. Much of the interaction between the reserve price and the ROFR can be understood by the impact of the reserve on the effective degree of asymmetry that the two bidders face. I show that endogenously setting the reserve price exacerbates the extent of the asymmetry which for low to intermediate degrees of asymmetry offsets the effect of a ROFR. Before proceeding, a brief discussion of the scope of the paper seems appropriate: The paper focuses its attention on the first-price auction with independent private values that are drawn from the uniform distribution for reasons of analytical tractability: It is well known (see Marshall et al., 1994 or Maskin and Riley, 2000) that in the asymmetric case of the first-price auction, bidding functions are characterized by a system of ordinary differential equations that most often cannot be solved analytically, unless some restrictions are put on the form of the type distribution. The uniform distribution allows me to obtain closed-form solutions, but makes it difficult to know how the findings extend to other model settings. Also, I restrict myself to the two-bidder case, which as the introductory examples show is a frequent scenario in actual procurement auctions, especially for ones of large magnitude. While generalizing the model to an n-bidder setting would seem like a natural extension it unfortunately leads to significant issues of analytical tractability, especially. Given the problems with tractability much of the literature on asymmetric n-bidder auctions such as Lebrun (1999) or Reny and Zamir (2004) has been focusing on existence. Cantillon (2005) is an exception that looks at the effect of bidder asymmetry on expected revenue, but much of her analysis is on the more tractable second-price auction. For the first-price auction she limits herself to the two-bidder case. As mentioned earlier, the ROFR proves to be a rather crude instrument and thus (in combination with the fixed choice of the auction format) cannot inform about an optimal mechanism. However, because of its simplicity and hence frequent use in practical situations the analysis of the ROFR provides a useful reference point in studying favoritism. Given these restrictions on the model this paper contributes to the understanding of favoritism in the asymmetric auction setting by providing results and intuition for an important benchmark case and the interrelationship between the ROFR and the reserve price.

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This paper is related to the literature on preference in asymmetric auctions and on the right of first refusal. The effects of granting a ROFR have been studied by Bikhchandani et al. (2005), Choi (2003), Manelli and Vincent (2003) and Arozamena and Weinschelbaum (2004). Bikhchandani et al. (2005) and Choi (2003) all look at symmetric auctions with a ROFR granted to a bidder. They all come to the conclusion that granting a ROFR is costly for the seller. They differ from this paper in that they look at settings with symmetric bidders and additionally allow for correlation in the bidders' private values. Manelli and Vincent (2003) discusses a ROFRlike feature in his paper on procurement design. His focus however is in determining conditions for social optimality rather than profit maximization or cost minimization as in this paper. Arozamena and Weinschelbaum (2004) also look at a symmetric auction setting, but additionally allow the ROFR to be granted with a probability of less than 1. Their work however focuses on the effect of favoritism on the bidding behavior rather than the auctioneer's expected benefit, as it leaves open whether the auctioneer is compensated in return for the preferential treatment granted. Rothkopf et al. (2003) as well as, Koc and Neilson (2004) and Burguet and Perry (2005) have explored issues regarding preferential treatment in asymmetric settings. Rothkopf et al. (2003) analyze asymmetric auctions in which preferential treatment comes in the form of adjusted bids. They find that a wide range of parameters that offering some degree of preferential treatment generally is beneficial to the auctioneer. Koc and Neilson (2004) look at an asymmetric auction setting, where the auctioneer offers each bidder preferential treatment in return for a bribe. Unlike in my model the preferential treatment is non-exclusive, that is available to any bidder willing to pay the bribe, and only takes effect when the paying bidder actually places the most competitive bid. Such a bidder only needs to pay the second highest bid. Burguet and Perry (2005) shares the most similarities with this paper in that they consider a private value asymmetric procurement auction where the auctioneer can decide to grant a ROFR. They make different assumptions about the cost distribution (relying on the family of power distributions), which leads them to conclude that granting a ROFR for free never benefits the auctioneer. However, they show that the auctioneer may gain if he auctions off the ROFR to the highest paying bidder beforehand. Because of the similarity of their model and difference in conclusions between my findings and their conclusions I provide some more detailed intuition in Section 3. I show how the con-

clusion is sensitive to the different assumptions made about the cost distributions and that the common conclusion that favoritism cannot benefit the auctioneer does not universally hold.10 The remainder of this paper is organized as follows: In Section 2 I describe the details of the model with asymmetric bidders and private values. Then, in Section 3 I first derive the equilibrium bidding functions for the model with no reserve price. Using the equilibrium bidding functions I obtain closed-form expressions for the expected payment when the buyer does or does not grant a ROFR. By comparing the expected payments under either regime, I characterize conditions that make it beneficial for the buyer to grant a ROFR. In Section 4 I allow the buyer to set the reserve price. Given an endogenous reserve price, I derive equilibrium bidding functions both when the auctioneer offers a ROFR and when he declines to do so. Again, I find closed-form expression for the expected payments by the buyer and compare them. Based on this comparison I describe the conditions under which a ROFR benefits the buyer. Finally, Section 5 summarizes the findings and concludes. 2. Model Suppose there is one buyer or auctioneer and two asymmetric sellers or bidders, weak (w) and strong (s). The buyer demands a single unit of a good, which he values at 1. The sellers are capable of providing the good at a cost θi, where i = w, s. The two sellers face each other in an asymmetric independent private value auction: Each bidder has private knowledge of his cost type θi. It is common knowledge that the cost types θi are distributed independently and uniformly on the (normalized) supports of [θ, 1] for the weak bidder and [0, 1] for the strong bidder, with Fi and fi being the distribution and density functions for each bidder. Notice that the support for the weak bidder is a subset of the support of the strong bidder. It is assumed that θ ∈ (0, 1), or that the weak bidder's cost distribution is first-order stochastically dominating and hence is the less efficient producer. Intuitively, it is convenient to interpret θ as the degree of asymmetry between the two bidders. The auctioneer can choose to grant a right of first refusal to one of the two bidders, which will become common knowledge. The sequence of actions is as follows: first, the auctioneer publicly announces whether to grant

10 The idea that the auctioneer may want to sell the ROFR to a bidder also can be found in Compte et al. (2000) who look at the effect of corruption on competition in government procurement auctions.

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one of the bidders a ROFR and if so, identifies the favored bidder. Then the two bidders submit bids in a first-price sealed bid auction (FPA). If no ROFR was granted, the winner of the auction is awarded the contract and is paid his bid. If a ROFR was granted and the favored bidder has the lowest bid, the favored bidder is awarded the contract and receives his own bid as payment. If the non-favored bidder submitted the best bid, the favored bidder is offered the opportunity to match the bid of the non-favored bidder and be awarded the contract. If the favored bidder declines to match, the non-favored bidder is awarded the contract and receives his own bid as payment.

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The expected payment in a first-price auction without favoritism is given by E ðPayment; noÞ ¼

2b ks ð1  h Þð1  b2 ks þ 2bks  ks Þ P

P

P

P

2b ks ð1  hÞð1 þ b2 ks  2 bks þ ks Þ pffiffiffiffiffiffiffiffi 2 arc tan ð ks ð b  1ÞÞ pffiffiffiffiffiffiffiffi  ks ks ð1  hÞ pffiffiffiffi 2 arc tan ð ks ð b  1ÞÞ pffiffiffiffi þ ks ks ð1  hÞ



P

P

P

P

P

P

P

P

where b ¼ 21 h .

3. No reserve price

P

P

Proof. See Appendix A. For the discussion of the model in this section, I assume that the auctioneer sets no reserve price. In Section 4, this assumption will be relaxed and the auctioneer will be able to set the reserve price r optimally. I derive conditions under which granting a ROFR benefits the auctioneer. This is done by comparing the expected payment by the auctioneer when assigning a ROFR to one of the bidders and when not offering one at all. I begin by deriving the benchmark case when the auctioneer does not favor either bidder. Obtaining the bidders' equilibrium bidding functions, allows me to generate the expression for the expected payment and to show that the larger the asymmetry, the larger the expected payment. I then proceed to derive the corresponding equilibrium expressions when assigning a ROFR to either bidder. It is shown that the auctioneer will always prefer to favor the weak bidder, if he is forced to assign a ROFR. However, assigning a ROFR is optional: He will only do so when asymmetry is sufficiently large.

□

Notice that the constants kw and ks equal 0 when the bidders are completely symmetric, θ = 0. In that case the equilibrium bidding functions are identical: θw(b) =θs(b) = 2b − 1 or b ¼ hþ1 2 . As θ increases, kw increases and ks falls. Compared to the symmetric case, a weak bidder therefore bids more aggressively (each 6 cost type places a lower bid if asymmetry increases) while a strong bidder bids less aggressively (also see Maskin and Riley, 2000 for general characteristics of bidding strategies in asymmetric first-price auctions). The figure below plots the expected payment against the level of asymmetry θ. Looking at Fig. 1 below it is clear that expected payment increases strictly in the degree of asymmetry θ. The intuition is as follows: any increase in asymmetry induces the weak bidder to bid more aggressively and lower his mark-up. A strong bidder knowing that his opponent is becoming weaker (because θ has increased) bids less aggressively and increases his mark-up. As asymmetry increases the scope of reducing the mark-up

3.1. No ROFR granted to either bidder I find for the equilibrium bidding functions for both bidders that: Proposition 1. If the auctioneer does not favor either bidder, then the unique inverse equilibrium bidding functions of the two bidders are hw ð bÞ ¼ 1 þ hs ðbÞ ¼ 1 þ

2ð b  1Þ 1 þ k w ð b  1Þ 2 2ð b  1Þ

and

1 þ k s ð b  1Þ 2

where kw ¼ ks ¼

hP2 þ2 Ph

ð1 h Þ P

2

. Fig. 1. Expected payment with no ROFR.

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becomes smaller and smaller for the weak bidder. These gains to the auctioneer get outweighed by the increasingly less aggressive bids of the strong bidder.

underbid his opponent and take full advantage of the difference in cost. This is because he is certain that the favored bidder will decline to match his bid. This can only occur if θ is sufficiently low, such that hþ1 2 b hi. Specifically this is the case, if the weak bidder is favored and the asymmetry between the two bidders is large enough such that θ N 0.5. □ P

3.2. Mandatory assignment of a ROFR Suppose now the auctioneer is forced to assign a ROFR to one of the two bidders. I first derive the equilibrium bidding functions for both the favored and the non-favored bidder and show why the auctioneer would assign the ROFR to the weak bidder, if he was required to grant such a right. The equilibrium bidding functions are given by, Lemma 1. In a first-price auction with ROFR the favored bidder i will always bid 1. The non-favorite −i's  equilibrium bidding function will be bi ðhÞ ¼ max hþ1 h i Þ, 2 ;P where θi is the lower end of the support of the favored bidder's cost distribution. Proof. First I show why the favored bidder has a weakly dominant strategy of bidding 1: If a favorite had bid b b 1 and won, then he could have done better by bidding more than b. If the non-favorite's bid b− i has been equal or less than b an increase in b doesn't change any payoff, because the favorite will always receive the offer to win by matching b− i. If b− i N b, then there is scope for increasing the winning bid b and thereby obtaining a higher payoff.11 The non-favored bidder's objective is to maximize:    1  b1 max ðbi  hÞ  min ;1 : 1  hi P

He maximizes the gross profit, given by the difference between his bid b− i and his cost θ, multiplied by the probability that he actually wins the contract, which is i equal to the probability 1b 1 h i that the favorite declines to 12 match the bid. The solution to the first-order condition of this problem b⁎− i provides the optimal bid as long as b⁎− i N θi, where θi is the favorite's lowest possible cost type. For sufficiently large values of θ the solution of the first-order condition is optimal and the non-favored bidder will mark-up his bid and bid hþ1 2 . Note that the amount of shading is independent of the characteristics of his opponent's cost distribution. If b⁎− i b θi the non-favored bidder can improve his profits by raising his bid. Hence he will just barely P

11

Note, how the rationale for bidding 1 is reminiscent of the reasoning for bidding one's own value in a standard second-price auction. 12 The min(.) function is to ensure that the probability that for b− i b θi the probability does not exceed 1.

Since the price that the auctioneer will pay is always the non-favored bidder's bid (paid either to the favored bidder if he matched the bid or the non-favored bidder if the favorite declined to match the bid), the expected payment by the auctioneer is solely determined by the non-favorite bidding function. I now show that declining to favor the strong bidder is preferable for the auctioneer, because the auctioneer can expect lower bids from a strong non-favored bidder rather than a weak one. The structure of the non-favored bidder's bidding function makes it necessary to compare the expected payments for two different cases: If asymmetry between bidders is low, that is θ ≤ 0.5, then the non-favored bidder follows bi ðhÞ ¼ hþ1 2 . The expected payment by the auctioneer when favoring bidder i is given by Z EðPayment; iÞ ¼

1 h i

bi ðhÞfi ðhÞdh

P

R

where − i, i = s, w. We find that E ðPayment; wÞ ¼ 01 1þh 2 R 1 1þh 1 1 h2 3 3 2ð 2  hþ2Þ dh¼ 4, while E ðPayment; sÞ¼ h 2 1h dh ¼ 1 h . The expected payment by the auctioneer is completely independent of the degree of asymmetry if the auctioneer favors the weak bidder. This is a result of a (strong) nonfavorite's bidding function that is independent of the characteristics of his opponent's cost distribution and the fact that a (strong) non-favorite's cost distribution is unaffected by the change in asymmetry. On the other hand, if the auctioneer favors the strong bidder then the degree of asymmetry will affect expected payment. The distribution of the non-favored (weak) bidder's bids is affected by any change in asymmetry and hence the expected payment reflects any changes in asymmetry as well. Comparing expected payments, I find E(Payment, s) − E(Payment,w) N 0 if and only if 14 hN0, which is true since θ N 0. Hence it is always more costly to favor the strong bidder and hence the auctioneer chooses to favor the weak bidder. If asymmetry between bidders is high, such that θ N 0.5, then a strong non-favored bidder bids bi ðhÞ ¼ hþ1 2 only some of time. If his cost type is θ b 2θ − 1, he will bid b− i(θ) = θ. This way he is able to take full advantage of his lower cost type as mentioned above. P

P

P

P

P

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Hence, the expected payment by the auctioneer if the weak bidder is favored, changes to Z 2 h1 Z 1 1þh dh E ðPayment; wÞ ¼ h dh þ 2 0 2 h1 P

P

P

¼ h2  h þ 1: P

P

As asymmetry increases, a larger portion of strong bidder can mark-up their bid to θ. Given that this minimum bid itself rises with greater asymmetry, expected payment increases quadratically. The expected payment by the auctioneer if the strong bidder is favored, E(Payment, s), remains unchanged, since a weak bidder always bids according to bi ðhÞ ¼ hþ1 2 . If I compare the expected payments from favoring either one bidder,2 I find that E(Payment,s) N E(Payment,w) N 0 because h  1 3 2ð 2  hþ2Þ N h2  h þ 1Þ. This condition simplifies to 1 h   1 hÞ h  14ÞN0, which is equivalent to θ N 0.5. Hence the expected payment is higher when favoring the strong bidder whenever the weak bidder is substantially weaker as characterized by θ N 0.5. Summarizing the results of the preceding paragraphs, I have hence shown that: P

P

P

P

P

P

P

Proposition 2. In an auction with mandatory assignment of a ROFR the auctioneer always prefers to grant the weak bidder w a ROFR. The (lowest) expected payment by the auctioneer is given by ( 3 1 4 ; if h V 2 : E ðPayment; wÞ ¼ 2 h  h þ 1; if h N 12 P

P

P

P

g

The intuition for this result closely follows what is known from the literature on optimal auctions, such as Myerson (1981). The auctioneer stands to gain from favoring the weaker bidder, because he is able to level the playing field. Eliciting more aggressive bids from the strong bidder by favoring his weak opponent yields more gains than vice versa, because the strong bidder has more scope for bidding more aggressively. In particular in this paper where the favoritism comes in the form of a ROFR, the expected payment is entirely determined by the bid of the non-favored seller. Declining to favor the strong bidder, who is the bidder with potentially lower cost, hence yields more low bids and a lower overall expected payment by the auctioneer.13

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In fact, the non-favorite's bidding function closely resembles the equilibrium bidding function for a symmetric (θ = 0) auction with no ROFR. As shown earlier, in such an auction the equilibrium bid is hþ1 2 . This is the same as a weak non-favorite or a strong non-favorite with low degrees of asymmetry. It implies that, at least for low degrees of asymmetry, granting a ROFR essentially achieves to induce the (strong) non-favored bidder to act as if he was facing an equal opponent. Hence, the auctioneer benefits when declining to favor the bidder who has the more favorable cost distribution. 3.3. Comparing expected payment with and without ROFR While I have determined that giving a ROFR to the weak bidder is the optimal choice when forced to grant a ROFR, I demonstrate that not granting a ROFR at all is even better for the auctioneer when asymmetry is low. To compare the expected payments of the two scenarios when the auctioneer grants or declines to grant a ROFR, see Fig. 2 below and the accompanying table in the appendix (Table A.1): Looking at Fig. 2, one sees that for values of θ b 0.419 the auctioneer prefers not to grant a ROFR, while for higher degrees of asymmetry he does better when favoring the weak bidder. The benefits from granting a ROFR are highest for medium degrees of asymmetry and tend to diminish as bidders become lopsidedly asymmetric. Furthermore, the largest reduction that the auctioneer can achieve by employing favoritism is about 5%. On the other hand, the benefits from foregoing any favoritism are largest when the bidders are symmetric or almost symmetric. I summarize the finding as

13

It is important to note that the ROFR can only effect a level playing field in a first-price auction setting. As noted earlier, from the point of view of the favored bidder, the ROFR transforms the auction into a situation reminiscent of the second-price auction for the favored bidder. Hence, in a secondprice setting the ROFR provides no actual benefits to the favored bidder — all bidders, favored or not, will continue to bid their own cost type.

Fig. 2. Expected payment with and without ROFR (no reserve price).

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Proposition 3. When the auctioneer has the option to assign a ROFR, he will only do so for degrees of asymmetry θ N 0.419. In that case, the auctioneer gains by favoring the weak bidder. To get a better intuitive sense of the results, it is helpful to consider the effect of an increase in asymmetry on the expected payment with or without a ROFR. In both settings an increase in asymmetry leads to higher expected payment. With a ROFR the increase in expected payment is delayed until asymmetry is sufficiently large, while without a ROFR asymmetry immediately impacts expected payment. Therefore rising asymmetry affects expected payment more directly without a ROFR. With a ROFR, the expected payment with a ROFR is unaffected by an increase in asymmetry (as long as PhV 12) because the non-favored (strong) bidder cannot exploit his cost advantage — the ROFR granted to his opponent induces him to bid as if he was facing an equally strong Only once asymmetry becomes sufficiently large bidder. 1 hN Þ 2 will the non-favored (strong) bidder start to P exploit the fact that in some cases he is the only possible bidder. Without a ROFR, on the other hand, any increase in asymmetry makes the strong bidder bids less aggressively, knowing that his (weak) opponent is becoming weaker (because θ has increased). Conversely, the weak bidder bids more aggressively as θ increases in order to compensate for his weakness. For any degree of asymmetry the benefits to the auctioneer from the more aggressive bids by the weak bidder are outweighed by the less aggressive bids from the strong bidder. Furthermore, any increase in asymmetry exacerbates these net losses to the auctioneer. Hence expected payment rises with asymmetry. Granting a ROFR eliminates the losses due to the lack of aggressiveness by the strong bidder at the price of subsidizing the weak bidder. At low levels of asymmetry this is an unfavorable trade-off for the auctioneer, while at higher levels of asymmetry this is beneficial. At this point, it is useful to contrast the above results with the findings of Burguet and Perry (2005). As mentioned in the introduction, they use the same framework but assume a different class of cost distributions whose main distinction is that they have identical supports regardless of the degree of asymmetry introduced. Given this type of cost distribution as asymmetry increases the expected surplus diminishes whether a ROFR is granted or not, just as in this paper. However, the expected surplus is always larger when the auction-

eer declines to offer a ROFR. This may be explained by the difference in how providing a ROFR interacts with an increase in asymmetry. In this paper an increase in asymmetry impacts the size of the support and affects the lower end of the support for the weak bidder. However it does not impact the shape of the distribution across the support. If the auctioneer grants a ROFR to one of the bidder, a change in asymmetry (at least for low to intermediate levels) does not change the bidding behavior of either bidder. The non-favored bidder in particular will see no reason to adjust his bidding behavior as the marginal trade-offs that he faces, which are in influenced by the shape of the opponent's cost distribution, are not altered by the increase in asymmetry. In contrast, in Burguet and Perry's paper an increase in asymmetry causes the cost type distribution to become more skewed towards the high end. That is, the shape and slope of the distribution distinctively change with asymmetry. If the auctioneer grants a ROFR the nonfavored bidder now faces different trade-offs. Hence his bidding function changes and the (strong) non-favored bidder bids less aggressively. While in my model the ROFR can completely offset any increase in asymmetry that is not possible in Burguet and Perry's setting. Under their model assumptions, offering a ROFR manages to dampen the loss in expected surplus as asymmetry increases, but this gain never manages to fully compensate for the cost of granting preferential treatment to one of the bidders. This difference in findings shows that whether granting a ROFR is beneficial or costly to the auctioneer depends on the distribution of the types of bidders. 4. Endogenous reserve price So far I have assumed that there was no reserve price. Now, I assume that the auctioneer can set a reserve price r in order to maximize his expected surplus. Again, I start by presenting the benchmark case when the auctioneer does not favor either bidder and then derive the corresponding equilibrium expressions when assigning a ROFR to either bidder. The auctioneer will always prefer to favor the weak bidder, if he is forced to assign a ROFR. However, when assigning a ROFR is optional, he only will do so when asymmetry is sufficiently large. Furthermore, granting a ROFR strictly improves the auctioneer's expected surplus only for intermediate levels of asymmetry. I show how control over the reserve price does improve expected surplus but also reduces the effectiveness of the ROFR.

J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424

4.1. No ROFR granted to either bidder I begin by computing the inverse equilibrium bidding functions for both bidders, while relegating the details of the derivations to Appendix A: Proposition 4. In a FPA, where the auctioneer optimally sets the reserve price r and no ROFR is granted, the two bidders' i = s, w bidding behavior is described by the following inverse bidding functions θi(b): hi ð bÞ ¼ 1 þ

ð r  1Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ b þ ki ðr  1Þ2 ðb  1Þ2 ðr  1Þ2

pffiffiffiffiffiffiffiffiffiffi ðr2Þrðh 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi and kw ¼ 1 4 . 2 ðr1Þ ks ðr1Þ ðr2Þrðh 2Þ h Additionally, the lowest bid placed by either bidder is Þr b ðrÞ ¼ ðr2 h 2 .

where ks ¼ 

P

P

P

P

P

Proof. See Appendix A.

□

   rh EðSurplus; noÞ ¼ 1  ð1  rÞ 1  1h P



½

b fw ðhw ðbÞÞ½1  Fs ðhs ðbÞÞ

Z þ

P

r

b r P

b fs ðhs ðbÞÞ½1  Fw ðhw ðbÞÞ

b

P

Looking at Fig. 3 I observe the following: the expected surplus decreases as the degree of asymmetry increases. The expected surplus never falls below 14 since the auctioneer can always turn the auction into a “takeit-or-leave-it” offer to the strong bidder. I show by numerical computation that this is in fact the preferred strategy if θ ≥ 0.607. The decrease in expected surplus is explained by the same intuition as in the model without an endogenous reserve price. When the auctioneer does not grant a ROFR any increase in asymmetry immediately translates into more aggressive bids by the weak bidder and less aggressive bids by the strong bidder. An increase in asymmetry reduces the support of the cost distribution for the weak bidder and removes the lowest cost types of the weak bidder from the bidding. Thus asymmetry raises the lowest bid placed. Even with more aggressive bids by the remaining cost types of the weak bidder this raises the expected payment by the auctioneer. 4.2. Mandatory assignment of a ROFR

The inverse bidding functions generate the optimal expected surplus to the auctioneer if he sets an optimal reserve price r⁎ N θ such that both bidders participate, which is given by

Z

1415

dhw ðbÞ  db db



dhs ðbÞ  db : db

While I obtain a closed-form expression for E(Surplus, no), it is not possible to derive a closed-form solution for r⁎ from the related first-order condition. I therefore compute r⁎ and E(Surplus, no) using numerically methods for various degrees of asymmetry θ. When the optimal reserve price is sufficiently low such that r⁎ ≤ θ the weak bidder declines to bid at all. The remaining strong bidder then faces a “take-it-orleave-it” situation. He then bids r⁎ as long as his cost type θs b r⁎ and not bid otherwise. The expected surplus to the auctioneer then becomes: Z r E ðSurplus; noÞ ¼ r  r dh:

Now, the auctioneer must assign a ROFR to one of the two bidders. I first derive the equilibrium bidding functions for both the favored and the non-favored bidder and determine that the auctioneer would assign the ROFR to the weak bidder. For the equilibrium bidding functions when a ROFR is assigned, I find that Lemma 2. In a first-price action with ROFR and a reserve price r b 1, the favored bidder i always bids r if his cost type θi ≤ r and he does not bid otherwise. The non

favored bidder −i bids bi ðhÞ ¼ min ðr; max hþ1 hi Þ 2 ;P if his cost type θ ≤ r and he does not bid otherwise. Proof. The bidding behavior of the favored bidder follows the same argument which I outlined in Proposition 1.

0

Straight-forward calculations show that in this case, r4 ¼ 12 and E ðSurplus; noÞ ¼ 14, which is the monopoly solution. For both r⁎ N θ and r⁎ ≤ θ, the figure below depicts the expected surplus to the auctioneer at the different levels of asymmetry.

Fig. 3. Expected surplus with no ROFR (endogenous reserve price).

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The favored bidder initially bids r and matches his opponent's bid if it is higher than his cost type and decline to match it otherwise, because any lower bid decreases his profits without improving his chances of winning. Since the favored bidder does not change his bidding behavior, the fundamental trade-offs that the non-favored bidder faces when making his bidding decision remain unchanged. Hence, his equilibrium bidding function does not change except for incorporating an additional constraint that truncates the upper portion of his bidding function. □ The somewhat surprising result for the non-favored bidder is that his equilibrium bidding function does not change other than that his bids are now “capped” at r. Generally, one would expect that a binding reserve price not only puts an upper bound on the bids submitted, but also changes the general structure of the bidding function such as the mark-up for any given cost type. Consider in comparison the equilibrium bidding function b(θ) of a (symmetric) first-price auction without ROFR with a reserve price r: bð hÞ ¼ h þ

2r  r2 þ hðh  2Þ : 2ð 1  hÞ

Here the mark-up on the cost depends on the reserve price and decreases as the reserve price r falls. With a lower reserve price a bidder is likelier to face more costefficient types and hence needs to bid more aggressively. This is because to win, a bidder has to beat his opponent's bid. Since some opponent types will not bid he needs to consider his opponent's distribution of only the cost types that actually bid, which is affected by the reserve price. However with a ROFR a non-favored bidder has to beat his opponent's cost type in order to win. Hence his optimal bid does not depend on the (truncated) distribution of just the cost types that actually bid. His mark-up is independent of the reserve price. The reason is that the ROFR essentially transforms a simultaneous bidding competition into a sequential bidding competition. The non-favored bidder just needs to focus on the opponent's cost type and not his bid. Using the non-favored bidder's equilibrium bidding function from above, I can construct and determine the expected surplus when favoring either the strong or weak bidder. Relegating the details of the derivations of the expected surplus functions to Appendix A, I find that Proposition 5. In an auction with mandatory assignment of a ROFR and an endogenous reserve price, the auctioneer always weakly prefers to grant the weak

bidder a ROFR. For intermediate degrees of asymmetry, θ ∈ [0.25,0.6389] the auctioneer strictly prefers to grant the weak bidder a ROFR. For very low and high degrees of asymmetry favoring either bidder yields identical expected surplus. The intuition here is, the optimal reserve is set in such a way that the ROFR essentially becomes irrelevant in terms of expected surplus (though not in terms of who actually gets chosen as the seller). For low asymmetry, the auctioneer sets a low enough reserve price such that the only bids actually placed by any bidder equal the reserve price. Hence expected payment is the same no matter who gets favored. In situations of large asymmetries, in a similar vein the reserve price is set in such a way as to shut-out all weak bidders and the ROFR becomes a moot point. Only for intermediate values of asymmetry does the ROFR provide the same “levelling of the playing field” as in the setting with no reserve price. 4.3. Comparing expected surplus with and without ROFR In the preceding section, I have determined that the auctioneer should grant a ROFR to the weak bidder when he can set a reserve price and is forced to assign a ROFR. When he can decline to grant a ROFR, I find that for intermediate degrees of asymmetry not granting a ROFR at all further improves expected surplus. Comparing the expected surplus under the two alternative regimes, granting a ROFR to the weak bidder and no ROFR at all, I find that Proposition 6. When the auctioneer has the option to assign a ROFR and can optimally set the reserve price, he will only do so for degrees of asymmetry θ ≥ 0.505. In that case the auctioneer weakly gains by favoring the weak bidder. The auctioneer strictly gains by favoring the weak bidder when 0.6389 ≥ θ ≥ 0.505. Fig. 4 below and the accompanying Table A.2 in Appendix A show that for low asymmetry, θ b 0.505, the expected surplus is higher when no ROFR is granted. For high levels of asymmetry, θ N 0.6389, the expected surplus with or without a ROFR is the same. Under either policy the auctioneer sets an optimal reserve price r ¼ 12 and shuts out the weak bidder. Hence the auctioneer strictly gains from granting a ROFR only when 0.505 ≤ θ ≤ 0.6389. Naturally the question arises, why the range of values in which a ROFR strictly improves expected surplus is intermediate.

J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424

Fig. 4. Expected surplus with and without ROFR (endogenous reserve price).

The intuition draws on the same arguments I outlined for Proposition 5. For low and high degrees of asymmetry, the auctioneer relies on the reserve price instead of the ROFR to improve expected surplus. For low degrees, the reserve price is set so low that any bidder that is still willing to place a bid bids the reserve price. Hence in terms of expected surplus the ROFR becomes irrelevant. For high degrees the reserve price renders the ROFR obsolete by turning the auction into a bilateral bargaining situation between the auctioneer and the strong bidder. For low asymmetry, the auctioneer can improve expected surplus even further by not granting a ROFR. This is possible because in the presence of a ROFR the reserve price has a dual purpose: not only does the reserve price help improve surplus, but it also counters the ROFR, which favors the weak bidder too much at low levels of asymmetry. The need to compensate for the ROFR forces the auctioneer to set the optimal reserve too low. Only for intermediate degrees of asymmetry, there no longer is a need to offset the ROFR. Now, the ROFR helps to counter the asymmetry between the two bidders. The ROFR complements the reserve price in this setting, both of them eliciting more aggressive bids. In contrast, without the ROFR the reserve price is not as effective in improving bids because of the asymmetry between the bidders. Given how the reserve price impacts the expected surplus as discussed above, it is useful to examine how and why the optimal reserve price changes as asymmetry increases: Fig. 5 below and Table A.2 in Appendix A depict the optimal reserve price when the auctioneer does or does not grant a ROFR. Under both policies the optimal

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reserve price rises as asymmetry θ increases until θ is sufficiently large. At that point, the optimal reserve reverts to r ¼ 12 as the auctioneer no longer tailors the reserve price to a two-bidder competition, but instead to a single bidder. For low levels of asymmetry the optimal reserve price is higher when no ROFR is granted. The intuition for the increase in the optimal reserve price as asymmetry increases is the same under both policies: Any increase in the reserve price raises the probability that the weak bidder places a bid. On the other hand, a higher reserve price also lowers the expected surplus to the auctioneer because of less aggressive bids by the (strong) bidder. As asymmetry increases, so does the marginal probability that the weak bidder places a bid when the reserve price is increased. Overall, the auctioneer therefore wants to increase his reserve price as asymmetry increases. The difference in the reserve price with or without ROFR is due to a difference in trade-offs: In an auction with ROFR a higher reserve price translates into a oneto-one increase in the bid placed by most types, since they bid the reserve if at all. In contrast when no ROFR is offered, only the most efficient cost type increases his bid one-to-one, while all other types' bids reflect the increased reserve price only partially because of bid-shading. An increase in the reserve price thus is more costly in terms of lost expected surplus when a ROFR is present. To offset the ROFR the auctioneer chooses a lower reserve price than when there is no ROFR. In summary, I find that granting a ROFR to the weak bidder improves expected surplus at intermediate levels of asymmetry. For low degrees of asymmetry the ROFR tilts the balance between the two bidders too much in

Fig. 5. Optimal reserve price with or without ROFR (excluding monopoly solution).

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favor of one side and the reserve price can be used to counterbalance the effect of a ROFR. For high degrees of asymmetry the ROFR is too weak and the reserve price replaces the ROFR as the more effective instrument to improve expected surplus. 4.4. Effective degree of asymmetry I have argued that the reserve price has two functions in an auction with a ROFR: to elicit generally more aggressive bids and to counterbalance the ROFR. In contrast, if there is no ROFR, the reserve price's purpose is limited to the former. The difference in the reserve price, when the auctioneer grants a ROFR or not, therefore reflects the attempt to offset the ROFR. I will demonstrate that this counterbalancing of the ROFR is best understood as a change in the degree of asymmetry that the bidders face. I start by defining the effective level of asymmetry: Definition 1. The effective asymmetry α is defined as h a ¼ r and measures the degree of asymmetry on the support of cost types for bidders that actually place bids. P

α has been defined in such a way as to enable comparisons between the case with and without a reserve price. Recall, in the model without a reserve price, θ served as a (relative) measure of asymmetry between the bidders. With an endogenous reserve price the support of cost types that bid has an upper limit of r. Hence, in order to measure the degree of effective asymmetry we normalize the value of θ by dividing it by the length of the support r. Similarly to θ in the model without a reserve price, the effective asymmetry α can take on values between 0, the lowest level of effective asymmetry, and 1, the highest level of asymmetry. α = 1 when the reserve price r it equals θ, or when no cost type of the weak bidder will actually place a bid. For the case when no reserve price is set by the auctioneer, r = 1 and hence α = θ. For different levels of θ, the effective degree of asymmetry are presented in Table 1 below where r⁎(no) is the optimal reserve price when the auctioneer does not favor any bidder, and r⁎(w) when a ROFR is granted to the weak bidder. α(no) and α(w) denote the effective asymmetry between the bidders when respectively a ROFR is granted or not. When I compare the degrees of effective asymmetry at each level of θ I find for θ b 0.4 that effective asymmetry is larger when a ROFR granted. For values θ N 0.4, effective asymmetry is smaller.

Table 1 Optimal reserve prices and effective degree of asymmetry θ

r⁎(no)

α(no)

r⁎(w)

α(w)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

0.5 0.51 0.53 0.54 0.55 0.57 0.58 0.6 0.61 0.63 0.65 0.66 0.68

0 0.1 0.19 0.28 0.36 0.44 0.52 0.59 0.65 0.72 0.77 0.83 0.88

0.42 0.44 0.45 0.46 0.48 0.5 0.53 0.57 0.6 0.63 0.67 0.7 0.73

0 0.11 0.22 0.32 0.41 0.5 0.56 0.62 0.67 0.71 0.75 0.79 0.82

If I interpret the difference in effective asymmetry, α(w) − α(no), as a way to offset the effect of the ROFR, then the findings nicely fit with the intuition offered previously: For low degrees of asymmetry θ the ROFR is harmful in that it tilts the playing field too heavily towards the weak bidder. Increasing effective asymmetry, such that α(w) N α(no), counters this tilt by increasing the relative cost advantage of the strong bidder. Once asymmetry surpasses θ = 0.4 assigning a ROFR to the weak bidder provides the desired “levelling of the playing field”. Decreasing effective asymmetry, such that α(w) b α(no), by choosing a slightly lower reserve price when granting a ROFR now allows the auctioneer to level the playing field even further. 5. Conclusion In this paper I have shown conditions under which an auctioneer deliberately wants to favor a bidder. I have demonstrated that favoring a weak bidder by offering him a ROFR benefits the auctioneer if the asymmetry between the two bidders is sufficiently large. With no reserve price assigning a ROFR strictly improves the expected surplus for sufficiently large asymmetry. When the auctioneer can set the reserve price optimally, assigning a ROFR weakly improves the expected surplus for sufficiently large degrees of asymmetry, with strict gains for intermediate levels of asymmetry. Whether the auctioneer can set a reserve price or not, the rationale for increasing the expected surplus is the same: Granting the weak bidder a ROFR eliminates the advantageous position of the strong bidder while allowing the weak bidder to bid less aggressively. If the asymmetry is sufficiently large, the gains from eliminating the strong bidder's advantage outweigh

J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424

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the losses to the weak bidder. When the auctioneer is also allowed to set the reserve price, the reserve price reduces the effectiveness of the ROFR. In settings of low and high asymmetry the reserve price in fact displaces the ROFR as the primary surplus-enhancing tool. For intermediate levels of asymmetry the auctioneer can combine the reserve price and the ROFR to increase expected surplus. The reserve price helps to counterbalance the effect of the ROFR by adjusting the effective degree of asymmetry between the two bidders. The notion that an auctioneer will prefer to assign a ROFR to the weaker bidder does nicely dovetail with the anecdotal evidence of the examples from the introduction. While in all of the examples it is difficult to know to what extent political considerations played a role, the observed assignments of the ROFR are consistent with the predicted behavior for an auctioneer seeking to minimize expenditure. In the bidding competition for the Queen Mary II, all sides clearly understood the favored bidder Harland & Wolff to be the weaker firm. Not only was it going through economically difficult times, but it also did not have a recent track record of building cruise liners on the scale of the Queen Mary II. Similarly, in the case of the A400M turbines, media reports had portrayed nonfavored Pratt & Whitney as the stronger contender, with early news reports predicting an award to Pratt & Whitney barring any political intervention. 14 Finally, in the contest for the Korean brewery Jinro, it seems reasonable to assume that given that Jinro collapsed in the wake of the Asian currency crisis of 1998 that all domestic firms including its suitor Oriental Brewery must have economically suffered as well. Hence, Oriental Brewery was likely to be in an economically weaker position than Coors to bid for Jinro. Thus in all three cases we see the weaker bidder obtaining a ROFR, which would benefit not only the favored bidder but the auctioneer as well. In more general terms, my results also confirm the finding of the literature on optimal auctions that the auctioneer does best when subsidizing the bidder who is more likely to be disadvantaged. The ROFR serves this purpose for sufficiently large degrees of asymmetry. For lower degrees of asymmetry the ROFR proves to be too simple and crude a tool because it excessively

handicaps the strong bidder. Nevertheless, the simplicity of the ROFR as a mechanism makes it an attractive tool for the auctioneer in order to increase his expected surplus. To demonstrate most clearly the effect of the ROFR on expected surplus, I used the simplest setting of two bidders with independent private values. Naturally questions arise regarding the robustness of the results presented, when there are for instance more than two bidders or when bidders have common values. I conclude by briefly outlining below how changes in either assumption would affect the findings of the paper. If there are more than two bidders, intuition suggests that when no ROFR is granted an increase in bidders will lead to more aggressive bids and increased expected surplus to the auctioneer. However, if a ROFR is granted, an increase in bidders will have no effect on the favored bidder and but will have an effect on the nonfavored bidders similar to the situation without ROFR. The reasoning here is that a non-favored bidder will only care about the ROFR conditional on him being the strongest among the n − 1 non-favored bidders. Thus all non-favored bidders are facing a scenario that implicitly has two “stages”: First an asymmetric auction without ROFR among the n − 1 non-favored bidders, implicitly followed by the “contest” against the favored bidder. Because of this analogy between the “first stage” of the auction with ROFR and the auction without ROFR, the main intuitions of the two-bidder model will likely still hold with n-bidders. If there are two bidders with (pure) common values, informal reasoning suggests that when a ROFR is granted the winner's curse will lead to more cautious bidding by the non-favored bidder and thus less expected surplus to the auctioneer. The idea here is that the favored bidder will be completely insulated from the winner's curse while the non-favored bidder faces an even worse winner's curse than without a ROFR present. The effect is most dramatic for a second-price auction as demonstrated by Bikhchandani et al. (2005) but would also obtain in a first-price auction. In an auction without ROFR the winner's curse would affect both bidders in less disparate fashion. Thus, with (pure) common values the auctioneer will be more cautious in granting a ROFR.15

14 Johnston, Lachlan, “Rolls spins ahead on Airbus contract”, Daily Telegraph, London (UK), May 7, 2003 and Aguera, Martin, “NonEuropean Engine May Power A400M”, Defense News, Apr 21, 2003.

15 Also see Bulow et al. (1999) for a common-value auction model with bidder asymmetries and favoritism, which share similarities with this paper.

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Appendix A Derivations for bidding function in Proposition 1 Either bidder i maximizes their expected profit given by max ðbi  hi Þ  ð1  Fi ðhi ðbi ÞÞÞ

In order to calculate the value of the constant ki, I first need to obtain the value of the lowest possible bid b. We know that this bid will be placed by the most efficient types of the strong and weak bidders. Hence I know that θw( b ) = θ and θs( b) = 0. Therefore     hw b  b hs b  b  2 2    ¼ b 1 or h  b  b ¼ b 1 : P

where F− i(·) is his opponent's cost distribution. The first-order conditions will be f− i(θ− i(b))·θ′− i(b)·(b − θi (b)) = 1 − F− i(θ− i(b)) or assuming uniform distributions hV i ðbÞ  ðb  hi ðbÞÞ ¼ 1  hi ðbÞ: If I add the left-hand and right-hand sides of these first-order conditions for i = w, s I get after rearranging terms,

P

P

P

P

P

P

P

P

1 Solving for b I get b ¼ 2h . I use this result with the solution found from the differential equation, which represents the inverse equilibrium bid function. Knowing that 

 2 b 1 hs b ¼ 1 þ 

2 ; 1 þ ks b 1 P

P

P

P

P

1 if I substitute b ¼ 2h and solve for ks I find that h2 2h ks ¼ 2 . One can analogously derive kw.

½hV w ðbÞhs ðbÞ þ hV s ðbÞhw ðbÞ  ½½hV w ðbÞb þ hw ðbÞ þ ½hV s ðbÞb þ hs ðbÞ ¼ 2 or ½hw ðbÞhs ðbÞV ½hw ðbÞb þ ½hs ðbÞb ¼ ½2b þ const:V:

P

P

P

ð1h Þ P

Derivations for expected payment in Proposition 1

Further by integration, I obtain ðhw ðbÞ  bÞðhs ðbÞ  bÞ ¼ b2  2b þ const:

The expected payment generally is written as Z 1 EðCost; noÞ ¼ bw ðhÞfw ðhÞ½1  Fs ðhs ðbw ðhÞÞÞdh hZ 1 þ bs ðhÞfs ðhÞ½1  Fw ðhw ðbs ðhÞÞÞdh: P

In order to find the value of the constant, I use the fact that θw(1) = θs(1) = 1. That is the highest possible bid 1 will be placed by the most inefficient type 1 of either bidder. In this case the constant is 1. Using the fact that (θw(b) − b)(θs(b) − b) = (b − 1)2, I rewrite the first-order conditions such that hV w ð bÞ ¼ hV s ðbÞ ¼

ðhw ðbÞ  bÞðhw ðbÞ  1Þ 2

ð b  1Þ ð hs ð bÞ  bÞ ð hs ð bÞ  1Þ ð b  1Þ

2

0

I perform a change of variables by substituting θw(b), θs(b) (obtained in Proposition 4) for θ respectively and w ðbÞ s ðbÞ substituting dhdb  db, dhdb  db respectively for dθ. The expression then becomes Z

1

E ðCost; noÞ ¼

and

b fw ðhw ðbÞÞ½1  Fs ðhs ðbÞÞ

b

Z

dhw ðbÞ  db db

P

:

þ

1

b fs ðhs ðbÞÞ½1  Fw ðhw ðbÞÞ

b

dhs ðbÞ  db; db

P

If I make a substitution of variable t = b − 1 then

hh 1h

where Fw ðhÞ ¼ and FG(θ) = θ. Evaluating the integral expression using Mathematica yields the expression stated in the proposition. P

P

ðhs ðbÞ  1  t Þðhs ðbÞ  1Þ hV : s ð bÞ ¼ t2 If I further define y = θs(b) − 1 then y′ = θ′s(b) and the Þy first-order condition becomes yV¼ ð yt t2 . Following Krishna (2002) and Griesmer et al. (1967), I know that the unique solution to this differential equation is y¼

2t 1 þ ks t 2

or

hs ð bÞ  1 ¼

2ð b  1Þ 1 þ k s ð b  1Þ 2

:

Table A.1 Expected cost with and without favoritism (no reserve price) θ

E(Payment,no)

E(Payment,w)

% Advantage of “favoritism”

0 0.05 0.1 0.15

0.667 0.675 0.684 0.693

0.75 0.75 0.75 0.75

−0.125 −0.111 −0.097 −0.082

J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424 Table A.1 (continued ) θ

matica, I find the solution to this differential equation as

E(Payment,no)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.9 1

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0.703 0.713 0.723 0.734 0.745 0.758 0.770 0.784 0.799 0.815 0.832 0.851 0.873 0.925 1

E(Payment,w)

% Advantage of “favoritism” −0.067 −0.052 −0.037 −0.022 −0.006 0.010 0.027 0.040 0.049 0.052 0.051 0.046 0.037 0.016 0

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.753 0.76 0.773 0.79 0.813 0.84 0.91 1

c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi or hs ðbÞ  1 t  c2 c2 þ t 2 k ð r  1Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ð b  1Þ þ k s ð r  1Þ 2  ð r  1Þ 2 þ ð b  1Þ 2

y¼

In order to calculate the value of the constant ki, I first need to obtain the value of the lowest possible bid b. We know that this bid will be placed by the most efficient types of the strong and weak bidders. Hence I know that θw(b) = θ and θs(b) = 0. Therefore     h w b  b hs b  b  2 ¼ b 1 ðr  1Þ2 or  2 ¼ b 1 ðr  1Þ2 : P

P

P

P



P

P

The initial steps of derivation for the inverse equilibrium bidding functions are identical to the derivations for Proposition 1 up to the point of determining the constant of the expression ðhw ðbÞ  bÞðhs ðbÞ  bÞ ¼ b2  2b þ const: The initial values needed to compute the constant are different for the endogenous reserve price case. The highest possible bid that either the weak or strong bidder will place is r and the associated cost type that will do so is θw , θs =r. Hence I know that the initial values are θw(r) =r and θs(r) = r. Therefore, (θw(r) − r)(θs(r) − r) = r2 − 2r + const. or const. = − (r2 − 2r). I now rewrite (θw(b) − b)(θs(b) − b) = (b − 1)2 − (r − 1)2 and use this equation to reformulate the first-order conditions (from Proposition 4) as

¼

ðhw ðbÞ  bÞðhw ðbÞ  1Þ 2

2

ð b  1Þ  ð r  1Þ ðhs ðbÞ  bÞðhs ðbÞ  1Þ ðb  1Þ2 ðr  1Þ2

and

hsVðbÞ

:

If I make a substitution of variable t = b − 1 and c = r − 1 then hsVðbÞ ¼

P



b



P

P

Derivation for Proposition 4

hwVðbÞ ¼

hb

ðhs ðbÞ  1  t Þðhs ðbÞ  1Þ : t 2  c2

If I further define y = θs(b) − 1 then y′ = θ′s (b) and the Þy first-order condition becomes y V¼ ðtyt 2 c2 . Using Mathe-

Þr Solving for b I get b ¼ ð2r 2h . I use this result together with the solution of the differential equation, which represents the inverse equilibrium bid function. Knowing that P

hs ðbÞ ¼ 1 þ

P

ðr  1Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðb  1Þ þ ks ðr  1Þ2 ðr  1Þ2 þðb  1Þ2

Þr if I substitute b ¼ ð2r 2h and solve for ks, I find pffiffiffiffiffiffiffiffiffiffi P

ks ¼ 

ðr2Þrð h1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi. ðr1Þ2 ðr2Þrðh 2Þ h P

P

P

One can analogously derive kw.

P

Derivation for Proposition 5 Table A.2 Expected surplus and optimal reserve prices with and without favoritism (endogenous reserve price) θ

E(Surplus,no)

r⁎(no)

E(Surplus,w)

r⁎(w)

% Advantage of favoritism

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0.417 0.406 0.395 0.383 0.371 0.360 0.347 0.333 0.319 0.304 0.288 0.271 0.252 0.25 0.25

0.5 0.513 0.526 0.539 0.553 0.567 0.582 0.597 0.612 0.629 0.646 0.664 0.684 0.5 0.5

0.385 0.375 0.365 0.355 0.344 0.333 0.322 0.313 0.303 0.295 0.287 0.278 0.263 0.25 0.25

0.423 0.437 0.452 0.468 0.484 0.5 0.533 0.567 0.6 0.633 0.667 0.7 0.733 0.5 0.5

−0.076 −0.076 −0.076 −0.075 −0.074 −0.073 −0.070 −0.061 −0.049 −0.030 −0.003 0.024 0.043 0 0

(continued on next page)

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J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424

Table A.2 (continued ) θ

E(Surplus,no)

r⁎(no)

E(Surplus,w)

r⁎(w)

0.75 0.8 0.9 1

0.25 0.25 0.25 0.25

0.5 0.5 0.5 0.5

0.25 0.25 0.25 0.25

0.5 0.5 0.5 0.5

% Advantage of favoritism 0 0 0 0

payment when the favored bidder does not participate (third line). The only difference now is that the nonfavored (strong) bidder may bid differently. If he has a low enough cost type, he can bid θ and still be ensured to win with this bid. The expected surplus to the auctioneer in this case is    rh EðSurplus; wÞ ¼ 1  ð1  rÞ 1  1h    Z 2r1 Z 1 Z r rh 1þh   2 h 1 h þ rdh þ rdh dh þ 1h 2 2h 1 2r1 r " !# Z 2r1 Z r  1r 1þh þ  2 h 1 h þ rdh dh þ 1h 2 2 h 1 2r1 P

P

Using the non-favored bidder's equilibrium bidding function, I construct an expression for the expected surplus when favoring either the strong or weak bidder. T h e s t r u c t u r e o f t h e b i d d i n g f u n c t i o n min r; max hþ1 2 ; h i ÞÞ makes it necessary to account for following cases: Depending on the value of θi, there may be bidders that mark-up their bids to θi. This is the case if θi N 0.5. If the auctioneer favors the weak bidder his expected surplus when asymmetry is low (θ ≤ 0.5) is given by

P

P

P

P

P

P

P

P

P

P

   rh E ðSurplus; wÞ ¼ 1  ð1  rÞ 1  1h " !# Z max ð0;2r1Þ Z 1 Z r rh 1þh   rdh þ rdh dh þ 1h 2 0 r max ð0;2r1Þ " !# Z r Z max ð0;2r1Þ 1r 1þh dh þ  rdh  1h 2 0 max ð0;2r1Þ P

P

P

P

P

The expression consists of three components, which are represented by the three lines of the equation above: the first line represents the expected value to the auctioneer from buying the good, which is equal to the probability that at least one of the two sellers places a bid. The second line comprises the part of the expected payment that occurs when the favored weak bidders places an initial bid of r. In this case the other bidder may bid as follows: Bid according to hþ1 2 , bid the reserve price r or do not bid at all (and hence the favored, weak bidder wins with his initial bid of r). Note that depending on the value of the reserve price r, some or all cost types of a non-favored bidder bid the reserve price. If the reserve price is low such that 2r − 1 b 0 all cost types that submit bids bid the reserve price. If the reserve price is higher such that 2r − 1 N 0, bidders with very low cost types bid below the reserve. Finally, the third line represents the part of the expected payment when the favored weak bidder does not bid at all. In this case the other bidder either bids according to hþ1 2 or bids the reserve price r. If asymmetry is high (θi N 0.5), E(Surplus, w) still has three components as shown below: the expected value (first line), the expected payment when the favored bidder participates (second line) and the expected

where the added (2θ − 1)θ terms reflect the part of the expected surplus when the non-favored (strong) bidder is assured the contract by bidding θ. Additionally, as discussed in Section 4.1, I have to consider the possibility that the auctioneer prefers to set a reserve price r ¼ 12 and Rr obtain an expected surplus of r  0 rdh ¼ 14. I derive the value of the optimal reserve price that maximizes E(Surplus, w) and distinguish four cases for the optimal reserve price: (i) When the two bidders are

 fairly equal in their cost distributions, P , the auctioneer sets a low ha 0; 14 q ffiffiffiffiffiffiffiffiffi reserve price of r ¼ 1  333 h . With 2r −1 b 0, all cost types of the non-favored (strong) bidder that do participate bid the reserve price. The intuition is as follows: the ROFR, as shown earlier, is not useful in order to reduce expected payment when asymmetry is low. In fact the ROFR is costly when bidders are similar. Hence, the auctioneer may want to set the reserve price in a way that neutralizes the ROFR. Since all participating bidders, whether favored or not, bid the reserve price, the ROFR loses its relevance in terms of expected payment. As asymmetry increases, the weak bidders are distributed more densely over a shrinking support. Any increase in the reserve price therefore has an increasingly larger marginal effect. The auctioneer wants to trade-off losses due to higher reserve prices against gains in the probability that the weak bidder participates in the bidding. Hence with asymmetry, the optimal reserve price will rise.

 (ii) For intermediate levels of asymmetry, h a 14 ; 12 , 1þ2 h the optimal reserve changes to r ¼ 3 . Once the 1 degree of asymmetry reaches h ¼ 4, some of the non-favored (strong) bidders take advantage of their low cost type and bid below the reserve price. This is now possible, because for these cost P

P

P

P

J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424

types (θ b 2r − 1) the reserve price no longer represents a binding constraint. Furthermore, for the auctioneer this means that an increase in the reserve price no longer uniformly applies to all cost types of the strong bidder as before. That is, the marginal loss due to higher reserve prices is lower  1 than under low levels of asymmetry h V 4 . Since the marginal gains to raising the reserve price still remain the same, the auctioneer adjusts the reserve price at a steeper rate than before.

 (iii) For high levels of asymmetry, h a 12 ; 1 I have to additionally consider the possibility that the auctioneer sets such a low reserve price that the (weak) favored bidder declines to bid with certainty. The reasoning is the following: if the auctioneer faces a situation when a bidder declines to bid with certainty, then the auctioneer faces a monopolist situation and his expected surplus is Rr given by r  0 rdh. The optimal reserve price to set is r ¼ 12, which yields a surplus of 14. Since only bidders with cost types larger than r refuse to bid, the only case in which an auctioneer can be sure that he is facing a single (strong) bidder is when h z 12. P

P

1423

1þ2 h

r ¼ 3 . However, the auctioneer experiences an increased rate of deterioration of the expected surplus due to an increase in asymmetry. (iii.b) For θ ∈ [0.6389] the auctioneer optimally sets r ¼ 1 2 and thereby makes sure that the only bidder to bid is the strong bidder. In this one-on-one situation the auctioneer has all the bargaining power and exploits this by making essentially a “take-it-or-leave-it” offer. P

Substituting the optimal reserve price into the auctioneer's expected surplus, I obtain the expected surplus when favoring the weak bidder w which is given by 8 > pffiffiffiffiffiffiffiffiffiffiffi > > 2 1h > > pffiffiffi > > > 3 3 > > > > > > > 43  32 h þ 16 h2 > < E ðSurplus; wÞ ¼ 108 > > > > 1  h 4 þ 23 h

> > > > > > 27 > > > >1 > > : 4

9 > > > > > > > > > > > > > >  > > 1 1 = if ha ; : 4 2 > >  > > > 1 > > if ha ; 0:6389 > > > 2 > > > > > ; if ha½0:6389; 1 >

 1 if ha 0; 4

P

P

P

P

P

P

P

P

P

P

On the other hand, the expected surplus when both bidder may bid with non-zero probability derives to be ð1h Þð4þ23 h Þ . Standard 27 pffiffiffi manipulation shows that  algebraic 1 as long as h V 46 19 þ 6 3 ¼ 0:6389 the auctioneer obtains a higher expected surplus by not shutting out the weak bidder with r ¼ 12. P

P

P

 (iii.a) For h a 12 ; 0:6389 the result and intuition for the optimal reserve prices remains unchanged from (ii). The only difference now is that the extent of asymmetry is so large that strong bidder with low cost types can exploit their cost advantage even more. The lowest cost types with θ b θ can mark their bids up to θ and matching the most efficient of the favored bidders. This change in bidding however, does not change the trade-offs that the auctioneer faces when setting the optimal reserve price. The marginal loss due to an increase in the reserve price remains unchanged, because the aforementioned change in bidding (bid θ and a large mark-up instead of bid hþ1 2 and a small markup) only affects low cost types of the strong (nonfavorite) bidder, for whom the reserve price is irrelevant anyway. The marginal gains to increasing the reserve also are the same, so the auctioneer keeps setting the optimal reserve according to P

ð1Þ The expected surplus decreases as the degree of asymmetry as measured by θ increases. The expected surplus never falls below 14 since the auctioneer can always turn the auction into a “take-it-or-leave-it” offer. I now consider the case when the auctioneer decides to grant the strong bidder a ROFR. The auctioneer's expected surplus in that case is given by the expression     ;r  h E ðSurplus; sÞ ¼ 1  ð1  rÞ 1  max 1 1h Z max ðh ;2r1Þ 1þh 1 dh  r 2 1h h Z r Z 1 1 1 þ r dh þ r dh 1h r max ð h ;2r1Þ 1  h Z max ðh ;2r1Þ 1þh 1  ð1  r Þ  dh 2 1h h Z r 1 þ dh : r max ðh ;2r1Þ 1  h P

½

½

P

P

P

P

P

P

ð

P

P

Þ

P

P

P

Þ

P

Similar to E(Surplus, w), it consists of three components – expected value to auctioneer, expected payment when the favored bidder places a bid and expected payment when he declines to bid – which are represented by the three lines of the equation above.

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J.-S. Lee / Int. J. Ind. Organ. 26 (2008) 1407–1424

When the auctioneer sets the reserve price r such that he maximizes E(Surplus, s) I find two different settings for the choice of the optimal reserve:

(i) If  the37 degree of asymmetry is sufficiently low h b 64 ¼ 0:578 , it best for the auctioneer to set ffiffiffiffiffiffiffiffiffi qis the reserve r ¼ 1  333 h . The rationale is the same as in the case when the auctioneers favors the weak bidder: The granting of the ROFR does more harm than good, so the auctioneer sets a reserve price that makes the ROFR more or less irrelevant in expected payment terms (the ROFR still does matter in terms of whom of the two bidders actually gets the contract). As before, an increase in asymmetry results in an increase in the reserve price. (ii) If asymmetry if sufficiently high, h z 37 64, the auctioneer stops considering the weak bidder as a potential competitor in the bidding contest. Instead he sets the reserve price r ¼ 12, which is optimal if he were only facing the strong bidder. Note, there is a discontinuity in the optimal reserve price at h ¼ 37 64. This is explained by the fact that as the auctioneer stops considering the weak bidder, the auctioneer is now operating under a different objective function. Because the auctioneer no longer benefits from soliciting bids from the weak bidder he reduces the auction to a single “take-it-or-leave-it” offer made to the strong bidder. P

P

P

P

By substituting the optimal reserve prices into E (Surplus, s) I find that if the strong bidder s is favored, the expected net surplus to the auctioneer is given by 8  9 pffiffiffiffiffiffiffi > 37 > > > 2 1 h > > if h a 0; > > < 3pffiffi3 64 = E ðSurplus; sÞ ¼ ð2Þ  >: > > > 1 37 > > > > if h a ;1 ; : 4 64 P

P

P

Examining the expected surplus functions (1) and (2) that for low degrees of asymmetry above shows 

h a 0; 14 as well as high levels of asymmetry (θ ∈ [0.6389,1]) favoring either bidder yields identical expected surplus. For intermediate values for θ ∈ (0.25,0.6389) the expected surplus is always higher when favoring the weak bidder. P

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