Christina Zou ’12

Harvard University, Department of Economics

**A key concern in market design is ensuring that behavior will be close to perfect competition rather than collusion. Much focus has been placed on creating methods of detecting explicit collusion and cartels, but tacit collusion has also plagued many historical and current markets, like the early US Federal Communications Commission (FCC) spectrum auctions for wireless broadband services. Based on Harrington (2005), four classes of methods for detecting tacit collusion in FCC spectrum auctions were presented. First, firm-license valuation equations were fit under the assumption of pairwise stability in matching, and in the resulting data, the Independence Test was found to be a robust detection method, while the Exchangeability test was not. Second, four potential collusive markers for structural breaks in firm bidding behavior were reviewed; price/bidding units ratio was found to be most robust. The third and fourth methods involved a Chow Test comparison of bid equation parameters of different competitors and a Bayesian approach to calculating posterior likelihoods of competitive and collusive models. Future assessments of new and emerging markets may use these methods in conjunction to aid in detecting and preventing tacit collusive behavior among participants.**

## Introduction

Competition is the mechanism through which assets sell for their maximal valuations among prospective bidders. In markets with imperfect competition, behavior deemed *collusive *can prevent assets from generating maximal potential revenue. Collusion has been observed in markets as diverse as those for school milk contracts (Pesendorfer, 2000), timber auctions (Baldwin, Marshall and Richard, 1997) and cattle auctions (Phillips, Menkhaus and Coatney, 2001). When it occurs on a large scale, collusion can mean millions of dollars in losses for market beneficiaries. For instance, in 2008, the Office of Fair Trading in the UK discovered collusive activities between 112 firms bidding for construction contracts worth over £3 billion, costing English taxpayers upwards of £300 million (Farmer, 2008).

Collusive activity can include combinations of price fixing, bid rigging, market division, and allocation schemes (DOJ, 2010). These bid rigging behaviors are sometimes displayed by bidding *cartels*, groups of bidders who decide to explicitly collude, bidding as a single entity rather than multiple bidders. The literature has elucidated many general market design aspects which encourage explicit collusion: small numbers of firms, market transparency, complete information, similarity among firms, product standardization, and barriers to entry (Heimler, 2007). Furthermore, much work has been done to test different methods of cartel detection. Though anything short of a wire-tap is not foolproof, and no test is sensitive to all types of cartel behavior, work by Bajari and Ye (2003), Abrantes-Metz, Froeb, Geweke and Taylor (2005), Porter and Zona (1993), Porter and Zona (1999), Ellison (1994), Banerji and Meenakshi (2004), and others has culminated in a toolkit of methods for cartel detection which Harrington (2005) categorizes into four general classes of approaches overviewed later in the paper.

Much attention has been given to cartel detection mechanisms partially because cartels are highly prosecutable by law. The US Sherman Antitrust Act, established in 1890 and effective over any firm whose actions affect US markets, deems explicit agreements to collude *per se *illegal—no circumstantiary evidence can overrule the act of explicit collusion. During the years 1980-1999, the US Department of Justice Antitrust Division convicted over 50 price-fixing crimes per year on average, more than 80% of which involved bid rigging (Connor, 2003). Designing markets hostile to sustainable cartels is a major area of study among economists and policymakers.

However, the interest of this paper is on a type of collusive behavior that is not explicitly agreed upon before an auction, termed *tacit collusion*. Because tacit collusion is not prosecutable, comparatively little work has been done on detection of tacit collusion in various auction markets. Nevertheless, tacit collusion can result in significant revenue loss and market inefficiencies just as explicit collusion does, and therefore economists should design markets that discourage tacit as well as explicit collusion. The goal of this paper is to test various approaches to detecting tacit collusion, which, if successful, can be used in the future to help economists determine whether certain existing markets might suffer from collusive behaviors. The specific focus of these tools will be upon FCC spectrum auctions, and in the following section we will explain this choice.

## Background

FCC spectrum auctions are government-run procedures for allocating airspace for wireless providers, with proceeds going to the US Treasury. Since 1994, FCC spectrum licenses, each covering a particular bandwidth and geographic region, have been sold off to private firms via a simultaneous ascending auction (SAA) format. This format, a natural generalization of the English auction, allows for multiple heterogeneous licenses to be sold at the same time, with bidders able to bid on any of the items. Bidding for licenses occurs online over an indefinite number of rounds, with the results of each round announced to bidders before the next round. Licenses are open for bidding as long as competitors continue submitting new bids for them. Though details of spectrum auctions are extensive, most important are the activity rules, which force bidders to purchase eligibility and either maintain certain levels of activity or risk losing eligibility for future rounds, and keep auctions moving quickly (McAfee and McMillan, 1996).

The FCC chose to implement a simultaneous ascending auction for its spectrum licenses because of its flexibility in allowing bidders to build privately efficient aggregations (“packages”) of complimentary licenses and for providing bidders with information about others’ license valuations during the course of auctions (Cramton and Schwartz, 2000). However, early FCC spectrum auctions were very prone to tacit collusion. One reason for this was the relatively small markets and light competition in earlier auctions (Cramton, 2004). Another was that their simultaneous and open nature meant bidders could observe each others’ bids after each round, coordinate collusive agreements and enforce them by punishing deviations.

Before the switch to click-box bidding in Auction 16, bidders communicated to each other through code bids, changing the last three digits of their bids to the IDs of the licenses they wanted. When bidders’ code bids weren’t respected, they punished their counterparties by placing retaliatory bids on *their* licenses and then immediately withdrawing them. Because all auctions before Auction 16 allowed unlimited withdrawals, bidders could use them to facilitate in collusion (Cramton and Schwartz, 2000). Bidders could also signal intent to bid aggressively on a license by opening with an intimidatory jump bid. Because bidder IDs were disclosed after each round until Auction 72, information disclosure facilitated punishment of deviations from tacit collusive “agreements.” Finally, because initial spectrum auctions didn’t have minimum opening bid requirements, bidders “parked” on many licenses by placing very low bids in order to 1) maintain high eligibility and 2) create the false appearance of competition. In the next section*,* anecdotal evidence of such forms of collusive behavior will be presented.

Over time, the FCC improved their spectrum auction design. Table 1 in the Appendix explains what policies were enacted during which auctions. Because of these policy changes, tacit collusion was much more difficult (and more rare) during later auctions than during earlier auctions. In their 2008 paper, Bajari and Yeo found that there were fewer retaliatory bids, jump bids, and more straightforward bidding in Auction 72 (700 MHz) compared to earlier auctions like PCS C Block (Auction 5) and PCS C&F Block Reauction (Auction 35).

The FCC spectrum auctions are unique among markets susceptible to tacit collusion in that they provide an abundant source of publically available bid data from an official website^{[1]}. In addition, verified evidence of tacit collusion has been observed in this market. FCC spectrum auctions are the prototype of a class of SAA auctions which has been gaining popularity in domestic and international markets; early auctions in Europe for 3G mobile wireless licenses raised nearly $100 billion, and electrical power and Treasury-bill auctions frequently feature similar structures (Milgrom, 2004). Many of these early markets are more likely to suffer from tacit collusion, as the FCC spectrum auctions did, than from cartels due to the infrequency of auctions, heterogeneity of assets, and lack of explicit communication between bidders. If we can develop a set of tools capable of detecting instances of tacit collusion from historical data on FCC spectrum auctions, they can be applied to improve market design of other similar markets.

Our basic research framework is drawn from Harrington’s comprehensive overview of cartel detection techniques, “Detecting Cartels” (Harrington, 2002). He outlines four general approaches: 1) Is firm behavior inconsistent with competition? 2) Is there a structural break in firm behavior? 3) Does the behavior of suspected colluding firms differ from that of competitive firms? 4) Does a collusive model fit the data better than a competitive model? We will adapt these general approaches for application to the complex structure of FCC spectrum auctions.

## Is firm behavior inconsistent with competition?

##### Auction data

The first problem of this approach is the estimation of a valuation equation for each firm on spectrum licenses as a function of exogenous variables on the firm and the licenses.

According to Cramton and Schwartz (2002), Auction 11, the DEF broadband PCS block auction, was susceptible to tacit collusion:

The DEF auction is especially well suited for a study of collusive bidding strategies in a simultaneous ascending auction. The auction featured both small markets and light competition. Small markets enhanced the scope for splitting up the licenses in the sense that each bidder can win many licenses… Light competition increased the possibility that collusive bidding strategies would be successful. Indeed, prices in the DEF auction were much lower than prices in the two earlier broadband PCS auctions.

After an investigation into Auction 11’s bid data, Cramton found that “six of the 153 registered bidders in the auction regularly signaled… they were 21 Century, AT&T, Mercury, North Coast, OPCSE, and US West,” which together account for 32 out of 37 recorded bouts of retaliation and code bidding. Of these six bidders, we choose to analyze the firm behavior of three—North Coast, US West, and AT&T. AT&T “was a large bidder with a reputation for retaliation. Bidders frequently bid substantially more for an identical license, rather than bid on a cheaper license held by AT&T. [For instance], AT&T used code bidding early in the auction to expel Powertel from Birmingham, AL” (Cramton and Schwartz, 2002). Meanwhile, North Coast had a history of colluding with NextWave, and US West had a history of colluding with McLeod. Some anecdotal evidence from Cramton for these claims is presented in Tables 1 and 2; USWest and McLeod contested Marshaltown, Waterloo and Rochester (block E)—283, 378, 452—and NextWave and NorthCoast contested Canton and Harrisburg (block F)—65 and 181.

Therefore, we choose to select five bidders to focus our attention on: AT&T, McLeod, NextWave, North Coast, and US West. Because we want to use data from licenses on which in which collusive bidding behavior was likely to have occurred to test the efficacy of collusion detection techniques, we create a statistic to proxy for the likelihood of collusive bids for a license, the *price/units (PU) *ratio: the ratio of a license’s final price to its bidding units (a FCC estimation of its value). If this ratio is particularly low, this suggests the license went for an abnormally low price which is evidence for possible collusion.

The average PU ratio in Auction 11 is 0.293, the 10^{th} percentile is 0.031, and the 20^{th} percentile is 0.062. From this 10^{th} percentile set of about 150 licenses, the median number of rounds to completion was 92 rounds. From the 20^{th} percentile of about 300 licenses, the median number of rounds to completion was 105. Across all licenses, the median number of rounds was 110. These statistics are in accordance with our expectations of collusion—there is less bidding (and a lower final price). Low PU ratios appear to be a good indication of collusive behavior. Then, for each of our five bidders, we select 15 of its winning licenses with the lowest PU ratios. Table 2 in the Appendix contains a list of selected licenses. There are a total of 908 bids made on the 75 licenses we have chosen, for an average of 12.1 bids per license (compare this to the total of 23157 bids for 1479 licenses for an average of 15.65 bids per license, and we see that there are fewer bids per license for the licenses we have chosen, as expected under collusion).

##### Valuation equations

The next step is to create a valuation equation for firm *i* on license *j*. The model we choose is

*value _{i,j }=_{ }β_{i1 }* population + β_{i2 }* familiarity + β_{i3 }* complementarity + β_{i4 }* fixed_effect_{j} + ε_{i,j}*

The population covered by each license is our *license-specific variable*, a bidder’s familiarity with a license and its complementarity with his bundle are *bidder-license interactions*, and fixed effects are *license-specific information* known to the public (revealed during the course of bidding during the auction, or known beforehand, like an incumbency or prospective merger) but not captured by the econometrician (Hong and Shum, 2003). The residual *ε _{i,j}* is the remaining unexplained portion of a bidder’s valuation on a license, and a key statistic in this test for collusion. This model is derived from that used by Bajari and Fox (2010), although they used a different data set and were concerned primarily with measuring efficiencies rather than detecting collusion (even if markets are efficient, in the sense that licenses go to the bidders who value them most, auction revenue may not have been maximized if collusion occurred). Because firms’ true valuations of licenses are quite complex, especially given the complicated nature of FCC spectrum auctions, inaccuracies in the assumptions we make about the form and inputs of valuation equations are a likely source of error in this test.

##### Calculating variable values

Firms will value licenses more when they are close to other licenses in their bundle, as they can reduce their operation costs when implementing licenses in the same geographic regions (Bajari and Fox, 2010). Therefore when calculating the value of each license, we include the value added by the closeness of that license to all other licenses won by the firm in this auction. Our proxy for complementarity will be a measurement called geocomplementarity, adapted from the well-known gravity equation in international trade (Fratianni, 2007). It has the desirable feature that any firm’s complementarities cannot decrease by adding licenses to a package. To simplify what would otherwise be a very complex process, we associate each license with its state, and then use the distances between state population centroids to proxy for distances between licenses.^{[2]}

The formula for geocomplementarity of license *m* with bidder *i*’s bundle *j* drawn from the total set of licenses *K*, adapted from Bajari and Fox, is

Because location is approximated by state population centroid, in cases where two licenses are from the same state, the distance is set to 130 miles, the approximate average radius of a state. The code for an Excel macro written to calculate the geocomplementarities of all 75 licenses in our sample with each of five bidders’ bundles is included in the Appendix as Script 1.

The familiarity of a firm with a license is defined to be the proportion of licenses won by the bidder in Auction 11 that were won in that license’s state. If a bidder wins a particularly high number of licenses for a certain state, we might infer that that bidder has a particular affinity for that state (existing infrastructure, closeness to firm headquarters, etc) which we call *familiarity*.

Out of 1479 licenses, 72 span more than one state—as this represents a small percentage, we simply take the first state listed. Northern Mariana Islands, US Virgin Islands and Guam are grouped with Hawaii as Pacific Islands. Table 3 in the Appendix contains the calculated familiarity values. The final variable, population covered by licenses, is provided in the data.

##### Fitting the valuation equations

In “Detecting Cartels,” Harrington overviews a procedure Bajari and Ye (2003) used to detect collusion in highway seal-coating contract auctions. Bajari and Ye use a parametric approach to fitting their valuation equations which incorporated bid prices. However, if collusion occurs, final firm bid prices may not reflect firms’ true values on licenses. Then using a parametric approach will generate incorrect parameters for valuation equations, and therefore we will be unable to extract unbiased residuals for each firm and license pair. Instead, we will use a nonparametric method developed in Bajari and Fox (2010).

Bajari and Fox define an auction outcome which is *pairwise stable in matches *to be one in which the total surplus valuation of any two bidders would not be increase by their swapping of one license each. *Pairwise stability in matches and prices, *a stronger condition,* *means bidders must not want to swap one each of their winning licenses at closing prices. Bajari and Fox argue that pairwise stability in prices and matches may not hold under models of simultaneous ascending auctions. If collusion occurred in FCC spectrum auctions, their final results may not have been stable in matches and prices. However, experimental evidence only supports pairwise stability in matches in FCC spectrum auctions. Cramton reports that in the first two years of PCS auctions, very little resale activity occurred post-auction. If auctions had not been pairwise stable in matches, bidders could have exchanged licenses and side payments after auctions to restore stability, so if they did not do so, we will take as true the assumption that Auction 11 was pairwise stable in matches. This implies that for every license A and B won by bidders *i* and *j* respectively, the following inequality must hold:

*Value _{i}(A) + Value_{j}(B) ≥ Value_{i}(B) + Value_{j}(A)*

Replacing our assumed model for value yields

*β _{i1 }pop(A) + β_{i2 }fam(i,A) + β_{i3} comp(i,A) + fixed_{effect} (A) + ε_{iA} + β_{j1} pop(B) + β_{j2} fam(j,B) + β_{j3} comp(j,B) + fixed_{effect} (B) + ε_{jB} *

*≥ β _{i1} pop(B) + β_{i2} fam(i,B)+β_{i3} comp(i,B) + fixed_{effect} (B)+ ε_{iB }+ β_{j1} pop(A) + β_{j2} fam(j,A) + β_{j3} comp(j,A)+fixed_{effect }(A)+ ε_{jA}*

We can observe that fixed license effects cancel out on both sides of the inequality. Note that we assume swapping licenses only affects the direct complementarities between the licenses swapped and other licenses. Swapping licenses does have an effect on the complementarity between every two licenses in a firm’s bundle, but if we assume each firm wins a large bundle of licenses, for simplicity we choose to ignore secondary effects of license swaps.

Note also that the residuals *ε _{iA}*,

*ε*,

_{jB}*ε*and

_{iB},*ε*are not, unfortunately, observable. In order to be able to use this approach, we might make assumption that

_{jA}*ε*+

_{iA}*ε*– (

_{jB}*ε*+

_{iB}*ε*) <

_{jA}*δ*for a small

*δ*in comparison to the other valuation equation terms. If this assumption holds, then the values of the markups should not significantly affect the estimation of accurate parameters using these inequalities. This is a strong assumption, as it requires all bidder markup distributions to be translations of “similar” shapes (within), but it is still compatible with both collusion and competition. Improving this step of the procedure is a topic for further work.

Because our sample includes 75 licenses, there are 2775 pairs of licenses and 2775 inequalities of the above form. The *maximal score estimator *is a function *m*(*β _{1}, β_{2}, β_{3}, β_{4}, β_{5}*) which counts the number of inequalities satisfied if

*β*are the input parameters. Then, our fitting procedure simply tries to find the five parameter vectors which maximize. Note that no information about bid prices is necessarily to estimate parameters using this approach, only information about which license was won by which bidder.

_{1}, β_{2}, β_{3}, β_{4}, β_{5}Computationally, a method called Differential Evolution^{[3]} was used to iterate over a parametric space to find a set of parameters which optimizes *m*. The code for an R script written to implement this approach is included in the Appendix as Script 2. Furthermore, full data from the optimization process can be found in Table 4 of the Appendix. The optimization process was time-intensive, requiring three or four hours for one run; this significantly limited the amount of bidder equations that could be fitted for the remainder of this paper.

About 72% of all inequalities were satisfied with this set of parameters. One challenge encountered while implementing this approach was that of coming up with a parametric space. We reason that magnitude is meaningless in our valuation equation, so arbitrarily choose an upper bound of 10 for all parameters, with the view that scaling can be adjusted after the parameters are fit. Negative values for the parameters were allowed in order to test our hypothesis that all parameters should be positive. A range of (-10, 10) was inputted for each parameter. All parameters returned were indeed positive.^{[4]}

Because fixed effects cancel out in the inequalities, there is no way to estimate them. For instance, if all bids for a license are lower than our value equation might predict, several things may have occurred: collusion, missing firm or license variables in the value model, or license-specific fixed effects unobservable to the econometrician but publically known by the bidders. Unfortunately, if we do obtain correlated residuals, special license effects (like incumbent or merger information) or collusion are both possible causes. For this purposes of this paper, fixed effects are assumed to be negligible. Realistically, if we chose good exogenous variables for the valuation equations, a proportion of licenses probably do have negligible special fixed effects.

##### Analyzing residuals

Now that we have our parameters, we find the predicted values of bidders for licenses. The average bid on these licenses was $129,000. Meanwhile, the valuation equations produce unitless values, so to scale them to proportion, each bid is divided by that average price of $129,000. Then, residuals were calculated as the difference between bid price and valuation using the fitted equations. Valuations were not calculated for licenses on which firms did not make bids. In the case of competition, one can only suspect that the bidder’s value of the license was lower than the minimum opening bid, but cannot be sure of what it is. For firms that did make bids on licenses, the final bid was used to proxy for value because if firms bid straightforwardly, they would be expected to bid up to their value and then drop out of the auction. However, this assumes competition and no bidder budget constraints, though they have been widely documented in practice (Bulow, Levin, and Milgrom, 2009).

##### Independence test

The aim of the independence test is to test for correlation between the unexplained parts (residuals) of bidders’ bids on the same license. If these residuals are correlated, then bidder behavior might not be compatible with a competitive model. If the residuals are independent, there is not sufficient evidence to suspect noncompetitive behavior. Tables 4 and 5 list residuals for AT&T vs. North Coast and AT&T vs. US West for licenses on which both pairs bid.

Of the 75 licenses, 19 were bid on by both AT&T and NorthCoast. From these 19 pairs of residuals an R^{2} of 41.85% was found. Twenty-two were bid on by both AT&T and US West, and of these pairs of residuals an R^{2} of 46.58% was found. Table 7 lists the same calculations for McLeod vs. US West and NextWave vs. North Coast.

McLeod and US West and NextWave and North Coast are two pairs of firms with a history of collusion. The R^{2} for these pairs, 80.49% and 63.05%, are both higher than those between AT&T and NorthCoast. This is indeed what we would expect: there has been general evidence of collusive and intimidatory behavior from AT&T, but McLeod and US West and NorthCoast and Next Wave are pairs of firms who have specifically colluded with each other. We would expect to see greater correlation between these residuals for those pairs.

It would be ideal to have estimated the bid equation for a firm not suspected of collusion, and finding its residuals’ correlations with one of these bidders. This could specifically entail estimating the bid equations from firms in later auctions, after policy changes had been enacted to counter collusion, or it could entail estimating bid equations for firms from the current auctions not suspected of colluding. Because this detection method makes several strong assumptions (the value model is correct, markup distributions are shared between bidders, fixed effects are negligible, final bids reflect bidder values) that could be sources of error, comparing the R^{2 }of pairs of firms not suspected of colluding with that of pairs suspected of colluding would have given us better context from which to interpret the correlation coefficients. Since fitting the bid equations is a lengthy process, this is left for future work. Furthermore, with more time it would have been desirable to apply the fitted bid equations to all licenses bid upon by both of two pairs of firms, rather than just those in our original sample of 75 licenses. Having more than ten to twenty data points for each pair of firms would have yielded more accurate correlation coefficients.

Following the reasoning in Bajari and Ye (2003), a Fisher transformation is used to infer the correlation coefficients for all licenses contested between pairs of firms from our sample:

where r is the sample correlation coefficient. Let *n* be the sample size; then the distribution of *Z* is approximately normal with

Hence *z = (Z – μ _{z})* is approximately standard normal. Because

*H*

_{0 }is r = 0, equivalently we test

*μ*= 0. Results are detailed in Table 8.

_{z}In accordance with the hypothesis, results allow us to reject the null hypothesis at a significance level of α = 0.05. Although p-values are very low, again, without a good benchmark, interpretation is difficult. Nevertheless, the results of the independence test are consistent with collusion as well as with fixed license effects or other license/firm characteristics not included in the value equation. While the results don’t imply collusion, we can say that, under the assumptions discussed in this section, they would be consistent with collusion and inconsistent with competition.

##### Exchangeability test

The Exchangeability test tests the belief that exogenous factors (population, familiarity, and complementarity) should enter every bidder’s value equation in the same way, absent collusion. The null hypothesis is *H*_{0: }*β _{ij} = β_{kj}* for firms

*i*and

*k*and variable

*j*(

*j*= 1, 2, 3). Our approach will be the Chow Test. For firms

*i*and

*k*this approach requires three sets of residuals:

*β _{i,j }=_{ }β_{i1 }* population + β_{i2 }* population+ β_{i3 }* complementarity + ε_{i,j}*

(1)

*β _{k,j }=_{ }β_{k1 }* population + β_{k2 }* population+ β_{k3 }* complementarity + ε_{k,j}*

(2)

*β _{ik,j }=_{ }β_{ik1 }* population + β_{ik2 }* population+ β_{ik3 }* complementarity + ε_{ik,j}*

(3)

We have already found fitted parameters for (1) and (2), and now rerun the maximal score estimator procedure to find the fitted parameters for (3), the case where one set of parameters is obtained for firm *i* and firm *k* together (while the exogenous variables retain their original values even in the combined equation.) For the scope of this paper, this procedure is attempted only for McLeod and West, two firms with a history of collusion. Table 5 in the Appendix for the DE runs results with parameter results for the combined equation. Note first that only 1905 of the inequalities (68.6%) were satisfied using the optimal parameters and, assuming McLeod and US West share parameters, compared to over 72% when parameters are estimated separately for the two firms. This evidence supports the belief that McLeod and US West may have different parameters.

Using the new parameters, combined residuals for McLeod and US are calculated and compared to the original separate residuals for McLeod and US West using the test statistic

where *S*_{c} is the sum of squared residuals under the combined model, *S*_{1} and *S*_{2} are the sum of squared residuals under separate models, *N*_{1} and *N*_{2} are the number of licenses in each group, and k is the total number of parameters (in this case, 3). *F* follows the *F* distribution with *k* and *N*_{1} + *N*_{2} – *2k *degrees of freedom. See Table 6 in the Appendix for the list of residuals and the statistical calculations.

The p-value we obtain is 0.84, so we fail to reject the null hypothesis. There are several possible explanations for this unexpected result. It is quite possible that firms are indeed colluding, but still respond similarly to exogenous variables. Alternatively, the small sample size from which these comparisons are drawn and the noisiness of the data could prevent us from uncovering actual significant discrepancies between bid parameters for different firms. Also, note that even if we had been able to reject the null hypothesis, firms could simply have different bid parameters and not be colluding. In the case of spectrum auctions, companies’ existing geographical spreads, flexibilities to adapt technologies and knowledge to new locations, and other factors can contribute to failure of the Exchangeability test independent of collusion.

The relatively high possibility of false negatives and positives (particularly of false positives) suggests that the Exchangeability test may be a weaker or less ideal test than the independence test as a tool for detection of collusion in spectrum auction markets.

##### Markers of collusion

Can we find structural breaks in firms’ behavior before and after policy changes in FCC spectrum auctions? Applying this common approach to spectrum auctions is made particularly challenging by the fact that there are no repeated auctions in our framework. Licenses are heterogeneous, every auction is for different blocks of airspace, auctions occur infrequently, different bidders participate in every auction, and bidders’ corporate structure and strategies change between auctions. In addition, many of the markers of cartel operation discussed in Harrington are not appropriate for detecting tacit collusion (for instance, the hypothesis that bids should be clustered in a particular way because of bidders’ efforts to prevent cheating from cartel agreements). For these and other reasons, careful attention must be paid to the application of standard collusive markers to these unique markets. Possible sources of Type I and Type II error are discussed alongside each of the following proposed tests.

##### License quantity variance

Because collusion implies lower quantities and higher value per license for colluders, and the total number of licenses is determined before the auction, larger quantities go to non-colluders if we can assume collusion is not all-inclusive. More likely is the scheme where certain firms are active colluders, placing intimidatory, code, and retaliatory bids (AT&T, US West, North Coast) whereas others are passive, colluding primarily when victimized (McLeod, NextWave) and otherwise placing competitive bids. We would expect active colluders to get comparatively high value for their won licenses compared to passive colluders and non-colluders, so *in general *they might take respectively fewer/more licenses under collusion than under competition and we would expect a larger variance in won license quantities. Table 9 shows the relevant standard deviations and their normalized values (normalized by the total number of licenses sold in the auction) for four significant auctions capturing differences in FCC spectrum policy (drawn from Table 1 in the Appendix). Previous literature suggests there was more collusive behavior in the earlier auctions, and would expect greater σ for those auctions. However, we find that σ actually increases for later auctions, and normalized σ shows no real pattern. License quantity variation appears to be a poor approach to detecting collusive behavior in markets like the FCC spectrum auctions.

Because every auction is for different blocks of airspace, auctions occur very infrequently, participating firms change between auctions, and firms evolve over time, we cannot consider different FCC auctions to be repeated auctions whose results are comparable. An explanation for the behavior observed is that for later auctions, as the wireless provider industry matured, certain firms emerged as dominant and purchased a larger percentage of the licenses available. In auction 5, the dominant 3 winners won 25.26% of licenses, in auction 35 they won 37.77% of licenses, in auction 66 thy won 34.64% of licenses, and in auction 73 they won 49.80% of licenses. An explanation involving collusion (false negatives) is that if collusive behavior correlates positively with firm bidding activity, then the large companies who would win more licenses under perfect competition might win fewer licenses under collusion (if, for instance, they feared detection and could now make the same profit margins with fewer licenses), reducing variance of license quantities won under collusion. Conversely, if we did find more license quantity variance in early auctions, we might observe that in early auctions many firms registered to enter FCC auctions with poor expectations of other bidders’ bidding functions, found that their valuations were not high enough to win licenses, and won no licenses. Lack of information about the markets in early auctions might create false positives.

##### PU ratio

The PU ratio was discussed earlier in the paper as a possible marker for licenses affected by collusive bids. Here, and in Figure 1, we attempt to use it as a test for collusion between auctions.

PU ratios were calculated for auctions 5, 35, 66, and 73 (corresponding with major auction structure changes, see Table 3 in the Appendix) as well as for auctions 4, 11, 17, and 22, which posted the highest net worths of licenses sold. From Figure 2 we see that PU ratio did increase with time as auction structure was improved to dissuade collusion. Note, however, that PU ratio is not a foolproof marker for collusive behavior. Ratios might have increased simply because the FCC bidding unit assignments did not accurately capture the increasing value of licenses as airspace supply decreased and demand increased in later auctions.

The remaining approaches will merely be outlined; implementation will be left for future work.

##### Bid probabilities and final bid rounds

For those firms whose behavior might be termed collusive, their chances of winning licenses in relatively early rounds are greater than they are for an average bidder. Using a combination of retaliation, intimidation (threatening), code bidding, etc., collusive firms can secure licenses quickly by pushing other bidders off them. Also, firms with a history of retaliation are less likely to face competition on their licenses. Therefore, we might expect that firms with high collusion rates win their average license at an earlier round than firms without a history of collusion. We also expect collusive firms to have higher probabilities of placing winning bids at early licenses. Therefore, it would be possible to compare the average round numbers of final bids between suspected colluders and suspected non-colluders; they should be lower for colluders. It would also be possible to plot and compare the bid win probability vs. final bid round number curves for suspected colluders and suspected non-colluders; we expect colluders’ curves to rise more steeply at early rounds compared to non-colluders’ curves.

Of course, it is possible to obtain false positives and negatives using final bid round and bid probability data. If we do find that final bid round numbers are on average lower for suspected colluders, they could also simply collectively prefer licenses which other participants do not want—for instance, colluders might be larger firms capable of providing service in remote locations inaccessible to non-colluders. The same possibility applies if we find a result that suspected colluders have higher bid win probabilities at early rounds of auction. Nevertheless, it is neither obvious nor necessary that collusion correlates with higher valuations of generally undesirable licenses. In general, these statistics are justifiable markers for collusive behavior when used in conjunction with the others mentioned in this paper.

##### Markups

Markups (residuals), the unexplained portions of companies’ bids, should be lower in the presence of collusive behavior. Colluders, on average, are able to purchase licenses for less than their estimated value. A viable marker for collusive behavior from a firm is a significantly lower average markup on its bid prices for won licenses compared to the auction average (or to behavior in a different auction which may be used as a baseline). This test is similar in theory to that of the PU ratio, but its added advantage is that markups capture bidders’ public information about values of licenses (due to process used to fit parameters, which relies on information about which licenses were won by which firms), while PU ratio only captures FCC’s estimates of license values, which may not reflect all public information.

##### Verifying suspected collusion

The remaining two approaches are primarily for verification, as opposed to screening, of collusive bidding. The first is behavioral in nature. Does behavior of suspected colluders differ from that of suspected non-colluders? Harrington suggests comparing bid functions for cartel members and non-cartel members, and the best way to apply this concept to spectrum auctions is to track the same firms across auctions before and after policy changes. The concept is similar to that used for the Exchangeability test, but while we previously fit bid equations using pairwise stability in matches because we wanted to retrieve accurate residuals, here the goal is to compare parameters, and actual final bid prices may be used^{[5]}. If we can assume competition after Auction 15 (compared to before Auction 15), then if firms were also competitive before Auction 15, their parameters should not have changed, and bid prices should reflect that exogenous variables enter into their valuation equations in the same way as before. If a researcher who detected collusion between two firms using the Exchangeability test wanted a *benchmark* with which to compare these results, he could compare a suspected colluder’s behavior before and after policy changes in spectrum auctions using this approach.

*Log*(*bid _{ij}*)

*=*

*α*

_{j}+ βX_{ij}+ ε_{ij}will be the bid equation, in which firm *i*’s final bid for license *j* is a function of *α _{j}*,

*β,*a fixed-effect license effect, a parameter of vectors, multiplied by a vector of exogenous variables which include license minimum opening bid, population, familiarity, and complementarity, and the residual

*ε*The approach would be to use the bid data to fit

_{ij }.*β*for all bids made by a firm suspected of collusion across auctions before and after a policy change (for instance, comparing auctions for A/B/C/D blocks across auction 15, a point of major policy change) and to fit

*β*separately for each auction. The Chow Test can again be used to test

*H*that

_{0}*β*is the same between the auctions of interest.

##### Comparing collusive and competitive models

If we have evidence from earlier approaches of collusion, a last method of finding support for that hypothesis is to run a horse race between competitive and collusive models for a firm’s bid behavior. A Bayesian approach may be used. Assume the prior distribution over A (the competitive model) and B (the collusive model) is uniform. Next, determine markups with respect to A and B. B is formed as a competitive model where two firms suspected of colluding act as one firm with one bid equation with one set of exogenous variable values for each license, taken as the optimal set of values across the two firms. The concept is that if two firms collude and act as 1 firm, their combined valuation of a license is the higher of their individual valuations—side payments between the two firms may occur after the auction concludes.

Note that this assumption limits the efficacy of this approach to cases where two firms are specifically suspected of frequent collusion with each other, and not with any other bidders. In this way, this concept may be more suitable for cartel detection than for detection of tacit collusion, which is generally more indiscriminate. For future work, the sensitivity of this approach may be tested on pairs like US West and McLeod or North Coast and NextWave, but for firms like AT&T, who simply placed frequent intimidatory, retaliatory, and code bids on their own licenses and those of any firm it fought against, this model will not be effective. Alternatively, one might try a model B which, rather than combining the values of two firms, weights one firms’ values of licenses *inversely* with the values of other firms for that license. Because the spectrum auctions are open, multiple-round auctions, we suspect firms gain information about each others’ valuations of licenses through bidding. If one firm suspects others may have an equal or higher valuation of a license compared to its own, its final bid price on the license may be lower than it otherwise would be.

After extracting the markups, the next step is to estimate the likelihood of these markups given models A and B. It is difficult to obtain knowledge of the markup distribution for both of these models. Bajari and Ye (2003) elicit markup distributions from experts in the highway seal-coating industry, but the wireless provider industry is likely more complex and these distributions would be more difficult to estimate. The competitive markup distribution may be drawn from a benchmark auction in which collusion most likely did not occur (in the case of FCC spectrum auctions, Auction 72 and onwards), and a separate distribution may be created for different blocks of airspace (A, B, C, D, E, and F). After these distributions are obtained, each bidder makes a random draw for each license, and then Bayes’ Rule can be used to estimate the posterior likelihoods of models, which suggest with what probability collusion occurred.

## Discussion

We present a variety of approaches to detecting possible tacit collusive behavior in American FCC spectrum markets. These approaches, themselves adapted from a framework provided by Harrington, may be modified for use in other SAA structure markets like spectrum markets in foreign countries, electricity markets, T-bill auctions, and certain online auctions.

First, we developed and tested two approaches to screening for behavior inconsistent with competition. We constructed and fit valuation equations for each firm on each license in the DEF block during auction 11 using the concept of pairwise stability in matching and optimizing the maximal score estimator using the computational method of Differential Evolution. From these equations, residuals were extracted and their correlations between pairs of firms were calculated to range from 41.8% to 80.6%. The Independence Test was applied with the null hypothesis that *r*, the correlation coefficient between pairs suspected of collusion, is 0. Obtained p-values were all under 0.01, allowing us to reject the null. The Exchangeability test was applied with the null hypothesis that exogenous variables enter different firms’ bid equations in the same way, in the case of competition. For the case of US West vs. McLeod, the Chow test gave a p-value which failed to reject the null.

The Independence Test and high correlation coefficients between residuals suggest the conclusion that noncompetitive behavior occurred in Auction 11. However, the result of the Exchangeability test was consistent with competition. Overall, one must be careful in interpreting these results. A number of assumptions were made within these models. We assumed firms held valuations on licenses based only on license complementarities (proxied roughly by state population centroids), license populations, and firm familiarity with license regions. Strong assumptions about the similarities between firms’ markup distributions had to be made, due to our inability to observe these markups in the auction results, in order to use inequalities to fit parameters. Final bid prices were used as a proxy for firms’ values on licenses, but this assumes that firms do not operate under budget constraints, which have been shown to influence their bids. If abnormally large markups were found, we could not entangle a collusive effect on bid prices from a license fixed effect such as an incumbency or prospective merger (while for the case of specific licenses, anecdotal information could be found, for an entire sample of licenses it was difficult to compile any data on license fixed effects), or from inaccuracy in the modeling of the valuation equation. Finally, this model was only run on five firms with a history of collusion. In the future, this model should also be applied to firms without a history of collusion, possibly the same firms suspected of collusion in early auctions but not in later ones (such as AT&T), to provide some sort of benchmark from which to interpret correlation coefficients and other result statistics.

The results of the Independence Test are incompatible with competition, so we might say they are consistent with collusion or with any of the above explanations of the modeling mechanism. The results of the Exchangeability test are compatible with competition. However, it is possible that small sample sizes (only about twenty licenses within our samples were bid upon by pairs of firms; future work should extend the set of licenses considered), inaccuracies in the bid equations, and general noisiness in the data explain the result of the Exchangeability test, rather than competition. Alternatively, it is possible that collusive behavior did occur in the auction, but did not largely affect the way they incorporate exogenous variables into their valuations. If we had found opposite results from the Exchangeability test, it is quite likely that exogenous variables might enter firms’ bid equations in different ways regardless of competition or collusion. The significant possibility of receiving false positives and negatives from the Exchangeability test weakens its power in detecting behavior inconsistent with competition.

A second approach to detecting tacit collusion was in testing for structural breaks in firm behavior. To this end, four collusive markers were introduced and two of them were tested on data across multiple FCC spectrum auctions. The first marker was license quantity variance—we tested the theory that license quantity variance should increase under collusion as compared to under competition, an idea drawn from Green and Porter (1984) on differently structured auctions. Using data from Auction 5, Auction 35, Auction 66, and Auction 72 (with historical evidence suggesting that the earlier auctions suffered from tacit collusion while the later ones did not) we did not find any correlation between license quantity variance and collusion. Because auctions happen so infrequently, are for heterogeneous blocks of airspace, and participants vary from auction to auction, it is not recommended that multiple auction results be compared in this way. This marker could report competition if quantity variance happened to increase in later auctions because as the industry matured, certain firms gained larger market shares, even if collusion had occurred. Alternatively, it could report collusion if quantity variance was larger in earlier auctions because more firms participated in auctions without good expectations of others’ valuation functions, were surprised by new information made public through rounds of bidding, and ended up not winning any licenses.

The next marker tested was price/bidding units ratio, a measure introduced earlier in the paper. One would expect that this ratio, a measure of whether licenses were sold cheaply or expensively, might be expected to be larger under competition than under collusion. Calculated over eight spectrum auctions, this marker did show an upwards trend and appears to be a justifiable indicator of collusive behavior. Nevertheless, because this measure depends on FCC estimates of license values, the FCC might have undervalued licenses in later auctions by not accurately incorporate increasing license demand, and thus firms could still have colluded in later auctions. The last two markers discussed were bid probabilities/final bid rounds and markups. Firms who colluded should have a greater probability of placing winning final bids at relatively early rounds in an auction compared to non-collusive firms, and markups for colluding firms on licenses they won should be relatively lower than for non-collusive firms. Testing these markers on available data is left for future work.

The third and fourth approaches were discussed only in theory. Both are intended as tools for verification of suspected collusion, rather than for general screening purposes. The third involves estimating bid equations for firms suspecting of colluding against those for the same firms during auctions in which they were not suspected of colluding. Unlike the first approach, this one only tests upon the parameters of bid functions; therefore, here we could use final bid prices to directly fit these parameters. A Chow Test may be applied to see if the resulting parameters are temporally different. If they were, we might infer collusion, unless there was reason to suspect a firm’s competitive bidding behavior would have had reason to change in between auctions. The fourth involves running a horse race between competitive and collusive models for firms suspected of collusion. A Bayesian approach can be applied on prior probabilities of the two models and the likelihoods of final markups given both models to calculate the posterior likelihoods of the models.

The tests described in this paper are meant to be applied together to auctions where tacit collusion might be suspected. Adapting and testing them on FCC spectrum auctions illustrates the difficulties of obtaining accurate results with any certainty for one test alone. Several approaches, like the license quantity variance marker and Exchangeability test, were unable to retroactively determine collusion where collusion was known to have occurred, in early FCC spectrum auctions. Because each test is susceptible to different forms of bias and sources of error, using all tests together minimizes the likelihood of inferring a false positive or negative from the available data.

In presenting the considerations we took into account when modifying Harrington’s four approaches to cartel detection to a relatively complicated market structure featuring simultaneous ascending auctions with multiple rounds and heterogeneous assets, we hope to have highlighted some of the challenges these unique markets present to attempts to model competitive or collusive firm bidding behavior. In addition, we hope to motivate application of these methods to other similarly designed markets, like spectrum auctions in foreign countries, electricity markets, Treasury bill markets, and so on. Furthermore, the work done in this paper should serve as a reminder to economists and policymakers that even though tacit collusion may not be illegal, as cartels are, *any* form of collusion carries severe consequences on revenue generation from large scale auctions. Increased efforts should be placed on identifying markets where bidders can easily collude and modifying those market designs to enforce competitive behavior.

## References

Abrantes-Metz, Rosa M., Froeb, Luke M., Geweke, John, and Taylor, Christopher T. “A Variance Screen for Collusion.” FTC Bureau of Economics Working Paper No. 275. Mar 2005.

Pesendorfer, Martin. “A Study of Collusion in First-Price Auctions.” Review of Economic Studies, 67 (2003): 381-411.

Bajari, Patrick and Fox, Jeremy T. “Measuring the Efficiency of an FCC Spectrum Auction.” Working Paper.

Bajari, Patrick and Yeo, Jungwon. “Auction Design and Tacit Collusion in FCC Spectrum Auctions.” Information Economics and Policy, Elsevier 21 (2009): 90-100.

Baldwin, Laura H., Marshall, Robert C., and Richard, Jean-Francois. “Bidder Collusion at Forest Service Timber Sales.” The Journal of Political Economy, 105 (1997): 657

Banerji, A., and Meenakshi, J. “Buyer Collusion and Efficiency of Government Intervention in Wheat Markets in Northern India: An Asymmetric Structural Auctions Analysis.” American Journal of Agricultural Economics, 86 (2004): 236-253.

Bulow, Jeremy, Levin, Jonathan, and Milgro, Paul R. “Winning Play in Spectrum Auctions.” NBER Working Paper No. w14765. Mar 2009.

Connor, John M. “Private International Cartels: Effectiveness, Welfare, and Anti-Cartel Enforcement.” Purdue University, unpublished manuscript. 2003.

Cramton, Peter. “Simultaneous Ascending Auctions.” University of Maryland – Working Paper. 2004.

Cramton, Peter and Schwartz, Jesse A. “Collusive Bidding: Lessons from the FCC Spectrum Auctions.” Journal of Regulatory Economics, 17 (2000): 229 – 252.

Ellison, Glenn. “Theories of Cartel Stability and the Joint Executive Committee.” RAND Journal of Economics, 25 (1994): 37-57.

Green, Edward J., and Porter, Robert H. “Noncooperative Collusion under Imperfect Price Information.” Econometrica, Econometric Society, 52(1984): 87-100.

Farmer, Ben. “Construction Cartel Cost Taxpayers 300 Million.” The Telegraph. 17 Apr 2008. <http://www.telegraph.co.uk/finance/newsbysector/constructionandproperty/2788332/Construction-cartel-may-have-cost-taxpayer-300-million.html>

Fratianni, Michele. “The Gravity Equation in International Trade.” Working Paper 307, Universita’ Politecnica delle Marche (I), Dipartimento di Economia. 2007,

Froeb, Luke M., and Shor, Mikhael. “Auctions, Evidence, and Antitrust.” The Use of Econometrics in Antitrust, American Bar Association. Jul 2002.

Harrington, Joseph E. “Detecting Cartels.” Working Paper, Johns Hopkins University. 2004.

Heimler, A. “What has Competition Done for Europe? An Interdisciplinary Answer.” Aussenwirtschaft – Zurich. 62 (2007): 419-454.

McAfee, R. Preston and McMillan, John. “Analyzing the Airwaves Auction.” Journal of Economic Perspectives, 10 (1996): 159-175.

Milgrom, P. “Putting Auction Theory to Work.” Cambridge University Press, 2004.

Phillips, Owen R., Menkhaus, Dale J., and Coatney, Kalyn T. “Collusive Practices in Repeated English Auctions: Experimental Evidence on Bidding Rings.” The American Economic Review, 93 (2003): 965 – 979.

“Price Fixing, Bid Rigging, and Market Allocation Schemes; What They Are and What to Look For.” US Department of Justice. Mar 2010. <http://www.justice.gov/atr/public/guidelines/211578.pdf>

Porter, Robert H., and Zona, J. Douglas. “Detection of Bid Rigging in Procurement Auctions.” Journal of Political Economy, 101 (1993): 79-99.

Porter, Robert H., and Zona, J. Douglas. “Ohio School Milk Markets: An Analysis of Bidding.” RAND Journal of Economics, 30 (1999): 263-288.

^{[1]} Compiled by the Center for Auctions, Procurement, and Competition Policy at http://capcp.psu.edu/FCC/index.html

^{[2]} Coordinates for each state’s population center are taken from the US Census in 2000 (http://www.census.gov/geo/www/cenpop/statecenters.txt).

^{[3]} A method from an R package found online called DEoptim was used for optimization. Information about the package is available here: http://cran.r-project.org/web/packages/DEoptim/index.html.

^{[4]} DEOptim takes parameters which include NP (the size of the set of possible parameter vectors re-generated at each iteration) and iteration (number of iterations). We ran the fit test 3 times, with NP = 5, NP = 30, and NP = 150. We observed that the higher NP was, the longer each iteration took but the more quickly the optimal maximal score estimator converged. Because DEOptim generates sets of parameter vectors randomly, we reran the test with different starting seed values and did not find a significant difference in results. In future work, the optimization process may be fine-tuned, perhaps with larger NP and number of iterations on a more powerful processor.

^{[5]} See page 15 for a discussion of the problems with using final bid prices.