Detecting bid rigging in public auctions for procuring infrastructure projects: formulating the reference scenario for decision-making

Abstract

Bid rigging is a fraudulent scheme in procurement auctions resulting in non-competitive bids awarded at prices above the competitive market. Bid rigging is a global problem that wastes public agencies’ resources and taxpayers’ money. While various methods and tools have been developed to detect bid rigging, it remains challenging for public agencies to identify what a competitive baseline auction looks like in the absence of collusion. Such a baseline is commonly known as a reference scenario, which involves analyzing the behavior of honest bids in previous auctions to determine if future bids could be collusive. Research on formulating reliable reference scenarios for bid rigging detection in the auctions of infrastructure projects has received limited attention. Hence, our paper analyses the key criteria required to develop a reference scenario for public agencies to detect bid rigging during an auction. Drawing on data from Brazilian public procurement and oversight agencies, a procedure for composing robust reference scenarios for detecting bid rigging in infrastructure first-priced auctions is presented and discussed. We then test our procedure’s generalizability using data from four countries (Australia, Brazil, Spain, and the United States) in two auction formats (i.e., capped and uncapped). A better understanding of the reference scenario formulation will allow public officials to increase the likelihood of detecting bid rigging when it exists and avoid flagging it as such when no collusive practices are involved.

Keywords:

• Bid rigging
• collusion
• infrastructure
• procurement
• reference scenario

Introduction

Public agencies representing governments worldwide purchase significant amounts of goods, services, and the construction of buildings and infrastructure from the private sector every year. In 2018, public procurement was estimated to represent US$11 trillion out of the world’s global Gross Domestic Product (GDP) of approximately US$90 trillion (Bosio and Djankov 2020). Similarly, governments devote 12% of their GDP to public procurement annually (Bosio and Djankov 2020). Naturally, with such significant sums of money being involved, the potential for fraud, theft, collusion, and corruption increases (Gottschalk and Smith 2016).

Collusion in public auctions involves a deceitful agreement or secret cooperation between two or more parties (sometimes not just bidders but also officials from the contracting agency) to limit open competition in a procurement auction (OECD 2010). Collusion is prohibited by competition law in most countries. However, it is one of the most difficult white-collar crimes to detect and prove in court (Signor et al. 2019). Collusion may also occur with corruption, having mutually reinforcing effects (OECD 2010).

Bid rigging is arguably the most typical collusive arrangement deployed in public procurement auctions. When collusion takes place, “bidders determine between themselves who should ‘win’ the tender and then arrange their bids – for example, by bid rotation, complementary bidding, or cover pricing – in such a way as to ensure that the designated bidder is selected by the purportedly competitive process” (OECD 2010, p. 9). Actual bid rigging cases in public infrastructure procurement have been abundant in countries such as Brazil, Japan, Italy, and Switzerland (García Rodríguez et al. 2022).

Unfortunately, as collusion and bid rigging are frequently difficult to detect, some organizations and cartels use those practices to grow their profits illegally. Procurement authorities and law enforcement agencies must remain vigilant to prevent this crime (IMF 2019, World Bank 2020, OECD 2021). Many methods for detecting bid rigging in auctions can be found in the literature (Rubinfeld and Steiner 1983, Porter and Zona 1993, Runeson and Skitmore 1999, Bajari and Ye 2003, Ballesteros-Pérez et al. 2015a, 2015b, Ballesteros-Pérez and Skitmore 2017, Imhof 2018, Huber and Imhof 2019, Signor et al. 2019, 2020a, 2020b, 2022, 2023, Bergman et al. 2020, García Rodríguez et al. 2020, 2022). However, most public agencies’ officials (e.g., procurement authorities, competition control organisms, or law enforcement agencies) do not possess the knowledge to apply them in practice effectively, even though they have decision-making power in the procurement process.

Consequently, the OECD (2021) has recommended that public procurement officials receive professional training when dealing with potentially collusive auctions. Yet, despite the OECD’s (2021) recommendation, few public agencies worldwide have undertaken such training. Instead, it is common for public agencies to form ad hoc committees to carry on the auctions and examine the likelihood of bid rigging. The upshot is that public officials forming part of such committees face a double-edged sword as they need to manage the entire auction while needing to know about detection methods to address the presence of criminal activity in real-time.

Collusion detection methods have traditionally relied on historical data from past auctions considered ‘honest’ to generate a reference scenario. A reference scenario represents a baseline to understand the dynamics of a competitive auction. This scenario can be compared with other auctions and be used to detect the occurrence of bid rigging. Despite the importance of counting on reliable reference scenarios, limited knowledge exists about their formulation. In addressing this gap, this paper proposes a method to formulate reference scenarios using actual data obtained from public auctions of different countries. It is envisaged that a better understanding of the reference scenario formulation will allow public officials to increase the likelihood of detecting bid rigging when it exists and avoid flagging it when no collusive practices are involved.

While there exists a wealth of research that has sought to detect collusion in public auctions procuring infrastructure projects (Ballesteros-Pérez et al. 2023), contemporary lines of inquiry tend to focus on two areas:

1. the role and use of artificial intelligence methods to detect bidding anomalies automatically and accurately. These approaches generally require large amounts of data for training the algorithms, and their outputs are also usually difficult to explain (García Rodríguez et al. 2020, 2022, Love et al. 2023); and

2. comparing auctions under suspicion with a reference scenario that describes honest bids. This approach requires significantly less data and is generally easier to explain (Signor et al. 2023).

Our paper aims to contribute to both streams of research by providing a vade mecum - guidelines - for formulating reference scenarios that different collusion detection methods could implement.

To this end, our paper commences with a brief literature review on bid rigging detection methods and the knowledge gap we aim to fill. Then, the theoretical foundation for formulating reference scenarios is presented. Next, we introduce a case study involving several capped first-priced auctions for asphalt paving gathered from a Brazilian public agency. Based on the results, we propose a process for formulating robust reference scenarios that can be used for bid rigging detection purposes. Then, we test our procedure using data from other countries in capped and uncapped auctions to determine its generalizability and discuss the implications of our work. Finally, we conclude our paper by identifying its contributions and limitations.

Literature review

The problems associated with competition avoidance and bid rigging have been known for centuries (Smith 1776). Indeed, it is common for some organizations to maximize their profits by changing, manipulating, and sometimes even breaking the rules (Rothkopf and Harstad 1994). However, it was not until the 19^th century that laws to protect consumers from predatory business practices were created to promote fair competition (Kleinwächter 1883; Sherman Antitrust Act 1890).

Despite the development of laws against anti-competitive arrangements worldwide, illegal practices such as bid rigging, price-fixing, and predatory pricing, remain common (Estache and Iimi 2008, Wells 2015, Imhof 2018, García Rodríguez et al. 2022). Arguably, one of the most prominent and mediatic examples of bid rigging came to light during the Brazilian Federal Police’s Operation Car Wash (Operação Lava Jato, in Portuguese) (Signor et al. 2019, 2020a, 2020b, 2022). During the investigation, it was discovered that a cartel comprising sixteen construction companies was colluding to ensure contracts were awarded only to those involved in their cooperative arrangement (US. DoJ 2016, Signor et al. 2019). Several collusion detection methods in public procurement auctions have been proposed to combat such scandals and provide a basis to detect the presence of bid rigging.

The first and simplest methods proposed to detect potential fraud in auctions are known as ‘structural’ models. They are used to observe the configuration and market conditions that favor collusion between competitors (Porter and Zona 1993, Huber and Imhof 2019, Bergman et al. 2020). Auctions that aim at procuring homogeneous products involving repetitive interactions with the same competitors, amongst other factors, are considered to facilitate collusion and bid rigging. While tackling these structural aspects of an auction and its market is extremely important to prevent collusion, public officials who need to detect a fraud ex-post have a limited understanding of the underlying dynamics required to address this problem (Signor et al. 2022, 2023).

More recently, we have seen researchers drawn to applying machine learning algorithms to detect bid rigging and collusion (Huber and Imhof 2019, García Rodríguez et al. 2020, 2022). Machine learning algorithms learn from massive data and, like other bid rigging methods, try to find relationships and understand what ‘honest’ auctions look like so that collusive behaviors can be flagged in upcoming auctions (García Rodríguez et al. 2022).

One of the advantages of machine learning applications in bid rigging detection is that they can detect anomalies and abnormal bidding patterns in real-time. Machine learning provides promising results for detecting bid rigging in different types of auctions and bidding contexts (Huber and Imhof 2019, García Rodríguez et al. 2020, 2022). Although some machine learning strategies offer an explainable output, frequently, the pathway of the algorithms to decide on a categorical outcome (e.g., a bid deemed or not as collusive) is almost unexplainable (Love et al. 2023). Consequently, it becomes difficult, if not impossible, to substantiate a public official’s legal decision should collusion be identified and have to be defended in court (Deeks 2019).

Hence, every collusion detection method must count on a sound theoretical basis, provide accurate classification results, and be accepted by scientific and legal communities. So far, econometric methods have been accepted as a tool for providing scientific evidence at the courts of the European Union and the United States (US) (Finkelstein and Levenbach 1983, Porter and Zona 1993, Howard and Kaserman 1989, Lanzillotti 1998, Baker and Rubinfeld 1999, National Research Council 2011, European Commission 2013). Providing specific details about the various econometric models proposed in the literature is beyond the scope of this paper. However, they usually use statistical inference to combine economic theory with observations of reality (Haavelmo 1944). Multiple linear regression is a common econometric model based on the observations of bidding data from past honest auctions. It predicts honest bids as a function of k predictor variables. This model can be generically described as follows: [Eq. 1] $$\widehat{y}=b_0+\mathrm{ }{b}_1{x}_1+b_2{x}_2+\dots +b_k{x}_k+\epsilon$$[Eq. 1]

As shown by Eq. 1, the prediction of each honest bid ($\widehat{y}$) is influenced by several variables. The most common independent variables include the project (economic) size, type and location, expected competition, economic environment, and competitors’ backlog. Significant deviations between the predicted and actual auction bids can indicate the presence of collusion.

However, the more variables considered, the more complex and prone to overfitting the structural model is, which can reduce its generalizability (Baker and Rubinfeld 1999, Signor et al. 2019). Moreover, not all possible influencing variables are of interest, especially those whose values cannot be inferred in real auction settings (Baker and Rubinfeld 1999). For example, competitors’ private financial and cost data might be crucial for deciding the amount of their bids but still need to be discovered by public procurement officials. Consequently, the effectiveness of some econometric models can become questionable as the simple (but inaccurate) versus complex (and accurate) paradox emerges (Love and Tenekedjiev 2022). As expected, econometric models will never incorporate all possible predictor variables (some subjective, such as the bidder’s impetus to win), nor will they be able to describe all economic phenomena accurately. Hence, an error ε is always expected for each predicted bid, as shown in Eq. 1.

Alternatively, other methods use data from previous honest auctions to formulate a reference scenario, expressing the honest bids’ values as probability distributions. This reference scenario makes it possible to assess the probability of occurrence within a given set of bids (Ballesteros-Pérez et al. 2015a, 2015b, Signor et al. 2020a, 2022) and/or the likelihood of the lowest bid being submitted (Signor et al. 2023). Generically, these methods are known as probabilistic methods.

Signor et al. (2023) recently proposed a probabilistic method to detect bid rigging based on order statistics, relying exclusively on the pre-tender estimate (PTE) and the number of bidders per auction. Signor et al. (2023) method represents a refinement of several previous probabilistic methods. This model demonstrated a higher level of detection accuracy among current probabilistic methods found in the literature. Moreover, Signor et al. (2023) method is easy to use, understand and automate. As such, Signor et al. (2023) method may be easily implemented by many public officials who form ad hoc auction committees to avoid collusion.

We will also use this method in this paper to emulate the public officials’ work and shed some light on increasing collusion detection. To do this, we will resort to different auction datasets and probability distributions representing the honest bids and analyze the results of other decisions. Finally, we will establish a routine to formulate a robust reference scenario required for applying Signor et al. (2023) modified order statistics method, improving the likelihood of detecting bid rigging during an auction.

However, the method developed by Signor et al. (2023) focused on uncapped auctions (i.e., auctions with no upper price limit for bidders). In this paper, we are initially concerned with capped auctions. This type of auction is primarily used in countries making a significant contribution to the world economy, such as Brazil and Spain (9^th and 15^th respectively in terms of their Gross Domestic Product), but also occasionally in other countries. However, for the sake of generalization, uncapped auctions from Australia and the US will also be analyzed after the Brazilian and Spanish-capped auctions.

In capped auctions, bidders’ bids are usually expressed as a percentage of the discount offered over a maximum contract price or price cap traditionally provided by an engineering estimate – the PTE. Therefore, moving forward, we need to modify Signor et al. (2023) method to consider discounts about the price limit (cap) rather than the bid/PTE ratio as it was conceived initially. In this case, Eq. 2 expresses the discount X offered by each bidder i. Eq. 3 represents the probability of the maximum discount x being observed by chance, based on the cumulative distribution function (CDF) that describes the reference scenario and the number n of bidders in the auction. The complementary probability of the latter is the Probability of Bid Rigging (PBR), as shown in Eq. 4. [Eq. 2] $${\mathit{Discount}}_i=X_i=1\text{-}\mathrm{ }\frac{{\mathit{bid}}_i}{\mathit{PTE}}$$[Eq. 2] [Eq. 3] $$F_{X_{\left(n\right)}}\left(x\right)=\mathit{Prob}\left(\max \hspace{0.17em}\left\{X_1,\dots ,X_n\right\}\le x\right)={\left[F_X\right(x\left)\right]}^n$$[Eq. 3] [Eq. 4] $$\mathit{PBR}={1-\left[F_X\right(x\left)\right]}^n$$[Eq. 4]

Reference scenario formulation

It is only possible to detect dishonest behavior in an auction by knowing what honest behavior looks like (European Commission 2013). The reference scenario is a conditio sine qua non (i.e., a necessary condition) for detecting bid rigging. Despite this condition, doubts remain about how to formulate this scenario properly. Thus, three questions need to be answered to develop a reference scenario:

1. What auctions are representative of honest (competitive) auctions?

2. What probability distribution better represents the bids of honest (competitive) auctions?

3. How can those probability distributions’ parameters be estimated in practice?

These questions are all relevant as a public official interested in formulating a reliable (representative) reference scenario can err in different ways, such as:

• Using a dataset that does not adequately represent honest auctions;

• Adopting a probability distribution that does not adequately represent honest bids;

• Misestimating the probability distribution parameters; or

• A combination of the errors above.

In real contexts, it is almost impossible to avoid all these errors altogether. The reference scenario is and must be intrinsically variable. What we see as ‘actual bids’ are nothing but a ‘sample’ from an underlying population, which will generally remain unknown. Consequently, public officials will have to resort to a representative approach that minimizes inherent sampling errors in the best possible way. It is, therefore, necessary to study to what extent their choices may interfere with the detection of bid rigging.

Selecting the reference scenario’s data

The reference scenario must represent the bidding behavior of honest auctions. In our attempt to formulate it, we must only use data from honest auctions. Herein lays the first problem, as a public official can never guarantee whether an auction was 100% honest (all bidders submitted truly competitive bids).

Different degrees of collusion, implicit or explicit, can exist between competitors, even if this does not compromise the entire auction or condition the winning bid. Several researchers have acknowledged this incertitude and attempted to propose a variable that could predict honesty. In this regard, the number of competitors is a key factor in facilitating or hindering the action of collusive groups (Selten 1973, Kuhlman and Johnson 1983, Brannman et al. 1987, Gómez-Lobo and Szymanski 2001, Gupta 2002, Pereira 2002, Lima 2010, Grega and Nemec 2015). As a result, the number of bidders has been used as a proxy to determine an auction’s honesty, defining a threshold beyond which collusion becomes unlikely.

Selten (1973) used a game theory approach to discover that up to four competitors adopting collusive practices are generally more beneficial to the bidders than submitting competitive bids. However, collusive arrangements are difficult to hold when bidders increase to five or six (and beyond), as they leave little profit for each cartel bidder. In the area of public infrastructure projects, multiple research studies have concluded that auctions from six to eight bidders also have a low probability of full collusion (Kuhlman and Johnson 1983, Brannman et al. 1987, Gómez-Lobo and Szymanski 2001, Gupta 2002, Pereira 2002, Lima 2010, Grega and Nemec 2015).

Naturally, when the structural conditions of the market are appropriate (such as constant auctioning, repeated competitors, and little room for innovation), cartels with a large number of companies can still arise, as we have seen in Brazil, Italy, and Japan (Conley and Decarolis 2016, Huber et al. 2020, Signor et al. 2022). In contrast, due to their size, complexity, or degree of innovation, large-scale engineering projects can only be carried out by a handful of contractors. In those cases, if collusion was present during an auction, we envision that a scientific method could not be readily applied when just winning discounts are considered. In those situations, the risks and uncertainties associated with procuring large-scale projects will invariably result in contractors increasing bids.

We hasten to note that collusion can be confirmed (not just detected) when perpetrators confess in court after identifying the auctions that were (or were not) collusive. A case in point occurred in Operation Car Wash (Signor et al. 2019, 2020a, 2020b, 2022). However, we assume that cases of this ilk can be treated as an exception and that the number of bidders can be used as a proxy for honesty in most large-scale engineering projects.

In addition to the number of bidders, the selection of the reference auctions must also reflect the characteristics of the auction to be tested. Focusing again on public infrastructure auctions, special attention must be given to features such as the type of project (e.g., building or paving), location (e.g., country, state, and municipality), auction procedure (e.g., capped/uncapped, open/sealed), economic size, the time of year, or market conditions. Naturally, it will be difficult to find perfectly homogeneous auctions to the one tested, but they at least need to be relatively compatible (Baker and Rubinfeld 1999). Therefore, the ideal reference scenario for public infrastructure auctions should be formulated with bid data from auctions with eight or more bidders whose characteristics are as close as possible to the auction being tested. Considering that this set of restrictions can leave very few (maybe none) auctions to the public official, this paper presents a practical case in which the impact of including auctions with fewer bidders (six or seven bids/auction) is analyzed when formulating the reference scenario.

Selecting the probability distribution

Once the dataset of auctions for formulating the reference scenario is available, it is necessary to select a probability distribution to represent the bidders’ honest discounts. This will also eventually allow the assessment of the bid rigging probability.

Ballesteros-Pérez and Skitmore (2017) summarized the probability distributions considered by 26 previous studies conducted between 1956 and 1986, demonstrating that most previous researchers had adopted the Normal and Log-normal distributions to represent bid variations. Then, Ballesteros-Pérez and Skitmore (2017) analyzed twelve auction datasets (ten uncapped and two capped) and only found small differences between nine possible distributions. They also concluded that the Log-normal distribution is usually among the best to represent bid dispersion as it consistently offers the best fit, irrespective of the testing approach. Similarly, Signor et al. (2023) adopted a Log-normal distribution to describe the reference scenario for honest discounts from 101 Brazilian first-price uncapped auctions, whereas Signor et al. (2020b) adopted a Triangular distribution in 187 Brazilian first-price capped auctions. Both pieces of research emphasized that other probability distributions would have provided nearly identical results in their analyses.

Instead of committing to a specific distribution while considering bid variability, resorting to non-parametric methods or even to the honest bid discounts’ frequencies might be possible. Yet, the bias-variance dilemma would have to be considered in this scenario. Hence, as the “data are always incomplete” (Tukey and Wilk 1966, p. 697), adopting a probability distribution is always advisable, convenient, and parsimonious (Box 1976).

Finally, when analyzing potential probability distributions, a public official must pay special attention to the distribution’s support. For capped auctions, the lower limit of possible distributions must comply with what is established by the procurement authority – when bids above the price cap are automatically disqualified, the lower theoretical limit for discounts must be zero. As for the upper limit, a bounded distribution is also recommended to represent abnormally high (irresponsible) discounts. However, irresponsible discounts do not usually have a predefined value. Thus, the upper limit for the candidate distributions does not need to be rigidly fixed. Indeed, outliers can happen and can be excluded by other means, such as an abnormally low bid criterion (Ballesteros-Pérez et al. 2015a, 2015b). Bearing this in mind, our research tests the impact of adopting different lower-bounded probability distributions, some of which will also be upper-bounded.

Assessing the probability distribution parameters

Once the set of honest auctions is identified and a probability distribution has been selected to represent the honest discounts, it is necessary to calculate its parameters. Parameter values of distributions can generally be calculated using different approaches (e.g., method of moments, least-squares, maximum likelihood). Hence, this paper will use the least-squares and maximum likelihood, the most commonly implemented methods in statistical software packages.

Notably, many common statistical goodness of fit tests (e.g., Kolmogorov-Smirnov and Shapiro-Wilk) cannot easily reject combinations of similar parameter values obtained by different methods when the number of data points (number of bidders per auction) is small (a range of 8 to 20 bidders per auction is still considered a small sample size) (Ballesteros-Pérez and Skitmore 2017).

Detecting collusive behavior in auctions is an arduous task, with public sector agencies generally possessing a limited understanding of how to do this effectively with available scientific methods. They should, in theory, be able to identify when collusion is affecting their auctions. Still, explicit guidelines outlining how they should perform this task have yet to be developed. Developing these guidelines from a practical point of view is the primary purpose of this paper.

Research approach

To recap, our paper aims to create guidelines for formulating reference scenarios that public officials can use in practice even when they are bound to use auctions with a suboptimal number of bidders, different distributions, or parameter estimates obtained with other approaches.

We adopt a case study approach as there is a clear need to understand the bid rigging detection process in a real-life context (Yin 2009). Data from Australia, Brazil, Spain, and the United States derived from previous studies where capped and uncapped auctions were available is used for comparative analysis and formulate reference scenarios – refer to the supplementary data file. Thus, our unit of analysis is an auction.

The first geographical context of this research is a Brazilian municipality, our unit of analysis being capped first-priced auctions for asphalt paving works. This research approach aligns with many previous studies examining bid rigging (Erfani et al. 2021, Li et al. 2021, Signor et al. 2023). In our case, ad hoc auction committee members heard about bid rigging cases in the news. They became suspicious of bids received in seven auctions for all the asphalt paving their municipality procured over three years. In these seven auctions, the public officials observed that only three paving contractors participated (submitted bids), even though others in the region could have provided a bid for the work. These seven auctions accounted for about one-third of the small municipality’s infrastructure expenditures in those three years. Table 1 summarizes the suspect auctions’ competitors and winning discounts. Due to the commercial and legal sensitivities associated with the data, all information that can disclose the identity of the contracting agency or the bidders has been anonymized.

Table 1. Summary of the seven suspect asphalt paving capped first-priced auctions.

Suspect Auction	Discounts
	Contractor A	Contractor B	Contractor C
SA 1	---	1.0%	---
SA 2	---	1.0%	---
SA 3	3.1%	2.0%	1.8%
SA 4	0.0%	---	---
SA 5	0.0%	---	---
SA 6	5.5%	5.4%	---
SA 7	---	2.0%	---

Our public officials’ suspicion grew stronger when they realized that their local asphalt paving market matched the structural conditions that favor collusion (Porter and Zona 1993, 1999):

• there are repetitive auctions, and the same companies tend to compete with each other;

• there are few potential competitors, and entry barriers seem to exist;

• companies have complete information on the major variables of the competition; and

• the product is homogeneous with little possibility for innovation.

Unfortunately, the public officials could not correct the above structural conditions favoring collusion in an ex-post analysis (once the auctions’ deadlines had passed and the bids were unsealed). Their only available option was to analyze the discounts from each auction. In this instance and, as anticipated earlier, we will mimic their steps and use the order statistics method that Signor et al. (2023) proposed to detect bid rigging but adapted to capped auctions.

Dataset

To formulate the reference scenario that Signor et al. (2023) method requires, public officials must choose a set of specific bids and describe it through a probability distribution. To do so, they could assess contemporaneous public data on nearby municipalities’ paving auctions, thus assuring the first condition on the adequacy of the dataset. Public data from about 2,500 paving auctions were available. As the second condition on the dataset’s adequacy, the officials learned that auctions with at least eight bidders could be considered honest, as described earlier. Only four auctions in the complete paving auction dataset meet this criterion. As these four auctions comprised more than 30 bids, some public officials could consider it enough to formulate the reference scenario. However, other officials could argue that four auctions may be too few to represent the competitive (honest) market adequately; hence, they decide to relax the number of bidders’ threshold and admit auctions with fewer bidders.

Eleven auctions had seven or more bidders, and 41 had six or more bidders (which can be considered sufficient in every aspect). Descriptive statistics for each of these reference datasets’ discounts are shown in Table 2, and their histograms are represented in Figure 1. These reference datasets were screened for possible outliers (i.e., in this case, discounts that are so high that they might involve irresponsible pricing). Therefore, using the Tukey (1977) criterion to exclude outliers, a general threshold of 35% was established. Only one discount in the ≥6 datasets exceeded that limit.

Figure 1. Discount histograms for the three reference datasets that could be used to formulate the reference scenario.

Table 2. Descriptive statistics for the three reference datasets.

Bidders	≥8	≥7	≥6
N° of auctions	4	11	41
N° of valid bids	36	85	264
Minimum Discount	1%	0%	0%
Q1	6%	6%	7%
Median Discount	11%	11%	13%
Q3	16%	15%	18%
Maximum Discount	29%	31%	34%

Table 2 indicates that the quartiles of the three possible reference datasets are similar. All the datasets present lower than higher discounts, which looks feasible in a competitive environment. Figure 1 shows that the ≥8 dataset has no discount above 30% and has a mode of discounts ranging between 5% and 10%. Datasets ≥7 and ≥6 display discounts in the maximum range of 30% to 35%, and their mode lies in the discount bin from 10% to 15%. These differences will likely condition both the probability distribution and its parameter values. As a result, the bid rigging detection capability offered by the reference scenarios using those auctions might also be affected. We will test all this later.

Probability distribution

Once the public officials have chosen a reference set of honest auctions, their next step is to select a probability distribution representing the discounts from those auctions. As previously noted, the probability distribution’s choice needs to consider that there is a lower limit of capped bids at a 0% discount. Therefore, only lower-bounded distributions can be considered. In addition, aiming to analyze the behavior of different distributions that the public officials could choose, we studied three candidates that were also upper-bounded and another which is not (Figure 2). The latter can better represent the uncertainty on the approximate location of the maximum irresponsible discount value. Hence, the tested probability distributions were:

Figure 2. Analyzed probability distributions.

• Uniform, which is a lower- and upper-bounded distribution that adopts both the zero-capped and the upper-irresponsible discounts, assuming a constant probability for every discount within this range;

• Triangular, which is a lower- and upper-bounded distribution that assumes that the zero-capped discount has the maximum probability of occurrence and that this probability reduces linearly till zero at the upper-irresponsible discount;

• Beta, which is a lower- and upper-bounded distribution that has a more flexible fit for the varying frequencies of honest discounts and assumes that both the zero-capped and the upper-irresponsible discounts have zero probability of occurrence; and

• Log-normal, a lower-bounded distribution with a flexible fit for the varying frequencies of honest discounts. It assumes that the zero-capped discounts have zero probability of occurrence and have no superior limit to represent irresponsible discounts.

Probability distribution parameters

The public officials’ last step in formulating the reference scenario is to assess the parameters of the probability distributions selected to represent honest discounts. We assume they would adopt the most common parameter estimation methods, such as the least squares or maximum likelihood estimation. Table 3 summarizes the parameters obtained using these two approaches for every combination of the dataset and probability distribution.

Table 3. Summary of probability distributions’ parameters (rejected distributions underlined).

Dataset [and KS critical value (α = 5%)]	Method	Probability distribution parameters [and KS statistic]
		Uniform (a; b)	Triangular (a; c; b)	Beta (α; β; A; B)	Log-normal (µ; σ)
≥8 [0.221]	Maximum likelihood	0.00; 0.26 [0.138]	0.00; 0.00; 0.33 [0.141]	1.24; 2.10; 0.00; 0.33 [0.104]	0.09; 0.83 [0.106]
	Least squares	0.00; 0.23 [0.133]	0.00; 0.00; 0.35 [0.126]	1.15; 2.27; 0.00; 0.35 [0.100]	0.10; 0.72 [0.112]
≥7 [0.148]	Maximum likelihood	0.00; 0.25 [0.165]	0.00; 0.00; 0.34 [0.102]	1.37; 2.76; 0.00; 0.36 [0.095]	0.09; 0.86 [0.141]
	Least squares	0.00; 0.22 [0.118]	0.00; 0.00; 0.35 [0.097]	1.37; 2.75; 0.00; 0.35 [0.078]	0.10; 0.67 [0.109]
≥6 [0.084]	Maximum likelihood	0.00; 0.26 [0.056]	0.00; 0.00; 0.35 [0.148]	1.38; 2.35; 0.00; 0.35 [0.065]	0.10; 1.03 [0.161]
	Least squares	0.00; 0.25 [0.057]	0.00; 0.00; 0.39 [0.105]	1.60; 2.65; 0.00; 0.35 [0.054]	0.12; 0.58 [0.111]

As Table 3 shows, for the ≥8 datasets, none of the distributions can be rejected according to the Kolmogorov-Smirnov (KS) test (α = 5%). In the case of the ≥7 datasets, only the Uniform distribution assessed through the maximum likelihood estimation is rejected. For the ≥6 datasets, the Triangular and the Log-normal distributions are rejected for both parameters’ assessment approaches. Figure 3 shows the adherence of all tested probability distributions and their parameters.

Figure 3. Q-Q plots for each distribution (≥8 bidders to ≥6 bidders, left to right) and parameters’ method of assessment [maximum likelihood estimation (MLE) above and least squares method (LSM) below]. Legend: Uniform (-); Triangular (∆); Beta (x) and Log-normal (+).

Assessing the probability of bid rigging

After having formulated the reference scenario (by selecting a dataset that represents the honest auctions and having chosen a specific distribution with some parameter values that maximize the fit to the set of actual honest discounts), the officials can calculate the probability of the existence of bid rigging for every auction listed in Table 1. We performed the calculations with Signor et al. (2023) order statistics method using the modifications shown in Eq.2-4.

More precisely, Signor et al. (2023) method rely on order statistic to assess the bid rigging probability. Hence, based on a dataset of honest bids for comparison, only the winning bid and the number of competitors in each suspect auction are sufficient to determine the probability of collusion. As a rule, the bid rigging probability is inversely proportional to the winning discount, and as the number of competitors increases, the method requires larger winning discounts to rule out the probability of bid rigging. Thus, the calculations were made for all possible combinations of honest auctions’ probability distributions in Table 3.

Table 4 identifies minor variations in the results obtained when different combinations of auctions, probability distributions, and parameters represent the reference scenario. Figure 4 allows comparing the same results graphically. Since the public officials would achieve similar results even when adopting different reference scenarios, they should feel confident when classifying all seven suspect auctions as most likely collusive. Thus, they can take the appropriate actions, such as sending this information to law enforcement agencies. The latter will then be able to analyze the suspect auctions along with other information (e.g., breaches of fiscal or telematic secrecy). These enforcement agencies will also search for other ‘plus factors’ that would prove, beyond a reasonable doubt, that the bid rigging results were effectively due to collusion and not from an unfortunate combination of coincidences (Lanzillotti 1998).

Figure 4. Graphical representation of the bid rigging probabilities for the suspect auctions.

Table 4. Probability of collusion existence in the suspect auctions according to Signor et al. (2023) order statistics method (for the non-rejected probability distributions).

Suspect Auction	N° of bidders	Winning discount	Dataset	Probability of bid rigging
				Maximum Likelihood Estimation				Least Squares Method
				Uniform	Triangular	Beta	Log-normal	Uniform	Triangular	Beta	Log-normal
SA 1	1	1.0%	≥8	96.1%	93.9%	96.9%	99.6%	95.6%	94.3%	95.9%	99.9%
			≥7		94.1%	97.4%	99.5%	95.4%	94.3%	97.3%	100.0%
			≥6	96.1%		97.8%		95.9%		98.7%
SA 2	1	1.0%	≥8	96.1%	94.0%	96.9%	99.6%	95.6%	94.3%	95.9%	99.9%
			≥7		94.1%	97.4%	99.5%	95.4%	94.3%	97.3%	100.0%
			≥6	96.1%		97.9%		96.0%		98.7%
SA 3	3	3.1%	≥8	99.8%	99.4%	99.8%	99.9%	99.8%	99.5%	99.7%	100.0%
			≥7		99.5%	99.9%	99.9%	99.7%	99.5%	99.8%	100.0%
			≥6	99.8%		99.9%		99.8%		100.0%
SA 4	1	0.0%	≥8	99.9%	99.8%	100.0%	100.0%	99.9%	99.9%	99.9%	100.0%
			≥7		99.9%	100.0%	100.0%	99.9%	99.9%	100.0%	100.0%
			≥6	99.9%		100.0%		99.9%		100.0%
SA 5	1	0.0%	≥8	99.9%	99.8%	100.0%	100.0%	99.9%	99.9%	99.9%	100.0%
			≥7		99.9%	100.0%	100.0%	99.9%	99.9%	100.0%	100.0%
			≥6	99.9%		100.0%		99.9%		100.0%
SA 6	2	5.5%	≥8	95.5%	90.7%	94.7%	92.6%	94.3%	91.6%	93.1%	95.6%
			≥7		91.2%	94.6%	92.2%	93.8%	91.6%	94.3%	96.6%
			≥6	95.5%		96.0%		95.2%		97.1%
SA 7	1	2.0%	≥8	92.2%	88.1%	92.8%	96.4%	91.2%	88.7%	91.0%	98.6%
			≥7		88.4%	93.4%	95.9%	90.8%	88.7%	93.2%	99.1%
			≥6	92.2%		94.5%		91.9%		96.1%

As further evidence of collusion, it is also possible to calculate the joint probability of several likely-bid rigged auctions occurring in a row. This approach is described in Signor et al. (2023). In the present case, the lowest probability of collusion (88.1%) was observed in Suspect Auction #7 from Table 4. With this result, it is also possible to conclude that the joint probability of winning seven out of seven auctions in a row was about 3.4 × 10⁻⁷, which would be, of course, nearly impossible to find in practice.

Validation

Public officials can adopt different reference datasets, probability distributions, and parameters to formulate the reference scenario representing honest bidding behavior. Then, apply a collusion detection method like Signor et al. (2023) method to develop a probability of collusion in one or several suspect auctions. However, the results of the subsequent bid rigging calculations will differ depending on those previous decisions.

Different results for seven suspect auctions were found in our case study using different reference scenarios. As these differences in bid rigging probabilities were small, it can be concluded that the results are convergent under similar conditions no matter what reference dataset, probability distribution, or parameters are chosen to formulate the reference scenario. Considering these results, we suggest that public officials adopt the basic procedure described in the previous section, summarized in Figure 5.

Figure 5. Proposed procedure flowchart to detect bid rigging by ad-hoc committees.

For validation purposes, we also analyze how the procedure summarized in Figure 5 performs in other types of auctions (i.e., capped and uncapped) and countries (e.g., Australia, Spain, and the United States, US).

Capped auctions

Our first validation test involves 51 capped auctions whose scope involved wastewater treatment plants and sewer systems in Spain and utilizes Ballesteros-Pérez et al. (2012) dataset. Following the previous guidelines, we estimate that Spanish public officials might want to test five auctions with a small number of bidders and low winning discounts, as shown in Table 5.

Table 5. Summary of the five suspect capped first-priced Spanish auctions.

Auction ID	Number of bidders	Winning discount
SP51 #6	5	8.0%
SP51 #23	4	6.8%
SP51 #24	1	0.5%
SP51 #40	1	2.0%
SP51 #41	3	5.8%

As the suspicious public officials in Spain would not be able to know in advance which auctions would be idoneous or not, they could establish the number of bidders as a criterion of honesty. Of these 51 Spanish-capped auctions, 31 had eight or more bidders, which would already be considered adequate under statistical aspects. Nevertheless, to test our guidelines, we will also test datasets with seven or more bidders (32 auctions) and six or more bidders (35 auctions), as shown in Table 6, which shows the basic statistics for each dataset. The Tukey (1977) criterion was adopted, and one outlier was excluded from each dataset.

Table 6. Descriptive statistics for the three Spanish reference datasets.

Bidders	SP51 ≥ 8	SP51 ≥ 7	SP51 ≥ 6
N° of auctions	31	32	35
N° of bids	570	577	595
N° of valid bids	569	576	594
Minimum Discount	0%	0%	0%
Q1	11%	11%	11%
Median Discount	17%	16%	16%
Q3	22%	22%	22%
Maximum Discount	37%	37%	37%

Following the proposed guidelines, the next step for the Spanish public officials would be to define statistical distributions for each dataset and calculate their respective parameters. Despite other distributions showing promising results, here in our tests, we kept the four original distributions (Uniform, Triangular, Beta, and Log-normal) for better comparison with the Brazilian results.

A positive point of this test is that while the Brazilian and Spanish auctions are capped, their histograms have markedly different characteristics: Spanish auctions have relatively fewer zero discounts. This significantly affects the Triangular distribution, which no longer presents its maximum at the zero discount as it did for the Brazilian auctions. Those differences can also be noticed when comparing Figure 1 and Figure 6. Namely, the latter shows the histograms for the ≥8, ≥7, and ≥6 bidders’ Spanish datasets. Table 7 also summarizes the parameters adopted for each Spanish dataset and distribution.

Figure 6. Discount histograms and candidate probability distributions for the three reference datasets that could be used to formulate the Spanish reference scenarios.

Table 7. Summary of probability distributions’ parameters for the Spanish datasets (rejected distributions underlined).

Dataset [and KS critical value (α = 5%)]	Method	Probability distribution parameters [and KS statistic]
		Uniform (a; b)	Triangular (a; c; b)	Beta (α; β; A; B)	Log-normal (µ; σ)
SP51 ≥ 8 [0.057]	Maximum likelihood	0.03; 0.31 [0.070]	0.00; 0.15; 0.38 [0.049]	1.88; 2.40; 0.00; 0.38 [0.051]	0.14; 0.76 [0.144]
	Least squares	0.04; 0.29 [0.072]	0.00; 0.14; 0.37 [0.045]	2.08; 2.27; 0.00; 0.35 [0.042]	0.16; 0.46 [0.090]
SP51 ≥ 7 [0.057]	Maximum likelihood	0.03; 0.30 [0.068]	0.00; 0.15; 0.38 [0.051]	1.86; 2.41; 0.00; 0.38 [0.050]	0.09; 0.76 [0.141]
	Least squares	0.04; 0.29 [0.075]	0.00; 0.14; 0.37 [0.047]	1.95; 2.15; 0.00; 0.35 [0.040]	0.16; 0.47 [0.088]
SP51 ≥ 6 [0.056]	Maximum likelihood	0.03; 0.30 [0.066]	0.00; 0.15; 0.38 [0.052]	1.81; 2.37; 0.00; 0.38 [0.050]	0.10; 0.78 [0.143]
	Least squares	0.04; 0.29 [0.076]	0.00; 0.13; 0.37 [0.045]	1.96; 2.19; 0.00; 0.35 [0.041]	0.16; 0.48 [0.090]

Once the probabilistic distributions and their parameters have been evaluated, it is time for public officials to calculate the probabilities of collusion for suspicious auctions. Table 8 and Figure 7 summarize the results.

Figure 7. Representation of the bid rigging probabilities for the Spanish suspect auctions.

Table 8. Individual probability results of collusion in the Spanish suspect auctions (for the non-rejected probability distributions).

Suspect Auction	N° of bidders	Winning discount	Dataset	Probability of bid rigging
				Maximum Likelihood Estimation				Least Squares Method
				Uni	Tri	Beta	LN	Uni	Tri	Beta	LN
SP51 #6	5	8.0%	≥8		100.0%	100.0%			100.0%	100.0%
			≥7		100.0%	100.0%			100.0%	100.0%
			≥6		100.0%	100.0%			100.0%	100.0%
SP51 #23	4	6.8%	≥8		100.0%	100.0%			100.0%	100.0%
			≥7		100.0%	100.0%			100.0%	100.0%
			≥6		100.0%	100.0%			100.0%	100.0%
SP51 #24	1	0.5%	≥8		100.0%	99.9%			100.0%	99.9%
			≥7		100.0%	99.9%			100.0%	99.9%
			≥6		100.0%	99.9%			99.9%	99.9%
SP51 #40	1	2.0%	≥8		99.3%	98.5%			99.2%	99.1%
			≥7		99.3%	98.5%			99.2%	98.8%
			≥6		99.3%	98.3%			99.2%	98.8%
SP51 #41	3	5.8%	≥8		100.0%	99.9%			100.0%	100.0%
			≥7		100.0%	99.9%			100.0%	99.9%
			≥6		100.0%	99.9%			100.0%	99.9%

Uncapped auctions

Recognizing that capped auctions are only common in some countries, we also tested the proposed guidelines for formulating reference scenarios in uncapped auctions. On this occasion, some Australian and US auction datasets were tested:

• Australian (AU161), which comprises 160 uncapped auctions by the specialist contractors’ bids for New South Wales Public Works and Housing (Runeson 1987);

• US (US62) comprises 58 uncapped auctions by the US Government agency building contracts (Brown 1986).

Trying to emulate what careful public officials would do in each country, auctions with few bidders and small discounts (two strong indicators of collusion) were selected for testing. From this point on, aiming to clarify the process, we will present the application of the recommended guidelines for each country separately.

(a) Australia

Table 9 summarizes five suspicious Australian auctions with few bidders and negative discounts (meaning that the winning bids were higher than their respective PTEs).

Table 9. Summary of the suspect Australian uncapped first-priced auctions.

Auction ID	Number of bidders	Winning discount
AU161 #41	4	−30.8%
AU161 #94	4	−24.6%
AU161 #97	5	−32.0%
AU161 #103	3	−24.4%
AU161 #113	2	−2.8%

As good reference scenarios are needed to tell honest from dishonest behavior, Australian public officials should use the proposed guidelines. Thus, adopting the number of bidders as a primary indicator of honesty, they could gather a statistically sufficient number of auctions with eight or more bidders. Yet here, we will also test the ≥7 and ≥6 bidders’ datasets for comparison. After excluding the outliers according to Tukey’s (1977) criterion, each possible dataset has the basic statistics described in Table 10.

Table 10. Descriptive statistics for every Australian reference dataset.

Bidders	≥8	≥7	≥6
N° of auctions	39	56	85
N° of bids	399	518	692
N° of valid bids	387	493	660
Minimum Discount	−72%	−68%	−64%
Q1	−15%	−17%	−16%
Median Discount	1%	−1%	0%
Q3	19%	15%	13%
Maximum Discount	70%	64%	57%

Table 10 shows that the discounts for the Australian uncapped auctions can be positive or negative, as the PTE does not limit the bids. For this reason, naturally, the Log-normal distribution cannot be used in this case. For convenience, we will replace it with the Normal distribution. Thus, statistical distributions were defined following the proposed guidelines, and their respective parameters were calculated for each dataset. Figure 8 shows the histograms and schematic representation of possible probability distributions that could represent the honest discounts, and Table 11 summarizes the parameters adopted for each Australian dataset.

Figure 8. Discount histograms and candidate probability distributions for the Australian reference scenario.

Table 11. Summary of probability distributions’ parameters for the Australian datasets (rejected distributions underlined).

Dataset [and KS critical value (α = 5%)]	Method	Probability distribution parameters [and KS statistic]
		Uniform (a; b)	Triangular (a; c; b)	Beta (α; β; A; B)	Normal (µ; σ)
AU161 ≥ 8 [0.069]	Maximum likelihood	−0.47; 0.49 [0.093]	−0.75; 0.02; 0.72 [0.084]	37.83; 16.53; −3.09; 1.37 [0.041]	0.01; 0.28 [0.051]
	Least squares	−0.37; 0.42 [0.093]	−0.63; 0.02; 0.66 [0.044]	37.80; 16.53; −3.09; 1.37 [0.041]	0.01; 0.28 [0.056]
AU161 ≥ 7 [0.062]	Maximum likelihood	−0.44; 0.43 [0.086]	−0.69; 0.00; 0.65 [0.072]	12.44; 8.79; −1.41; 0.99 [0.040]	0.01; 0.25 [0.038]
	Least squares	−0.37; 0.36 [0.083]	−0.60; 0.01; 0.67 [0.033]	12.40; 8.82; −1.41; 0.99 [0.040]	0.00; 0.25 [0.038]
AU161 ≥ 6 [0.053]	Maximum likelihood	−0.41; 0.39 [0.086]	−0.65; 0.02; 0.58 [0.063]	18.82; 11.46; −1.67; 1.00 [0.043]	0.01; 0.23 [0.040]
	Least squares	−0.33; 0.32 [0.095]	−0.56; 0.01; 0.52 [0.038]	18.83; 11.50; −1.67; 1.00 [0.042]	0.01; 0.23 [0.034]

The proposed guidelines’ steps were followed, and the probabilities of every Australian suspect auction were assessed. The individual results are shown in Table 12, and Figure 9 graphically summarizes the comparison of the results. This first test on uncapped auctions produced logical results aligned with the prevalent theory: high collusion probabilities when the winning discounts are low and a smaller probability of bid rigging for the one auction with only two bidders and a winning bid value near the PTE. Moreover, the results are convergent (especially when the probability of collusion is high), meaning that the guidelines proposed in this paper are robust for collusion detection.

Figure 9. Graphical representation of the bid rigging probabilities for the Australian suspect auctions.

Table 12. Individual probability results of collusion in the Australian suspect auctions.

Suspect Auction	N° of bidders	Winning discount	Dataset	Probability of bid rigging
				Maximum Likelihood Estimation				Least Squares Method
				Uni	Tri	Beta	Normal	Uni	Tri	Beta	Normal
AU161 #41	4	−30.8%	≥8			100.0%	100.0%		100.0%	100.0%	100.0%
			≥7			100.0%	100.0%		100.0%	100.0%	100.0%
			≥6			100.0%	100.0%		100.0%	100.0%	100.0%
AU161 #94	4	−24.6%	≥8			99.9%	99.9%		99.9%	99.9%	99.9%
			≥7			99.9%	99.9%		99.9%	99.9%	99.9%
			≥6			99.9%	99.9%		99.9%	99.9%	99.9%
AU161 #97	5	−32.0%	≥8			100.0%	100.0%		100.0%	100.0%	100.0%
			≥7			100.0%	100.0%		100.0%	100.0%	100.0%
			≥6			100.0%	100.0%		100.0%	100.0%	100.0%
AU161 #103	3	−24.4%	≥8			99.5%	99.4%		99.4%	99.5%	99.4%
			≥7			99.5%	99.5%		99.4%	99.4%	99.6%
			≥6			99.6%	99.6%		99.6%	99.6%	99.6%
AU161 #113	2	−2.8%	≥8			81.9%	80.1%		81.3%	81.8%	80.1%
			≥7			79.6%	77.8%		79.0%	79.0%	79.3%
			≥6			79.0%	78.0%		78.9%	78.7%	78.0%

(b) United States

Aiming to strengthen the generalist nature of the guidelines proposed in our paper, a final test was carried out for uncapped auctions for the US. Once again, all steps described in the flowchart of Figure 5 were followed. Only basic information will be presented in the Tables and Figures below for the sake of brevity. Table 13 summarizes the suspicious US auctions.

Table 13. Summary of the suspect US uncapped first-priced auctions.

Auction ID	Number of bidders	Winning discount
US62 #4	4	−53.6%
US62 #10	2	−4.7%
US62 #26	4	−17.6%
US62 #28	3	−7.2%
US62 #54	5	−34.5%

A particularity of the US case is that public officials in that country would find only 16 auctions with eight or more bidders when looking for honest auctioning data. If auctions with seven bidders were also admitted, the dataset would have 24 auctions. Finally, when considering all auctions with six or more bidders, would be 39 auctions available to compose the reference dataset. Here, we will test all options for better comparison. After excluding the outliers according to Tukey’s (1977) criterion, each possible dataset has the basic statistics displayed in Table 14.

Table 14. Descriptive statistics for every US reference dataset.

Bidders	≥8	≥7	≥6
N° of auctions	16	24	39
N° of bids	165	221	311
N° of valid bids	151	199	289
Minimum Discount	−47%	−45%	−46%
Q1	−13%	−10%	−11%
Median Discount	−3%	−1%	0%
Q3	7%	8%	8%
Maximum Discount	38%	39%	41%

Once again, the Log-normal distribution cannot be used and will be replaced with the Normal distribution. Figure 10 shows the histograms and schematic representation of the possible probability distributions that could represent the honest discounts, and Table 15 summarizes the parameters adopted for each US dataset.

Figure 10. Discount histograms for the three reference datasets that could be used to formulate the US reference scenario and candidate probability distributions for honest bids.

Table 15. Summary of probability distributions’ parameters for the US datasets (rejected distributions underlined).

Dataset [and KS critical value (α = 5%)]	Method	Probability distribution parameters [and KS statistic]
		Uniform (a; b)	Triangular (a; c; b)	Beta (α; β; A; B)	Normal (µ; σ)
US62 ≥ 8 [0.111]	Maximum likelihood	−0.34; 0.25 [0.122]	−0.52; 0.01; 0.39 [0.082]	3883.4; 14.45; −175.71; 0.61 [0.060]	−0.05; 0.17 [0.087]
	Least squares	−0.25; 0.19 [0.121]	−0.43; −0.01; 0.33 [0.064]	3889.0; 14.45; −175.72; 0.61 [0.054]	−0.03; 0.16 [0.060]
US62 ≥ 7 [0.096]	Maximum likelihood	−0.30; 0.25 [0.110]	−0.48; 0.01; 0.40 [0.093]	312.93; 38.73; −8.37; 1.01 [0.052]	−0.02; 0.16 [0.064]
	Least squares	−0.23; 0.20 [0.095]	−0.38; 0.01; 0.31 [0.044]	314.00; 38.75; −8.37; 1.01 [0.053]	−0.02; 0.15 [0.050]
US62 ≥ 6 [0.080]	Maximum likelihood	−0.31; 0.27 [0.103]	−0.50; 0.02; 0.41 [0.074]	196.46; 20.04; −7.77; 0.77 [0.044]	−0.02; 0.17 [0.061]
	Least squares	−0.25; 0.22 [0.094]	−0.42; 0.03; 0.33 [0.050]	196.60; 20.10; −7.75; 0.77 [0.046]	−0.01; 0.16 [0.045]

The same steps were followed, and the probabilities of every US suspect auction were assessed. The individual results are shown in Table 16. Figure 11 graphically compares the results.

Figure 11. Graphical representation of the bid rigging probabilities for the US suspect auctions.

Table 16. Individual probability results of collusion in the US suspect auctions.

Suspect Auction	N° of bidders	Winning discount	Dataset	Probability of bid rigging
				Maximum Likelihood Estimation				Least Squares Method
				Uni	Tri	Beta	Normal	Uni	Tri	Beta	Normal
US62 #4	4	−53.6%	≥8		100.0%	100.0%	100.0%		100.0%	100.0%	100.0%
			≥7		100.0%	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%
			≥6		100.0%	100.0%	100.0%		100.0%	100.0%	100.0%
US62 #10	2	−4.7%	≥8		78.5%	79.1%	74.3%		78.9%	79.3%	79.1%
			≥7		81.1%	82.2%	81.3%	81.9%	83.0%	82.7%	81.6%
			≥6		81.2%	82.9%	80.9%		83.0%	83.0%	83.3%
US62 #26	4	−17.6%	≥8		99.6%	99.8%	99.7%		99.8%	99.8%	99.9%
			≥7		99.8%	99.9%	99.9%	100.0%	99.9%	99.9%	100.0%
			≥6		99.8%	99.9%	99.9%		99.9%	99.9%	99.9%
US62 #28	3	−7.2%	≥8		92.8%	93.5%	91.0%		93.5%	93.6%	93.8%
			≥7		94.2%	95.2%	94.8%	95.0%	95.6%	95.5%	95.2%
			≥6		94.2%	95.4%	94.5%		95.4%	95.4%	95.7%
US62 #54	5	−34.5%	≥8		100.0%	100.0%	100.0%		100.0%	100.0%	100.0%
			≥7		100.0%	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%
			≥6		100.0%	100.0%	100.0%		100.0%	100.0%	100.0%

This second test on uncapped auctions also produced logical results aligned with the prevalent theory. Although results vary when using different datasets, probability distributions, and parameters for the reference scenario, collusive behavior will likely be flagged when abnormally low discounts (more expensive bids) are found in auctions, even considering the number of bidders. Moreover, the results are convergent (especially when the probability of collusion is high), meaning (again) that the guidelines proposed in this paper are robust for collusion detection.

Discussion

Automating the calculation process proposed and tested in this paper in other real auction contexts is possible. Similarly, creating a computational routine that calculates the bid rigging probabilities and implementing our approach would be equally feasible. This would offer the user the same results and allow taking the necessary measures in the presence of potential fully colluded auctions.

Hence, after the extensive validation exercise, a major practical implication is that public officials in charge of flagging bid rigging should have a reasonable degree of freedom when formulating the reference scenario representing honest bidding behavior. Also, when an honest auction dataset is available, we advocate that Signor et al. (2023) order statistics collusion detection method should provide sufficiently reliable results for flagging potentially collusive auctions. We tested this approach for different types of auctions (capped and uncapped) and countries (Brazil, Spain, Australia, and the US), and our results are consistent when the probability of collusion is high.

In this paper, we have provided public agencies with much-needed practical knowledge to apply detection methods effectively. Ensuring fairness and honesty during the bidding process provides the platform for mitigating opportunistic behaviors, claims, and delays. Collusion increases construction costs and reduces product quality and the ability to innovate, all at the expense of excessively driving up the contractor’s profitability (Friedman 1956, Li and Love 1999, OECD 2010). As Angel Gurria (2018), former OECD Secretary-General, cogently remarked, “integrity, transparency, and the fight against corruption [collusive behaviors] have to be part of the culture. They have to be thought of as fundamental values”. Detecting bid rigging in public auctions is just one of many hurdles that must be conquered to eliminate corruption in construction. Still, it is the first hurdle that must fall as it enables misbehaviors to be launched and cascade and manifest in projects (Signor and Love 2024).

It is important to point out that we identified potentially collusive auctions in old datasets from different countries in the tests we carried out. However, there are cases in which behaviors compatible with collusion can be verified in honest auctions and explained by rational economic decisions (Rubinfeld and Steiner 1983). For this reason, it is emphasized that these collusion probabilities should be seen only as warnings or alarms in the administrative phases of auctions where public officials do not have access to investigative tools. Confirmation of collusion (purposeful action) in these auctions can only be done through ‘plus-factors’ or a ‘smoking gun’ (Lanzillotti 1998), which generally can only be achieved in procedures such as police investigations or leniency agreements of antitrust agencies in each country.

Conclusions

The public sector relies on competitive bidding market mechanisms to achieve better value for money. However, lower prices, improved quality products, services, and innovation can only occur when companies genuinely compete. In the presence of collusion, the benefits of a competitive marketplace are undermined. Nevertheless, governments worldwide are confronted with bid rigging in public procurement.

While many governments and organizations, such as the Organization for Economic Co-operation and Development, have produced robust guidelines to combat bid rigging in public procurement processes, public officials must understand collusive auctions’ dynamics and nuances. This knowledge gap impacts their ability to detect the existence of collusive practices in the auctions they are handling. Possessing a basic understanding of the reference scenario that represents honest competition can go a long way toward helping them eventually detect the presence of bid rigging.

In this paper, we have generated alternative reference scenarios from real auctions and observed how their different characteristics impact (or not) the eventual bid rigging detection capability. Namely, we have analyzed the potential influence of the number of bidders as a proxy to define the honesty of a set of auctions, the probability distributions adopted to describe the honest bids, and the estimation method of these probability distributions’ parameters. The approach we used to analyze the impact of different reference scenarios on bid rigging detection was the order statistics method recently proposed by Signor et al. (2023), whose accuracy is superior to most collusion-detection methods published in the past. Furthermore, preliminary tests by the authors show that the proposed guidelines for formulating reference scenarios should also have similar results for other methods.

Our results and analyses all point towards the same outcome: when auctioneers are faced with various alternatives for building a reference scenario and when the probability of full collusion is high, different reference scenarios will likely lead to similar detection results. In this paper, we have suggested a particular approach that can be followed to build alternative (but still reliable) reference scenarios.

Hence, our developed reference scenarios act as a vade mecum to enable different methods to detect collusive bidding behavior. Tests show this conclusion is valid for other countries and types of auctions. The results obtained for capped auctions in Brazil and Spain and uncapped auctions in Australia and the US allow us to conclude that the proposed guidelines for constructing reference scenarios are robust and generalizable. As this is the first in-depth analysis of the formation of reliable reference scenarios that depict honest competition in infrastructure auctions, the authors trust that the first results reported here will be of value to all researchers and practitioners dealing with public procurement auctions daily.

While we understand that our findings are valid for most engineering project auctions, we need to point out that they have been dependent on reliable data stemming from honest bids. This point can be particularly limiting in cases where only a handful of companies can carry out the project in question or when there are not enough honest auctions with the minimum number of six to eight bidders or other characteristics necessary to compose the reference scenario. Thus, there is a need for future research addressing these scenarios.

Finally, as our work only analyzed sealed, first-price auctions, it is also expected that future studies will look at other forms of bidding, such as second and below-average prices. Also, testing our approach in other datasets comprising different (perhaps more recent) periods, locations, projects, and bidding forms may refute or, as we hope, strengthen our findings.

Acknowledgments

The authors would like to thank the Editor, Dr. Florence Phua, and Associate Editor, Professor Islam El-adaway, and the four anonymous referees for their constructive and insightful comments on an earlier version of this manuscript.

Disclosure statement

The authors report that there are no competing interests to declare.

Data availability statement

All data and examples supporting this study’s findings are available in the attached supplementary file and from the corresponding author upon request.