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Nonparametric Identification of First-Price Auction with Unobserved Competition: A Density Discontinuity Framework
\paragraph{Motivations.} There exists a large literature on nonparametric identification of auction models; see, e.g., Athey and Haile (2007) or Hendricks and Porter (2007) for a review. In the case of sealed-bid first-price auctions, a vast majority of work assumes that the analyst can observe all of the bids, or both the winning bid and the number of competitors. This may not always be observed. In French timber auctions, for example, only the winning bid may be available to researchers to preserve bidder anonymity (Lamy, 2012). Indeed, it is common practice in many markets that can be treated as auctions for only the winning bid (i.e. the transaction price) to be recorded. For instance, a company soliciting price quotes for a task to be completed is implicitly organizing a first-price auction. While the company may not record all quotes or the number of responses, the price paid to the winning bidder is likely to appear in accounting records. “Bidding wars” are becoming commonplace in housing markets, where houses are sold through a competitive bidding process resembling an informal first-price auction, as noted in Han and Strange (2015). Governments may offer subsidies to attract firms, as recently considered by Kim and Yu (2024) and Slattery (2020) using an auction framework. Observing all the subsidy offers may be difficult, because states or firms may both have some interest in confidentiality. Hence, in many economic situations of interest, the records may only contain the final winning bid. Therefore, the ability to identify auction primitives solely from winning bid data may enlarge the scope of auction theory applications.
A second motivation stems from misspecification considerations. Indeed, structural estimation procedures of auctions crucially depend on the number of active buyers, which may differ from the observed number. For instance, Laffont, Ossard and Vuong (1995) consider an application where bidders are agents of several retail sellers, in which case the number of bids underestimates the level of competition. Given 11 observed bids, these authors estimate the number of active buyers to be 18, causing important changes in the estimated structural parameters. Alternatively, some buyers may enter an auction simply to gain information, in which case their bids would be dominated and never impact the winning bid. Some bidders may collude and place phantom bids in an attempt to hide their cartel memberships; counting these bids would be misleading as it does not account for collusion. For instance, in our application as in many other procurements, sellers can be legally constrained to stop auctions with less than three bidders in attendance. In this case, two buyers may be tempted to contact a third to submit a cover bid but ultimately allow the auction to go through. Econometric methods that check the effectiveness of bidder attendance can be a useful tool for regulators before proceeding to further possibly costly investigations. More generally, using only winning bids provides a robust approach for identifying primitives of interest when the truthfulness of observed bids is dubious.
Last, participation is a parameter of interest in itself. As noted by Bulow and Klemperer (1996), increasing competitive participation would yield higher seller expected revenues than choosing an optimal reserve price under the symmetric independent private value paradigm. In our setup, the number of active buyers is viewed as a latent random variable which can vary across auctions. Estimating its distribution and comparing it with the observed number of bids when available may be useful to detect participation anomalies.
\paragraph{Baseline model.} Under the independent private value paradigm, buyers bid below their private values. The mechanism of the Bayesian Nash equilibrium implies in turn that the bid probability density function (pdf hereafter) is positive throughout its support, illustrating the attractiveness of such profitable bids. As a consequence, the bid pdf is positive at its upper bound and zero thereafter, resulting in a discontinuity at this point.
We develop a new approach for identification of first-price auction models that exploits associated discontinuities in the winning bid pdf. First, we identify the distribution of unknown competition. In particular, we build on an important restriction that first-price auction models impose on the data: the bid quantile function must be strictly increasing with respect to the number of bidders. Therefore, the upper boundary of the bid distribution, conditioning on the number of bidders, is strictly increasing, as well. As bid densities are discontinuous at these locations, this creates jumps at these upper boundaries in the winning bid pdf. A novel result of the paper is that these jumps identify the distribution of the number of bidders.
Second, we identify the value distribution function by iteratively exploiting two equilibrium mappings that relate the value and bid quantile functions. Based on the location of the discontinuities in the winning bid density function, we create a sequence of expanding quantile intervals over which the private value quantile is identified. For every iteration, we start by identifying the bid quantile function in the most competitive auction, which has the largest number of bidders. This information can then be used to identify the value quantile function in the same quantile interval and further calculate the corresponding bid quantile for other competition levels.
\paragraph{Buyer uncertainty and unobserved heterogeneity.} The paper considers several extensions of the baseline model. Section (ref) focuses on buyer uncertainty and auction heterogeneity. As the econometrician in the baseline model, buyers may also face unknown competition, as considered in Harstad, Kagel and Levin (1990) and Kong (2020), among others. Such uncertainty may arise due to the presence of a reserve price, as considered in Guerre, Perrigne and Vuong (2000), or entry costs, as in Li and Zheng (2009) or Marmer, Shneyerov and Xu (2013). We consider a setup where the number of potential bidders, $N$, is known by the buyers but not the effective one. We allow $N$ to vary across auctions, and give conditions on its support ensuring identification of its distribution together with the private value one and the probability that a potential buyer bids.
First-price auctions with unobserved heterogeneity affecting the auctioned good are considered in Krasnokutskaya (2011), and is especially challenging in our framework. Indeed, the presence of a continuous unobserved heterogeneity component washes discontinuities out of the winning bid density. Fortunately, considering its first and second derivatives allows for the identification of the participation distribution. We also show that some features of the unobserved component distribution can be recovered, so that it can be parametrically identified, raising hope for nonparametric identification of the private value distribution.
\paragraph{Estimation and application.} Following Chernozhukov and Hong (2004), Ibragimov and Has'minski (1981), Hirano and Porter (2003) who established efficiency of Bayesian methods for irregular parametric models as arising in our setup, we devise a Bayesian estimation method implemented with importance sampling. Preliminary simulation experiments show that a too large support for the number of potential buyers causes bias when estimating the private value parameter. We therefore estimate a specification for each support candidate and retain the ones with the highest posterior probability. The participation prior is, conditionally on the considered support, the non informative Jeffrey's Dirichlet prior with parameter $1/2$, which is often used for mixture models (Fr\"{u}hwirth-Schnatter, 2006). Prior for the private value parameter are uniform distributions derived from preliminary estimation of this parameter for each support candidate.
We apply this methodology to a new dataset of IT procurements for the Shanghai local government. A general China regulation for these kind of procurements is that they should be close if less than three bidders attend. Using winning bids to estimate participation suggests that this regulation is only effective in $10\%$ of our sample. A counterfactual analysis shows that the corresponding expected loss ranges from $10\%$ of a budget forecast for small contracts to $2\%$ for bigger ones. As most of the procurements in our sample are for small contracts, this loss can be substantial and better understanding the reasons for such low participation would help to improve these auctions.
\paragraph{Related literature.} Allowing for unknown competition started early in the empirical auction literature. Laffont et al. (1995) estimate the number of buyers $N$ as a parameter that they take to be constant across auctions. Paarsch (1997) treats unknown competition as a nuisance parameter, which is eliminated using conditional likelihood estimation. For ascending eBay auctions, Song (2004) shows that the private value distribution and a constant number of buyers are identified from winning and second-highest bids, but not from winning bids alone when $N$ is random. More pertinent to our paper is the misclassification approach of An, Hu and Shum (2010), who study identification from the winning bid using a proxy $N^* \leq N$ for the number of buyers and an instrument that can be a discretized second bid. Shneyerov and Wong (2011) suppose that only winning bids and the number of active bidders are observed. Some recent work on ascending auctions with unobserved heterogeneity uses the winning bid plus additional information for identification when the number of bidders is imperfectly observed or unobserved. More specifically, Freyberger and Larsen (2022) and Luo and Xiao (2023) use additional bids, while Hern\'andez, Quint and Turansick (2020) assumes that a “participation shifter” is available.
The present paper contributes to the literature on nonparametric identification of finite mixtures; see for instance the review of Compiani and Kitamura (2016). Existing identification results require either exclusion restrictions or multiple independent measurements. A first-price auction example of the latter can be found in D'Haultf\oe uille and F\'{e}vrier (2015), who recover the distribution of an unobserved continuous auction characteristic from three bids. Our approach only uses the winning bid.
The discontinuity design (DD) literature has expanded rapidly in recent years; interested readers are encouraged to refer to review papers by Imbens and Lemieux (2008), Kleven (2016) and Jales and Yu (2017). Recent auction applications include Coviello and Marinello (2014), Choi, Neisheim and Rasul (2016) and Kawai, Nakabayashi, Ortner and Chassang (2023). As in the DD literature, this paper employs jump sizes for identification purposes --- more specifically, to identify the participation distribution. However the structural nature of our paper departs from this literature.
\paragraph{Remainder of the paper.} The next section states our identification results for the baseline model while Section 3 focuses on buyer uncertainty and unobserved heterogeneity. Section 4 reports our estimation results and Section 5 concludes the paper. Appendix A describes the estimation procedure and reports the results of related simulation experiments. Appendix B groups some remaining proofs.
In this section, we start by describing the benchmark auction model and introduce two equilibrium mappings that are convenient for describing our discontinuity identification strategy. Next, we derive the restrictions that the model imposes on the observed winning bids, especially with respect to the formation of discontinuities. Finally, we describe our identification strategy in two steps. First, we identify the distribution of the number of buyers from the discontinuities in the winning bid density function. Second, we identify the value distribution function using the two equilibrium mappings iteratively.
Suppose there is a single item for sale with $N$ active symmetric buyers bidding for the item. All buyers observe $N$. In contrast, the analyst does not observe $N$, which causes auction-specific unobserved heterogeneity. Each buyer $i$ also observes her private valuation $V_{i}$, which is unknown to other buyers. The private values $V_{i}$ are $i.i.d.$ draws from a distribution $F\left(\cdot\right) $, which is known to all the buyers and is independent of $N$. The buyers are risk neutral and their bids $B_{i}$ are formed according to a symmetric best-response strategy. In sum, the primitives are the distribution of the number of buyers $N$ and the private value distribution.
We assume that the analyst only observes the winning bid $W$, i.e., the maximum bid among the $N$ buyers in the set $\mathcal{N}$ of active buyers
Hence, the analyst observes draws from the unconditional cumulative probability distribution of the winning bid $G(\cdot)$, which is a mixture of the conditional winning bid distributions given $N$:
where $G_n(\cdot)$ is the conditional bid distribution given $N=n$.
The following two assumptions introduce some additional conditions for the distribution of $N$ and for the private value distribution $F\left( \cdot\right) $.
Assumption N.\ The number of active buyers $N$ \ is a discrete random variable with support $\left\{ \underline{n} ,\ldots,\overline{n}\right\} $ for some integers $2\leq \underline{n}\leq\overline{n}<\infty$, i.e., $p_{n}=\mathbb{P}\left( N=n\right) >0$ for $n=\underline{n},\underline{n}+1,\ldots, \overline{n}$ \textit{with }$\sum_{n=\underline{n}}^{\overline{n}}p_{n}=1$.
Assumption IPV. Buyers' private values $V_{i}$ are $i.i.d.$ draws from a common knowledge distribution $F\left(\cdot\right)$ and are unknown to competitors. The cumulative distribution function $F\left(\cdot\right)$ has a compact support $\left[ \underline{v},\overline{v}\right] $. Its probability density function $f\left( \cdot\right) $ is continuous and strictly positive over $\left[ \underline{v},\overline {v}\right] $.
Both theoretical and empirical literatures adopt the assumption of a private value distribution with compact support. In particular, it rules out multiple asymmetric equilibria; see Maskin and Riley (1984, Remark 2.3), who also establish that symmetric Bayesian Nash Equilibrium bids are given by a strictly increasing and continuously differentiable function of private values.
For our discontinuity approach, the compact support assumption ensures the existence of discontinuities in the density of unconditional winning bids that we exploit in this paper. In particular, the winning bid densities $g_n (\cdot)$ given $N=n$ stay bounded away from $0$ at the upper boundary of its support; see ((ref)) below.
That the private value p.d.f $f(\cdot)$ is positive over $[\underline{v},\overline{v}]$ is a current assumption in the auction literature, as seen in Maskin and Riley (1984), Lebrun (1999), Guerre et al. (2000), among others. It is used here to identify the lowest number $\underline{n}$ of active buyers; see Lemma (ref)-(iii) below. The alternative identification method of Section (ref) allows us to relax this condition as noted in Footnote (ref).
In this subsection, we describe two equilibrium mappings that are repeatedly used in our identification procedure. Specifically, there is an equilibrium mapping from the value distribution to the bid distribution, and vice versa. Our discontinuity identification strategy is conveniently described using the quantile framework as in Guerre, Perrigne and Vuong (2009), Liu and Luo (2017), and Guerre and Gimenes (2022), that we recount below.
Let $V\left( \alpha\right)=F^{-1}\left( \alpha\right) $ represent the private value quantile function, where $\alpha \in \left[ 0,1\right] $ is the quantile level. Let $ B_{n}\left( \alpha\right) $ denote the bid quantile function given that $n$ buyers participate in the auction, and set $B_n^{(1)} (\alpha) = \frac{\partial B_n (\alpha)}{\partial \alpha}$. Following Milgrom (2001)'s exposition of the identification strategy of Guerre, Perrigne and Vuong (2000), the private value quantile function $V\left( \cdot\right) $ can be viewed as the common valuation function of buyers who receive independent uniform private signals
which determines their private values $V_{i}=V\left( A_{i}\right) $. By Assumption IPV, $B_{i}=\beta_{n}\left( A_{i}\right) $ for all $i$, where $ \beta_{n}\left( \cdot \right)$ is strictly increasing and continuously differentiable. It follows that for any $b$ in the range of $\beta_{n}\left( \cdot\right) $,
because $A_{i}$ is uniformly distributed over $\left[ 0,1\right] $. Hence, the best-response strategy is the bid quantile function
Now, let us relate the bid and private value quantile functions. Suppose that buyer $i$ receives signal $\alpha$ but makes a generic bid $B_{n}\left( a\right) $ for some $a \in \left[ 0,1\right] $. Since her opponents bid $B_{n}\left( A_{j}\right) $, the probability that her bid $ B_{n}\left( a\right) $ wins the auction is given by $\mathbb{P}(\max_{j\neq i}A_{j}\leq a)$, which is equal to $a^{n-1}$ as the signals of the $ n-1$ opponents $A_{j}$, where $j \neq i$, are independent and uniform. It follows that the expected payoff of buyer $i$ is $\left( V\left( \alpha\right) -B_{n}\left( a\right) \right) a^{n-1}$, which is maximized when $a=\alpha$. Since
setting this derivative to $0$ gives \footnote{It gives equivalently that $B_n^{(1)} (\alpha)=\left(V(\alpha) -B_n (\alpha)\right)\cdot \frac{n-1}{\alpha} <\infty$. As $g_n(b)=1/B_n^{(1)}\left[G_n (b)\right]$, it follows that a Bayesian Nash equilibrium bid pdf $g_n(\cdot)$ cannot vanish on $(\underline{v},\overline{b}_n]$, as mentioned in the introduction.}
This constitutes the equilibrium mapping from the bid quantile function to the value quantile function, which is the basis of the identification of $V(\cdot)$ with knowledge of $B_n(\cdot)$.
Now, let us consider the inverse of the mapping ((ref)). Indeed, ((ref)) is equivalent to $\frac{\partial \left[ B_{n}\left( \alpha\right) \alpha^{n-1} \right] }{\partial \alpha}=V\left( \alpha\right) \left( n-1\right) \alpha^{n-2}$, and it follows
For convenience of identification that will be clarified later on, let us introduce the conditional bid upper bound \[ \overline{b}_n = B_n (1) = (n-1) \int_0^{1} t^{n-2} V(t) dt , \] which gives
This constitutes the equilibrium mapping from the value quantile to the bid quantile function. The two mappings represented in ((ref)) and ((ref)) are repeatedly used in our identification procedure.
The structure of winning bid distributions compatible with a first-price auction where buyers observe $N$ follows from the mixture expression of $G(\cdot)$ in Equation ((ref)) and the best-response differential equation ((ref)).
In short, a c.d.f $G(\cdot)$ as in Proposition (ref) is a mixture with components constrained by compatibility conditions driven by the best response differential equation ((ref)). The compatibility conditions of Proposition (ref)-(ii) reflect that the mixture components $G_n (\cdot)$ are generated by the same private value distribution, an important feature for identification. In particular, our identification results rely on the constraints it imposes on the extremities of the conditional bid p.d.f $g_n (\cdot)$, as illustrated in the next corollary. Recall $\overline{b}_n = B_n (1)$, $\underline{v}=V(0)=\underline{b}_n$, and $\overline{v}=V(1)$.
Equation ((ref)) implies that $g_{n}\left( \overline{b}_{n}\right)$ is strictly positive. It turns out from ((ref)) that this causes discontinuities in the winning bid p.d.f $g(\cdot)$ at each $\overline{b}_{n}$, as studied in the next section. As it follows that $\overline{b}_{n}$ is identified, ((ref)) shows that $g_{n}\left( \overline{b}_{n}\right)$, where $n\in \{\underline{n},\ldots,\overline{n}\}$, are determined by the common unknown parameter $\overline{v}$. We employ this consequence of the compatibility conditions of Proposition (ref)-(ii) later on to identify the distribution of $N$.
In this subsection, we introduce a numerical example to illustrate the discontinuity features of the winning bid p.d.f that follows from Corollary (ref). This example will also be useful for introducing our identification procedure. A general lemma completes the example.
Consider the private value c.d.f $F(v) = v^2$ for all $v$ in $[0,1]$ and a number of buyers $N=\{2,3\}$ with equal probability. As $V(\alpha)=\alpha^{1/2}$, it follows that \[ B_n (\alpha)= \frac{n-1}{\alpha^{n-1}} \int_0^{\alpha} t^{n-2+\frac{1}{2}}dt = \frac{n-1}{n-\frac{1}{2}}\alpha^{1/2}. \] Hence, $\overline{b}_n=\frac{n-1}{n-\frac{1}{2}}$ yields the conditional bid p.d.f $g_n (b)$, given $N=n$, is equal to $2b/\overline{b}_n^2$ on $[0,\overline{b}_n]$ and vanishes outside this interval. Note that the support of the conditional density function increases with the number of buyers. Both densities jump to zero at their upper boundaries as expected from ((ref)).
Let us now turn to the winning bid, the observation of the analyst. As expected, the unconditional p.d.f $ g(b) = \frac{1}{2} \cdot 2 G_2(b)g_2(b) + \frac{1}{2} \cdot 3 G_3^2(b)g_3(b) $ displayed in Figure (ref) is discontinuous at $\overline{b}_2$ and $\overline{b}_3$, with jump sizes $\Delta_2$ and $\Delta_3$, respectively. The resulting winning bid c.d.f exhibits kinks at these values, as illustrated in Figure (ref). In this example, Figure (ref) exhibits two discontinuities (and Figure (ref) exhibits two kinks) because $N$ takes two potential values here.
The increasing support property of the conditional bid p.d.f. and the winning bid p.d.f. discontinuities in Figure (ref) are generic, as shown in the upcoming lemma. Lemma (ref)-(i) states more generally that bids increase with competition --- a key feature of first-price auctions that does not hold in ascending ones, or when buyers do not observe $N$. Lemma (ref)-(ii) focuses on the winning bid p.d.f discontinuities and its jumps. The identification of $\underline{n}$ in Lemma (ref)-(iii) uses that the lower tail index of $G(b)$ is the one of $\left(G_{\underline{n}} (b)\right)^{\underline{n}}$, which is $\underline{n}$.
Lemma (ref) is an important building block for identifying the competition distribution. Part (iii) is a tail identification result for $\underline{n}$ similar to Hill and Shneyerov (2013) who considered a common value framework. Lemma (ref)-(ii) shows that the jumps in the winning bid p.d.f identify $\mathbb{P}(N=n)$ up to the unknown $\overline{v}$.
Here, we first focus on identification of the participation distribution and then turn to the private values.
In this subsection, we describe the identification of the support of $N$ and its distribution using the discontinuity points and jump sizes. To identify the support, we exploit two implications of Lemma (ref): (a) the minimum number of buyers $\underline{n}$ is identified from the winning bid distribution tail near the lower boundary; (b) each number of buyers $n$ generates a discontinuity in the winning bid distribution, which identifies the difference $\overline{n}-\underline{n}$. More specifically, Lemma (ref)-(ii) identifies $\underline{n}$ and $\overline{n}$ through $\underline{n}=\lim_{t\downarrow0}\frac{\log G\left( \underline{v}+t\right) }{\log t}$ and \[ \overline{n}=\underline{n} +\mathsf{Card}\left\{ b;g\left( \cdot\right) \text{ is discontinuous at }b\right\} -1. \] This also identifies the support of the distribution of $N$ as $\mathbb{P}(N=n)>0$ for all $n$ with $\underline{n} \leq n \leq \overline{n}$ by Assumption N.
Next, we exploit the jumps in the p.d.f to identify $p_n=\mathbb{P} (N=n)$. Recall that Equation ((ref)) identifies $p_n$ up to the private value upper bound $\overline{v}$,
But $\sum_{\underline{n}}^{\overline{n}}p_{n}=1$ implies
Hence, $p_{n}$ satisfies
and is identified because the discontinuity points $\overline{b}_k$ and jump sizes $\Delta_k$ are identified. We summarize these identification results in the next lemma.
The identifying equations ((ref)) and ((ref)) can also be used to derive inequality constraints satisfied by the jumps sizes $\Delta_n$, discontinuity locations $\overline{b}_n$, and the lowest and largest numbers of bidders $\underline{n}$ and $\overline{n}$. Indeed $\overline{v}>\overline{b}_{\overline{n}}$ and $0 \leq p_n \leq 1$ are equivalent to the following inequalities
given that $\Delta_n>0$ must also hold by Lemma (ref)-(ii). A violation of any of these inequalities indicate that the model is misspecified.
We first return to the numerical example to illustrate our iterative identification procedure for the private value distribution.
\paragraph{Numerical example (cont'd).} By Lemma (ref), $\underline{n}=2$, $\overline{n}=2$ and $p_1=p_2=\frac{1}{2}$ are identified. Let us now turn to the identification of the private value distribution, which is based on the winning bid c.d.f \[G(b)=\frac{1}{2}\left( G_2^2(b)+G_3^3(b)\right)\] displayed in Figure (ref). Since $G_2^2(b)=1$ on $[\overline{b}_2,\overline{b}_3]$, \[ G_3 (b) = \left(2G(b)-1\right)^{\frac{1}{3}}, \quad b \in [\overline{b}_2,\overline{b}_3]. \] It follows that $B_3 (\cdot)$ is identified on $[\alpha_1,1]$, where $\alpha_1=G_3 (\overline{b}_2)$, using the top portion of the winning bid distribution; see Figure (ref) when $G(b) \in [\frac{1}{2}(1+G_3^3(\overline{b}_2)),1]$. Using the mapping ((ref)) from the bid quantile function to the private value one gives \[ V(\alpha) = B_3 (\alpha) + \frac{1}{2}\alpha B_3^{(1)} (\alpha), \] and $V(\cdot)$ is identified on $[\alpha_1,1]$. Additionally, using the mapping ((ref)) from the private value quantile function gives \[ B_2 (\alpha) = \frac{1}{\alpha} \left[ \overline{b}_2 - \int_{\alpha}^{1} V(t) dt \right] \] so that $B_2(\cdot)$ is also identified on $[\alpha_1,1]$. The identified $B_2 (\alpha)$, $B_3 (\alpha)$, and $V (\alpha )$, where $\alpha \in [\alpha_1,1]$, are displayed in blue in Figure (ref).
Next, we enlarge the interval $[\alpha_1,1]$ over which $V(\cdot)$ is identified. For this purpose, let $\beta_1=B_2 (\alpha_1)<\overline{b}_2$ and observe that $G_2(b)$ is identified for $b \geq \beta_1$. Given that \[ G_3 (b) = \left(G_2^2(b)-2G(b)\right)^{\frac{1}{3}}, \] $G_3 (b)$ is identified for $b \geq \beta_1$, as $B_3 (\alpha)$ is identified for $\alpha \geq \alpha_2=G_3 (\beta_1)$. Figure (ref) shows that $\alpha_2<\alpha_1$ and arguing as above gives us identification of $V(\cdot)$ and $B_2 (\cdot)$ on $[\alpha_2,1]$. Three portions of $V(\cdot)$, $B_3 (\cdot)$, and $B_2(\cdot)$ are identified through three iterations and plotted in Figure (ref) in purple, red, and orange, respectively. Furthermore, Figure (ref) suggests that additional iterations of this identification procedure should allow us to recover any $V(\alpha)$.
\paragraph{The general case.} The iterative identification described above can be easily generalized. Showing the convergence of the quantile-level sequence $\{\alpha_k\}$ to $0$ can be done using the important fact that the bid quantile functions $B_n (\cdot)$ decrease with $n$ and only cross at the origin. This implies identification of the private value distribution when only observing the winning bid, as stated in the next general result.
\paragraph{Setup and assumptions.} Consider an environment where there are $N$ potentially active buyers, who submit a bid with probability $d$, an additional parameter to be identified. Buyer uncertainty then arises from the fact that the total number of active buyers is not known but follows a binomial distribution of parameter $(N,d)$ given $N$, assuming from now on that the bidding decisions and $N$ are independent. This auction setup is summarized in the next assumption.
Assumption BU.\ There are $N$ potentially active buyers, observable to the buyers but not the econometrician. Given $N$, each buyer decides to participate in the auction with probability $d$ in $(0,1)$, privately and independently of the other buyers. Each active buyer draws a private value from the common knowledge distribution $F(\cdot)$ and the seller reserve price is $\underline{v}>0$. The econometrician observes the winning bid $W$ or that the auction fails if none of the buyers attend.
In addition, the private value distribution $F(\cdot)$ satisfies Assumption IPV, and $N$ can vary across auctions with $p_n = \mathbb{P} (N=n)$, $n=\underline{n},\ldots,\overline{n}$ as in Assumption $N$. This setup covers in particular the case where there is a reserve price $r$ constant across auction, $F(\cdot)$ denoting the truncated private value distribution $F(\cdot|V \geq r)$ and $d=P(V\geq r)$ being the probability of entering the auction.
\paragraph{Model primitives.} Under Assumption BU, the minimal bid is $\underline{b}=\underline{v}>0$ when at least one bidder attends the auction. Let $G_n (\cdot|d)$ be the bid distribution given $N=n>0$ with support $[\underline{b},\overline{b}_n]$.
As a convention, the winning bid is set to $0$ when there is no bid. It follows that the winning bid c.d.f, given that no buyers attend the auction and $N=n$, can be written as $G_n^0 (\cdot|d)$ over $[0,\infty)$. Since the winning bid distribution given $N=n$ and $0<m\leq n$ buyers attend the auction is $G_n^m (\cdot|d)$, the unconditional winning bid distribution is
A key difference with the baseline model is that \[ G (\underline{b}|d) = \sum_{n=\underline{n}}^{+\overline{n}} p_n \left(1-d \right)^n \] is now the probability that no buyers attend the auction, which is positive. The lowest number of bidders $\underline{n}$ cannot be identified from the lower tail of $G(\cdot|d)$ as done in the baseline model, where $G(\cdot|d)$ was vanishing and behaving locally as a power $(b-\underline{b})^{\underline{n}}$ near $\underline{b}$.
Some closed-form expressions can be easily obtained for the equilibrium bid quantile function $B_{n} (\cdot|d) =G_{n}^{-1} (\cdot|d)$. Given $N=n$ and assuming that a buyer with private value $V(\alpha)=F^{-1} (\alpha)$ attends the auction, the expected profit generated by a bid $B_{n} (a|d)$ is
This expression is obtained noting that, when $n$ bidders are potentially active and at least one is, the distribution of the remaining number of bids is a binomial with parameter $(n-1,d)$. The first-order condition characterizing the Bayesian Nash Equilibrium is then \[ \frac{d}{d \alpha} \left[ \left(1-d+d \cdot \alpha\right)^{n-1} B_n (\alpha|d) \right] = (n-1) d \left(1-d+d \cdot \alpha\right)^{n-2} V(\alpha), \] implying, given $B_n(0|d) = \underline{v}$,
which are the counterparts of ((ref)) and ((ref)). In particular, integrating by parts in ((ref)) shows that \[ B_n (\alpha|d) = V(\alpha) - \int_{0}^{\alpha} \left( \frac{1-d+d \cdot t}{1-d + d \cdot \alpha} \right)^{n-1} V^{(1)} (t) dt, \] which implies that $B_n (\alpha|d)$ is strictly increasing with respect to $n$ for all quantile levels $\alpha>0$. Hence the largest bids $\overline{b}_n = B_n (1|d)$ satisfy $\overline{b}_{\underline{n}} < \overline{b}_{\underline{n}+1} < \cdots < \overline{b}_{\overline{n}}<\overline{v}$ as in the baseline model.
The next lemma describes implications of ((ref)) and ((ref)) for the conditional bid p.d.f $g_n (b|d) = \frac{d}{db} G_n (b|d)$, which parallels Corollary (ref). Henceforth, we shall consider the additional parameter \[ \overline{v}^{(1)} =V^{(1)} (1) = \frac{1}{f(\overline{v})}. \]
As the expression of the winning bid distribution ((ref)) gives a p.d.f of \[ g(b|d) = \sum_{n=\underline{n}}^{+\overline{n}} n p_n \cdot d \cdot \left(1-d+d \cdot G_n (b|d) \right)^{n-1} g_n (b|d), \] ((ref)) shows that $g(\cdot|d)$ exhibits downward jumps $\Delta_n$ at each $\overline{b}_n$, $n=\underline{n},\ldots,\overline{n}$ with \[ \Delta_n =\lim_{t\downarrow 0}\left( g(\overline{b}_n-t|d)-g(\overline{b} _n+t|d) \right) = \frac{np_n}{(n-1)(\overline {v}-\overline{b}_{n})}, \] an expression identical to the jumps of the baseline model ((ref)). As ((ref)) ensures that $g(\cdot|d)$ is continuous at other points of $(\underline{b},\overline{b}_{\overline{n}})$, this can be used to identify $\overline{n}-\underline{n}$ and $p_{n}=\frac{n-1}{n}\Delta_{n}\left( \overline{v}-\overline{b}_{n}\right)$ up to $n$ and $\overline{v}$.
Equation ((ref)) shows that the conditional bid p.d.f $g_n (\cdot|d)$ and the winning bid p.d.f $g(\cdot|d)$ both diverge with a $-\frac{1}{2}$ power rate at the lowest bid $\underline{b}=\underline{v}$. This is intuitively due to buyer uncertainty, as a bidder can win with a very low bid if no others attend the auction. As a consequence, this can be used to check the presence of bidder uncertainty. On the other hand, it does not allow for the identification of $\underline{n}$ using the lower tail behavior of $G(\cdot|d)$ as simply as in the baseline model.
\paragraph{Discontinuities and identification strategies.} A possible way to recover the minimal number $\underline{n}$ of potentially active bidders relies on the derivative p.d.f discontinuities, as permitted by ((ref)). The winning bid derivative p.d.f is
which, by ((ref)), is continuous over $(\underline{b},\overline{b}_{\overline{n}})$, with the possible exception of the identified $\overline{b}_n$, $n=\underline{n},\ldots,\overline{n}$, where it may exhibit jumps
by ((ref)), ((ref)), and a little algebra. To take advantage of the fact that $n-\underline{n}$ is identified as the rank of $\overline{b}_n$,\footnote{This follows from $\overline{b}_{\underline{n}}<\cdots<\overline{b}_{\overline{n}}$, defining the rank of $\overline{b}_{\underline{n}}$ as $0$, the one of $\overline{b}_{\overline{n}}$ being $\overline{n}-\underline{n}$. }, set \[ m=n-\underline{n}, \quad \overline{b} (m) = \overline{b}_{\underline{n}+m}, \quad \varrho (m) = \frac{\Delta^{(1)}_{\underline{n}+m}}{\Delta_{\underline{n}+m}}, \] which are all identified. It then follows from the expression of $\Delta^{(1)}_{n} $ and $\Delta_{n}$ given the above that
As this equation includes the three unknowns $\underline{n}$, $\overline{v}$, and $\overline{v}^{(1)}/d$, three of these equations are, in principle, needed for identification from ((ref)). However, implementing this strategy may be hard due to the nonlinear nature of ((ref)).
A simple overidentification strategy introduces as additional unknowns some well chosen nonlinear functions, such as $\underline{n} \left(\overline{v}-\overline{b} (0)\right)^2$, $ \left(\overline{v}-\overline{b} (0)\right)^2$, and $\underline{n} \left(\overline{v}-\overline{b} (0)\right)$, to transform ((ref)) into a linear equation. This allows us to obtain a condition ensuring identification of the considered auction specification as stated in the next proposition.
As $\varrho (m)$ and $\overline{b}_n$ can be estimated from the data, the overidentification condition $\det(I_{\mathcal{M}}) \neq 0$ is testable. Proposition (ref) holds under the condition $\overline{n}-\underline{n}\geq 5$, a condition that can be weakened by introducing fewer additional unknowns in ((ref)) and by taking into account that $\underline{n}$ is an integer number.\footnote{Note that Proposition (ref) also applies when buyers are certain about participation (i.e. $d=1$), in which case $\underline{n}$ is identified without relying on the tail argument used in Lemma (ref)-(iii). }
\paragraph{Buyer uncertainty and number of buyers.} To some extent, the entry probability $d$ in (ref) can depend upon the number of potential bidders $n$. For instance, the parameters $\beta_0$ and $\beta_1$ in $d_n =1/(\beta_0 + \beta_1 \cdot n)$ can be identified using a straightforward modification of Proposition (ref).
Entry cost models as considered in Li and Zheng (2009) also generates buyer uncertainty with a parameter $d$ depending upon $n$ and an entry cost $c$. If in addition the private value distribution depends upon a finite-dimensional $\theta$, then the buyer uncertainty parameter is determined by an indifference condition which ensures $d=d(n,c,\theta)$, assuming $c$ is constant across auctions. Equation (ref) can then be used to identified $c$ and $\theta$, using only that the largest bid $\overline{b}_n$ increases with $n$. Indeed, a difficulty there is that, as noted by Li and Zheng (2009), bidding strategies may not increase with the number of potential bidders. If $c$ varies across auction, nonparametric identification can be restored if the entry cost can be small enough to allow entry of all bidders, suggesting that cost variation can be useful.
\paragraph{Setup and assumptions.} Consider now a setup with auction heterogeneity, where for each buyer $i$,
$\chi$ being an auction-specific variable which is not observed by the econometrician but common knowledge to buyers, and $V_i$ is an i.i.d. private value component drawn from $F(\cdot)$ satisfying Assumption IPV.
Assumption UAH.\ The unobserved auction heterogeneity component $\chi$ is independent of $N$ and all the private values $V_i$. The p.d.f $\varphi (\cdot)$ of $\chi$ has a compact support $[0,\overline{\chi}] \subset [0,\infty)$, over which it is strictly positive and twice continuously differentiable. $\overline{\chi} \neq \overline{b}_n - \overline{b}_m$ for all $\underline{n} \leq n,m \leq \overline{n}$.
The restriction on $\overline{\chi}$ shortens some proofs but can be easily removed.
\paragraph{Model primitives.} Under ((ref)), the bids $\widetilde{B}_i$ are equal to $\chi + B_i$, where the conditional quantile function $B_n(\cdot)$ of the i.i.d. $B_i$ given $N=n$ is given by ((ref)) and satisfies ((ref)). It follows that the winning bid $\widetilde{W}$ is now \[ \widetilde{W} = \chi + W, \text{ where } W = \max_{i \in \mathcal{N}} B_i. \] The conditional c.d.f of $W$ is the one of the baseline model, so that, $\Phi (\cdot)$ being the c.d.f of $\chi$,
as the p.d.f of $W$ is $n G_n^{n-1} (b) g_n (b)$. It follows that the p.d.f of $\widetilde{W}$ is, by Assumption N and recalling that $\chi$ belongs to $[0,\overline{\chi}]$,
noting that $g(\cdot)$ is the winning bid p.d.f of the baseline model.
\paragraph{Winning bid p.d.f derivatives discontinuities.} Integrating out $g(\cdot)$ in ((ref)) gives a smooth p.d.f $\widetilde{g} (\cdot)$. However discontinuities arise when differentiating $\widetilde{g} (\cdot)$. It indeed holds that applying the Liebnitz rule for integral differentiation to ((ref)) yields
As the integral expression above remains continuous, ((ref)) implies that the discontinuities of $\widetilde{g}^{(1)} (\cdot)$ arise at each $\overline{b}_n$ and $\overline{b}_n+\overline{\chi}$, with jumps that are opposite in sign but of proportional magnitude. The next lemma summarizes some properties of $\widetilde{g} (\cdot)$ and its first and second derivatives. Recall that $\overline{v}^{(1)}=V^{(1)}(0)=1/f(\underline{v})$.
\paragraph{Identification of the participation distribution $p_n$.} Lemma (ref)-(i) ensures that $\underline{n}$ is identified while (ii) implies that the number of jumps of $\widetilde{g}^{(1)} (\cdot)$ above $\underline{b}$ is $2(\overline{n}-\underline{n})$ under the restriction on $\overline{\chi}$ of Assumption AUH, so that $\overline{n}$ can also be recovered. It also shows that $\overline{b}_{\underline{n}}<\cdots<\overline{b}_{\overline{n}}$ are identified as locations of downward jumps. As upward jumps are located at $\overline{b}_{\underline{n}}+\overline{\chi}<\cdots<\overline{b}_{\overline{n}}+\overline{\chi}$, the unobserved heterogeneity upper bound support $\overline{\chi}$ is also identified. This may help identify parametric heterogeneity distributions that depend upon a unique parameter.
Identifying the participation distribution is more difficult than in the baseline model because the downward jumps $\widetilde{\Delta}_{n}$ in Lemma (ref)-(ii) now depend upon two unknown parameters, $\overline{v}$ and $\varphi (0)$, unless the latter is identified. Introducing the upward jumps do not help, as they depend on the unknown $\overline{v}$ and $\varphi (\overline{\chi})$. To address this issue, one can try to identify $\overline{v}$ using the discontinuity ratio $\widetilde{\varrho}_n = \frac{\widetilde{\Delta}^{(1)}_n}{\widetilde{\Delta}_n}$, which, by Lemma (ref)-(ii,iii), satisfies
This equation involves three unknowns, $\overline{v}$, $\overline{v}^{(1)}$, and the ratio $\varphi^{(1)}(0)/\varphi(0)$. Three of these equations are, in principle, needed for identification. Similar to equation ((ref)), it can be used as for (over)identification purposes, considering several values of $n$ and introducing extra variables that are nonlinear functions of the initial unknowns to back out $\overline{v}$, $\overline{v}^{(1)}$, and $\varphi^{(1)}(0)/\varphi(0)$ as the unique solution of an extended linear system.
Proposition (ref) gives a testable condition, ensuring that the participation distribution can be identified. It also leaves the door open for identification of parametric private value distributions with a parameter in a one-to-one correspondence with $\left(\underline{v}, \overline{v},\overline{v}^{(1)}\right)$, as the latter can be identified.
Similarly, the parametric unobserved heterogeneity distribution that can be uniquely recovered from $\left(\overline{\chi}, \varphi (0), \varphi^{(1)} (0), \varphi (\overline{\chi}), \varphi^{(1)} (\overline{\chi}),\right)$ can also be identified. As the expression of the winning bid p.d.f $\widetilde{g} (\cdot)$ in ((ref)) shows that it is the convolution of the baseline winning bid p.d.f $g(\cdot)$ by $\varphi (\cdot)$, the deconvolution technique of Krasnokutskaya (2011) using the identified $\varphi(\cdot)$ allows for the recovery of $g(\cdot)$. If so, the identification procedure developed for the baseline model can be applied to nonparametrically recover the private value distribution.
We study public procurement auction data collected from Shanghai, China. When conducting procurement activities with fiscal funds, all governmental organizations in China must abide by the government's procurement guidelines. Contracts are awarded through various methods, including competitive negotiation, public tendering, and bid by invitation. To illustrate our methodology, we shall concentrate on public tender.
Although nationwide data are available for collection, we study procurement data from Shanghai for several reasons. First, while many organizations publish the resulting procurement auction data, only some, such as the ones in Shanghai, publish the reserve price. Second, the Shanghai government procures many contracts. Third, the projects are categorized, allowing further control of auction heterogeneity. We restrict the sample to IT services, whose project type codes are consistently coded, and single item auctions.
The procurement guidelines impose several requirements. First, the paid price should be lower than the average market price, for an equivalent if not better quality. This can be ensured by imposing a reserve price or by increasing participation. Second, the procuring organization is then required to repeat the bidding procedure if fewer than three bidders are qualified or submit proposals.
However, the number of bidders is not released to the public. The published information records the winning bid if the auction was successful, the winner identity, the reserve price or an estimated budget given by the procurement manager, “budget” or “appraisal budget” hereafter. While budget is available for all auctions, only $ 54\%$ have a reserve price. In practice, bids are always smaller that the announced budget, suggesting that this amount acts as a reserve price when the latter is not available. The reserve price is identical or slightly smaller than budget. In the sequel, the budget variable will be redefined as the minimum of the reserve price and the budget when both are available.
The sample covers 886 auctions running from 20th December 2020 to 24th April 2023. 95 were failed due to less than three bids. Budget for failed auctions tend to be lower than for successful ones. The winning bid was equal to the budget for 10 of the remaining ones, perhaps due to rounding as also observed in Table (ref). These 105 auctions will be excluded of the analysis. Table (ref) also reports statistics for winning markup bids, ie the winning bid divided by the appraisal budget. It shows that the winning bid for the remaining auctions is around $97 \%$ of the appraisal budget in mean, with a small standard deviation $.06$, suggesting a high correlation between bidder values and contract sizes.
As seen from Figure (ref), the appraisal budget standardized by its standard deviation is highly concentrated near the origin, but also with possible large values. The range of the winning markup bid also decreases with budget. This can be caused either by a greater dispersion of private values, or greater competition for procurement with small budget. Symmetrically, for bigger contracts, a markup private value distribution concentrated over 1, or lower competition, can reduce the winning markup bid range.
We consider a two-stage model where a bidder decides first to participate to the procurement after observing the appraisal budget given by the procurement manager. Participating bidders then draw independently a private cost, given by the auction budget times a markup. After observing participation, each bidder decides a markup bid, and bids this markup multiplied by the appraisal budget. The number of potential bidders, $\overline{n}$, is assumed to be constant across auctions.
The Bayes estimation procedure is similar to the one detailed in the simulation experiment Appendix A. Estimating a model for a large number of maximal bidders $\overline{n}$ does not seem to work well in the simulations. We then proceed by estimating a model for several $\overline{n}$ and choose the one with the highest posterior probability. Model specification and prior distributions are detailed below, starting with participation. Participation and private costs will depend upon budget, noting that our identification result easily extends to this case by conditioning.
\paragraph{Participation distribution.} In view of the appraisal budget, a bidder can decide whether the contract can fit in his order book. Less demanding contracts can be attractive for those with a small available capacity, while more costly ones can suit those with an empty order book. As we do not have access to bidder individual data, we do not attempt to model the participation decision and just assume that it depends on budget in the following way:
where $X$ is budget standardized with its standard deviation, and $\sum_{n=2}^{\overline{n}} \pi_{1n} =1$ and $\sum_{n=2}^{\overline{n}} \pi_{2n} =0 $. These two conditions ensure that $\mathbb{P} \left( \left. N=n\right| X \right) = \pi_{1n} + \pi_{2n} \cdot X$ when all these numbers are between $0$ and $1$.
Given $\overline{n}$, the priors for $\boldsymbol{\pi}_{1} = \left( \pi_{1,2},\ldots,\pi_{1,\overline{n}} \right)$ and $\boldsymbol{\pi}_2= \left( \pi_{2,2},\ldots,\pi_{2,\overline{n}} \right)$ are independent, the baseline participation distribution $\boldsymbol{\pi}_{1}$ having a Dirichlet distribution of parameter $0.5$ and dimension $\overline{n}$. The deviation $\boldsymbol{\pi}_2$ from the baseline takes value in the simplex $\left\{\boldsymbol{\pi}_2; \sum_{n=2}^{\overline{n}} \pi_{2n} = 0\right\}$, so that the $\pi_{2n}$ can be positive or negative, allowing $X$ to have an increasing or decreasing impact on $\mathbb{P} \left( \left. N=n\right| X \right)$ depending on the sign of $\pi_{2n}$. The prior for $\boldsymbol{\pi}_2$ is given by $\pi_{2n} = \lambda \cdot \left( p_{2n} - 1/\overline{n} \right)$, where $\boldsymbol{p}_2$ has a Dirichlet distribution of parameter $0.5$ and dimension $\overline{n}$. The scale parameter $\lambda$ is independently drawn from a uniform over $[0,2]$.
\paragraph{Markup private cost distribution.} Given the appraisal budget, we assume that the private cost takes the form $\text{budget} \times V$, where $V$ is the markup private cost distributed over $[0,1]$. The distribution of $V$ is the same than in the simulation experiment, of Appendix A, but with a parameter $\theta$ depending on the standardized budget $X$, that is
The priors for $\theta_0$ and $\theta_1$ are independent uniforms with support given by a preliminary estimation of this parameters for various $\overline{n}$. Let $\underline{b}_n (x|\boldsymbol{\theta})$ be the minimal bid when $n$ bidders effectively compete, the value of the covariate being $x$ and the parameters $\theta_0$ and $\theta_1$ being grouped in $\boldsymbol{\theta}$. Following Korostelev and Tsybakov (1993, Chap. 7), a preliminary estimation of $\boldsymbol{\theta}$ can be obtained minimizing the markup bid support area, that is $\overline{x}$ being the maximal value of the standardized budget,
under the constraints that the support contains all the observations, ie $w_{\ell} \geq \underline{b}_{\overline{n}} (x_{\ell}|\boldsymbol{\theta})$, $\ell=1,\ldots,781$. The results of such a preliminary estimation of $\theta_0$ and $\theta_1$ are reported in Table (ref) for a number $\overline{n}$ of maximal bidders between 2 and 6.
While the optimized area sharply decreases when $\overline{n}$ grows from 2 to 3, the decrease is less than $.001$ when $\overline{n}$ grows from 5 to 6. The maximal value of $\overline{n}$ is set to 5, and the prior for this parameter is taken proportional to $\overline{n}-1$, the number of values that can be taken by $\underline{n}$. The prior for $\theta_i$ is then a uniform over $\left[\widehat{\theta}_{i,2}-\widehat{\Delta}_i/2,\widehat{\theta}_{i,5}+\widehat{\Delta}_i/2\right]$ where $\widehat{\Delta}_i=\widehat{\theta}_{i,5}-\widehat{\theta}_{i,2}$, that is $[0.0562,2.7570]$ for $\theta_0$ and $[-0.0359,1.0153]$ for $\theta_1$.
The parameters are estimated using 1 million importance sampling draws for each pair $(2,\overline{n})$, $\overline{n}=2,\ldots,5$, as well as the marginal likelihood of $\overline{n}$.\footnote{The large number of parameters and the presence of the covariate suggests that the considered model may be more difficult to estimate than the one used in the simulation experiment section. } As the value obtained for $\mathbb{P}\left(N=2|\overline{n}\right)$ is very high for all values of $\overline{n}$, it does not appear necessary to estimate the model for $\underline{n}$ distinct from 2. The maximal $\overline{n}$-marginal likelihood is $1.038 \times 10^{-6}$ which is achieved for $\overline{n}=5$, the other likelihoods being negligible. Hence any reasonable priors for $\overline{n}$ generate a posterior concentrated at $\overline{n}=5$, which then gives our final estimates.
The concentration parameter $\theta_0$ and $\theta_1$ estimates of the private cost distribution are $1.5164$ and $0.8326$. Figure (ref) shows that the minimal bid boundary for $n=5$ is mostly below its counterpart derived from the preliminary estimates in Table (ref). Table (ref) reports the Bayesian estimates of the participation distribution parameters $\boldsymbol{\pi}_1$ and $\boldsymbol{\pi}_2$ from (ref).
While the estimates of the parameters $\pi_{1,5}$ and $\pi_{2,5}$ look small, Figure (ref) shows that only the participation probability for $n=2$ and $n=5$ remain mostly positive. The probability that only two bidders really participate is never below .9. This suggests that the regulation constraint of having at least three bidders in each procurement does not ensure an effective participation, except may be in few auctions with bigger contracts where the probability of having five effective bidders can be close to $.1$. The right of Figure (ref) nevertheless shows some bids above the minimal bid frontier for $n=2$, due to a large number of small auctions.
Figure (ref) completes our estimation results with the winning markup bid pdf for three appraisal budget quantile levels. Going from the extreme quantile level $.01$ to the budget median does not cause important shape changes, up to a shrinking winning bid support due to the lower probability of having three or four bidders. Going from the budget median to the extreme quantile level $.99$ generates much more qualitative changes. First, the 2-bidder winning bid component is much more concentrated due to a strong increase of the parameter $\theta (X)$. Second, the 5-bidder component is more noticeable due to a higher probability of having five bidders effectively participating to the procurement.
To conclude this application section, combining Figures (ref) and (ref) show that the appraisal budget increases participation but decreases bidder ability to place competitive bids. This suggests that an optimal choice of the contract size could ensure a higher average markup for the seller. However Figure (ref) shows that the estimated expected markup bid decreases with budget.
Hence the participation increase does not compensate for the lesser bidder's competitiveness. Selling smaller contracts, as observed in the data, is probably a way to optimize the procurement procedure, assuming that the procurement institution can choose the contract size in some circumstances. However, running more frequent procurements can be more costly. Using small contracts to run bigger projects can also generate costs that should also taken into account.
Understanding the low participation is also important. Figure (ref) shows that the expected markup bid generated by three bidders is substantially smaller than the model one. The difference ranges from $12\%$ of the appraisal budget for small contracts to $2\%$ for the biggest ones. As shown by Figure (ref), the three bidder participation rule does not seem effective. It can signal that this rule is not respected, possibly due to bidders inviting fake opponents or sellers not enforcing it.
This paper shows that, under the independent symmetric private value paradigm, the first-price winning bid is sufficient to identify model primitives when competition is observed by the buyers but not the econometrician. This is suitable in the presence of phantom bids or when the number of observed bids does not reflect participation. To some extent, buyers can be uncertain about their competitors and auction-specific unobserved heterogeneity can be present.
Our theoretical results shed light on new identification arguments for discrete mixture models, which are widely used in economic applications, in particular when unobserved heterogeneity is plausible. In our model, the mixture components are generated by the same function. The components are ordered according first-order stochastic dominance and their supports are nested. These two features may appear in other relevant economic mixtures. Investigating how essential these features are for identification can also be of interest for future extensions.
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