Structural Estimation of Auction Models

From Auction Equilibrium to an Estimable Model

Auction data record outcomes such as entry decisions, bids, and winners; structural estimation uses a model of bidders’ incentives to connect those observations to less directly observed quantities, such as value distributions and entry costs. The central step is to translate the equilibrium strategy into a probability model for the observed data. A likelihood or a set of moment conditions is only as credible as the equilibrium and distributional assumptions that produce it.

For a first-price auction, a bid is not generally a bidder’s value. It is an equilibrium choice determined by the bidder’s value, the auction rules, and beliefs about rivals. If the equilibrium bidding function is known under a proposed set of primitives, it can be used to express the distribution of observed bids in terms of those primitives. Participation can contribute further information, but it also changes which values appear among bidders who submit bids.

Definition: A structural auction model specifies (1) primitives such as value distributions, entry costs, and risk preferences; (2) the auction and participation rules; and (3) an equilibrium mapping from those primitives to observable outcomes. A structural likelihood assigns probabilities or densities to observed auction data using that mapping. A structural moment condition is an equation implied by the model that relates observable data to its primitives.

Throughout the examples, bids and values are measured in dollars. Unless stated otherwise, bidders are risk-neutral, values are private, and auction outcomes are independent across auctions. These assumptions are substantive: changing risk preferences, value dependence, or the auction rules can change the equilibrium bidding function and hence the likelihood.

Turning an Equilibrium Bid into a Density

Suppose two bidders draw independent values uniformly from \([0,\bar v]\), both participate, and the auction is a first-price auction with no reserve. In the symmetric equilibrium, a bidder with value \(v\) bids \(b(v)=v/2\). Because this function is strictly increasing, observing a bid identifies the corresponding value under the model: \(v=2b\). The density of bids follows by changing variables from the value density.

f_B(b)=f_V\!\left(v(b)\right)\left|\frac{dv(b)}{db}\right| =\frac{1}{\bar v}\times 2 =\frac{2}{\bar v}, \qquad 0\leq b\leq \frac{\bar v}{2}.

The factor \(2\) is the Jacobian, the absolute derivative of the inverse bidding function. The density is zero outside the equilibrium bid support. For independent auctions with observed bids \(b_1,\ldots,b_m\), the likelihood is the product of their bid densities, provided the model’s assumptions apply to each auction.

Worked Example: A Likelihood for Bids Under a Uniform Value Model

Suppose three independent hypothetical auctions each have two participating bidders. In every auction, values are independent uniform draws from \([0,\bar v]\), and bidding follows the symmetric first-price equilibrium \(b=v/2\). The recorded bids, one per auction, are \(0.15\), \(0.20\), and \(0.25\) dollars. Find the maximum-likelihood estimate of \(\bar v\) under this model.

For any proposed \(\bar v\), every recorded bid must satisfy \(0\leq b_i\leq \bar v/2\). Thus \(\bar v\geq 2\max_i b_i=0.50\). On this support, each bid has density \(2/\bar v\), so the likelihood is

L(\bar v)=\prod_{i=1}^{3}\frac{2}{\bar v} =\left(\frac{2}{\bar v}\right)^3, \qquad \bar v\geq 0.50.

This likelihood decreases as \(\bar v\) increases, so its maximum over the permitted parameter values occurs at the smallest permitted value:

\widehat{\bar v}_{\mathrm{ML}}=2\max_i b_i =2(0.25)=0.50\text{ dollars}.

At this estimate, the density for each bid is \(2/0.50=4\) per dollar, and the likelihood is \(4^3=64\) per dollar cubed. This is a support-boundary maximum: the largest observed bid reaches the model’s upper bid limit. It is an estimate under the stated uniform-value and equilibrium assumptions, not direct evidence that the true population maximum value is exactly \(0.50\) dollars.

Participation Changes the Relevant Distribution

When entry depends on a bidder’s value, the distribution of values among entrants is not the same as the distribution among all potential bidders. For example, suppose potential bidders’ values are uniform on \([0,1]\), and a bidder enters exactly when \(v\geq t\). The probability of entry is \(q=1-t\). Conditional on entry, the value distribution is uniform on \([t,1]\), with cumulative distribution function

F_t(v)=\frac{v-t}{1-t},\qquad t\leq v\leq 1.

That conditional distribution matters for bidding once the set of entrants is known. If exactly two such bidders compete in a first-price auction with no reserve, the symmetric equilibrium bid is \(b(v)=(v+t)/2\). It follows that a bidder at the lower end of the entrant-value support bids \(b(t)=t\). It is not correct to use the equilibrium bid function or rival-value distribution for the original, unselected population without checking the participation and bidding sequence.

A cutoff condition can connect participation to an entry cost. Suppose there are two potential bidders, each independently entering when \(v\geq t\), and entry costs \(c\). Entry decisions are made before bidding; after entry is observed, bidders bid in a first-price auction. Let \(q\) be the probability the rival enters. At the cutoff \(v=t\), the bidder’s gross expected surplus is \(t\) if the rival stays out: the bidder is the only entrant and can win with a zero bid. If the rival enters, the cutoff type is at the bottom of the entrant-value support, so its probability of beating that rival is zero. Consequently, the expected gross surplus at the cutoff is \((1-q)t\), and the indifference condition is \(c=(1-q)t\). This calculation conditions on rival entry; it does not treat an entrant’s value as if it were drawn from the original population.

Worked Example: An Entry Cutoff with Selected Values

Consider two potential bidders with independent uniform values on \([0,1]\). Each enters if and only if \(v\geq t\), and entry costs \(c\). Assume a first-price auction with no reserve follows, and entrants observe who entered before bidding. If \(t=0.50\), calculate the entry probability and the cutoff bidder’s gross expected surplus. Then state the entry cost consistent with cutoff indifference.

A bidder enters with probability \(q=1-t=0.50\). Conditional on entering, the rival’s value is uniform on \([0.50,1]\), not on \([0,1]\). When both bidders enter, the two-bidder equilibrium bid function for that conditional distribution is \(b(v)=(v+0.50)/2\). At \(v=0.50\), the cutoff bidder bids \(0.50\), and the probability of beating an entrant is zero: a rival’s value is at least \(0.50\), with equality having probability zero.

If the rival does not enter, the cutoff bidder is alone, wins with a zero bid, and earns gross surplus \(0.50\) dollars. If the rival enters, expected gross surplus is zero. Therefore,

\text{cutoff gross surplus} =(1-q)t+q(0) =(1-0.50)(0.50)+0.50(0) =0.25\text{ dollars}.

Indifference between entering and staying out requires \(c=0.25\) dollars. In particular, it would be wrong to assign probability \(0.50\) to beating an entering rival by using the original uniform distribution. Under the entry rule, the rival’s value is selected into \([0.50,1]\), and the cutoff bidder is at its lower endpoint.

Joint Likelihoods for Entry and Bids

If data record both participation and bids, the model can use both sources of information. A likelihood contribution has to respect the different types of observations: entry indicators are discrete, while bids are continuous. In the cutoff model above, each potential bidder enters with probability \(1-t\). If both enter, their conditional values are independently uniform on \([t,1]\), and their equilibrium bids satisfy \(b=(v+t)/2\). The inverse is \(v=2b-t\), with derivative \(dv/db=2\). Therefore, conditional on both entering, each bid has density

f_B(b\mid \text{entry},t)=\frac{2}{1-t}, \qquad t\leq b\leq \frac{1+t}{2}.

A sole entrant bids zero in this example. Thus the model assigns probability to the entry pattern and, when there are two entrants, a conditional density to their bids. If the cost is also an unknown primitive and the cutoff is an equilibrium, the model imposes the additional restriction \(c=(1-q)t\), with \(q=1-t\) here.

Worked Example: Evaluating a Joint Entry-and-Bid Likelihood

Suppose two hypothetical auctions each have two potential bidders, values are independent uniform draws on \([0,1]\), and the cutoff is \(t=0.20\). In the first auction both bidders enter and submit bids of \(0.35\) and \(0.40\) dollars. In the second, bidder 1 enters and bidder 2 does not. Evaluate the likelihood contribution of these observations at \(t=0.20\), treating the named entry patterns as observed.

Each potential bidder enters with probability \(1-t=0.80\). The probability of both entering in the first auction is \(0.80^2=0.64\). Conditional on entry, bids must lie between \(0.20\) and \((1+0.20)/2=0.60\) dollars. Both observed bids are in that support, and each has density

f_B(b\mid \text{entry},t=0.20) =\frac{2}{1-0.20} =2.50\text{ per dollar}.

The second auction’s specified pattern has probability \(0.80(0.20)=0.16\). Multiplying the discrete entry probabilities by the two conditional bid densities gives

L =(0.64)(2.50)(2.50)(0.16) =0.64\text{ per dollar squared}.

The numerical value is a likelihood contribution, not a probability for the exact pair of continuous bids. A density can exceed one and carries inverse units; here the two bid densities contribute units of inverse dollars squared. At this cutoff, the implied entry cost under cutoff indifference is \(c=t^2=(0.20)^2=0.04\) dollars, because \(q=1-t\). The likelihood calculation alone does not establish that this equilibrium restriction or the assumed uniform distribution is true.

Moment Conditions and What Identifies the Primitives

A structural model need not be estimated only by maximizing a likelihood. It can also imply moments relating the distribution of bids to its primitives. In the two-bidder uniform model with all bidders participating, \(b=v/2\) and \(E[V]=\bar v/2\). Hence the model implies \(E[B]=\bar v/4\). Replacing the population bid mean by the sample mean gives a method-of-moments estimate. This use of a moment depends on the same equilibrium and distributional assumptions as the likelihood approach.

Worked Example: A Moment Estimate of the Value Support

In a separate hypothetical sample of three independent two-bidder first-price auctions, suppose one bid per auction is recorded as \(0.10\), \(0.20\), and \(0.30\) dollars. Assume all bidders participate, values are independent uniform draws on \([0,\bar v]\), bidders are risk-neutral, and the equilibrium bid is \(b=v/2\). Estimate \(\bar v\) by matching the sample mean bid to the model-implied mean.

The sample mean bid is

\bar b=\frac{0.10+0.20+0.30}{3}=0.20\text{ dollars}.

Since \(E[B]=E[V/2]=\bar v/4\), the moment equation \(\bar b=\bar v/4\) yields

\widehat{\bar v}_{\mathrm{MM}}=4\bar b=4(0.20)=0.80\text{ dollars}.

The observed bids are all within the implied bid support \([0,0.40]\) dollars at this estimate. This is a check that the estimate does not contradict the support restriction, not proof that the uniform distribution is appropriate.

Identification means that distinct primitive values imply distinct observable distributions or moments, under the specified model. In the uniform example, the mean bid identifies \(\bar v\) because the model fixes the relation \(E[B]=\bar v/4\). In a cutoff-entry model, entry frequencies can inform \(t\), while the conditional bid distribution can provide additional restrictions. The equilibrium relation can then link \(t\) to the entry cost \(c\). These conclusions are conditional on assumptions such as the value distribution, independence across bidders, known auction rules, and the proposed equilibrium. If those assumptions change, the same observed bid or entry frequency can correspond to different primitives.

Common Pitfalls and Practical Interpretation

The most consequential error in structural auction estimation is applying a distribution or strategy to the wrong population. When entry selects bidders by value, the values of entrants follow a conditional distribution. For a uniform population on \([0,1]\) with cutoff \(t\), that distribution is uniform on \([t,1]\). Its lower endpoint is \(t\), so a cutoff bidder does not have a one-half chance of beating an entrant. The correct probability at the cutoff is zero. Likewise, a bidding function derived for a different support or a different number of active bidders cannot be inserted without justification.

A second pitfall is treating a structural estimate as if it were assumption-free. A bid density is derived from a value density and an equilibrium bidding function; it does not reveal the value distribution independently of those ingredients. Entry data also reflect both incentives and the payoff consequences of the auction. A complete specification should make clear which primitives are fixed, which are estimated, how participation is modeled, and which equilibrium restrictions connect these elements.

Key takeaway: To construct an auction likelihood or moment condition, first state the primitives and participation rule, then obtain the relevant equilibrium mapping and the distribution conditional on selection. Only after these steps should observed bids and entry outcomes be translated into probabilities, densities, or moments.

Check Your Understanding

Answer using the assumptions stated in each question.

With two participating bidders, values uniform on \([0,\bar v]\), and equilibrium bids \(b=v/2\), what is the bid support and density?
Potential bidders have uniform values on \([0,1]\) and enter if \(v\geq t\). What is the conditional distribution of an entrant’s value?
In the cutoff example with \(t=0.50\) and rival entry probability \(q=0.50\), why is the cutoff bidder’s probability of beating an entering rival zero?
For two bidders with values uniform on \([0,\bar v]\) and bids \(b=v/2\), derive the moment relation between \(E[B]\) and \(\bar v\).
Why does a likelihood contribution involving exact continuous bids represent a density rather than the probability of observing precisely those bid values?