Auction Entry, Participation, and Selection

Why Entry Changes the Auction Data

Auction records commonly contain bids only from people who chose to participate. But participation is a decision, not a random sampling rule. A bidder weighs the cost of entering against the profit expected if they enter, and that expected profit depends on their value and on the likelihood that others enter. As a result, the bidders represented in the data may differ systematically from the population of potential bidders.

To study this selection, it helps to separate three stages: the population of potential bidders, the decision to enter, and the bids submitted by entrants. The goal here is to model the entry decision and understand what conditioning on participation does to the data. The examples use private values and, where a transparent payoff calculation is useful, a second-price auction. The entry logic also applies to other auction formats, although their continuation payoffs differ.

Definition: A potential bidder is an individual who could participate in an auction. Let \(V_i\) denote bidder \(i\)'s private value and \(C_i\) their cost of entering. Let \(D_i=1\) if bidder \(i\) enters and \(D_i=0\) otherwise. An entrant has \(D_i=1\); a submitted bid is observed only for an entrant who reaches the bidding stage. The participation rate is \(P(D_i=1)\).

Assume for now that bidders observe their own value and entry cost before deciding whether to enter. Entry costs are paid when a bidder enters, regardless of whether they win. In a symmetric equilibrium, a bidder enters when the expected payoff from entering, before subtracting the entry cost, is at least that cost. The expected payoff depends on the other bidders’ entry and bidding strategies, so the participation decision and the auction’s continuation behavior must be consistent with one another.

Entry as a Comparison of Expected Payoff and Cost

Write \(\Pi_i^{\mathrm{in}}(v,c)\) for bidder \(i\)'s expected payoff from entering, before subtracting the entry cost, when their value is \(v\) and cost is \(c\). This expectation is taken over uncertainty about the other bidders and their behavior. The net payoff from entering is \(\Pi_i^{\mathrm{in}}(v,c)-c\). If not entering gives payoff zero, the entry rule is:

D_i = \begin{cases} 1, & \Pi_i^{\mathrm{in}}(V_i,C_i)-C_i \geq 0,\\ 0, & \Pi_i^{\mathrm{in}}(V_i,C_i)-C_i < 0. \end{cases}

When a bidder is indifferent, either decision may be consistent with best response; if the type distribution is continuous, a single cutoff type usually has probability zero. The expected payoff is an equilibrium object: it depends on how likely rivals are to enter and, if they do, on how they bid.

A simple payoff expression makes the role of rival participation concrete. Consider two potential bidders in a second-price auction with no reserve price. If bidder \(i\) is the only entrant, they receive the object and pay zero. If the rival enters, the entrant’s payoff before entry cost is \((v-V_j)_+\), where \((z)_+=\max(z,0)\). Suppose the rival enters with probability \(q\), independently of bidder \(i\)'s value, and the rival’s value has distribution \(F\). Then:

\Pi_i^{\mathrm{in}}(v) = (1-q)v + q\,\mathbb{E}\!\left[(v-V_j)_+\mid D_j=1\right].

The first term is the payoff when the rival stays out; the second is the expected surplus when the rival enters. The expectation uses the rival’s value distribution among entrants, not automatically the distribution among all potential bidders. This distinction matters whenever rival participation depends on value or cost.

Worked Example: Entry Payoff When a Rival May Stay Out

Suppose two bidders have independent values distributed uniformly on \([0,1]\). The auction is second-price with no reserve, and a sole entrant pays zero. A potential bidder has value \(v=0.60\) and entry cost \(c=0.25\). Assume the rival enters with probability \(q=0.50\), and, conditional on entering, the rival’s value is uniform on \([0,1]\). Should this bidder enter?

If the rival does not enter, the bidder’s gross payoff is \(0.60\). If the rival enters, the expected surplus is \(\mathbb{E}[(0.60-V_j)_+]=\int_0^{0.60}(0.60-x)\,dx\). Evaluating the integral gives \(0.60^2/2=0.18\).

Therefore, the expected payoff before entry cost is \((1-0.50)(0.60)+0.50(0.18)=0.30+0.09=0.39\). The net payoff from entering is \(0.39-0.25=0.14\), which is positive. Under these assumptions, entry is a best response.

For comparison, if a rival were certain to enter, the gross payoff would be \(0.18\), giving net payoff \(0.18-0.25=-0.07\). Thus the same bidder can find entry worthwhile when rivals may stay out but not when a rival is certain to compete.

A Symmetric Entry Cutoff

A cutoff strategy is a rule under which bidders enter above a value threshold and stay out below it. The cutoff is not imposed independently of the auction: it must make each bidder’s decision optimal given the other bidders’ cutoff rule. The next derivation gives a compact example of that equilibrium condition.

Worked Example: Deriving the Equilibrium Entry Threshold

There are two potential bidders. Their values are independent and uniform on \([0,1]\), each pays the same entry cost \(c=0.25\) if entering, and they compete in a second-price auction with no reserve. A sole entrant pays zero. Consider a symmetric cutoff strategy: enter if and only if the bidder’s value is at least \(t\). Derive \(t\) and check that the strategy is an equilibrium.

If the rival uses cutoff \(t\), the rival stays out with probability \(t\) and enters with probability \(1-t\). At a bidder’s own cutoff value \(v=t\), an entering bidder earns \(t\) if the rival stays out. If the rival enters, their value is at least \(t\), so the bidder with value \(t\) earns zero (apart from a probability-zero tie). Thus the cutoff type’s gross expected payoff is \(t\cdot t=t^2\). Indifference at the cutoff requires:

t^2-c=0 \qquad\Longrightarrow\qquad t=\sqrt{c}=\sqrt{0.25}=0.50.

To check the incentive on each side of the threshold, first take \(v<0.50\). A rival who enters has value at least \(0.50\), so the bidder’s payoff when that rival enters is zero. The gross expected payoff from entering is then \(0.50v\), and the net payoff is \(0.50v-0.25<0\). Staying out is better.

Now take \(v>0.50\). Even if the rival stays out, the bidder earns \(v\); if the rival enters, the bidder’s payoff is nonnegative. Hence the gross expected payoff is at least \(0.50v\), and the net payoff is at least \(0.50v-0.25>0\). Entry is better. At \(v=0.50\), the net payoff is zero. Therefore, the cutoff rule with \(t=0.50\) is a symmetric equilibrium. The implied entry probability for each potential bidder is \(1-t=0.50\).

The example’s symmetry and simple auction rules make the cutoff calculation possible in closed form. In a first-price auction, or with heterogeneous costs, values, or bidder characteristics, the continuation payoff generally changes and the cutoff condition may require a different calculation. The general principle remains: entry is determined by net expected equilibrium payoff, not by value alone.

What Bid Data Represent

Let \(D\) indicate entry and let \(V\) be a potential bidder’s value. The value distribution among entrants is the conditional distribution \(P(V\leq v\mid D=1)\). It can differ from the population distribution \(P(V\leq v)\). For example, if \(D=1\) exactly when \(V\geq t\), and values have continuous distribution function \(F\), then for \(v\geq t\):

P(V\leq v\mid D=1) = \frac{P(t\leq V\leq v)}{P(V\geq t)} = \frac{F(v)-F(t)}{1-F(t)}.

For \(v<t\), this conditional probability is zero. The formula shows directly that the entrant sample omits the part of the population below the threshold. If entry also depends on costs, observable characteristics, or unobserved factors, the selected distribution is instead shaped by all variables that affect \(D\). Selection need not always favor higher values: for instance, if costs are related to values, the direction and size of the difference depend on that relationship.

Submitted bids are not necessarily a direct record of values. In a second-price private-value auction, truthful bidding makes the bid equal to the bidder’s value under the standard assumptions of that auction. In a first-price auction, bids are strategic choices and generally reflect both values and the expected competition from other entrants. In either format, the sample of submitted bids is conditional on participation. Interpreting it as if it were an unselected sample of all potential bidders can therefore distort conclusions about the population.

Worked Example: How a Bid Sample Can Overstate the Population Mean

Suppose a large hypothetical population has values uniformly distributed on \([0,1]\), with entry costs \(c=0.25\) in the two-bidder second-price model above. Its equilibrium entry threshold is \(0.50\). To make the selection visible in a finite hypothetical data set, consider ten potential bidders with values \(0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85,\) and \(0.95\). Assume the cutoff rule determines entry, and entrants bid truthfully.

The mean value among all ten potential bidders is \((0.05+0.15+0.25+0.35+0.45+0.55+0.65+0.75+0.85+0.95)/10=5.00/10=0.50\). Only the five bidders with values at least \(0.50\) enter. Their mean value—and, in this auction, their mean submitted bid—is \((0.55+0.65+0.75+0.85+0.95)/5=3.75/5=0.75\).

The entrant mean is \(0.25\) higher than the potential-bidder mean, in value units. Treating these five bids as though they were an unselected sample would overstate the population mean in this example. The example does not imply that entrant bids always overstate values: that conclusion depends on the entry rule and, in formats such as first-price auctions, on the relationship between bids and values.

Why Selection Matters

Participation affects both the economic interpretation of auction records and the information available for analysis. If nonentrants are absent from the bid file, a researcher may observe neither their values nor their entry costs. The missing bidders are not generally interchangeable with the observed bidders: their decision not to enter may reveal that their expected profits were low relative to their costs. Even when bid data are abundant, the distribution among participants need not recover the distribution among all potential participants without assumptions or additional information about the entry process.

A useful way to organize the issue is to keep two questions separate. First, what determines whether a potential bidder enters? Second, conditional on entering, how does that bidder choose a bid? Entry selection concerns the first question; strategic bid formation concerns the second. Conflating them can lead to treating selected bids as direct, representative measurements of values.

Common Mistakes

Equating potential bidders with observed bidders. A bid file usually records entrants, not everyone who could have entered. The participation decision determines which bidders appear.
Ignoring entry costs in the decision rule. A bidder enters when expected payoff before the cost is at least the cost, not merely when the bidder’s value is positive.
Assuming a rival’s participation is irrelevant. Entry changes the probability and severity of competition, which changes the expected payoff from entering.
Assuming selection must raise observed values. Entry may favor high values in a simple cutoff model, but costs and other factors can change or reverse the pattern.
Reading a submitted bid as a value without checking the auction format. Truthful bidding in the second-price example does not make bids equivalent to values in a first-price auction.

Key takeaway: Auction participation is an equilibrium choice based on expected continuation profits and entry costs. Bid data are conditional on that choice, so conclusions about the population of potential bidders require attention to selection as well as to how entrants choose bids.

Check Your Understanding

Use the entry and selection logic in this tutorial to answer the following questions.

In the two-bidder second-price example, what components make up an entrant’s expected payoff when a rival enters with probability \(q\)?
Why does a bidder at the equilibrium cutoff earn zero net payoff in the cutoff example?
If entry occurs exactly when \(V\geq t\), how does \(P(V\leq v\mid D=1)\) differ from the population distribution for values below \(t\)?
Give one reason why the entrant sample need not always have a higher mean value than the potential-bidder population.
Why does observing bids only from entrants create a different issue from the strategic relationship between bids and values?