Estimating Private-Value Distributions from Bids

Why Estimate Values From Bids?

In a first-price auction, a bid is a strategic choice, not a direct measurement of the bidder’s willingness to pay. Under a symmetric private-value model, however, equilibrium bidding is monotone: higher values submit higher bids. That ordering allows the observed bid distribution to reveal the underlying private-value distribution. The practical task is to estimate the bid distribution and its density well enough to apply the inversion without allowing boundary noise or sampling error to dominate.

The method in this tutorial is nonparametric in the sense that it does not impose a particular parametric family, such as a normal or lognormal distribution, on private values. It does still rely on a structural model of bidding. An estimated distribution is therefore a model-based recovery of values, not simply a relabeling of observed bids.

Model and Inversion

Consider repeated first-price auctions with a fixed number \(N\geq 2\) of bidders. In each auction, bidders are risk neutral and have independent, identically distributed private values with cumulative distribution function \(F\) and density \(f\) on a common support \([\underline v,\overline v]\). Assume \(f(v)>0\) in the interior of the support. Bidders know the distribution and choose bids to maximize expected payment-adjusted winning payoff. There are no unmodeled reserve prices, entry decisions, or bidder-specific value distributions in the model considered here.

Suppose the symmetric equilibrium bid function \(b(v)\) is differentiable and strictly increasing. Let \(G\) and \(g\) denote the CDF and density of an individual bidder’s bid. Monotonicity implies that a bidder with value \(v\) bids \(b(v)\), so the bid and value CDFs obey \(G(b(v))=F(v)\). To use the density change-of-variables identity in the interior, impose the additional regularity condition \(b'(v)>0\) at the points where the inversion is applied. Then \(g(b(v))b'(v)=f(v)\) at those points. Strict increase alone does not justify division by \(b'(v)\); the positive-derivative condition is essential.

To see the strategic source of the formula, a bidder with value \(v\) who submits bid \(x\) wins when all \(N-1\) opponents bid below \(x\). Under independent values and a common increasing strategy, this happens with probability \(G(x)^{N-1}\). The expected payoff is \((v-x)G(x)^{N-1}\). At an interior optimum where the relevant density is positive, differentiating with respect to \(x\) and setting the derivative to zero gives

For an interior bid with \(G(x)>0\) and \(g(x)>0\), divide by \(G(x)^{N-2}\) and rearrange. At the equilibrium bid \(x=b(v)\), this yields

This first-order condition explains the economic adjustment: the value exceeds the bid by a term determined by the probability of facing a lower opponent bid relative to the local bid density. It is an interior condition, not a boundary formula.

From a Bid Sample to an Estimated Value Distribution

Suppose \(J\) independent auctions are observed, with the same \(N\) bidders in each auction and all bids recorded. Write the bid observations as \(B_{aj}\), where \(a\) indexes auctions and \(j\) indexes bidders. Under the model, the pooled bids have CDF \(G\) and density \(g\). A basic nonparametric procedure estimates both and substitutes them into the inversion.

For example, a kernel estimator of the bid density is \(\widehat g_h(x)=\dfrac{1}{JN h}\sum_{a=1}^{J}\sum_{j=1}^{N}K\!\left(\dfrac{x-B_{aj}}{h}\right)\), where \(K\) is a kernel density function and \(h>0\) is a bandwidth. The corresponding smoothed CDF is \(\widehat G_h(x)=\dfrac{1}{JN}\sum_{a=1}^{J}\sum_{j=1}^{N}\mathcal K\!\left(\dfrac{x-B_{aj}}{h}\right)\), where \(\mathcal K(t)=\int_{-\infty}^{t}K(u)\,du\). The plug-in value assigned to an observed bid \(B_{aj}\) is

The empirical CDF of these pseudo-values is a direct estimate of the value distribution: \(\widehat F(v)=\dfrac{1}{JN}\sum_{a=1}^{J}\sum_{j=1}^{N}\mathbf{1}\{\widehat V_{aj}\leq v\}\). A pseudo-value is an estimate generated by the inversion, not an observed value. The resulting CDF is useful for describing the distribution and estimating features such as quantiles, but its accuracy depends on the bid-density estimate as well as on the structural assumptions.

In practice, an estimator can use a leave-one-out version of \(\widehat G_h\) and \(\widehat g_h\) when transforming a bid, so that the bid does not contribute to its own density estimate. This can reduce finite-sample self-influence. It does not remove the need to choose a bandwidth or to check that the estimated density is sufficiently stable.

Support Boundaries and Density Instability

The inversion is most defensible in the interior of the bid support. Near a boundary, kernel estimators can place weight outside the feasible support, producing bias in both \(\widehat G\) and \(\widehat g\). More importantly, the inversion divides by \(\widehat g\). When that estimate is close to zero, a small estimation error in the density can create a very large change in the pseudo-value.

At the lower endpoint, \(G(b)\) can approach zero while the density estimate is also small, so the ratio may be numerically unstable even when its theoretical interior limit is well behaved. At the upper endpoint, sparse observations create a similar problem. A general boundary estimate should not be obtained by substituting a boundary bid into an interior formula without establishing the appropriate limit and regularity conditions.

A practical analysis should therefore identify the bid support, inspect the estimated density, and state which interior region is used for inversion. Boundary corrections or local boundary estimators may reduce kernel bias, but they cannot create information where the data are extremely sparse. Trimming bids with very small estimated densities is one possible stability measure; it changes the region for which values are reported and should be disclosed. If the value support is known, it can also provide a useful check on implausible tail estimates.

Sampling Uncertainty and Interpretation

The estimated distribution varies across samples, and uncertainty comes from estimating both \(G\) and \(g\), bandwidth choice, and the nonlinear division in the inversion. A useful practical procedure is to resample whole auctions with replacement, keeping each auction’s bids together, and then repeat the entire estimation procedure in every resample. For each replicate, re-estimate the bid CDF and density, recompute pseudo-values, and calculate the target quantity, such as \(\widehat F(v)\). Percentiles of the replicate estimates can be used to form a bootstrap interval when the procedure and sample size support that approximation.

Resampling whole auctions respects the sampling unit when bids from the same auction are observed together. Independently resampling individual bids would discard their auction grouping. Bootstrap intervals should not be treated as a repair for model misspecification, boundary instability, or a density estimate that is effectively zero. The bandwidth and any trimming rule should be held to a clearly specified procedure and applied again in each replicate.

Assumptions, Checks, and a Common Pitfall

The recovered distribution has a structural interpretation only if the model is credible. In particular, the procedure assumes a common symmetric bidding strategy, independent private values drawn from a common distribution, risk-neutral first-price bidding, and a fixed known number of participating bidders. It also requires adequate data in the bid region being inverted, positive bid density there, and a sufficiently regular equilibrium bid function. The formula does not by itself test these assumptions.

A common mistake is to say, “The equilibrium bid function is strictly increasing, so differentiating the CDF identity lets us divide by \(b'(v)\) everywhere.” That conclusion is not valid. A differentiable strictly increasing function can have derivative zero at a point; for example, \(b(v)=v^3\) is strictly increasing but has \(b'(0)=0\). The correction is to impose \(b'(v)>0\) at the interior points where the density change-of-variables and inversion are used, and to avoid applying the pointwise formula where that condition or a positive bid density fails.

A second practical warning is that an unusually large estimated value may reflect a tiny estimated density rather than compelling evidence of an extreme value. Report the bid region, bandwidth, density behavior, and uncertainty procedure alongside distribution estimates. These choices are part of the estimation, not cosmetic details.