First-Price Auctions with Asymmetric Bidders

Why Asymmetry Changes the Bidding Problem

In a first-price auction, a bidder who wins pays their own bid. Bidding more can raise the chance of winning, but it also reduces the surplus earned if the bidder wins. The trade-off depends on the probability of beating the other bidders. When bidders have different value distributions, they generally face different distributions of opponents’ bids, so the trade-off—and thus the equilibrium bidding function—can differ by bidder.

This distinction matters when studying bids as evidence about values. In a symmetric model, a common bidding function may be inverted to translate a bid into a value. With asymmetric bidders, the relevant inverse is bidder-specific. Pooling bids or using the wrong inverse can therefore produce incorrect value distributions.

Model and assumptions: There are \(N\) bidders in a sealed-bid first-price auction for one item, with a fixed set of participants and no reserve price. Bidder \(i\) has private value \(v_i\), drawn independently from a continuous distribution \(F_i\) with density \(f_i\). Bidders are risk-neutral, know the distributions, and choose bids to maximize expected profit. Ties have probability zero under the continuous bid distributions considered here. Bidder \(i\)'s increasing equilibrium strategy is \(b_i(v_i)\).

Write \(G_j(b)\) for bidder \(j\)'s bid cumulative distribution function (CDF), and \(g_j(b)\) for its density. If bidder \(j\)'s strategy is strictly increasing, a bid \(b\) corresponds to the value \(b_j^{-1}(b)\). Consequently,

G_j(b)=F_j\!\left(b_j^{-1}(b)\right).

This identity is bidder-specific: both the value distribution and the strategy used to translate between values and bids may differ across bidders.

The Asymmetric Equilibrium Condition

Suppose bidder \(i\), with value \(v\), considers a bid \(b\). The bidder wins when every opponent bids below \(b\). Independence implies that the probability of winning is the product of the opponents’ bid CDFs. Expected profit is therefore

\pi_i(b\mid v)=(v-b)\prod_{j\ne i}G_j(b).

For an interior bid at which the relevant densities are positive, differentiating with respect to \(b\) gives the first-order condition below. It is necessary for an interior optimum; the objective and any bid-support boundaries must also be checked to establish a global best response.

-\prod_{j\ne i}G_j(b) +(v-b)\sum_{j\ne i}g_j(b)\prod_{\substack{k\ne i\\k\ne j}}G_k(b)=0.

Dividing by the positive product \(\prod_{j\ne i}G_j(b)\) gives a convenient form. In equilibrium, bidder \(i\) chooses \(b_i(v)\), so the condition becomes

\bigl(v-b_i(v)\bigr) \sum_{j\ne i}\frac{g_j(b_i(v))}{G_j(b_i(v))}=1.

The ratios \(g_j(b)/G_j(b)\) are the bid-distribution hazards relevant to the probability of beating each opponent. This equation is coupled across bidders: bidder \(i\)'s strategy depends on opponents’ bid distributions, which in turn depend on their strategies and value distributions. In general, there is no single symmetric bid function to apply to everyone.

Worked Example: A Two-Bidder Asymmetric Equilibrium

Consider two bidders with independent values. Let bidder 1's value distribution be uniform on \([0,1.5]\), and let bidder 2's CDF be \(F_2(v)=(v/2)^2\) on \([0,2]\). Both distributions have the same lower endpoint, but they differ in shape and upper support. We verify the following proposed strategies:

b_1(v)=\frac{2v}{3},\qquad b_2(v)=\frac{v}{2}.

Both strategies map their value supports onto the bid interval \([0,1]\). To find the induced bid distributions, invert each strategy separately. For bidder 1, a bid \(b\) corresponds to value \(v=3b/2\), so

G_1(b)=F_1(3b/2)=\frac{3b/2}{1.5}=b,\qquad 0\le b\le 1.

For bidder 2, a bid \(b\) corresponds to value \(v=2b\), giving

G_2(b)=F_2(2b)=\left(\frac{2b}{2}\right)^2=b^2,\qquad 0\le b\le 1.

Given bidder 2's bid CDF, bidder 1's expected profit from bidding \(b\in[0,1]\) with value \(v\) is \((v-b)b^2\). Its derivative is \(-b^2+2b(v-b)\), which is zero at \(b=2v/3\). For \(v\in[0,1.5]\), this bid lies in \([0,1]\). The profit is zero at \(b=0\) and at \(b=1\) when \(v=1.5\); for lower values, the interior critical point maximizes the objective. Equivalently, the derivative is positive below \(2v/3\) and negative above it. Bids above 1 cannot improve on bidding 1, because they have the same probability of winning and cost more. Thus the proposed bid is a global best response, including at the top of the value support.

For bidder 2, expected profit from a bid \(b\in[0,1]\) is \((v-b)b\). Its derivative is \(v-2b\), so profit increases up to \(b=v/2\) and decreases thereafter. This bid lies in \([0,1]\) for \(v\in[0,2]\); bids above 1 again cannot improve on bidding 1. The two strategies are therefore mutual best responses, including their support boundaries, and form an equilibrium.

The strategies differ because each bidder faces a different opponent bid distribution: \(G_1(b)=b\), whereas \(G_2(b)=b^2\). The inversions also give different value interpretations for the same bid. For example, a bid of 0.4 corresponds to bidder 1's value \(3(0.4)/2=0.6\), and bidder 2's value \(2(0.4)=0.8\). These values are consistent with the distributions: \(F_1(0.6)=0.4\) and \(F_2(0.8)=0.16\).

Why Bidder-Specific Inversion Matters

In an increasing-strategy model, observed bids reveal values only through the assumed equilibrium strategies. If bidder identities are observed and the strategies are known, the value associated with bidder \(i\)'s bid \(b\) is \(b_i^{-1}(b)\). If the strategies are estimated rather than known, recovering values also requires learning how each bidder type maps values into bids.

Worked Example: Recovering Values From Labeled Bids

Use the equilibrium above. Suppose a hypothetical auction dataset records a bid of 0.4 from a bidder known to be bidder 1, and a bid of 0.4 from a bidder known to be bidder 2. Find the implied values and compare their CDF positions.

For bidder 1, solve \(b_1(v)=2v/3=0.4\). The inverse is \(v=3b/2\), so the implied value is \(3(0.4)/2=0.6\). Since bidder 1 is uniform on \([0,1.5]\), the CDF value is \(F_1(0.6)=0.6/1.5=0.4\).

For bidder 2, solve \(b_2(v)=v/2=0.4\). The inverse is \(v=2b\), so the implied value is \(2(0.4)=0.8\). The CDF value is \(F_2(0.8)=(0.8/2)^2=0.16\).

The same numerical bid does not correspond to the same value or the same value percentile for the two bidders. Applying bidder 1's inverse to bidder 2's bid would assign value 0.6 instead of 0.8. That mistake would also incorrectly place bidder 2 at the 0.4 CDF position rather than the 0.16 position.

With sufficiently informative data, bidder labels let an analyst estimate bidder-specific bid distributions \(G_i\). But a bid distribution is not itself a value distribution. Recovering \(F_i\) requires an equilibrium restriction connecting the bidders' bid distributions to their strategies and the \(F_i\)'s. In the asymmetric case, these restrictions are coupled, rather than a collection of separate inversions that ignore how each bidder responds to the others.

Identification Challenges in Auction Data

Identification asks whether distinct underlying value distributions or model parameters can generate the same observable bid data under the model. Asymmetry creates several related challenges:

Bidder identities matter. Without labels or observed bidder characteristics that identify types, a bid cannot generally be assigned the correct bidder-specific bid distribution or inverse strategy.
The equilibrium is coupled. The best response of each bidder depends on the bid distributions of all opponents. Treating each bidder's bid data as if it came from an independent symmetric auction discards this strategic link.
Bid and value variation are entangled. A difference in observed bids could reflect differences in value distributions, differences in bidding strategies induced by those distributions, or both. Recovering values requires assumptions that separate these sources.
Support and data coverage matter. Inversions rely on the strategy being increasing and on the observed bid lying in the relevant strategy's range. Sparse data near the extremes can make the corresponding value tails especially difficult to learn.

The following calculation shows how loss of bidder labels changes even a simple descriptive object. It does not solve the structural identification problem; it illustrates why pooling can conceal information needed for that problem.

Worked Example: What Pooling Does to Bid Ranks

Suppose a hypothetical dataset combines equal numbers of bids from the two bidder types in the equilibrium above, but does not record bidder identity. For bids between 0 and 1, the pooled bid CDF is the equal-weight mixture of \(G_1\) and \(G_2\):

G_{\mathrm{pool}}(b) =\frac{1}{2}G_1(b)+\frac{1}{2}G_2(b) =\frac{b+b^2}{2}.

At \(b=0.4\), this gives \(G_{\mathrm{pool}}(0.4)=(0.4+0.16)/2=0.28\). That pooled rank does not tell us whether the bid came from bidder 1, whose bid CDF at 0.4 is 0.4, or bidder 2, whose bid CDF there is 0.16. In the known model, the implied values are 0.6 and 0.8, respectively. Without the bidder label, the pooled CDF alone does not select between those interpretations. Treating the pooled rank as a bidder-specific rank would therefore be unjustified.

From Equilibrium Restrictions to Empirical Work

An applied analysis must distinguish what is observed from what is imposed by the model. The data may contain bids, auction outcomes, bidder identities, and observed bidder characteristics; values are usually unobserved. The equilibrium condition supplies restrictions linking those observables to unobserved values, but those restrictions do not remove the need to specify what is known about the auction environment.

In particular, the analyst must state which bidders are modeled as participating, how bidder types are identified, and what is assumed about dependence among values. This tutorial's independent-private-value setup excludes common-value information and participation selection. Those are distinct sources of complications and should not be folded silently into an asymmetric private-value interpretation.

The asymmetric first-order conditions can help organize estimation, but they are not by themselves proof that every observed bid pattern identifies a unique set of value distributions. Identification depends on the available labels and covariates, support and regularity conditions, the maintained equilibrium model, and how much the observed bids reveal about the coupled strategies. Structural estimation must respect these restrictions rather than applying a symmetric inversion separately to each bidder.

Common Mistakes

Using the wrong inverse. If \(b_i(v)=2v/3\), then \(v=3b/2\), not \(2b\). Substitute the inverse into \(F_i\) to obtain \(G_i(b)\).
Assuming different value distributions always imply visibly different bids at every value. The equilibrium response depends on opponents' bid distributions and the auction environment. The general result is that bidder-specific distributions generally produce bidder-specific strategies, not that every pair must differ at every point.
Treating a first-order condition as a complete equilibrium proof. Check that the proposed bid is feasible and globally optimal, including at boundaries or outside an opponent's bid support.
Interpreting a pooled bid rank as a value rank. A pooled CDF combines bidder-specific distributions; it is not automatically the relevant rank for any individual bidder.

Key takeaway: In an asymmetric first-price auction, each bidder's incentive depends on opponents' bid distributions. Value recovery therefore requires bidder-specific strategy inversions and an equilibrium model that respects the strategic links among bidders.

Check Your Understanding

Answer each question using the assumptions and definitions in this tutorial.

Write bidder \(i\)'s expected profit from bidding \(b\) when opponents' bids are independent, and state the interior first-order condition.
If \(b_i(v)=2v/3\), what value corresponds to a bid of 0.4? Show the inversion.
In the worked equilibrium, why does bidder 1's bid CDF equal \(b\), while bidder 2's equals \(b^2\)?
Why does observing the same bid from two different bidder types not necessarily identify the same value percentile?
Give two reasons why a pooled bid distribution may be insufficient for recovering bidder-specific value distributions.