Counterfactual Auction Design and Revenue Evaluation

Why Counterfactual Revenue Requires a Model

An auction designer may ask whether a higher reserve, a different payment rule, or a new participation policy would raise expected revenue. The answer is not obtained by changing one line in an observed data set: bidders may change their bids, entry, or decision to participate when the rules change. A counterfactual prediction therefore combines estimated primitives with a model of how bidders respond to the proposed design.

The object of interest is expected seller revenue under a specified rule and a specified environment. It is not revenue conditional on a sale unless that conditioning is explicitly part of the question. For a proposed design, revenue must account for outcomes such as no bidder entering, no bidder meeting the reserve, and the identity and payment of the winner.

Definition: Let $d$ denote an auction design, including its format, reserve price, and participation rules. Let $\theta$ denote the primitives of a structural model, such as value distributions and entry costs. The counterfactual expected revenue is $$ R(d;\theta)=E_{\theta}\!\left[\text{seller revenue generated by design }d\text{ in equilibrium}\right]. $$ A plug-in prediction replaces the unknown primitives with estimates: $\widehat R(d)=R(d;\hat\theta)$. The expectation includes all modeled states, including states with no sale, for which seller revenue is typically zero.

This definition makes the role of equilibrium explicit: the distribution of bids, allocation, and payments under design $d$ must be the one implied by bidder behavior under that design. A calculation that holds observed bids fixed while changing the reserve is generally not a structural counterfactual, because those bids may no longer be equilibrium bids.

A Reserve-Price Counterfactual

Consider a second-price auction with two potential bidders. Suppose their private values are independent and uniformly distributed on $[0,1]$, both bidders participate, and the winner pays the larger of the reserve $r$ and the losing bidder’s bid. Under truthful bidding, bids equal values. If neither value reaches $r$, there is no sale and revenue is zero.

Worked Example: Revenue With a Reserve of One-Half

All values in this example are hypothetical and measured in value units. Let $V_1,V_2$ be independent uniform draws on $[0,1]$. Set $r=1/2$.

There are two possible sale cases. If exactly one value is at least $r$, the winner pays $r$. The probability of this event is $2(1-r)r$, so its contribution to expected revenue is $2r^2(1-r)$. If both values are at least $r$, the winner pays the lower value. The density of the lower of two independent uniform values is $2(1-v)$, so this case contributes $\int_r^1 2v(1-v)\,dv$. Thus

R(r)=2r^2(1-r)+\int_r^1 2v(1-v)\,dv.

Substituting $r=1/2$,

R(1/2)=2(1/2)^2(1/2)+\int_{1/2}^1 2v(1-v)\,dv =\frac14+\left[v^2-\frac{2}{3}v^3\right]_{1/2}^1 =\frac14+\frac16=\frac{5}{12}\approx 0.4167.

Without a reserve, the winner pays the lower of the two values. Its expected value is

R(0)=\int_0^1 2v(1-v)\,dv=\frac13\approx 0.3333.

The exact expected-revenue difference is

\frac{5}{12}-\frac13=\frac{1}{12}\approx 0.0833

value units per auction. This is an unconditional revenue comparison: auctions with no sale contribute zero, rather than being omitted from the calculation.

For this benchmark, the reserve can also be optimized rather than merely evaluated at one value. Simplifying the revenue expression gives $R(r)=1/3+r^2-(4/3)r^3$. Its derivative is $R'(r)=2r-4r^2=2r(1-2r)$. Revenue rises for $0<r<1/2$ and falls for $1/2<r\leq 1$. Thus $r=1/2$ maximizes expected revenue on $[0,1]$ in this particular model. This conclusion depends on the assumed value distribution, number of participating bidders, and auction rules; it is not a universal optimal reserve.

Result: In a second-price auction with $n\geq2$ participating bidders, independent private values drawn from a continuous distribution $F$, and reserve $r$, revenue is $$ R(r)=E\!\left[\max\{r,V_{(n-1)}\}\,1\{V_{(n)}\geq r\}\right], $$ where $V_{(n)}$ is the highest value and $V_{(n-1)}$ is the second-highest. The indicator makes revenue zero when no value reaches the reserve. The formula presumes truthful bidding, which is an equilibrium in this second-price private-value setting.

Comparing Auction Rules

A counterfactual may change the payment rule rather than the reserve. Under the standard private-value benchmark, a first-price auction and a second-price auction can produce the same expected seller revenue when they implement the same allocation and satisfy the revenue-equivalence conditions. These include risk-neutral bidders, independent private values from the same distribution, the same participation and allocation rules, and the same expected payoff for the lowest eligible type. Revenue equivalence concerns expected revenue under these conditions; it does not say that bids or payments are identical in each realized auction.

Worked Example: First-Price Revenue With the Same Reserve

Use the hypothetical two-bidder uniform-value setting above, with reserve $r=1/2$, but use a first-price auction: the highest eligible bidder wins and pays their own bid. Both bidders participate, values are independent, and bidders are risk-neutral. In the symmetric equilibrium, an eligible type $v\geq r$ bids

b(v)=v-\frac{\int_r^v F(t)^{n-1}\,dt}{F(v)^{n-1}}.

Here $n=2$ and $F(v)=v$, so

b(v)=v-\frac{\int_r^v t\,dt}{v} =v-\frac{v^2-r^2}{2v} =\frac{v}{2}+\frac{r^2}{2v}.

The highest value has density $2v$ on $[0,1]$. A sale occurs when that value is at least $r$, so expected revenue is

R_{\mathrm{FP}}(r)=\int_r^1 b(v)\,2v\,dv =\int_r^1(v^2+r^2)\,dv.

At $r=1/2$, substituting the reserve and evaluating the integral gives

R_{\mathrm{FP}}(1/2) =\left[\frac{v^3}{3}+\frac{v}{4}\right]_{1/2}^1 =\left(\frac13+\frac14\right)-\left(\frac{1}{24}+\frac18\right) =\frac{5}{12}\approx 0.4167.

This matches the second-price revenue in the previous calculation. It is a comparison of expected revenues under the specified symmetric private-value model, not a claim that first-price bids equal values or that the two formats have the same outcome in every auction.

Participation Is Part of the Counterfactual

The number of active bidders matters directly for revenue. If a proposed reserve or format changes entry or participation, it is generally incorrect to predict revenue by evaluating a bidding model at the old number of bidders. A credible calculation needs a participation model, or a justified assumption that participation is unchanged. It must also distinguish potential bidders from actual participants and account for any selection associated with who enters.

Worked Example: Allowing the Number of Participants to Vary

Suppose two potential bidders independently participate with probability $q=0.8$, regardless of their values and of the reserve. Conditional on participating, their values are independent uniform draws on $[0,1]$. Compare a second-price auction with reserve $1/2$ to one with no reserve. Assume a single participant with a reserve pays $1/2$ if their value reaches the reserve; with no reserve, a single participant pays zero.

The probabilities of exactly one and exactly two participants are

P(N=1)=2q(1-q)=2(0.8)(0.2)=0.32,\qquad P(N=2)=q^2=0.64.

With exactly one participant, expected revenue under the reserve is $rP(V\geq r)=(1/2)(1/2)=1/4$. With two participants, the worked calculation above gives revenue $5/12$. With no participants, revenue is zero. Therefore,

R(1/2)=0.32\left(\frac14\right)+0.64\left(\frac{5}{12}\right) =0.08+0.2667\approx 0.3467.

Without a reserve, one participant generates zero revenue, and two participants generate $1/3$ in expectation. Hence

R(0)=0.64\left(\frac13\right)\approx 0.2133, \qquad R(1/2)-R(0)\approx 0.1333.

The result depends on the stated assumption that participation probability remains $0.8$ under both designs and is independent of values. If the reserve discourages entry, this calculation does not capture that response and the predicted revenue difference may change.

From Estimated Primitives to a Design Evaluation

In an empirical application, primitives such as value distributions, entry costs, or bidder heterogeneity are estimated from auction data. A design evaluation uses those estimates to compute equilibrium behavior and then averages seller revenue across the modeled distribution of market states. A disciplined workflow makes clear where the estimate enters and which behavioral response is being simulated.

Specify the proposed design.
State the format, reserve, eligibility, and participation rules. Define whether the target is revenue per potential auction or revenue conditional on a sale.

State the equilibrium and participation model.
Describe how bidders choose whether to participate and how they bid after observing the rules. Identify the equilibrium concept and any assumption selecting among multiple equilibria.

Compute outcomes using estimated primitives.
For each modeled market state, apply the equilibrium implied by the new design, record allocation and payments, and include zero revenue when there is no sale.

Average and compare.
Average simulated revenues over the target distribution of market states. Compare designs using the same target population and report units and the scope of the comparison.

A plug-in estimate $\widehat R(d)=R(d;\hat\theta)$ treats the estimated primitives as fixed. It does not by itself describe sampling uncertainty. To reflect parameter uncertainty, one can recompute revenue across bootstrap estimates or draws from an estimated sampling distribution, using the same procedure for each design. The resulting uncertainty summaries concern the model-based revenue functional under the stated specification. They do not automatically account for misspecification, an incorrect equilibrium model, or a shift in the target population.

Conditions for Credible Revenue Predictions

A revenue prediction is only as credible as the assumptions that connect observed auctions to the proposed design. Three groups of assumptions deserve explicit attention.

Equilibrium assumptions. The model must describe bidders’ information, objectives, and strategic responses under the new rules. If multiple equilibria are possible, the prediction must specify which equilibrium is used and why it is relevant. A model that fits observed bids under one format need not predict behavior under another without appropriate structural restrictions.
Participation assumptions. The analyst must model who can enter, who actually participates, and whether the design changes those decisions. If observed bidders are selected from a larger pool, treating them as a random sample of potential bidders can distort the inferred value distribution and predicted revenue.
Extrapolation assumptions. The proposed reserve, bidder composition, or format may lie outside the conditions represented in the observed data. Predictions then rely more heavily on functional-form and behavioral assumptions. Changes in market composition or institutional rules may also make the estimated primitives unrepresentative of the counterfactual environment.

These qualifications are especially important when comparing designs with different entry incentives. A higher reserve might increase the payment conditional on a sale while reducing the probability of a sale or discouraging participation. The relevant comparison is the change in unconditional expected revenue after all modeled responses, not the isolated change in a payment threshold.

Common Mistake

Pitfall: “The observed bids are unchanged, so I can apply the new reserve to them and call the resulting total the counterfactual expected revenue.” This treats bids and participation as fixed even when the new rule changes bidders’ incentives. It may also omit auctions with no sale or condition implicitly on an unchanged set of participants.

Correction: State the equilibrium and participation assumptions for the proposed design, predict the resulting bids and entry under those assumptions, and average revenue over all relevant states, including no-sale outcomes. If behavior is deliberately held fixed, describe the result as a mechanical calculation under fixed behavior, not as an equilibrium counterfactual.

Key takeaway: Counterfactual auction revenue is a model-based expectation over equilibrium outcomes. Estimated primitives supply inputs, but credible predictions also require explicit assumptions about equilibrium selection, participation, and the relevance of extrapolating beyond observed auction environments.

Check Your Understanding

Answer each question by referring to the model assumptions as well as the calculation.

In the two-bidder uniform-value example, why does the no-reserve revenue equal the expected lower of the two values?
What does the indicator $1\{V_{(n)}\geq r\}$ ensure in the second-price revenue expression?
Why can equal expected revenue across first-price and second-price formats coexist with different bids in an individual auction?
In the variable-participation example, which assumption would fail if a higher reserve discouraged entry?
Give one reason a revenue prediction for a new auction format may rely on extrapolation beyond observed data.