Auction Formats and Private-Value Benchmarks

Why Auction Rules and Strategic Behavior Must Be Separated

An auction format specifies how bids or actions determine the winner and the payment. It does not, by itself, tell us what bidders will do. In a first-price auction, for example, the highest bidder wins and pays their own bid, but a bidder’s optimal bid depends on the distribution of opponents’ bids and their own value. A useful benchmark therefore separates the rules of the auction from the equilibrium behavior those rules induce.

This tutorial compares four standard formats for a single object. The benchmark is independent private values: each bidder knows their own value, and a bidder’s value does not depend on information held by other bidders. That setting makes the formats comparable, but revenue equivalence requires additional assumptions that should be made explicit.

Definition: In an independent private-values model, bidder $i$ has value $v_i$ for the object; that value is the bidder’s own information and does not depend on other bidders’ values. Values are independent draws from a common continuous distribution $F$ on $[0,\bar{v}]$. A bidder who wins and pays $p$ receives monetary payoff $v_i-p$; a bidder who loses receives zero.

Four Auction Formats

In a first-price sealed-bid auction, each bidder submits a bid without seeing the others’ bids. The highest bidder wins and pays their own bid. Because the winner’s payment is their bid, bidding one’s full value can leave no surplus when that bid wins. In the symmetric private-value equilibrium, bidders shade their bids below their values; the amount of shading depends on the environment.

In a second-price sealed-bid auction, each bidder submits a concealed bid. The highest bidder wins but pays the highest bid submitted by another bidder. With private values, bidding one’s value is weakly dominant: it cannot make a bidder worse off than another bid, regardless of the other bids.

An English auction is an ascending-price auction. The price rises while bidders remain active; a bidder exits when the price exceeds the bidder’s value. Under the private-value benchmark, staying active up to one’s value gives the same outcome as bidding one’s value in a second-price auction: the highest-value bidder wins and pays a price determined by the strongest rival. The equivalence concerns the benchmark incentives and outcomes, not every detail of how an actual auction is run.

A Dutch auction is a descending-price auction. The price starts high and falls until a bidder accepts; that bidder wins and pays the current price. The first bidder to accept effectively chooses a price at which to bid. In the standard private-value setting, this decision has the same strategic trade-off as choosing a bid in a first-price auction: either way, the winner pays the price they chose.

Format	How the winner is determined	Winner’s payment	Private-value benchmark relationship
First-price sealed bid	Highest submitted bid wins	Winner’s own bid	Bid shading in symmetric equilibrium
Second-price sealed bid	Highest submitted bid wins	Highest losing bid	Bidding one’s value is weakly dominant
English	Last active bidder wins	Price when the rival exits, in the standard benchmark	Exit at one’s value parallels second-price bidding
Dutch	First bidder to accept wins	Price accepted by the winner	Strategically equivalent to first-price bidding

Worked Example: Comparing Outcomes Under Four Formats

Suppose three bidders have private values of 18, 42, and 67 dollars. Assume, for illustration, that the first-price bids are strictly increasing in value, as in a symmetric monotone equilibrium, and that the auction has no reserve price. The values alone are not enough to calculate those first-price bids; that requires knowing the value distribution and the equilibrium bidding rule.

Second-price auction. If bidders bid their values, the bidder with value 67 wins and pays the highest rival bid, 42 dollars. Their payoff is $67-42=25$ dollars.

English auction. Under the private-value benchmark, the bidder with value 18 exits first, at 18 dollars, and the bidder with value 42 exits next, at 42 dollars. The bidder with value 67 wins at approximately 42 dollars and earns a payoff of $67-42=25$ dollars, ignoring any tie or increment convention.

First-price auction. The bidder with value 67 wins if bids increase with values, but pays their own equilibrium bid, which is below 67 dollars when the bid is shaded. The payoff is 67 dollars minus that bid. The other values do not identify the bid without the distribution and equilibrium strategy.

Dutch auction. Under the standard strategic equivalence with first-price bidding, the bidder with value 67 is the first to accept in equilibrium. They pay the chosen stopping price, the analogue of their first-price bid. It is not generally correct to set that price equal to their value.

Why Truthful Bidding Works in a Second-Price Auction

Consider one bidder with value $v$, facing a highest rival bid $c$. If the bidder wins, they pay $c$; if they lose, their payoff is zero. The bidder’s own bid determines whether they win, but, except in a tie, it does not determine the price paid. That separation is the reason truthful bidding is weakly dominant.

If $c<v$, winning gives payoff $v-c>0$, while losing gives zero. A bid equal to $v$ wins in this case. If $c>v$, winning gives payoff $v-c<0$, while losing gives zero; a bid equal to $v$ loses. Thus truthful bidding selects the preferred outcome in both cases. A different bid can sometimes produce the same outcome, but cannot improve on truthful bidding for any rival bid. At a tie, the result can depend on the tie-breaking rule, without changing the weak-dominance conclusion.

Worked Example: Checking a Second-Price Bid

A bidder’s value is 60 dollars, and the highest rival bid is 45 dollars. Compare bids of 52, 60, and 75 dollars. Assume the bidder wins when their bid exceeds 45 dollars.

All three bids win, and the winner pays 45 dollars. In each case the bidder’s payoff is $60-45=15$ dollars. Bidding 52 or 75 does not improve the payoff over bidding truthfully.

If instead the rival bid is 65 dollars, a bid of 52 or 60 loses and gives zero. A bid of 75 wins but gives $60-65=-5$ dollars. Truthful bidding avoids that loss. This shows why shading below value is unnecessary in a second-price auction, even though it can be useful in a first-price auction.

Revenue Equivalence: Statement and Justification

The standard revenue-equivalence benchmark compares expected seller payments, not the payment in every realized auction. For the comparison, suppose there are $n$ bidders; values are independent and identically distributed from a common continuous $F$ on $[0,\bar{v}]$; bidders are risk-neutral; and the object is allocated to the bidder with the highest value. Assume also that the lowest-value type has zero expected utility in each format, as with no reserve price and no entry fee. The auctions being compared must implement the same allocation rule and satisfy these incentive and boundary conditions.

Under these assumptions, the symmetric equilibrium of each format has the same expected payment from each bidder conditional on their value. Summing over bidders gives the same expected total seller revenue. In particular, the benchmark predicts equal expected revenue for the standard first-price, second-price, English, and Dutch formats. This does not say that a given seller receives the same amount in each realized auction.

Theorem: In an incentive-compatible auction with allocation probability $q(v)$ for a bidder of value $v$, let $m(v)$ be that bidder’s expected payment conditional on value $v$, and let $U(v)=v q(v)-m(v)$ be expected utility. If $U(0)=0$, then $$ U(v)=\int_0^v q(t)\,dt \qquad\text{and}\qquad m(v)=v q(v)-\int_0^v q(t)\,dt. $$ Consequently, auctions with the same allocation probability at every value and the same lowest-type utility have the same conditional expected payments, and hence the same expected revenue.

The formula follows from incentive compatibility. A type $v$ that reports as though it were type $z$ receives expected utility $v q(z)-m(z)$. Truthful reporting requires this to be no larger than $U(v)$. Applying the same condition to type $z$, who could report $v$, gives $$ U(v)\geq vq(z)-m(z)=U(z)+(v-z)q(z). $$ For $v>z$, the two incentive inequalities imply $$ (v-z)q(z)\leq U(v)-U(z)\leq (v-z)q(v). $$ The allocation probability $q$ is nondecreasing when higher types are more likely to win. Taking increasingly fine partitions of the interval from $0$ to $v$, the lower and upper sums converge to the same integral, so $U(v)-U(0)=\int_0^v q(t)\,dt$. Using $U(0)=0$ and $U(v)=v q(v)-m(v)$ gives the stated payment formula.

In the symmetric independent private-values benchmark without a reserve, any bidder’s probability of winning with value $v$ is $$ q(v)=F(v)^{n-1}, $$ because each of the other $n-1$ independent values must be below $v$. Each standard format allocates the object to the highest value in equilibrium, so it has this same $q(v)$. The theorem then gives the same conditional expected payment for each format. The equivalence between auction formats rests on this common allocation rule and the boundary condition, not on all formats charging the winner in the same way.

Worked Example: Uniform Values and Benchmark Revenue

Suppose two bidders’ values are independent and uniform on $[0,100]$ dollars, with no reserve price. In a second-price auction with truthful bids, the seller’s revenue is the lower of the two values: the winner pays the losing bidder’s value.

Let $M$ denote the lower of the two values. For $0\leq x\leq 100$, $$ P(M>x)=\left(\frac{100-x}{100}\right)^2, $$ because both bidders’ values must exceed $x$. For a nonnegative random variable bounded above by 100, its expectation is the integral of its survival probability. Therefore, $$ E[M]=\int_0^{100}\left(\frac{100-x}{100}\right)^2\,dx =\frac{1}{10000}\int_0^{100}(100-x)^2\,dx =\frac{100}{3} \approx 33.33\text{ dollars}. $$

Thus expected revenue in the second-price auction is approximately 33.33 dollars. The revenue-equivalence benchmark predicts the same expected revenue in the first-price, English, and Dutch formats under the stated assumptions. It does not predict that each format collects 33.33 dollars in every auction; actual revenue varies from one pair of values to another.

When the Benchmark Does Not Apply

Revenue equivalence is a conditional result, not a universal ranking of auction formats. If a comparison changes the number of participants, introduces different reserve prices or entry costs, changes the allocation rule, or violates the private-value and independence assumptions, the theorem’s conditions may fail. A revenue difference then requires analysis of the altered model; it cannot be inferred merely from the auction’s name.

For example, with common values, the object’s value is uncertain to each bidder and bidders’ information is related through that shared value. An English auction can reveal information as bidders exit. That information channel is absent from the independent private-values benchmark, so the simple equivalence between English and second-price outcomes need not hold. Likewise, if a bidder’s value depends on others’ values, a bidder’s own value alone may not determine the preferred action.

Common Mistakes

Claiming that all four formats have the same payment rule. They do not: first-price and Dutch winners pay their own chosen price, while second-price and standard English winners pay a price tied to the strongest rival. Revenue equivalence concerns expected revenue under assumptions.
Assuming the highest-value bidder pays their value in every format. In a second-price auction the winner pays a rival’s bid; in a first-price auction the winner typically shades below their value.
Using truthful bidding as a rule for every auction. Truthful bidding is weakly dominant in the private-value second-price benchmark. It is generally not the equilibrium bidding rule in first-price or Dutch auctions.
Applying revenue equivalence after changing the environment. The result requires a common allocation rule and the same lowest-type utility, alongside the benchmark assumptions. If these conditions change, equal expected revenue is not guaranteed.
Reading equal expected revenue as equal realized revenue. The theorem compares averages over the model’s value draws, not the payment collected in each individual auction.

Key takeaway: Under symmetric, risk-neutral independent private values, the standard formats allocate to the highest-value bidder and have the same lowest-type utility. Those shared features support revenue equivalence. The formats still differ substantially in their bidding rules and in the payments they generate auction by auction.

Check Your Understanding

Use the benchmark assumptions carefully when answering.

In a second-price auction, a bidder’s value is 80 dollars and the highest rival bid is 55 dollars. What is the bidder’s payoff if they win?
Why is a Dutch auction strategically equivalent to a first-price auction in the standard independent private-values setting?
State two assumptions, in addition to independent private values, that are needed for the revenue-equivalence comparison described here.
For two independent uniform values on $[0,100]$, what is the expected second-price revenue, and what does revenue equivalence predict for the other formats?
Does revenue equivalence imply equal payments in every realized auction? Explain the distinction between realized and expected revenue.