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Descending order statistics list a sample from greatest to least. If $V_{(1)}\le \cdots \le V_{(n)}$ uses the usual increasing convention, then $V_{[k]}=V_{(n-k+1)}$. This notation distinguishes the next-highest value used by revenue equivalence from the second-smallest sample value.

Assume $n$ risk-neutral players have independent identically distributed atomless nonnegative values and use an increasing symmetric equilibrium in a rank-order contest with unit effort costs. Let prizes satisfy $w_1\ge \cdots \ge w_n=0$, and normalize the lowest type's equilibrium utility to zero. For descending order statistics $V_{[1]}\ge \cdots \ge V_{[n]}$,To derive it, decompose the prize vector into layers: layer $k$ gives $w_k-w_{k+1}$ to each of the best $k$ players. A truthful multi-unit auction implementing that allocation charges each winner $(w_k-w_{k+1})V_{[k+1]}$, hence total $k(w_k-w_{k+1})V_{[k+1]}$. Revenue equivalence transfers its expected payment to the all-pay contest, since the interim allocation and lowest-type utility agree. Summing the layers proves the formula. The symmetry and common-value-distribution assumptions are needed to identify rank allocation from the bid ranking.

The SIPV model concerns a fixed number of bidders competing for one object. Each bidder knows their own private valuation; the value of winning depends on that valuation alone. Valuations are independent random variables with the same atomless distribution $F$, known to everyone, and bidders have identical roles and access to the same strategies. The standard revenue equivalence setting also assumes risk neutrality, quasilinear utility, and noncooperative bidding: expected utility is valuation times winning probability minus expected payment.

The revenue equivalence theorem states that auctions with the same equilibrium allocation probability $p(v)$ for each type and the same expected utility for the lowest type have the same expected payment by each type, and therefore the same expected seller revenue. In particular, if the object goes to the highest valuation and the lowest type has zero expected utility, then all such auctions have the same expected revenue. In a strictly increasing symmetric equilibrium, $p(v)=F(v{)}^{n-1}$.

For a bidder of valuation $v$, the profit as a function of winning probability is $vp-e(p)$. At an interior stationary point $p(v)$,The chain rule then gives the first two identities:Integrating from zero and using integration by parts gives the interim payment identityWith the usual zero-payment normalization $e(p(0))=0$, this becomesStationarity alone does not determine the integration constant: a fixed participation fee would add that constant. The stated formula uses the lowest-type normalization from revenue equivalence.

For the lowest-price auction with independent uniform distributions on $[0,1]$, consider the symmetric increasing equilibrium, so that $p(v)=v^{n-1}$. The zero type can bid zero, lose almost surely, and pay nothing. The interim payment identity therefore yieldsSum this expected payment over the $n$ bidders and average over their uniform distributions:This uses $n\ge 2$, as appropriate for this auction format.

For three bidders, assume the common bid is $Av$ with $A>0$. If a bidder deviates to $Az$, $0\le z\le 1$, they win precisely when both opposing valuations $u,w$ are below $z$. On that event they pay $A\min (u,w)$, so their expected payment and profit areFor $0<v<1$, stationarity at $z=v$ gives $2v^2-A{v}^2=0$, henceThis is a global best response, not merely a stationary point: for $A=2$, $\partial U/\partial z=2z(v-z)$ is positive for $0<z<v$ and negative for $v<z\le 1$. Any bid above $2$ gives the same sure-win payment as $z=1$ and cannot improve the payoff.

The uniform lowest-price auction equilibrium also verifies the revenue calculation constructively for any $n\ge 2$. If opponents bid $(n-1)u$, a deviation to $(n-1)z$ has profitThus $z=v$ is optimal and $b(v)=(n-1)v$ is a symmetric Bayesian Nash equilibrium. The realized price is $(n-1)$ times the minimum valuation; this order statistic has expectation ${\int }_0^1(1-t{)}^{n}dt=1/(n+1)$, giving the same expected revenue.

The resulting revenue comparison for two uniform private values isThus the optimally reserved ascending auction is best, followed by the optimal posted price, followed by the ascending auction without a reserve. These values are in the normalized monetary units used for the valuations.

The revenue equivalence theorem for symmetric independent private values applies to mechanisms with the same allocation probabilities at every type and the same expected utility for the lowest type, under the usual incentive and risk-neutrality assumptions. The reserved and unreserved auctions have different allocation rules. Without a reserve, a bidder of valuation $v$ wins with probability $G(v)=v$. With reserve $r$, that probability is zero for $v<r$ and $v$ for $v>r$. In particular the car is withheld when even the highest valuation is below $r$.

The interim payment identity makes the difference explicit. With lowest-type utility zero, the ordinary auction has expected payment $m(v)=v^2/2$. The reserved auction hasSumming the two bidders' ex ante payments gives $2{\int }_r^1{m}_r(v)dv=1/3+r^2-4r^3/3$, exactly the revenue calculated above. Hence there is no conflict with revenue equivalence: its equal-allocation hypothesis does not hold.

Assume the usual continuous nonnegative valuation distribution, so ties occur only on zero-probability events. Put $t=F(v)$. In a monotone symmetric Bayesian Nash equilibrium, a type $v$ is first with probability $t^{n-1}$ and second with probability $(n-1)(1-t)t^{n-2}$. Its rank-order expected prize allocation in $G_1$ is thereforeThe all-pay effort identity gives $b_1(v)={\int }_{\underline{v}}^{v}s{a}_1^{\prime }(s)ds$. It also verifies equilibrium directly: a type $v$ imitating type $z$ has utility $v{a}_1(z)-b_1(z)$, whose derivative is $(v-z)a_1^{\prime }(z)$, so the true type is a best response.

In the first version of $G_2$, the two contests have expected allocationsThere is no common effort budget, and quasilinear utility makes the two effort choices separable. Since $a_{21}+a_{22}=a_1$, adding their all-pay effort identities yieldsThe equality holds type by type for aggregate effort, rather than only after taking expectations. The within-player correlation of the two efforts does not enter these additive expected payoffs.

For the second version of $G_2$, let $V_{[1]}\ge \cdots \ge V_{[n]}$ denote descending order statistics. The expected effort in a rank-order contest with prize vector $(2,1,0,\dots )$ isEquivalently, decompose the allocation into a unit award to the best player and a unit award to each of the best two players, then use revenue equivalence: the corresponding total auction payments are $V_{[2]}$ and $2V_{[3]}$. Two separate first-place contests with prize values one and two instead generateConsequentlyFor a nondegenerate continuous distribution, the inequality is strict. No regularity of virtual valuations is needed for this comparison. With a uniform distribution on $[0,1]$, the two totals are $(3n-5)/(n+1)$ and $3(n-1)/(n+1)$, giving a difference of $4/(n+1)$.

Use the standard risk-neutral quasilinear utility model: utility is value received minus payment. A bidder may abstain for utility zero, payments to the seller are nonnegative, and the zero-value type has utility zero. The last normalization is necessary for the requested revenue formula; if arbitrary subsidies were allowed, an additive payment constant would remain undetermined. The independent private values model is symmetric, and each bidder has a unit-demand valuation in the two-item part. Participation satisfies individual rationality.

Let $G(\theta )$ be the interim probability of receiving an item and $m(\theta )$ the interim expected payment of a type $\theta$. Write $u(\theta )=\theta G(\theta )-m(\theta )$. By mimicking the Bayesian Nash equilibrium bid of a type $z$, a type $\theta$ could obtain utility $\theta G(z)-m(z)$. Equilibrium therefore gives, for $\theta >z$,The lower inequality uses type $\theta$'s incentive constraint, and the upper uses type $z$'s. Since $G$ is continuous in these auctions, the inequalities give $u^{\prime }(\theta )=G(\theta )$. With $u(0)=0$,This is the interim payment identity underlying revenue equivalence.

For one item, independence and the uniform distribution give $G(\theta )={\theta }^{n-1}$: all other values must be below $\theta$. Hence the expected payment from a bidder conditional on its value isThis is an unconditional-in-winning interim payment, not the amount paid conditional on winning. In the first-price sealed-bid auction, $m(\theta )=b(\theta )G(\theta )$, so the Bayesian Nash equilibrium bid iswith $b(0)=0$. For $n=1$, the seller's revenue is zero and a zero bid suffices under the stipulated allocation rule.

For two items and three unit-demand bidders, a type $\theta$ wins if at most one of the other two values exceeds it. ThereforeThe interim payment identity gives $m(\theta )={\theta }^2-2{\theta }^3/3$. Dividing by the winning probability gives the symmetric Bayesian Nash equilibrium bidTo check that this is an Bayesian Nash equilibrium rather than just a necessary formula, its derivative is $b^{\prime }(\theta )=2(1-\theta )(3-\theta )/[3(2-\theta {)}^2]$, positive for $0\le \theta <1$. A type $\theta$ mimicking a type $z$ receives utility $\theta G(z)-m(z)$, whose derivative in $z$ is $(\theta -z)G^{\prime }(z)$. It is positive before $z=\theta$ and negative after it, so the truthful type-matching bid is globally optimal. Bids above the highest Bayesian Nash equilibrium bid can only increase payment without increasing the winning probability; the usual nonnegative bid range covers the lower boundary.

By symmetry and the law of total expectation, the seller's expected two-item revenue isFor one item and the same three bidders,More generally the single-item revenue with $n$ bidders is $(n-1)/(n+1)$. Allocating a second item lowers competition enough that it adds no expected revenue in this particular three-bidder uniform model.

The final mechanism is a direct revelation mechanism because the message submitted by bidder $i$ is its valuation report, in the same type space $[0,1]$, and allocation and payment are explicit functions of those reports. It is the Clarke pivot mechanism for selecting two unit-demand winners. Truthful reporting is a dominant strategy: bidder $i$'s utility equals its true allocation value plus the other bidders' reported allocation values, minus a term depending only on the others' reports. Reporting its true value makes the efficient allocation maximize the first two terms, while the last is unaffected by its report.

Let the ordered values be ${\theta }_{(1)}\le {\theta }_{(2)}\le {\theta }_{(3)}$. For a winner $i$, the maximum welfare achievable by the other two bidders is their combined value; in the actual allocation only the other winner receives an item. Thus the difference defining its payment is the excluded bidder's value ${\theta }_{(1)}$. For the loser, both other bidders receive items already, so its payment is zero. Consequently both winners pay the lowest valuation, andThis equals the first-price sealed-bid auction revenue above. A conditional check gives the same interim payment identity: the minimum of the other two values has density $2(1-z)$, and bidder $i$ wins exactly when that minimum is below $\theta$. Its conditional expected pivot payment is ${\int }_0^{\theta }2z(1-z)dz={\theta }^2-2{\theta }^3/3=m(\theta )$. The mechanisms have equal expected revenue, even though their realized payments need not coincide.

A reserve price is the seller's minimum acceptable price. In an English auction it changes both the sale rule and the payment rule: the highest valuation must exceed the reserve and the winner pays at least the reserve. Mechanisms with different reserves generally have different allocation probabilities, so revenue equivalence need not equate their expected revenues.

For two independent uniform valuations on $[0,1]$, optimal posted price revenue is $2/(3\sqrt{3})$, attained at $p=1/\sqrt{3}$. An English auction without reserve earns the expected minimum, $1/3$. With reserve $r\in [0,1]$, revenue is $r\mathrm{Pr}(\text{exactly one valuation exceeds }r)$ plus the expected smaller valuation when both exceed $r$, namely $1/3+r^2-4r^3/3$. It is maximized at $r=1/2$, giving $5/12$. Different reserve-dependent allocation probabilities explain why this differs from the no-reserve revenue without violating revenue equivalence.

The independent private values model is symmetric when bidders have identical roles and draw their valuations independently from the same distribution. For revenue equivalence in the usual single-object setting, bidders additionally have risk neutrality and quasilinear utility. Under a strictly increasing symmetric equilibrium bidding rule, a type $v$ wins with probability $F(v{)}^{n-1}$.

// Target: mathematical-optimization.bigb