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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.

An equilibrium outcome of a mechanism can be reproduced by asking for types and then sending the messages prescribed by the original equilibrium strategies. Truthful reports are then a Bayesian Nash equilibrium: a profitable false report would induce a profitable original deviation. This reduces optimization over indirect mechanisms to direct revelation mechanisms with Bayesian incentive compatibility, under the same information and participation assumptions.

With $n$ independent uniform types, $m<n$ equal prizes of scale $w(m/n)$ and unit effort costs, a symmetric Bayesian Nash equilibrium has total expected effort $R_m=m(n-m)w(m/n)/(n+1)$. The all-pay effort identity yields $b(v)=w(m/n){\int }_0^{v}t{q}_m^{\prime }(t)dt$. The allocation derivative is a Beta distribution density with parameters $n-m,m$, so integration gives the formula.