Auction Design without Commitment

Second, by (11), mi(θ_-i,θ^k) ≥ θ^k for all m_i(p,θ_-i) = θ^k for all (θ^k,θ-ə ∈
Yi∩supp(p) such that ζθk^-1 ,θ_-i¢ ∈ Yi. By VETO-IC and this property,

^P    [θi^k - mi(θi^k,θ-i)]p(θi^k,θ-i)                           (14)

(θi^k,θ-i)∈Yi

≥ ^P    max{θ_i^k - mi (θ_i^k-1 , θ-i), 0}p(θ_i^k , θ-i)

(θ_i^k,θ-i)∈Yi

= ^P     max{θ_i^k - mi(θ_i^k-1, θ-i), 0}p(θ_i^k, θ-i).

(θ_i^k-1,θ-i)∈Yi

By the induction hypothesis (9),

θ_i^k - mi (θ_i^k-1 ,θ-i)=θ_i^k - m^E (i, θ-i ,p), for all (θ_i^k-1 , θ-i) ∈ Yi ∩ supp(p).   (15)

By (15) and (12),

^P         max{θ_i^k - mi(θ_i^k-1, θ-i), 0}p(θ_i^k, θ-i)

(θ_i^k-1,θ-i)∈Yi∩supp(p)

= ^P        [θi^k - m^E(i, θ-i,p)]p(θi^k,θ-i)

(θ_i^k-1,θ-i)∈Yi∩supp(p)

= ^P     [θ_i^k - m^E(i, θ-i, p)]p(θ_i^k , θ-i).

(θ_i^k-1,θ-i)∈Yi

Thus,

^P     max{θ_i^k - mi (θ_i^k-1 , θ-i), 0}p(θ_i^k , θ-i)

(θ_i^k-1,θ-i)∈Yi

≥ ^P    [θi^k -m^E(p,θ-i)]p(θi^k,θ-i).

(θ_i^k-1,θ-i)∈Yi

This together with (14) imply that

^P    [θi^k - mi(θi^k,θ-i)]p(θi^k,θ-i) ≥ ^P     [θi^k - m^E(i, θ-i,p)]p(θi^k,θ-i).

(θi^k,θ-i)∈Yi                                                    (θ_i^k-1,θ-i)∈Yi

Thus, by (13),

^P    [θi^k - mi(θi^k,θ-i)]p(θi^k,θ-i) ≥ ^P    [θi^k-m^E(i,θ-i,p)]p(θi^k,θ-i),

(θ_i^k,θ-i)∈Yi                                                    (θ_i^k,θ-i)∈Yi

and hence

^P    mi(θi^k,θ-i)p(θi^k,θ-i) ≤ ^P    m^E(i, θ-i,p)p(θi^k,θ-i).     (16)

(θi^k,θ-i)∈Yi                                         (θi^k,θ-i)∈Yi

Finally, (16) implies that (11) holds as equality, as desired.

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