Auction Design without Commitment



Second, by (11), mi(θ-ik) ≥ θk for all mi(p,θ-i) = θk for all (θk,θ-ə
Yisupp(p) such that ζθk-1 -i¢ Yi. By VETO-IC and this property,

P    [θik - mi(θik-i)]p(θik-i)                           (14)

(θik-i)Yi

P    max{θik - mi (θik-1 , θ-i), 0}p(θik , θ-i)

(θik-i)Yi

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

(θik-1-i)Yi

By the induction hypothesis (9),

θik - mi (θik-1 -i)=θik - mE (i, θ-i ,p), for all (θik-1 , θ-i) Yi supp(p).   (15)

By (15) and (12),

P         max{θik - mi(θik-1, θ-i), 0}p(θik, θ-i)

(θik-1-i)Yisupp(p)

= P        [θik - mE(i, θ-i,p)]p(θik-i)

(θik-1-i)Yisupp(p)

= P     [θik - mE(i, θ-i, p)]p(θik , θ-i).

(θik-1-i)Yi

Thus,

P     max{θik - mi (θik-1 , θ-i), 0}p(θik , θ-i)

(θik-1-i)Yi

P    [θik -mE(p,θ-i)]p(θik-i).

(θik-1-i)Yi

This together with (14) imply that

P    [θik - mi(θik-i)]p(θik-i) ≥ P     [θik - mE(i, θ-i,p)]p(θik-i).

(θik-i)Yi                                                    (θik-1-i)Yi

Thus, by (13),

P    [θik - mi(θik-i)]p(θik-i) ≥ P    [θik-mE(i,θ-i,p)]p(θik-i),

(θik-i)Yi                                                    (θik-i)Yi

and hence

P    mi(θik-i)p(θik-i) ≤ P    mE(i, θ-i,p)p(θik-i).     (16)

(θik-i)Yi                                         (θik-i)Yi

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

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