The name is absent

where the last expression is greater than zero because δ < 1.

However, from equation (9), ъд(к—2)+v_b(к—2) < max(δ^k~²Zj.δ^ ^k+2Z_b).
Since к ≥ J₀ + 1 implies к — 2 ≥ J₀ — 1, and by definition of J₀, δ⁷° Z_a <
δ^m~²°Z_b, we know δ^j°-¹ZA < δ^m-^j°-^γZB, and hence δ^{k 2}Z.₁ < N° ^lZ.₁ <
δ^m-^°^-1ZB < δ^m~^kZb. Hence, since both δ^{k 2}Z.₁ and δ^ ^k '²Z_b are strictly
less than δ^m~^kZ_b, we have a contradiction to v_A(k — 2)+v_b(к 2) δ"' ^kZ_B >
0. Hence, v_A^ — 1) = 0. It immediately follows that v_B(к) = δ^{m k}Z_b.

We have hence showed by induction that for every к ≥ J₀, v_A(^ = 0 and
for every к ≥ J₀ + 1,v_B (к) = δ^m~^kZ_b. An immediate consequence is that

vb (Jo ) = δvB (J0 —1) + δ max(0,VB (J0 + 1)+ va(J0 + 1) — va(J0 — 1) — Vb (J0 —1))

Since v_a(J₀) = 0 implies J(J₀ — 1) — J(J₀ + 1) < 0, the maximand in the
expression is nonnegative and

^{(7) v}B ^(J0)  =  ^δ[^vB ^(J0 ^{+ 1}) ^{+ v}A^(J0 ^{+ 1}) ^{— v}A^(J0 ^{— 1)]}
= δ[δ^m→°⁺¹)ZB — va(J0 — 1)]

Since from (25) v_a(J₀ — 1) = δγW" ²Z_a — v_B(J₀)], we have a system of
two linearly independent equations in two unknowns. These have a unique
solution which is

F ¹ Za — δ^m^²°⁺¹ZB

1 — δ²

6 References

Amann, E. and W. Leininger, 1995, Expected revenue of all-pay and first-
price sealed bid auctions with affiliated signals, Journal of Ecnomics - Zeits-
chrift für Nationalokonomie 61 (3), 273-279

Amann, E. and W. Leininger, 1996, Asymmetric all-pay auctions with
incomplete information: The two-player case, Games and Ecnomic Behavior,
14 (1), 1-18.

Arbatskaya, M., 2003, The exclusion principle for symmetric multi-prize
all-pay auctions with endogenous valuations, Economics Letters, 80 (1), 73-
80.

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