The name is absent

Now we confirm that the effort choices in the candidate equilibrium are
indeed mutually optimal replies. For interior states j < m — 1, a deviation
b⁰(j) > 0 makes B win the battle, instead of A. It leads to j + 1, instead
of j — 1, but V в (j + 1) = v_b (j — 1) = 0 . Hence, this deviation reduces B’s
payoff by b⁰(j) compared to b(j) = 0. For A, for j < m — 1, contestant A
reaches j = 0 along the shortest possible series of battle victories and does
not spend any effort. Any positive effort can therefore only decrease A’s
payoff. For j = m — 1, the battle either leads to j = m where B finally
wins the prize, or to j = m — 2. The values the players attribute to reaching
these states are Vj(m) = 0, v_b(m) = Z_b, and v_A(m — 2) = δ^m^²ZA and
v_b (m — 2) = 0. Using the results in Hillman and Riley (1989) and Baye,
Kovenock and deVries (1996) on a complete information all-pay auction with
prizes δ'7r' ²Z_a — 0] = δ"'"'' ⁱZ_a for A and 5[Z_b — 0] = 5Z_b for B, it is
confirmed that (20) and (21) describe the unique equilibrium cumulative
distribution functions of effort for this all-pay auction. ■

Proposition 2 shows that a very strong player has a positive continuation
value regardless of the interior state in which the tug-of-war starts and wins
with probability 1 without expending effort for every interior state except
j = m — 1.

So far we have ruled out the case of equality of continuation values at
interior states, and we turn to this case now which exhausts the set of possible
cases.

Proposition 3 The tug-of-war with δ^ioZ_a = δ(^{m 7}'°) Z_b ≡ Z for some j₀ ∈
{2,...(m — 1)} has a unique subgame perfect equilibrium in which players
spend a(j) = b(j) = 0 in all interior states j = j₀. They choose efforts
a(j) and b(j) at j = j₀ from the same uniform distribution on the range
[0,Z]. Payoffs are Vj(j) = δ⁷Za  and Vb(j) = 0 for j < jo, Vj(j) = 0 and

vb(j) = δ^m~⁷Zb for j > jo and Vj(j) = Vb(j) = 0 for j = jo.

Proof. We again construct an equilibrium to demonstrate existence.
Uniqueness follows from arguments similar to those appearing in the Ap-
pendix.

In the candidate equilibrium each contestant expends zero effort at any
state j = j₀ and expends effort at j = j₀ according to a draw from the
distribution

^fw = ½ 1 far '∈∈^Z                  (22)

1 foi ʃ > z .

17

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