At j = j0 the expected effort of each player equals Z∕2, and each wins this
battle with a probability of 1/2 and, in this case, eventually wins the overall
contest j0 — 1 or (m — j0) — 1 periods later, respectively, without spending
any further effort. This determines the continuation values in the candidate
equilibrium. These continuation values are
va = vb = O
if j = j0
if j < j0
if j > j0.
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va = δj Za and vb = 0
va = 0 and vb = δm~j Zb
It remains to show that the candidate equilibrium describes mutually
optimal replies. Consider one-stage deviations for A and B for some state
j < j0. A choice α0(j ) > 0 will not change the equilibrium outcome in the
battle in this period and hence will simply reduce A’s payoff by α0(j). A
choice b0(j) > 0 will make B win. If j < j0 — 1, following the candidate
equilibrium A will simply win a series of battles until final victory occurs.
Hence, b0(j) > 0 reduces B’s payoff by this same amount b0(0) of effort. If
j = j0 — 1, B’s battle victory will lead to j = j0, and candidate equilibrium
play from here on will yield a payoff equal to zero to B. Accordingly, the
deviation b0(j) > 0 yields a reduction of B’s payoff by this same amount.
Consider one-stage deviations for A and B in some state j > j0. The same
line of argument applies, with A and B switching roles. Finally, consider
one-stage deviations for A and B at j = j0. Any such deviation for A must
be a choice α0(j) > Z. Compared to α(j) = Z, this choice makes A win with
the same probability 1, but yields a reduction in A’s payoff by α0(j) — Z,
compared to α(j) = Z. The same argument applies for deviations by B at
this state. ■
The intuition for Proposition 3 is as follows. The two contestants enter
into a very strong fight whenever they reach the state j = j0. In this state
they are perfectly symmetric and they anticipate that the winner of the battle
in this state moves straight to final victory. In the battle that takes place in
this case, they dissipate the maximum feasible rent from winning this battle.
This maximum rent is what they get if they can move from there through
a series of uncontested battles to final victory. Once one of the contestants,
say A , has acquired some advantage in the sense that the contest has moved
to j < j0, the only way for B to reach victory passes through the state with
j = j0. As all rent is dissipated in the contest that takes place there, B
is simply not willing to spend any effort to move the contest to that state.
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