The name is absent

j₀ — 1 = m/2. Accordingly, a battle with positive expected efforts takes
place at m/2 if δɪ⁺¹Z_a = δ^m '(^2^⁺¹⁾Z_b. No expected effort is expended in
this case only if δ~⁺¹Z_a = δ^m '(~⁺¹) Z_b. However, the set of values of g
for which δ~2 ⁺¹Z_a = δ^m '⁽^^⁺¹⁾Z_b holds has a measure of zero. A similar
argument applies for g ∈ (δ²,1), again with A and B and j = 0 and j = m
switching roles. ■

Proposition 5 characterizes conditions on the asymmetry in the valuations
of the prize that are sufficient to make the tug-of-war evolve peacefully if it
starts in the symmetric state j = m/2. A sufficient condition for this to
happen is that j = m is not a tipping state. If tipping states are j₀ and j₀ — 1
with j₀ — 1 > m/2 then the equilibrium process moves from j = m further
away from the tipping states towards the terminal state j = 0. If Z_a < Z_b,
and, hence, tipping states are j₀ and j₀ + 1, with j₀ + 1 < m/2, then the
equilibrium process moves further away from these states and towards the
terminal state j = m.

Note that the number of states is irrelevant for whether the tug-of-war
that starts in state j = m is resolved peacefully or not, provided that m > 2.
Whether the tug-of-war is resolved peacefully or not depends only on the ratio
of the two prizes and the discount factor. For a given continuous distribution
of g, as the discount factor becomes large, the tug-of-war is resolved almost
surely peacefully.

Of course, offsetting the potential gains from the tug-of-war in promoting
the peaceful resolution of resource contests are the potential costs of delay
arising from the multi-stage nature of the conflict. The all-pay auction is re-
solved in a single stage (m = 2) and hence reduces this delay to the minimum
attainable in a non-trivial contest. On the other hand, from Proposition 5
it is apparent that adding more states beyond m = 4 does not increase the
chance of peaceful resolution and only adds potential delay when a peaceful
outcome arises. Moreover, if ^m is a tipping state, for a given draw of Z_a and
Z_b the sum of expected payoffs at this state is simply

δ^m∕2

----—2 ^ma^χ{(^zA ^{— z}b), ^(ZB ^{— Z}A^{)}               (24)}
1 — δ

which is a strictly decreasing function in m. We state this as

Proposition 6 The sum of expected payoffs in the tug-of-war with m > 4
which have a symmetric state m is maximized at m = 4.

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