j0 — 1 = m/2. Accordingly, a battle with positive expected efforts takes
place at m/2 if δɪ+1Za = δm '(^2^+1)Zb. No expected effort is expended in
this case only if δ~+1Za = δm '(~+1) Zb. However, the set of values of g
for which δ~2 +1Za = δm '(^^+1)Zb 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 Za < Zb,
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 Za and
Zb the sum of expected payoffs at this state is simply
δm∕2
----—2 maχ{(zA — zb), (ZB — ZA)} (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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