Using Proposition 5 we may compare the cases m = 2 and m = 4. When
m = 4 we know that there is a probability Γ(δ2) + (1 — Γ(δ 2)) that the
allocation is peaceful and a probability of Γ(⅛ 2) — Γ(δ2) that the allocation
is violent. We know that in the case m = 2 the allocation is always violent
and the sum of the players’ payoffs is δ(Z(i) — Z(2) ), where Z(1) ≡ max(ZA,Zβ )
and Z(2) ≡ min(ZA,Zβ). Ignoring discounting, the loss due to conflict, Z(2),
can be decomposed into the expected loss due to effort expended, 1 Z(2)[1 +
(Z(2)∕Z(1) )], and the expected loss due to misallocation of the prize, ɪ Z(2) [1 —
(Z(2)∕Z(1))]. The loss due to delay then comes when the factor δ is applied.
In the case where m = 4 the allocation is peaceful when (Z(1) ∕Z(2) ) > δ 2.
In this case, starting from the state y = 2 it takes two periods for the player
with the higher value to win and no effort is expended. Hence, the sum of the
payoffs of the two players in this case is δ2Z(1). The only inefficiency in this
case is due to delay. For realizations of (Za,Zb) satisfying (Z(1)∕Z(2)) > δ 2.
the tug-of-war is more efficient than the all-pay auction if and only if δ2Z(1) >
δ(Z(1) — Z(2)) or, equivalently, (Z(1)∕Z(2)) < (1 — δ ) 1. Since by assumption
we are in the range where the tug-of-war is peaceful, (Z(1)∕Z(2)) > δ 2. Note
that δ 2 > (1 — δ ) 1 if δ < δ+ ≡ x 52 1. In this case, the all-pay auction is
more efficient than the tug-of-war in this range of values of (Z(1)∕Z(2)). For
δ > δ+. δ 2 < (1 — δ) 1 and the tug-of-war is more efficient than the all-pay
auction for values of (Z(1)∕Z(2)) in the interval (δ 2. (1 — δ) 1 ) and the all-pay
auction is more efficient for values of (Z(1)∕Z(2)) in the interval ((1 — δ) 1. ∞).
When 1 ≤ (Z(1)∕Z(2)) < δ 2, in the tug-of-war the state y is a tipping
state and the allocation involves active effort expenditure. The expected
sum of the payoffs in this case can again be compared to those in the all-pay
auction. For m = 4 the expected sum of the payoffs in the tug-of-war can be
calculated from equation (24) and is equal to y-^2j2 (Z(1) — Z(2)). Comparing
this to the expected sum of payoffs in the all-pay auction we find that for
z(1) = z(2) , 1⅛2 (Z(1) — Z(2)) > δ(⅞) — Z(2)) if and only if δ2 + δ — 1 > 0
or δ > δ+ ≡ x 52 1. Therefore, when the realization of (Za,Zb) is such that
the initial state y is a violent state in the tug-of-war, the tug-of-war is more
efficient than the all-pay auction when δ > δ+ and the all-pay auction is
more efficient than the tug-of-war when δ < δ+.
We may summarize these results in the following proposition:
Proposition 7 When δ < δ+ = x52 1, for any realization of (Za,Zb), the
all-pay auction (m = 2) is more efficient than a tug-of-war with m = 4. When
δ > δ+ the tug-of-war with m = 4 is more efficient than the all-pay auction
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