0 ≤ VA(j) ≤ δj Za (7)
0 ≤ vb (j ) ≤ δm- Zb (8)
and
va(j) + vb(j) ≤ max(δjZA,δm-jZb) (9)
We can now prove the following
Proposition 1 Consider a tug-of-war with m > 3. Suppose jo ∈ {2, ...m—1}
exists such that
δj°-1Za > δm-(70-1)ZB and δj° Za < δm-j°Zb . (10)
Then a unique Markov perfect equilibrium exists which is characterized as
follows:
For all interior states j ∈ {jo — 1,jo}, the equilibrium effort choices are
a(j) = b(j) = 0. Only at j0 — 1 and j0 does a battle with a positive probability
of stricty positive effort choices take place. Payoffs for A in the continuation
game at j are δZa for j < jo — 1, [δj°-1ZA — δm (j° 1) Zb] for j = jo — 1,
and 0 for j > jo; payoffs for B are δm-Zb for j > jo, (-^ [δm-j°Zb—δj°Za]
for j = jo and 0 for j ≤ jo — 1.
Proof. We consider existence here and relegate the proof of uniqueness
to the Appendix. We consider the following candidate equilibrium: For all
interior states j ∈ {jo — 1,jo}, the effort choices are a(j) = b(j) = 0. At jo — 1
and jo players choose efforts according to cumulative distribution functions
Fj and Gj for players A and B in states j as follows:
Fj°-1(a) =
—s a ⅛ for
δ λ9° J
(1-δ2) δBA
1 for
a ∈ [0, (¾]
. ^δB°a
a > (1-<S2)
(11)
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