The name is absent

0 ≤ VA(j) ≤ δ^j Z_a                         (7)

0 ≤ v_b (j ) ≤ δ^m- Z_b                       (8)

and

va(j) + vb(j) ≤ max(δ^jZA,δ^m-jZb)                 (9)

We can now prove the following

Proposition 1 Consider a tug-of-war with m > 3. Suppose j_o ∈ {2, ...m—1}
exists such that

δ^j°^-1Za > δ^m-(^70-1)ZB and δ^j° Za < δ^m-j°Zb .           (10)

Then a unique Markov perfect equilibrium exists which is characterized as
follows:

For all interior states j ∈ {j_o — 1,j_o}, 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,      [δ^j°^-1ZA — δ^{m (j}° ¹⁾ Zb] for j = jo — 1,

and 0 for j > jo; payoffs for B are δ^m-Zb for j > jo, (-^ [δ^m-j°Zb—δ^j°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 ∈ {j_o — 1,j_o}, the effort choices are a(j) = b(j) = 0. At j_o — 1
and j_o players choose efforts according to cumulative distribution functions
Fj and G_j for players A and B in states j as follows:

—s ^a ⅛  for

δ λ9°     J

(1-δ²) ^δBA

1 for

a ∈ [0, (¾]
.  ^^δB°_a

^a > (1-<S²)

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