The name is absent



0 VA(j) δj Za                         (7)

0 vb (j ) δm- Zb                       (8)

and

va(j) + vb(j) max(δjZAm-jZb)                 (9)

We can now prove the following

Proposition 1 Consider a tug-of-war with m > 3. Suppose jo {2, ...m1}
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 e
ffort choices take place. Payoffs for A in the continuation
game at j are δ
Za for j < jo 1,      [δj°-1ZA δm (j° 1) Zb] for j = jo1,

and 0 for j jo; payoffs for B are δm-Zb for j > jo, (-^ [δm-j°Zbδj°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  for

δ λ9°     J

(1-δ2) δBA

1 for

a [0, (¾]
.  ^δB°a

a > (1-<S2)

(11)


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