The name is absent



vA(k + 2) + vb(к + 2) Z'Za = Z 1 δ)vB(к + 1) > 0. However, by (9)
vA(k + 2) + vb(к + 2) max(δk+2Za.Z" (k'2Zb). Moreover, by definition of
0,Z" i Za > Z" u" i'Zb, which, since к + 2 J0, implies that δk'2 iZa >
Z
" (k'2''1 Za, or δkZa > δm~(k+2)ZB. This in turn implies that vA(k + 2) +
vb (k + 2) max(δk+2Za.Z" (k'2 Zb) < δkZa.contradicting the claim that
vA(k + 2) + vb(k + 2) δkZa = Z 1 δ')vB(k + 1) > 0 Hence, vb(k + 1) = 0.
which immediately implies that vA(k) = δvA(k — 1) = δkZa.

This induction argument therefore shows for all k ≤ J0 1, vb(k) = 0
and for all k J0 2.vA(k) = δkZa . An immediate consequence is that

VA(J0 1) = δ[(vA(j0 2) + vB (j0 2)) vB (j0)] = δfδ7" 2ZA vB (j0)] (25)

This equation will be used in the continuation to derive values at J0 1 and
j0.

To address equilibrium behavior in states greater than or equal to J0.we
start in state
m. Note that since m is a terminal state vb (m) = Zb and
vA(m) = 0.If J0 = m — 1. then by definition δ^' iZa < δZB and δ^' 2Za >

δ2ZB. Moreover, by Claim 1 vb(m2) = 0 and by (25) vA(m2) = δ^' 2Za
δvB (m 1). Hence,

vA(m 1) = δvA(m) + δ max(0.vA(m 2)+ vb (m 2) vA(m) vb (m))

= 0 + δ max(0.J (m 2) Zb )

δmax(0. niaxZ”' 2ZaZ2Za) Zb)

0

where the final inequality follows from the fact that δ"' 2Za = δj" iZa <
δ
m-j"-1ZB = δm-(m-1-1ZB = Zb. Hence, VA(m 1) = 0. It follows that

vb (m1) = δvB (m2)+δ max(0.ZB vA(m2)vb (m2)) = δ(ZBvA(m2))

Hence,

vA(m 2) = Z" 2ZaδvB (m 1) and

vb (m 1) = δ(ZB VA(m 2))

Solving simultaneously implies

25



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