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(m—2) = 0 and by (25) vA(m—2) = 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 <
δm-j"-1ZB = δm-(m-1-1ZB = Zb. Hence, VA(m — 1) = 0. It follows that
vb (m—1) = δvB (m—2)+δ max(0.ZB —vA(m—2)—vb (m—2)) = δ(ZB—vA(m—2))
Hence,
vA(m — 2) = Z" 2Za — δvB (m — 1) and
vb (m — 1) = δ(ZB — VA(m — 2))
Solving simultaneously implies
25
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