The name is absent



vA(m - 2)


Sm-2 Za - S2Zb


1 - S2


and


SZb -    Za

vB (m - 1) =---------2---- 0

1 - S

where both inequalities follow from J0 = m - 1.

This completes the case for J0 = m - 1. So suppose J0 {3,..., m - 2}
and look at к > J0.

Claim 9 In any subgame perfect equilibrium in stationary Markov strategies,
for all
кJ0, vA (к) = 0 and for all кJ0 + 1, vB(к) = Sm-kZB.

Proof. By an argument similar to that above, vB (m) = Zb and vA(m) =
0
imply J(m) J(m-2), which in turn implies vA(m-1) = 0+S max(0,J(m-
2) - Zb) = 0.

Suppose now that for some к, J0 + 1 к < m, vA(Z) = 0 for all I к and
vb (Z) = Sm-ZB for all Z к + 1. We will now demonstrate that this implies
that v
B(к) = Sm-kZb and vA^ - 1) = 0. Since the supposition holds for
к
= m - 1, this will then prove claim 2 by induction.

So assume that for some к, J0 + 1 к < m, vA(Z) = 0 for all Z к
and
vb(Z) = Sm-ZB for all Z к + 1. Since va (к) = 0, we know that
va^ - 1) + vB(к - 1) - va^ + 1) - vB(к + 1) 0, so that

vB(к) = SvB(к - 1) + S max(θ,vA(fc + 1) + vB(к + 1) - vA^ - 1) - vB(к - 1))
=
S[va^ + 1)+ vb(к + 1) - va(& - 1)]

= S[Sm-(k+1)ZB - va^ - 1)]

Moreover, vA^-1) = SvA(^ + Smax(0,vA^-2)+ vB(к-2)-vA(к)vB(к)).
Since
va(^) = 0 by assumption and vB(к) = Sm-kZB - SvA^ - 1), we have
va^ - 1) = S max(0,vA^ - 2) + vB(к - 2) - Sm-kZb + SvA^ - 1)). Suppose
by way of contradiction that
va (к - 1) 0. Then

va(& - 1) = S[va^ - 2) + vb(к - 2) - Sm-kZb + Sva^ - 1)] 0.

or

va(£ - 1)[S-1 - S] = va (к - 2) + vb (к - 2) - Sm-k Zb0.

26



More intriguing information

1. The name is absent
2. CHANGING PRICES, CHANGING CIGARETTE CONSUMPTION
3. Migrant Business Networks and FDI
4. TOWARDS THE ZERO ACCIDENT GOAL: ASSISTING THE FIRST OFFICER MONITOR AND CHALLENGE CAPTAIN ERRORS
5. The name is absent
6. Les freins culturels à l'adoption des IFRS en Europe : une analyse du cas français
7. Fiscal Sustainability Across Government Tiers
8. The name is absent
9. Examining Variations of Prominent Features in Genre Classification
10. The Effects of Reforming the Chinese Dual-Track Price System