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 к ≥ J0, vA (к) = 0 and for all к ≥ J0 + 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 vB(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 Zb > 0.
26
More intriguing information
1. The name is absent2. Surveying the welfare state: challenges, policy development and causes of resilience
3. Business Networks and Performance: A Spatial Approach
4. An Efficient Secure Multimodal Biometric Fusion Using Palmprint and Face Image
5. The name is absent
6. AN EMPIRICAL INVESTIGATION OF THE PRODUCTION EFFECTS OF ADOPTING GM SEED TECHNOLOGY: THE CASE OF FARMERS IN ARGENTINA
7. Placentophagia in Nonpregnant Nulliparous Mice: A Genetic Investigation1
8. Automatic Dream Sentiment Analysis
9. THE UNCERTAIN FUTURE OF THE MEXICAN MARKET FOR U.S. COTTON: IMPACT OF THE ELIMINATION OF TEXTILE AND CLOTHING QUOTAS
10. The name is absent