A.Prove that, given any stage game *W*, *ˆ*δ *such that if *δ *≤ ˆ*δ *and *γ *is a Nash equilibrium of *R**δ *(*W*), then for every history *ht*−1 that has positive *ex ante* probability (according to *γ *), *γ *(*ht*−1) = *α* * for some Nash equilibrium *α* * of *W* (possibly dependent on *ht*−1)*. *Is this conclusion also true if *ht*−1 has zero *ex ante* probability?

**B. **Consider the chain-store game with an infinite horizon, as described in Section 8.4. Compute the minimum discount rate¯*δ *that is consistent with the fact that the (constant) strategies described in (8.10) and (8.11) define a Nash equilibrium of the repeated game. Is the lower bound ¯*δ *affected if those strategies are required to define a subgame-perfect equilibrium?