In class, we learned that there is a unique subgame perfect Nash equilibrium for the infinite Rubinstein bargaining with alternating offers (and some discount factor). My question is: Why is it not a perfect subgame equilibrium if
- Player 1 always asks for the entire money and rejects every offer
- Player 2 always asks for the entire money and rejects every offer
Then both players would receive $0$ but if only one player changed his strategy, he couldn't do better since the other player would reject every proposal.
For it to be a subgame perfect Nash equilibrium, you need a Nash equilibrium in every subgame. If you look at the subgame right after P1 makes a positive offer, P2 chooses between accepting or rejecting. Rejecting gives him $0$, since both players are playing a "always reject" strategy. Accepting means getting some positive amount. So this subgame is not in Nash equilibrium.
You're correct that no player has an incentive to deviate unilaterally - so your strategy profile is a Nash equilibrium, just not a subgame perfect Nash equilibrium.