Alternating offer bargaining game In an infinite-period alternating offer bargaining game if we suppose the discount factor is $\delta=1/2$, and the increment is $0.01$, what will be the subgame perfect Nash Equilibria?

It seems that the "proposer proposes $0.66$ and responder accepts" will be a subgame perfect Nash Equilibrium, but are there any other equilibria? How about the case when the proposer offers $0.67$?
Consider the following strategies. In any period a player makes an offer, they offer a share of $0.33$ to the other player, keeping $0.67$ for themselves. In any period a player accepts or rejects an offer, they accept any offer that gives them at least a share of $0.33$ and rejects anything worse.

Clearly, the outcome of these strategies will lead to the immediate acceptance of the offer. All that remains is to check that these strategies form an equilibrium. We do so by appealing to the one-shot deviation principle.

Consider the perspective of a player deciding to accept or reject. They will not want to deviate from the prescribed strategies only if

$$ 0.33 \ge 0.67 \delta \iff \delta \le \frac{33}{67} $$

However, this violates the assumption that $\delta = \frac{1}{2}$. Thus, the given strategies cannot form an equilibrium.

For characterising other equilibria, see van Damme, Selten and Winter (1990).