Subgame-perfect Nash equilibrium perfect information
The game is indeed an $8\times 4$-game. A strategy formally specifies what to do at each information set, including those information sets that can never occur under the strategy. In that sense, a strategy is not simply a &quot;plan of action.&quot; This point has been famously made in (see Section 2):
Rubinstein, Ariel. &quot;Comments on the interpretation of game theory.&quot;
Econometrica: Journal of the Econometric Society (1991): 909-924.