On the (rather strange) notion of strategies in repeated games As undergraduates often note, the definition of a strategy appears to be rather strange in the context of repeated games. To illustrate, consider a very simple game in which one player needs to make a decision twice. Every time they need to make the decision, they have 2 possible choices -- call them A or B. Ordinarily, we would think of this person as having $4$ possible (pure) strategies: choose A twice, choose B twice, choose A then B, or choose B then A. According to the standard definition, however, we would say that there are 2 possible histories (A or B); and that a strategy must specify what happens following every possible history. As a result, the player really has $2^3 = 8$ pure strategies! Moreover, each of these strategies specifies what would happen in an impossible situation; namely, if the person were to have made a different choice in the first stage than the one they planned to make.
In case this doesn't seem strange to you, let me provide an example of a pure strategy (I'll put it verbally to convey the full force of the strangeness):

In the first round, I will choose action A. After choosing A, I will then choose A again. However, if I had chosen B in the first round, then I would then have chosen B again in the second round.

My question is why we need to include this last sentence.
Before finally posing my question, I also want to note that this question is not specific to the rather trivial game above. Obviously, one can devise similar examples for more complex multi-player games.
Question: Why does the standard notion of 'strategy' force players to specify what they would do in situations that they know will never arise given their choices earlier in the game? And can we prove standard results (e.g. existence of a Nash equilibrium in extensive games) without making use of this rather un-natural concept of a strategy?
Your proposed notion of what a strategy should be in repeated game can be adapted to all extensive form games has been called a plan of action by Ariel Rubinstein, who discusses this and related issues in the paper

Rubinstein, Ariel. &quot;Comments on the interpretation of game theory.&quot;
Econometrica: Journal of the Econometric Society (1991): 909-924.

Some concepts become harder to define, such as backward induction in games of perfect information. However, one can still do pretty much all of game theory. The book &quot; Rationality in extensive form games&quot; by Andres Perea uses the definition as a plan of action throughout.