I am in my first course in grad level game theory. While I was reading through Fudenberg and Tirole's Game Theory, I constantly come into contact with the word 'continuation' to describe some games and strategies but I was never able to find where the authors exactly define what they are referring to. Some sources say continuation game is a game starting at a given information set and including all edges and vertices/nodes until the terminal nodes. But isn't a subgame defined as a part of game tree which starts at a single node and contains all the succeeding nodes and wholly containing the information sets associated with them? Any help in understanding the term would be highly appreciated.
Consider a game with private information such as a privately known willingness-to-pay or any other type. We usually model this as a game in which at first "Nature" draws the type and then players make their moves. Such games do not have proper subgames because a proper subgame never splits up an information set and Nature's first move connects the entire game that follows into a subgame. That is why in such games we do not look for subgame perfect equilibrium, but a perfect Bayesian or sequential equilibrium.
A continuation game is kind of like a subgame that can start at an information set larger that just one decision node and it also assigns beliefs (probabilities) about at which decision node the player is.