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.
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$\begingroup$ Can you give an example (with page number) where Fudenberg and Tirole talk about continuation games? $\endgroup$– Michael GreineckerCommented Jan 10, 2021 at 2:09
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$\begingroup$ It shows up in many places, you can find it in the introduction of Chapter 8, on Equilibrium Refinements for instance. $\endgroup$– David KimCommented Jan 12, 2021 at 20:11
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$\begingroup$ The term is there in scare-quotes, which suggests it is not really a formal term. $\endgroup$– Michael GreineckerCommented Jan 12, 2021 at 20:17
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$\begingroup$ Yes, but I think I have seen this term in many other places as well. Could you elaborate on what the author is trying to deliver here? I have added a comment on the answer posted by Bayesian as well. $\endgroup$– David KimCommented Jan 12, 2021 at 20:22
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1$\begingroup$ I think they just want to behavior to be optimal given what will happen "from then on", even when no new subgame starts. The various refinements formalize how this can be interpreted. $\endgroup$– Michael GreineckerCommented Jan 12, 2021 at 20:28
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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.
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$\begingroup$ If I understood correctly, do you mean a continuation game is just a one of "game" which branches out from the Nature's move? If that is so, can a continuation game contain broken information set which is not from the move of the Nature? (e.g., containing only a part of information set of some player) If the answer is no, can I understand a continuation game as just a 'subgame' which allows for broken information only from the Nature? $\endgroup$ Commented Jan 12, 2021 at 20:19
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$\begingroup$ No, not it does not necessarily branch out from Nature. I just used this as an example because such games don't have subgames at all. A subgame always starts at a singleton information set (decision node). A continuation game starts at an entire information set with beliefs attached to each node it contains. $\endgroup$– BayesianCommented Jan 13, 2021 at 10:03