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How should I call a "part" of an extensive form game which is neither (i) a subgame, nor (ii) a stage in a repeated game?

For instance, consider the class of games constructed by "stacking" 2 Stackelberg games, but where the identity of the player who choses first in the second part may depend on the actions in the first part.

I believe these kinds of games are not strictly speaking repeated games (right?). If that's correct, I cannot really speak of the first "part" of the game as the first "stage" (or can I?).

Do you know of any standard terminology to talk about such "parts" of games which are not strictly speaking stages of repeated games?

Edit: if you can substantiate your answer with examples of papers where the terminology you propose is used, that would be even better.

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  • $\begingroup$ The game at a particular node? $\endgroup$ – ChinG Feb 16 '16 at 17:18
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    $\begingroup$ I would call it a stage game, as it is a stage of a dynamic game, where the order of moves, among other things, is one of the state variable. You may even index the stages by $t$ and denote the game at each stage as $\Gamma_t$, for example. $\endgroup$ – Herr K. Feb 16 '16 at 17:55
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    $\begingroup$ I don't think the use of stage game is quite correct here. More generally, anything of the form *** - game cannot be correct, since if it is not a subgame, then it is not a game at all. $\endgroup$ – HRSE Feb 17 '16 at 5:16
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A "part" of an extensive form game that is not a proper subgame because it does not start at a single node but an entire information set would be called "continuation game". This terminology is fairly standard (Perfect Bayesian Equilibrium).

However, I think what you are after is a stochastic game which consists of several states. Each state corresponds to a different game. In your example there would be two states: One state for each player $i\in\{1,2\}$ being the Stackelberg leader. Then you also need a transition process that maps the period-1 state and the period-1 action profile into a probability distribution over period-2 states. You then find equilibria for all state games in period 2 and proceed to solve for equilibria in period 1 with the corresponding (probability-weighted sum of) continuation payoffs.

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