Is there a way to characterize the distinction between simultaneous vs sequential games? I'm trying to describe a situation where players can only take actions without knowledge of other players' actions in a given stage of the game (i.e. where the actions taken in any stage of the game are simultaneous), but where repeated games are permitted. I've read Gibbons (1992) but am nowhere wiser.
In the standard theory of games, simultaneous and sequential games are distinguished by the means of something called "Information Sets". Alluding to the origins of Game Theory (von Neumann and Morgenstern 1944), simultaneous move games can be thought of as special cases of sequential move games. In their landmark work, vN-M give an explicit method to construct simultaneous move version a sequential game. For them, every game is essentially a sequential move game. But that's just history.
From a theoretical standpoint, all that matters is what does a player "know" about the previously played moves while making a decision in a game. This notion of the "information that a player has before making a move" is characterized in sequential move games (and their game trees) via Information sets.
Consider, for example, prisoner's dilemma. But suppose that it is played sequentially, in two different ways:
- Player 1 moves. Player 2 learns Player 1's move. Player 2 moves.
- Player 1 moves. Player 2 does not know about Player 1's move, but knows that Player 1 has moved. Player 2 moves.
In most cases, 2. is essentially a simultaneous move game.
Here's how we conventionally represent a simultaneous move game in extensive form. In the above image, the dotted line joining the two decision nodes of Player 2 signify that while making her decision, Player 2 does not know at which node she actually is. If there were no dotted line, it would mean that Player 2 knows at which node she is, and hence has learned about Player 1's choice. In the pictured version, all that Player 2 knows that if she plays C, she would get either 3 or 1, but that's it. Note that this is exactly similar to the situation we have in a simultaneous move game. A repeated version of this one shot simultaneous move game is then easy to imagine.
In GT jargon, we say that these two nodes which are joined are contained in the same Information set. So, an Information set is the collection of nodes among which a player cannot differentiate.