# Sequential Game is Extensive form game?

Dynamic, Sequential, Stochastic, Extensive form and Evolutionary games.

I know what all of them mean very roughly.

I want to clarify them.

As I understand Extensive form is a description of a game, not the type of a game. But Some use Extensive form game as a type of game.
So, is Extensive form game sequential game?

Can anybody clarify the definitions of all these terms and relationships/differences?

First of all, you can differentiate between static (essentially all players move simultaneously and only once) and dynamic (essentially non-static) games.

An extensive-form game is essentially a game tree. This form of presentation makes sense when looking at games where players move sequentially. However, you could also represent a simultaneous-move game in a game tree. It just doesn't add any insights over the normal form (matrix).

Game trees are a great tool to present dynamic games such as standard sequential-move games with perfect and complete information (player 1 moves first, player 2 observes the move and reacts, and so on), and they can also be used to present games of incomplete information or games of imperfect information.

Repeated games are dynamic games in which the same static stage game is repeated multiple times. Such games are dynamic but usually not represented in extensive-game form. If the stage game is not always exactly the same, but instead the game to be played in each period depends on a state, we speak of a stochastic game. How the state transitions from one state to the other can depend on the actions played before (and pure chance). Strictly speaking, A repeated game is a stochastic game with only a single state.

I have never heard the term "evolutionary game" without "theory" following. I am not sure if there is a definition of "evolutionary games". However, I see evolutionary game theory as a subfield of game theory that deals with certain forms of dynamic games and also different underlying assumptions. In particular, there is often no "common knowledge of rationality" and often players find their "optimal" strategy by trial and error without really knowing what they are doing.

• Not only is there no common knowledge of rationality in evolutionary game theory, there is usually no rationality at all, just learning/imitation/selection dynamics (replicator dynamics). Jun 6, 2019 at 20:37
• That is true. Feel free to edit that in! However, we should stress that "evolutionary game theory = replicator dynamics" is not true. I am tempted to say that anyone who wants to explain biological/evolutionary phenomena with game theory does evolutionary game theory. Even if it is explaining the peacock's tail with signaling, which requires some rationality. Jun 6, 2019 at 20:47
• Then, do you agree with the definition of Extensive form I made in the question, "Extensive form is a description of a game, not a type of game"? Jun 7, 2019 at 0:29
• Yes, the "extensive form" is a way to describe a game. However, when people talk about "extensive-form games" they almost always mean sequential games. See Fudenberg & Tirole "Game Theory", chapter 3 (section 3.3 for a formal definition which boils down to my "essentially a game tree"). Jun 7, 2019 at 8:56