Representation of a game with simultaneous movements

Question

In Game Theory, can you use a tree structure to represent a game with simultaneous movements or you have to use a matrix form? In a sequential game it is logical to use a tree, as every node represents a move so by following the branches you can follow the moves one after another, but I can't see how it would be logical if it could be used in a non-sequential game. In such a game there would be 2 trees, one for player 1 staring the game and one for player 2 starting the game, but both of them would not represent the same game as in reality both players start at the same time.

Answer 1

An extensive-form game can represent simultaneous moves via the use of imperfect information, that is, the fact that two players move at the same time is captured by one of them (either of them) moving first and subsequently the other player moves without learning what the first player to move did. For example, the following bimatrix game:

can be represented by the following imperfect-information extensive-form game:

In this extensive-form game, player 1 moves and then player 2 moves without knowing which move player 1 chose. Note, we could as well have just had player 2 move first in the extensive-form game, and then have player 1 move second, uninformed about player 2's move. Finally, note that if we dissolve the non-singleton information set, then we have a commitment (aka Stackelberg) game, as follows, where now the order of the players moving does matter.

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