I have some difficulties in representing the following game in the standard form.
Game: two players game is represented as a game tree (in extensive form), a game tree is a full binary tree, both players make decision consequently, a game tree has $n$ layers and $2^n$ leafs as outcomes.
Task:represent the given game in the standard form, how many rows and columns standard form table has.
Appoach:
Extensive form game is a game tree when players make decision one by one, so the time matters, and players play sequentially.
Normal-form game is a game when players decide simultaneously, time is meaningless.
I am confused by the rules how to converse extensive form game to the normal-form game.
Ideas: if $n$ is even then there $n/2$ layers of the tree when player 1 makes decision and player 2 makes decision. Player 1 start the game, the root gives 2 strategies (binary tree) and all consequent node of the player 1 (nodes at even layers) contribute additional 2 strategies, so in total there are $2^{n/2}$ strategies for player 1. Arithmetic for player 2 is less intuitive and I still don't have it.
I will appreciate any help in constructing normal form game for the given extensive form.
Addendum: one crucial point that I cannot get in convertion the extensive form in normal form is information set.
Addendum 2: Joseph Malkevitch thank you very much for the reference to the great book, after reading I want to make few conclusions. Every node in the tree has it's own information set, just because players make decisions consecutively, therefore the node that currently should make decision should have learned all previous decisions of all nodes on the path.
For example, if we take a game with $n=2$, only two rounds, 4 outcomes, 2 strategies for the player 1 and 4 strategies for the player 2.
When there $n$ rounds, player 1 has $\prod_{i=1,i \space is \space odd}^{n} 2^i$ strategies, player 2 has $\prod_{i=1,i \space is \space even}^{n} 2^i$ strategies in the normal-form game/
I would appreciate is you can approve or disapprove it.
Best Answer
You might find Chapter 3 of David Kreps book Game Theory and Economic Modelling (oxford, 1990) of use here. He refers to extensive and strategic forms of games.