LectureFeb10_MATH4321_12S

# LectureFeb10_MATH4321_12S - Goal: Give a complete...

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Goal: Give a complete mathematical description of the Goal: Give a complete mathematical description of the game game Example: Example: 1. 1. Pick up bricks: A pile of 5 bricks has been stacked on the Pick up bricks: A pile of 5 bricks has been stacked on the ground. Two players take turn to pick up either one or two ground. Two players take turn to pick up either one or two bricks from the pile. The player who picks up the last brick bricks from the pile. The player who picks up the last brick loses \$1 to the other player. loses \$1 to the other player.

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The Extensive Form of a Game The extensive form, is built on the basic notions of position and move.

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In the extensive form, games are sequential , interactive processes which moves from one position to another in response to the wills of the players or the whims of chance . It will justify and give meaning to more abstract concepts such as strategy. Three new concepts make their appearance in the extensive form of a game: the game tree , chance moves , and information sets .
The Game Tree. The extensive form of a game is modeled using a directed Graph that we have seen in combinatorial games. A directed graph is a pair (T,F) where T is a nonempty set of vertices and F is a function that gives for each x T a subset, F(x) of T called the followers of x. The vertices represent positions of the game. The followers, F(x), of a position, x, are those positions that can be reached from x in one move.

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A path from a vertex t 0 to a vertex t 1 is a sequence, x 0 , x 1 , . . . , x n , of vertices such that x 0 = t 0 , x 0 = t 1 and x j is a follower of x j-1 for i = 1, . . . , n. For the extensive form of a game, we deal with a particular type of directed graph called a tree. The path from a vertex to one of its followers is called an edge and we may think of edges to represent moves from a given position.
Definition. A tree is a directed graph, (T,F) in which there is a special vertex, t 0 , called the root or the initial vertex, such that for every other vertex t T, there is a unique path beginning at t 0 and ending at t. The existence and uniqueness of the path implies that a tree is connected, has a unique initial vertex, and has no circuits or loops.

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: When a tree is used to define a game, the vertices are interpreted as positions and the edges as moves . Each nonterminal position is assigned either to the player
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## This note was uploaded on 03/13/2012 for the course MATH 4321 taught by Professor Cheng during the Spring '12 term at HKUST.

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LectureFeb10_MATH4321_12S - Goal: Give a complete...

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