14.123 Intro to Game Theory Lecture Summary

14.123 Intro to Game Theory Lecture Summary - Chapter 7...

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Chapter 7 Preliminary Notions in Game Theory The games can be represented in two forms: 1. The normal (strategic) form, 2. The extensive form. I fi rst describe these representations illustrate how one can go from one representation to the other. 7.1 Normal form De fi nition 15 (Normal form) An n-player game is any list G = ( S 1 , . . . , S n ; u 1 , . . . , u n ) , where, for each i N = { 1 , . . . , n } , S i is the set of all strategies that are available to 61
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62 CHAPTER 7. PRELIMINARY NOTIONS IN GAME THEORY player i , and u i : S 1 × . . . × S n R is player i ’s von Neumann-Morgenstern utility function. Notice that a player’s utility depends not only on his own strategy but also on the strategies played by other players. Moreover, each player i tries to maximize the expected value of u i (where the expected values are computed with respect to his own beliefs); in other words, u i is a von Neumann-Morgenstern utility function. We will say that player i is rational i ff he tries to maximize the expected value of u i (given his beliefs). It is also assumed that it is common knowledge that the players are N = { 1 , . . . , n } , that the set of strategies available to each player i is S i , and that each i tries to maximize expected value of u i given his beliefs. When there are only two players, we can represent the (normal form) game by a bimatrix (i.e., by two matrices): 1 \ 2 1,1 4,1 3,2 left right up 0,2 down Here, Player 1 has strategies up and down, and Player 2 has the strategies left and right. In each box the fi rst number is Player 1’s payo ff and the second one is Player 2’s (e.g., u 1 ( up,left ) = 0 , u 2 ( up,left ) = 2 .) I will use the following notational convention throughout the course. Given any list X 1 , . . . , X n of sets with generic elements x 1 , . . . , x n , I will write X = X 1 × · · · × X n and designate x = ( x 1 , . . . , x n ) as the generic element, write X i = Q j = i X j and designate x i = ( x 1 , . . . , x i 1 , x i +1 , . . . , x n ) as the generic 6 element for any i , and write ( x i 0 , x i ) = ( x 1 , . . . , x i 1 , x 0 i , x i +1 , . . . , x n ) . For example, S = S 1 × · · · × S n is the set of strategy pro fi les s = ( s 1 , . . . , s n ) , S i is the set of strategy pro fi les s i = ( s 1 , . . . , s i 1 , s i +1 , . . . , s n ) other than player i , and
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7.2. EXTENSIVE FORM 63 ( s 0 i , s i ) = ( s 1 , . . . , s i 1 , s 0 i , s i +1 , . . . , s n ) is the strategy pro fi le in which i plays s i 0 and the others play s i . 7.2 Extensive form The extensive form contains all the information about a game explicitly, by de fi ning who moves when, what each player knows when he moves, what moves are available to him, and where each move leads to, etc. In contrast, these are implicitly incorporated in strategies in the normal form. (In a way, the normal form is a ‘summary’ representation.) We fi rst introduce some formalisms. De fi nition 16 A tree is a directed graph (i.e. a set of nodes with directed edges that connect some of the nodes) such that 1. there is an initial node, for which there is no incoming edge; 2. for every other node, there is one incoming edge; 3. every node is connected to the initial node by a unique path.
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