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lecture3_6-Auction

# lecture3_6-Auction - 14.12 Game Theory Lecture Notes...

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14.12 Game Theory Lecture Notes Lectures 3-6 Muhamet Yildiz We will formally de f ne the games and some solution concepts, such as Nash Equi- librium, and discuss the assumptions behind these solution concepts. In order to analyze a game, we need to know who the players are, which actions are available to them, how much each player values each outcome, what each player knows. Notice that we need to specify not only what each player knows about external parameters, such as the payo f s, but also about what they know about the other players’ knowledge and beliefs about these parameters, etc. In the f rst half of this course, we will con f ne ourselves to the games of complete information, where everything that is known by a player is common knowledge. 1 (We say that X is common knowledge if These notes are somewhat incomplete – they do not include some of the topics covered in the class. Some parts of these notes are based on the notes by Professor Daron Acemoglu, who taught this course before. 1 Knowledge is de f ned as an operator on the propositions satisfying the following properties: 1. if I know X, X must be true; 2. if I know X, I know that I know X; 3. if I don’t know X, I know that I don’t know X; 4. if I know something, I know all its logical implications. 1

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everyone knows X, and everyone knows that everyone knows X, and everyone knows that everyone knows that everyone knows X, ad in f nitum.) In the second half, we will relax this assumption and allow players to have asymmetric information, focusing on informational issues. 1R e p r e s e n t a t i o n s o f g a m e s The games can be represented in two forms: 1. The normal (strategic) form, 2. The extensive form. 1.1 Normal form De f nition 1 (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 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 f hetr iestomax im izetheexpectedva lueo f u i (given his beliefs). 2 It is also assumed that it is common knowledge that the players are N = { 1 } , that the set of strategies available to each player i is S i ,andthateach 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 \ 2l e f t r i g h t up 0,2 1,1 down 4,1 3,2 2 We have also made another very strong “rationality” assumption in de f ning knowledge, by assuming that, if I know something, then I know all its logical consequences.
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lecture3_6-Auction - 14.12 Game Theory Lecture Notes...

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