Lecture 3

# Lecture 3 - We will formally define the games and some solution concepts such as Nash Equilibrium and discuss the assumptions behind these solution

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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 payoffs, but also about what they know about the other players’ knowledge and beliefs about these parameters, etc. In the first half of this course, we will confine 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 defined 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 everyone knows X, and everyone knows that everyone knows X, and everyone knows that everyone knows that everyone knows X, ad infinitum.) In the second half, we will relax this assumption and allow player to have asymmetric information, focusing on informational issues. 1 Representationsofgames The games can be represented in two forms: 1. The normal (strategic) form, 2. The extensive form. 1.1 Normal form Definition 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 iff he tries to maximize the expected value of u i (given his beliefs). 2 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

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## This note was uploaded on 04/05/2012 for the course ECON 406 taught by Professor Sjostrom during the Spring '12 term at Rutgers.

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Lecture 3 - We will formally define the games and some solution concepts such as Nash Equilibrium and discuss the assumptions behind these solution

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