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# Advanced Microeconomic Theory ECON 104

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This 9 page Class Notes was uploaded by Dr. Janiya Bernier on Thursday October 22, 2015. The Class Notes belongs to ECON 104 at University of California - Berkeley taught by S. Kariv in Fall. Since its upload, it has received 17 views. For similar materials see /class/226704/econ-104-university-of-california-berkeley in Economcs at University of California - Berkeley.

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Date Created: 10/22/15

Advanced Microeconomics Economics 104 Fall 2008 Introduction Topics 7 Terminology ancl notations functions preferences utility representation and pro les 7 Games and solutions strategic vs extensive games and perfect vs imperfect information 7 Rationality a rational agent and boundedly rational agent 7 Formalities a strategic game of perfect information Terminology and notations OR 17 Sets 7 R is the set of real numbers 7 R is the set of nonnegative real numbers 7 R is set of vectors on n real numbers 7 R3 is set of vectors of n nonnegative real numbers For 3531 6 R 90 2 y 42gt 90239 2 21239 forallz39 xgty4gtxi2yiandacjgtyj for all 239 and some j as gtgt 31 42gt an gt y for all 239 Functions A function f R 7gt R is 7 increasing if gt Whenever an gt y 7 non decreasing if 2 Whenever an gt y and 7 concave if ax 1 7 am 2 am 1 7 WW V1735 E R and Va 6 01 Let X be a set The set of maximizers of a function f X 7gt R is given by arg maxigx fac Preferences i e a binary relation on some set of alternatives A g R From i we derive two other relations on 7 strict performance relation agtb 4gt aibandnotbia 7 indifference relation fl 4 a i b and b i a i is said to be 7 complete if a i b or b i a VaJ E A 7 transitive if aibandbicthenaic Vabc E A Utility representation A function u A gt R is a utility function representing i if for all a b E A ail 4gt ua 2ub i can be presented by a utility function only if it is complete and transitive rational i is said to be 7 continuous preferences cannot jump if for any sequence of pairs ak bkz 1 With 11 i bk7 and 11 gt a and bk A 117 a N 7 strictly quasieconcave if for any I E A the upper counter set a E A a i b is strictly convex These guarantee the existence of continuous wellebehaved utility function representation Pro les Let N be a the set of players MLeN or simply 90239 7 a pro le7 ie7 a collection of values of some variable7 one for each player xjjENi or simply 357i 7 the list of elements of the pro le at acjjEN for all players except 239 Li M 7 a list and and an element xi Which is the pro le LEN Games and solutions 0 1 OR 11713 A game is a model of interactive multi7person decision7making We distinguish between 7 noncooperative and cooperative games 7 the units of analysis are individuals or sub groups 7 strategic normal form games and extensive form games 7 players move simultaneously or precede one another and 7 Gams With perfect and imperfect information 7 players are perfectly or imperfectly informed about characteristics events and actions A solution is a systematic description of outcomes in a family of games 7 Nash equilibrium 7 Subgame perfect equilibrium 7 extensive games With perfect informa7 tion 7 Perfect Bayesian equilibrium 7 games With observable actions 7 Sequential equilibrium and re nements 7 extensive games With im7 perfect 1nformat1on The classic references are Von Neumann and Morgenstern 1944 Luca and Raiffa 1957 and Schelling 1960 see R and OR Rational behavior and bounded rationality O 12 OR 14 16 Consider 7 a A set of actions 7 a C set of consequences 7 a consequence function g A 7gt C and 7 a preference relation i on the set C Given any set B Q A of actions a rational agent chooses an action aquot E B such that for all a E B And When i are speci ed by a utility function U C 7gt R 1 E arg maXaEB Uga With uncertainty about 7 the environment 7 events in the game or 7 actions of other players and their reasoning A rational agent is assumed to have in mind 7 a state space 9 7 a subjective probability measure over 9 and 7 a consequence function g A X Z 7gt C A rational agent is an expected UNM utility ugaw maximizer Formalities O 2 OR 21 A strategic game of perfect information a nite set N of players7 and for each player 239 E N 7 a noneempty set Ai of actions 7 a preference relation iion the set A XJENAJ of possible outcomes We Will denote a strategic game by N7 Ai7 539 or by N7 Ai7 1 When iican be represented by a utility function ui A A R A twoplayer nite strategic game can be described conveniently in a bi matrix For example7 consider the 2 X 2 game

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