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# Microeconomic Theory III ECON 6110

Cornell

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This 45 page Class Notes was uploaded by Dr. Shawn Emmerich on Saturday September 26, 2015. The Class Notes belongs to ECON 6110 at Cornell University taught by A. Guerdjikova in Fall. Since its upload, it has received 37 views. For similar materials see /class/214350/econ-6110-cornell-university in Economcs at Cornell University.

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Date Created: 09/26/15

Ani Guerdjikova Fall 2008 Am Guerdykma 9 Game theory relaxes the main assumptions of general equilibrium theory 9 Game theory relaxes the main assumptions of general equilibrium theory a no market power 9 Game theory relaxes the main assumptions of general equilibrium theory a no market power a no strategic interaction 9 Game theory relaxes the main assumptions of general equilibrium theory a no market power a no strategic interaction 0 no external effects 9 Game theory relaxes the main assumptions of general equilibrium theory no market power no strategic interaction no external effects no information deficits coco 9 Game theory relaxes the main assumptions of general equilibrium theory a o o a 9 Main questions no market power no strategic interaction no external effects no information deficits 9 Game theory relaxes the main assumptions of general equilibrium theory a no market power a no strategic interaction 0 no external effects a no information deficits 9 Main questions a Equilibrium What behavior should we expect in a situation of strategic interaction 9 Game theory relaxes the main assumptions of general equilibrium theory a no market power a no strategic interaction 0 no external effects a no information deficits 9 Main questions a Equilibrium What behavior should we expect in a situation of strategic interaction a Outcomes What outcomes can we expect to result in a given situation of strategic interaction 0 Game theory relaxes the main assumptions of general equilibrium theory a no market power a no strategic interaction 0 no external effects a no information deficits 9 Main questions a Equilibrium What behavior should we expect in a situation of strategic interaction a Outcomes What outcomes can we expect to result in a given situation of strategic interaction 0 The form of the game How can we design a game so as to induce certain behavior Game theory can be viewed as a theory of rational interactive behavior 0 guideline for institutional design 0 framework for developing economic theories E V lr mu M m l 0 Game theory analyzes strategic interaction among rational agents 9 Rationality means goaloriented behavior under given constraints 9 A game must therefore specify o the motivation of the playerstheir payoff function a the conditions under which interaction takes place the rules of the game a the information of the players 0 Extensive form games represent all possible moves in the game 0 Extensive form games represent all possible moves in the game 9 Strategic form games represent all possible strategies in the game 0 Extensive form games represent all possible moves in the game 9 Strategic form games represent all possible strategies in the game e Coalitional form games represent all possible outcomes of the game The Enemwe Firm aims The set of players The set of nodes The predecessor function The set of actions N UNgt NUoo for all n E N there is a number k such that 7quot n o A The action function X N o gt A The set ofend nodes TNn NiU 1n The set of decisions nodes D N NTN Player partition NIE is a partition of D N Payoff function r T N gt 1R i Am GuerdJikova Theorem In the game of Chess 0 either White can force a Win 0 or black can force a Win 0 or both black and White can force a draw The Enemive Form ofthe ame De nition Information set Information sets are sets of decision nodes among which a player cannot distinguish Notation Let u be an information set of player i E I and let U u i 1L be the set of information sets of this player A07 1077 im 0407 is the set of actions at node n It must be that for all i E I i ull Q N for all I lL if U is a partition of M iii if n and n 6 ul then An Anl for all I lL iii implies that we can denote the set of actions at an information set ll E U by Au An for all n E u Am Guerdjikova To summarize we can describe a game in extensive form by r I Ma Aux NixE was Altugtueue r ui lforall uiE Uiand aIIiEI then we say that Fis a game with perfect information o If ui gt 1 for some i 6 land some u E U then we say that Fis a game with imperfect information 0 A game Fhas perfect recall if no two nodes in an information set ull of any player i can be traced to two distinct actions at an earlier information set ull of this player 0 If all elements of Fare common knowedgethen we say that F is a game with complete information 0 If not all elements of Fare common knowledge then we say that F is a game with incomplete information Remember 0 A strategy is a complete plan of a player about how to play the game 0 A strategy must specify the behavior of the player even in situations nodes which are not reached a A strategy combination a speci cation of a strategy for every player determines a unique path along the game De nition Pure Strategy A pure strategy of player i assigns to each information set of this player an action which is available at this information set Formally 5 U gt A St 5ui E Au for all u E U l I a A pure strategy can be written as S Si S39 o The set of pure strategies of player i is denoted by 5 o A strategy combination or a strategy pro le 5 51s is a vector consisting of one pure strategy for each player a The set of strategy combinations is 551xx5 0 An i incomplete strategy combination is the vector of strategies for all players different from i 5 51n5i71 5i15l O The set of all i incomplete strategies is 5 51 X gtlt 51 X 5 X X S l I Remarks 9 Each strategy combination determines unique path through the game 0 If the game is finite each strategy combination determines a unique end node that is reached if this strategy combination is played hence there is a function nSHTN which assigns to each strategy combination 5 E S a unique end node n 0 Hence each strategy combination can be assigned a unique payoff for each player i E I De nition Mixed Strategy A mixed strategy is a probability distribution over the pure strategies of a player Formally for S m is a mixed strategy with milf denoting the probability with which player i chooses the pure strategy slk milf satisfy K E m 1 m e 01 for all k 1K k1 l I Notation 0 Let X be a nite set X X1XK K AX ltp1pK 6 1Rquot 2 pk 1 k1 is the set of probability distributions on X o The set of mixed strategies of a player i is M A 5 o A mixed strategy combination speci es a mixed strategy for each player In m1m o The set of mixed strategy combinations is MM1gtltgtltM 0 Each mixed strategy combination implies a probability distribution over the set of pure strategy combinations 5 ms m151 ms 0 The expected payoffof player i E I given a mixed strategy combination m is Pm Episms 56 De nition Behavioral Strategy A behavioral strategy assigns to each information set of a player a probability distribution over the actions available at this information set Formally for U is a behavioral strategy if bfl e A M110 is a probability distribution on A l l Notation o Denote by B quot1 A A quot0 the set of probability distributions over the actions available to player i at information set ul 0 The set of behavioral strategies of player i is BBu gtltgtltBltuL 0 A behavioral strategy combination is a vector consisting of one behavioral strategy for each player bb1b o The set of behavioral strategy combinations is BBlgtltgtltB l I o In nite games a behavioral strategy combination b b1b implies a probability distribution over the set of end nodes 0 The probability that the end node n E TN is reached under the behavioral strategy b is denoted by qb n o The expected payoFF of a player i given the behavioral strategy combination b is Rib Z rnqbn nETN Am GuerdJiKWa lf L 7 l nilwi De nition Perfect Recall A game Fhas perfect recall if no two nodes in an information set ulll of any player i can be traced to two distinct actions at an earlier information set ul of this player if L t l nilwni De nition Perfect Recall A game Fhas perfect recall if no two nodes in an information set ulll of any player i can be traced to two distinct actions at an earlier information set ul of this player Theorem In games with perfect recall there exists for each mixed strategy combination m a behavioral strategy combination b such that both strategies imply the same probability distribution over the end nodes ms qb n for alls E S De nition Game in Strategic Form A game in strategic form is described by a the set of players I 0 the sets of strategies of the playerS Siiel 0 the payoff functions pmE F I Si39g i Piiel39 l I Remarks 6 Each extensive form game can be represented as a game in strategic form 0 There might be several games in extensive form which represent the same strategic form game 0 Games in strategic form can be interpreted as games in which all players choose simultaneously 0 Games in strategic form with two players and finite number of strategies can be represented by a matrix n Strateg Firm Player 2 Am Guerdykma Matching pennies Player 2 Am Guerdyknva Equilibrim n quwllilbf Hum 39 0 O Initially game theorists were searching for a quotsolutionquot of a game and for quotoptimal strategiesquot In a game the optimal strategy of a player in general depends on the strategies chosen by the other players Optimal strategies can be only determined given the expectations ofa player about the behavior of his opponents Equilibrium concepts impose consistency between expected strategy choices and actually played strategies Example Matching pennies Player 2 Am Guerdyknva Equmbrmm De nition Dominant Strategy is a strictly dominant strategy if Pi 554 Pi SHLI for a Si E Si15i7 i and all 571 E 5 l I De nition Equilibrium in Dominant Strategies A strategy combination s 33 is an equilibrium in dominant strategies if is a dominant strategy for each player i E I

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