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,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Dec,2006,Fudan University,Game Theory-Lecture 1,*,Dec,2019,Fudan University,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Static Games of Complete Information-Lecture 1,Nash Equilibrium-Pure Strategy,Yongqin Wang,yongqinwangfudan.edu,CCES,Fudan University,Dec,2006,Fudan University,1,Game Theory-Lecture 1,Outline of Static Games of Complete Information,Introduction to games,Normal-form(or strategic-form)representation,Iterated elimination of strictly dominated strategies,Nash equilibrium,Applications of Nash equilibrium,Mixed strategy Nash equilibrium,Dec,2006,Fudan University,2,Game Theory-Lecture 1,Agenda,What is game theory,Examples,Prisoners dilemma,The battle of the sexes,Matching pennies,Static(or simultaneous-move)games of complete information,Normal-form or strategic-form representation,Dec,2006,Fudan University,3,Game Theory-Lecture 1,What is game theory?,We focus on games where:,There are at least two rational players,Each player has more than one choices,The outcome depends on the strategies chosen by all players;there is strategic interaction,Example:Six people go to a restaurant.,Each person pays his/her own meal,a simple decision problem,Before the meal,every person agrees to split the bill evenly among them,a game,Dec,2006,Fudan University,4,Game Theory-Lecture 1,What is game theory?,Game theory,is a formal way to analyze strategic interaction among a group of rational players(or agents),Game theory has applications,Economics,Politics,Sociology,Law,etc.,Dec,2006,Fudan University,5,Game Theory-Lecture 1,Classic Example:Prisoners Dilemma,Two suspects,held in separate cells,are charged with a major crime.However,there is not enough evidence.,Both suspects are told the following policy:,If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail.,If both confess then both will be sentenced to jail for six months.,If one confesses but the other does not,then the confessor will be released but the other will be sentenced to jail for nine months.,-1,-1,-9,0,0,-9,-6,-6,Prisoner 1,Prisoner 2,Confess,Mum,Confess,Mum,Dec,2006,Fudan University,6,Game Theory-Lecture 1,Example:The battle of the sexes,At the,separate,workplaces,Chris and Pat must choose to attend either an opera or a prize fight in the evening.,Both Chris and Pat know the following:,Both would like to spend the evening together.,But Chris prefers the opera.,Pat prefers the prize fight.,2,1,0,0,0,0,1,2,Chris,Pat,Prize Fight,Opera,Prize Fight,Opera,Dec,2006,Fudan University,7,Game Theory-Lecture 1,Example:Matching pennies,Each of the two players has a penny.,Two players must,simultaneously,choose whether to show the Head or the Tail.,Both players know the following rules:,If two pennies match(both heads or both tails)then player 2 wins player 1s penny.,Otherwise,player 1 wins player 2s penny.,-1,1,1,-1,1,-1,-1,1,Player 1,Player 2,Tail,Head,Tail,Head,Dec,2006,Fudan University,8,Game Theory-Lecture 1,Static(or simultaneous-move)games of complete information,A set of players(at least two players),For each player,a set of strategies/actions,Payoffs received by each player for the combinations of the strategies,or for each player,preferences over the combinations of the strategies,Player 1,Player 2,.Player,n,S,1,S,2,.,S,n,u,i,(,s,1,s,2,.,s,n,),for all,s,1,S,1,s,2,S,2,.,s,n,S,n,.,A static(or simultaneous-move)game consists of:,Dec,2006,Fudan University,9,Game Theory-Lecture 1,Static(or simultaneous-move)games of complete information,Simultaneous-move,Each player chooses his/her strategy without knowledge of others choices.,Complete information(on games structure),Each players strategies and payoff function are common knowledge among all the players.,Assumptions on the players,Rationality,Players aim to maximize their payoffs,Players are perfect calculators,Each player knows that other players are rational,Dec,2006,Fudan University,10,Game Theory-Lecture 1,Static(or simultaneous-move)games of complete information,The players cooperate?,No.Only non-cooperative games,Methodological individualism,The timing,Each player,i,chooses his/her strategy,s,i,without knowledge of others choices.,Then each player,i,receives his/her payoff,u,i,(,s,1,s,2,.,s,n,),.,The game ends.,Dec,2006,Fudan University,11,Game Theory-Lecture 1,Definition:normal-form or strategic-form representation,The,normal-form,(,or strategic-form,),representation,of a game,G,specifies:,A finite set of players 1,2,.,n,players strategy spaces,S,1,S,2,.S,n,and,their payoff functions,u,1,u,2,.u,n,where,u,i,:,S,1,S,2,.S,n,R,.,Dec,2006,Fudan University,12,Game Theory-Lecture 1,Normal-form representation:2-player game,Bi-matrix representation,2 players:Player 1 and Player 2,Each player has a finite number of strategies,Example:,S,1,=,s,11,s,12,s,13,S,2,=,s,21,s,22,Player 2,s,21,s,22,Player 1,s,11,u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),s,13,u,1,(,s,13,s,21,),u,2,(,s,13,s,21,),u,1,(,s,13,s,22,),u,2,(,s,13,s,22,),Dec,2006,Fudan University,13,Game Theory-Lecture 1,Classic example:Prisoners Dilemma:normal-form representation,Set of players:,Prisoner 1,Prisoner 2,Sets of strategies:,S,1,=,S,2,=,M,um,C,onfess,Payoff functions:,u,1,(M,M,)=-1,u,1,(M,C,)=-9,u,1,(C,M,)=0,u,1,(C,C,)=-6;,u,2,(,M,M)=-1,u,2,(,M,C)=0,u,2,(,C,M)=-9,u,2,(,C,C)=-6,-1,-1,-9,0,0,-9,-6,-6,Prisoner 1,Prisoner 2,Confess,Mum,Confess,Mum,Players,Strategies,Payoffs,Dec,2006,Fudan University,14,Game Theory-Lecture 1,Example:The battle of the sexes,Normal(or strategic)form representation:,Set of players:,Chris,Pat,(=Player 1,Player 2),Sets of strategies:,S,1,=,S,2,=,O,pera,Prize,F,ight,Payoff functions:,u,1,(O,O,)=2,u,1,(O,F,)=0,u,1,(F,O,)=0,u,1,(F,O,)=1,;,u,2,(,O,O)=1,u,2,(,O,F)=0,u,2,(,F,O)=0,u,2,(,F,F)=2,2,1,0,0,0,0,1,2,Chris,Pat,Prize Fight,Opera,Prize Fight,Opera,Dec,2006,Fudan University,15,Game Theory-Lecture 1,Example:Matching pennies,Normal(or strategic)form representation:,Set of players:,Player 1,Player 2,Sets of strategies:,S,1,=,S,2,=,H,ead,T,ail,Payoff functions:,u,1,(H,H,)=-1,u,1,(H,T,)=1,u,1,(T,H,)=1,u,1,(H,T,)=-1,;,u,2,(,H,H)=1,u,2,(,H,T)=-1,u,2,(,T,H)=-1,u,2,(,T,T)=1,-1,1,1,-1,1,-1,-1,1,Player 1,Player 2,Tail,Head,Tail,Head,Dec,2006,Fudan University,16,Game Theory-Lecture 1,Example:Tourists&Natives,Only two bars(bar 1,bar 2)in a city,Can charge price of$2,$4,or$5,6000 tourists pick a bar randomly,4000 natives select the lowest price bar,Example 1:Both charge$2,each gets 5,000 customers and$10,000,Example 2:Bar 1 charges$4,Bar 2 charges$5,Bar 1 gets 3000+4000=7,000 customers and$28,000,Bar 2 gets 3000 customers and$15,000,Dec,2006,Fudan University,17,Game Theory-Lecture 1,Example:Cournot model of duopoly,A product is produced by only two firms:firm 1 and firm 2.The quantities are denoted by,q,1,and,q,2,respectively.Each firm chooses the quantity without knowing the other firm has chosen.,The market price is,P,(,Q,)=,a-Q,where,Q=q,1,+q,2,.,The cost to firm,i,of producing quantity,q,i,is,C,i,(,q,i,)=,cq,i,.,The normal-form representation:,Set of players:,Firm 1,Firm 2,Sets of strategies:,S,1,=,0,+),S,2,=,0,+),Payoff functions:,u,1,(,q,1,q,2,)=,q,1,(,a-,(,q,1,+q,2,)-,c,),u,2,(,q,1,q,2,)=,q,2,(,a-,(,q,1,+q,2,)-,c,),Dec,2006,Fudan University,18,Game Theory-Lecture 1,One More Example,Each of,n,players selects a number between 0 and 100 simultaneously.Let,x,i,denote the number selected by player,i.,Let,y,denote the average of these numbers,Player,i,s payoff=,x,i,3,y,/5,The normal-form representation:,Dec,2006,Fudan University,19,Game Theory-Lecture 1,Solving Prisoners Dilemma,Confess always does better whatever the other player chooses,Dominated strategy,There exists another strategy which always does better regardless of other players choices,-1,-1,-9,0,0,-9,-6,-6,Prisoner 1,Prisoner 2,Confess,Mum,Confess,Mum,Players,Strategies,Payoffs,Dec,2006,Fudan University,20,Game Theory-Lecture 1,Definition:strictly dominated strategy,-1,-1,-9,0,0,-9,-6,-6,Prisoner 1,Prisoner 2,Confess,Mum,Confess,Mum,regardless of other players choices,s,i,”,is strictly better than,s,i,Dec,2006,Fudan University,21,Game Theory-Lecture 1,Example,Two firms,Reynolds and Philip,share some market,Each firm earns$60 million from its customers if neither do advertising,Advertising costs a firm$20 million,Advertising captures$30 million from competitor,Philip,No Ad,Ad,Reynolds,No Ad,60,60,30,70,Ad,70,30,40,40,Dec,2006,Fudan University,22,Game Theory-Lecture 1,2-,player game with finite strategies,S,1,=,s,11,s,12,s,13,S,2,=,s,21,s,22,s,11,is strictly dominated by,s,12,if,u,1,(,s,11,s,21,),u,1,(,s,12,s,21,),and,u,1,(,s,11,s,22,),u,1,(,s,12,s,22,),.,s,21,is strictly dominated by,s,22,if,u,2,(,s,1,i,s,21,),u,2,(,s,1,i,s,22,),for,i,=1,2,3,Player 2,s,21,s,22,Player 1,s,11,u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),s,13,u,1,(,s,13,s,21,),u,2,(,s,13,s,21,),u,1,(,s,13,s,22,),u,2,(,s,13,s,22,),Dec,2006,Fudan University,23,Game Theory-Lecture 1,Definition:weakly dominated strategy,1,1,2,0,0,2,2,2,Player 1,Player 2,R,U,B,L,regardless of other players choices,s,i,”,is at least as good as,s,i,Dec,2006,Fudan University,24,Game Theory-Lecture 1,Strictly and weakly dominated strategy,A rational player never chooses a strictly dominated strategy.Hence,any strictly dominated strategy can be eliminated.,A rational player may choose a weakly dominated strategy.,The order of elimination does not matter for strict dominance elimination(pin down the same equilibrium),but does for weak one.,Dec,2006,Fudan University,25,Game Theory-Lecture 1,Iterated elimination of strictly dominated strategies,If a strategy is strictly dominated,eliminate it,The size and complexity of the game is reduced,Eliminate any strictly dominated strategies from the reduced game,Continue doing so successively,Dec,2006,Fudan University,26,Game Theory-Lecture 1,Iterated elimination of strictly dominated strategies:an example,1,0,1,2,0,1,0,3,0,1,2,0,Player 1,Player 2,Middle,Up,Down,Left,1,0,1,2,0,3,0,1,Player 1,Player 2,Middle,Up,Down,Left,Right,Dec,2006,Fudan University,27,Game Theory-Lecture 1,Example:Tourists&Natives,Only two bars(bar 1,bar 2)in a city,Can charge price of$2,$4,or$5,6000 tourists pick a bar randomly,4000 natives select the lowest price bar,Example 1:Both charge$2,each gets 5,000 customers and$10,000,Example 2:Bar 1 charges$4,Bar 2 charges$5,Bar 1 gets 3000+4000=7,000 customers and$28,000,Bar 2 gets 3000 customers and$15,000,Dec,2006,Fudan University,28,Game Theory-Lecture 1,Example:Tourists&Natives,Bar 2,$2,$4,$5,Bar 1,$2,10,10,14,12,14,15,$4,12,14,20,20,28,15,$5,15,14,15,28,25,25,Payoffs are in thousands of dollars,Bar 2,$4,$5,Bar 1,$4,20,20,28,15,$5,15,28,25,25,Dec,2006,Fudan University,29,Game Theory-Lecture 1,One More Example,Each of,n,players selects a number between 0 and 100 simultaneously.Let,x,i,denote the number selected by player,i.,Let,y,denote the average of these numbers,Player,i,s payoff=,x,i,3,y,/5,Dec,2006,Fudan University,30,Game Theory-Lecture 1,One More Example,The normal-form representation:,Players:player 1,player 2,.,player,n,Strategies:,S,i,=0,100,for,i,=1,2,.,n.,Payoff functions:,u,i,(,x,1,x,2,.,x,n,)=,x,i,3,y,/5,Is there any dominated strategy?,What numbers should be selected?,Dec,2006,Fudan University,31,Game Theory-Lecture 1,New solution concept:Nash equilibrium,Player 2,L,C,R,Player 1,T,0,4,4,0,5,3,M,4,0,0,4,5,3,B,3,5,3,5,6,6,The combination of strategies(B,R)has the following property:,Player 1 CANNOT do better by choosing a strategy different from,B,given that,player 2 chooses R.,Player 2 CANNOT do better by choosing a strategy different from,R,given that,player 1 chooses B,.,Dec,2006,Fudan University,32,Game Theory-Lecture 1,New solution concept:Nash equilibrium,Player 2,L,C,R,Player 1,T,0,4,4,0,3,3,M,4,0,0,4,3,3,B,3,3,3,3,3.5,3.6,The combination of strategies(B,R)has the following property:,Player 1 CANNOT do better by choosing a strategy different from,B,given that,player 2 chooses R.,Player 2 CANNOT do better by choosing a strategy different from,R,given that,player 1 chooses B,.,Dec,2006,Fudan University,33,Game Theory-Lecture 1,Nash Equilibrium:idea,Nash equilibrium,A set of strategies,one for each player,such that each players strategy is best for her,given that all other players are playing their equilibrium strategies,Dec,2006,Fudan University,34,Game Theory-Lecture 1,Definition:Nash Equilibrium,Given others choices,player,i,cannot be better-off if she deviates from,s,i,*,(cf:dominated strategy),Prisoner 2,Mum,Confess,Prisoner 1,Mum,-1,-1,-9,0,Confess,0,-9,-6,-6,Dec,2006,Fudan University,35,Game Theory-Lecture 1,2-,player game with finite strategies,S,1,=,s,11,s,12,s,13,S,2,=,s,21,s,22,(,s,11,s,21,),is a Nash equilibrium if,u,1,(,s,11,s,21,),u,1,(,s,12,s,21,),u,1,(,s,11,s,21,),u,1,(,s,13,s,21,)and,u,2,(,s,11,s,21,),u,2,(,s,11,s,22,),.,Player 2,s,21,s,22,Player 1,s,11,u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),s,13,u,1,(,s,13,s,21,),u,2,(,s,13,s,21,),u,1,(,s,13,s,22,),u,2,(,s,13,s,22,),Dec,2006,Fudan University,36,Game Theory-Lecture 1,Finding a Nash equilibrium:cell-by-cell inspection,1,0,1,2,0,1,0,3,0,1,2,0,Player 1,Player 2,Middle,Up,Down,Left,1,0,1,2,0,3,0,1,Player 1,Player 2,Middle,Up,Down,Left,Right,Dec,2006,Fudan University,37,Game Theory-Lecture 1,Example:Tourists&Natives,Bar 2,$2,$4,$5,Bar 1,$2,10,10,14,12,14,15,$4,12,14,20,20,28,15,$5,15,14,15,28,25,25,Payoffs are in thousands of dollars,Bar 2,$4,$5,Bar 1,$4,20,20,28,15,$5,15,28,25,25,Dec,2006,Fudan University,38,Game Theory-Lecture 1,One More Example,The normal-form representation:,Players:player 1,player 2,.,player,n,Strategies:,S,i,=0,100,for,i,=1,2,.,n.,Payoff functions:,u,i,(,x,1,x,2,.,x,n,)=,x,i,3,y,/5,What is the Nash equilibrium?,Dec,2006,Fudan University,39,Game Theory-Lecture 1,Best response function:example,If Player 2 chooses L then Player 1s best strategy is M,If Player 2 chooses C then Player 1s best strategy is T,If Player 2 chooses R then Player 1s best strategy is B,If Player 1 chooses B then Player 2s best strategy is R,Best response:the best strategy one player can play,given the strategies chosen by all other players,Player 2,L,C,R,Player 1,T,0,4,4,0,3,3,M,4,0,0,4,3,3,B,3,3,3,3,3.5,3.6,Dec,2006,Fudan University,40,Game Theory-Lecture 1,Example:Tourists&Natives,what is Bar 1s best response to Bar 2s strategy of$2,$4 or$5?,what is Bar 2s best response to Bar 1s strategy of$2,$4 or$5?,Bar 2,$2,$4,$5,Bar 1,$2,10,10,14,12,14,15,$4,12,14,20,20,28,15,$5,15,14,15,28,25,25,Payoffs are in thousands of dollars,Dec,2006,Fudan University,41,Game Theory-Lecture 1,2-,player game with finite strategies,S,1,=,s,11,s,12,s,13,S,2,=,s,21,s,22,Player 1s strategy,s,11,is her best response to,Player 2s strategy,s,21,if,u,1,(,s,11,s,21,),u,1,(,s,12,s,21,)and,u,1,(,s,11,s,21,),u,1,(,s,13,s,21,).,Player 2,s,21,s,22,Player 1,s,11,u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),s,13,u,1,(,s,13,s,21,),u,2,(,s,13,s,21,),u,1,(,s,13,s,22,),u,2,(,s,13,s,22,),Dec,2006,Fudan University,42,Game Theory-Lecture 1,Using best response function to find Nash equilibrium,In a 2-player game,(,s,1,s,2,),is a Nash equilibrium if and only if player 1s strategy,s,1,is her best response to player 2s strategy,s,2,and player 2s strategy,s,2,is her best response to player 1s strategy,s,1,.,-1,-1,-9,0,0,-9,-6,-6,Prisoner 1,Prisoner 2,Confess,Mum,Confess,Mum,Dec,2006,Fudan University,43,Game Theory-Lecture 1,Using best response function to find Nash equilibrium:example,M is Player 1s best response,to,Player 2s strategy L,T is Player 1s best response,to,Player 2s strategy C,B is Player 1s best response,to,Player 2s strategy R,L is Player 2s best response,to,Player 1s strategy T,C is Player 2s best response,to,Player 1s strategy M,R is Player 2s best response,to,Player 1s strategy B,Player 2,L,C,R,Player 1,T,0,4,4,0,3,3,M,4,0,0,4,3,3,B,3,3,3,3,3.5,3.6,Dec,2006,Fudan University,44,Game Theory-Lecture 1,Example:Tourists&Natives,Bar 2,$2,$4,$5,Bar 1,$2,10,10,14,12,14,15,$4,12,14,20,20,28,15,$5,15,14,15,28,25,25,Payoffs are in thousands of dollars,Use best response function to find the Nash equilibrium.,Dec,2006,Fudan University,45,Game Theory-Lecture 1,Example:The battle of the sexes,Opera is Player 1s best response to Player 2s strategy Opera,Opera is Player 2s best response to Player 1s strategy Opera,Hence,(,Opera,Opera,)is a Nash equilibrium,Fight is Player 1s best response to Player 2s strategy Fight,Fight is Player 2s best response to Player 1s strategy Fight,Hence,(,Fight,Fight,)is a Nash equilibrium,2,1,0,0,0,0,1,2,Chris,Pat,Prize Fight,Opera,Prize Fight,Opera,Dec,2006,Fudan University,46,Game Theory-Lecture 1,Example:Matching pennies,Head is Player 1s best response to Player 2s strategy Tail,Tail is Player 2s best response to Player 1s strategy Tail,Tail is Player 1s best response to Player 2s strategy Head,Head is Player 2s best response to Player 1s strategy Head,Hence,NO Nash equilibrium,-1,1,1,-1,1,-1,-1,1,Player 1,Player 2,Tail,Head,Tail,Head,Dec,2006,Fudan University,47,Game Theory-Lecture 1,Definition:best response function,Player,i,s best response,Given the strategies chosen by other players,Dec,2006,Fudan University,48,Game Theory-Lecture 1,Definition:best response function,Player,i,s best response to other players strategies is an optimal solution to,Dec,2006,Fudan University,49,Game Theory-Lecture 1,Using best response function to define Nash equilibrium,A set of strategies,one for each player,such that each players strategy is best for her,given that all other players are playing their strategies,or,A stable situation that no player would like to deviate if others stick to it,Dec,2006,Fudan University,50,Game Theory-Lecture 1,Cournot model of duopoly,A product is produced by only two firms:firm 1 and firm 2.The quantities are denoted by,q,1,and,q,2,respectively.Each firm chooses the quantity without knowing the other firm has chosen.,The market priced is,P,(,Q,)=,a-Q,where,a,is a constant number and,Q=q,1,+q,2,.,The cost to firm,i,of producing quantity,q,i,is,C,i,(,q,i,)=,cq,i,.,Dec,2006,Fudan University,51,Game Theory-Lecture 1,Cournot model of duopoly,The normal-form representation:,Set of players:,Firm 1,Firm 2,Sets of strategies:,S,1,=,0,+),S,2,=,0,+),Payoff functions:,u,1,(,q,1,q,2,)=,q,1,(,a-,(,q,1,+q,2,)-,c,),u,2,(,q,1,q,2,)=,q,2,(,a-,(,q,1,+q,2,)-,c,),Dec,2006,Fudan University,52,Game Theory-Lecture 1,Cournot model of duopoly,How to find a Nash equilibrium,Find the quantity pair(,q,1,*,q,2,*,)such that,q,1,*,is firm 1s best response to Firm 2s quantity,q,2,*,and,q,2,*,is firm 2s best response to Firm 1s quantity,q,1,*,That is,q,1,*,solves Max,u,1,(,q,1,q,2,*)=,q,1,(,a-,(,q,1,+q,2,*)-,c,),subject to,0,q,1,+,and,q,2,*,solvesMax,u,2,(,q,1,*,q,2,)=,q,2,(,a-,(,q,1,*,+q,2,)-,c,),subject to,0,q,2,+,Dec,2006,Fudan University,53,Game Theory-Lecture 1,Cournot model of duopoly,How to find a Nash equilibrium,Solve Max,u,1,(,q,1,q,2,*)=,q,1,(,a-,(,q,1,+q,2,*)-,c,),subject to,0,q,1,+,FOC:,a-2q,1,-q,2,*-,c,=0,q,1,=,(,a-q,2,*-,c,)/2,Dec,2006,Fudan University,54,Game Theory-Lecture 1,Cournot model of duopoly,How to find a Nash equilibrium,SolveMax,u,2,(,q,1,*,q,2,)=,q,2,(,a-,(,q,1,*,+q,2,)-,c,),subject to,0,q,2,+,FOC:,a-2q,2,q,1,*,c,=0,q,2,=,(,a q,1,*,c,)/2,Dec,2006,Fudan University,55,Game Theory-Lecture 1,Cournot model of duopoly,How to find a Nash equilibrium,The quantity pair(,q,1,*,q,2,*,)is a Nash equilibrium if,q,1,*,=,(,a q,2,*,c,)/2,q,2,*,=,(,a q,1,*,c,)/2,Solving these two equations gives us,q,1,*,=q,2,*,=,(,a,c,)/3,Dec,2006,Fudan University,56,Game Theory-Lec
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