产业组织理论授课课件-IO-slides-introtogames.ppt

1、Non-Cooperative Game Theory To define a game,you need to know three things:The set of players The strategy sets of the players(i.e.,the actions they can take)The payoff functionsGame AColumn Player Row Player RedBlackRed2,25,0Black0,53,3PlayersStrategy sets for each playerPayoffs for each player,for

2、 each possible outcomePayoff to RowPayoff to ColumnGame BColumn Player Row Player RedBlackRed4,40,1Black1,01,1 What happened in these two games?What were the strategies?What were the outcomes?Why did we get these outcomes?Should we have expected these outcomes?In other words-How do we solve these ga

3、mes?Solving Games We are looking for the equilibrium.What is equilibrium?Equilibrium is a strategy combination where no one player has an incentive to change her strategy given the strategies of the other players.Huh?Game AColumn Player Row Player RedBlackRed2,25,0Black0,53,3Nash Equilibrium(NE)Form

4、ally,a set of strategies forms a NE if,for every player i,i(si,s-i)i(si*,s-i).Note that the equilibrium is defined in terms of strategies,not payoffs.Why is this a solution?Because its a rest point-no incentive for one player to change unilaterally.How Do We Find NE?Elimination of Dominated Strategi

5、es.A player has a dominated strategy if there is one action/strategy which always provides a lower payoff than another strategy,no matter what other players do.If you cross off all dominated strategies,sometimes you are left with only NE.Game AColumn Player Row Player RedBlackRed2,25,0Black0,53,3Rep

6、eated elimination can find the NELeftCenterRightTop3,25,44,3Middle1,64,22,5Bottom1,36,35,4 Elimination of dominated strategies only works if the strategies are strictly dominated Always worse,not just equal to or worse LeftCenterRightTop3,25,44,3Middle3,64,22,5Bottom1,36,35,4Sometimes there arent do

7、minated strategies so you have to check for NE cell by cell “Battle of theSexes”ScreamThe BeachScream2,10,0TheBeach0,01,2Sometimes there arent any NE“Evens and Odds”1 finger2 fingers1 finger1,-1-1,12 fingers-1,11,-1We can use the“Normal”or matrix form if:There are only 2(sometimes 3)players There ar

8、e a finite number of strategies Actions approximately simultaneousIf actions are sequential,must use another form,the“Extensive”form:Still only really feasible for 2 or 3 players,although can accommodate“chance”Still must have finite number of strategiesExtensive Form Games Use a game“tree”to depict

9、 the order in which players make decisions and the choices that they have at each decision point.Decision points are called“nodes”.Players strategies or choices branch off from each decision node.At the end of each branch on the game tree are the payoffs the players would receive if that branch were

10、 the path followed.US vs.Saudi Arabia Oil“Game”QuotaTariffNothingRRRNNN 90,80 100,60 75,50 100,60 40,80 50,100USSaudi ArabiaSolving Extensive Form Games Nash Equilibrium has the same meaning in extensive form games as in normal form games.There is also another solution concept in extensive form game

11、s,the Subgame Perfect Equilibrium(SPE)strategy which has some advantages over Nash Equilibrium.US vs.Saudi Arabia Oil“Game”QuotaTariffNothingRRRNNN 90,80 100,60 75,50 100,60 40,80 50,100USSaudi ArabiaSubgame=part of larger game that can stand alone as a game itself.Sub-Game Perfect Equilibrium A sub

12、game can be defined for any node other than a terminal(payoff)node,and includes all of the subsequent“branches”of the tree that emanate from that node.For a strategy to be a Subgame Perfect Equilibrium(SPE)strategy,it can only contain actions that are optimal for their respective subgames.QuotaTarif

13、fNothingRRRNNN 90,80 100,60 75,50 100,60 40,80 50,100USSaudi ArabiaQuotaTariffNothingRRRNNN 90,80 100,60 75,50 100,60 40,80 50,100USSaudi ArabiaTo find all of the Subgame Perfect Equilibria:For each subgame,determine the optimal strategy.Find the optimal strategy for the“pruned”tree.QuotaTariffNothi

14、ngRRRNNN 90,80 100,60 75,50 100,60 40,80 50,100USSaudi ArabiaCompare Subgame Perfect Equilibria(SPE)to NE:NE can include incredible threats,along as unilateral changes are not optimal.Example:Quota;R if Quota or Tariff,N if Nothing Another ExampleEnterStay OutHigh PLow PHigh PLow P 2,2 -1,0 0,5 0,0

15、EntrantIncumbentFind optimal strategy for each subgame(prune the tree).Find Entrants optimal action.Repeated Games In repeated games,strategies are much richer.In a one-shot Prisoners Dilemma game,players can either cooperate or defect.In a repeated game,players choose whether to cooperate or defect

16、 each period.Players can have strategies that are contingent on the other players actions.Cooperate if the other player cooperated last period.Defect if the other player has ever defected.Note:In repeated games,must discount future payoffs.(1/(1+r)t =t is the discount factor for period t.Solving Rep

17、eated Games If the game has a finite horizon(that is,it ends after a specified number of rounds),you use backwards induction.Start by finding the optimal strategy in the last period Move to the next to the last period,and find the optimal strategy,recognizing the effects on the final round.If the ga

18、me has an infinite horizon,you cant use backwards induction because there is no last period.To solve infinite horizon games,you check different strategies to see if they meet the requirements of equilibrium.For each player,changing strategies unilaterally will not make the player better off.Solving

19、Repeated Prisoners Dilemma Games For finite horizon repeated PD,use backwards induction.In the last period,always optimal to defect If your action in the next-to-the-last period does not affect the optimal strategy in the last period,you do better by defecting in the next to the last period And so o

20、n.For finite horizon repeated PD,collusion is never optimal.Solving Repeated Prisoners Dilemma Games For infinite horizon repeated PD,consider different strategies.“Grim Trigger”strategy:Cooperate as long as other player cooperates,but once he defects,defect forever.His defection“triggers”the punish

21、ment.“Grim”because punishment lasts forever.To check if there is a symmetric equilibrium with trigger strategies:Make sure that cooperating is better than defecting if other player has cooperated.Make sure that“punishment”is a credible threat,that you will actually go through with it.Prisoners Dilem

22、maColumn Player Row Player DefectCooperateDefect2,25,0Cooperate0,53,3When Are Trigger Strategies are NE?Assume other player also using a trigger strategy.If neither has defected,both cooperate this period.If you follow the trigger strategy,i.e.cooperate,you get C this period(the payoff from cooperat

23、ion)and you get C each period in the future.An infinite stream of payments of C can be written as 1/(1-)*C.If you defect,you get D this period(the increased payoff from unilateral defection)but in all future periods you get P(the punishment payoff level)Total earnings thus are D+/(1-)*P.Thus followi

24、ng the strategy is optimal if:1/(1-)*C D+/(1-)*P.When Are Trigger Strategies are NE,cont?The condition 1/(1-)*C D+/(1-)*P can be rewritten as:(D -C)/(D-P)So the discount factor,must be sufficiently large for collusion to be sustainable.How do we interpret this?A high discount factor means that payof

25、fs in the future are relatively important.You are willing to forsake immediate,but transitory gains from defection for higher payoffs in the future.When Are Trigger Strategies are NE,cont?Is punishment a credible threat?Once again,assume other player also using a trigger strategy.If either has defec

26、ted,both will punish this period.If you follow the trigger strategy,i.e.punish,you get P this period and you get P each period in the future.If you dont punish,you will get a lower payoff,since defecting is a best response to other players playing defecting.Therefore the punishment is a credible threat.