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Dynamic Game Theory and the Stackelberg Model

Dynamic Game Theory and the Stackelberg Model. Dynamic Game Theory. So far we have focused on static games . However , for many important economic applications we need to think about the game as being played over a number of time- periods , making it dynamic .

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Dynamic Game Theory and the Stackelberg Model

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  1. DynamicGameTheoryandtheStackelberg Model

  2. DynamicGameTheory • So far wehavefocused on staticgames. However, formanyimportanteconomicapplicationsweneedtothinkaboutthegame as beingplayedover a number of time-periods, making it dynamic. • A game can be dynamicfortworeasons; interactionandrepetation.

  3. DynamicOne-offGames • Therearetwofirms (A, B) consideringtoenter a new market. Unfortunatelythe market is onlybigenoughtosupportone of thetwofirms. Ifbothfirmsenterthe market, theywillbothmake a loss of $10m. Ifonlyonefirmenters, thatfirmwillearn a profit of $50m andtheotherfirmwilljust break even. Firm B observeswhetherfirm A has enteredthe market before it decideswhatto do.

  4. Since firm B observeswhatfirm A doesanddecides, it has 4 strategies. In a staticgamethereareonly 2 strategies. • Whatarethe 4 strategies of firm B? • What is theresult of thisgame?

  5. Thesolution • Thereare 3 pure-strategyNashequilibria: • Firm B threatensalwaystoenterthe market irrespective of whatfirm A does. Iffirm A believesthatthreat, it willstayout of the market. • Firm B promisesalwaystostayout. Iffirm A believesthepromise, it willalwaysenter. • Firm B promisesalwaysto do theopposite of whatfirm A does. If A believesthispromise, it willalwaysenter.

  6. Insuch a gamecredibility is a keyissue. • InthisgamefirmB’sthreatsandpromisesare not creadible. (Why?) • Since weassumethattheplayersarerational, incrediblestatementswillhave no effect on otherplayers’ behaviour. • Hencetheresult of thisgame is that A willalwaysenterand B willalwaysstayout.

  7. SubgamePerfectNashEquilibrium • GametheoristsarguethatNashequilibriumconcept is tooweak. • SubgameperfectNashequilibrium is a strongerconceptthatdoes not allownoncrediblethreatstoinfluencebehaviour. • SubgameperfectionwasintroducedbyReinhardSelten (1965). • A subgame is a smallergameembedded in thecompletegame. • A subgameperfectNashequilibriumrequiresthatthepredictedsolutionto be a Nashequilibrium in everysubgame.

  8. Continuingfromthesameexample • Therewere 3 Nashequilibria. • Firm B threatensalwaystoenterthe market irrespective of whatfirm A does. Iffirm A believesthatthreat, it willstayout of the market. • Firm B promisesalwaystostayout. Iffirm A believesthepromise, it willalwaysenter. • Firm B promisesalwaysto do theopposite of whatfirm A does. If A believesthispromise, it willalwaysenter.

  9. Let’sseeifthesestrategiesaresubgameperfect • Thisstrategy has twosubgames. First is; firm A entersandfirmBenters. Thesecond is firm A staysoutandfirm B enters. Consideringthesubgamesonlythesecond is a Nashequilibrium. Hencethisstrategy is not subgameperfect. • Not subgameperfect • Subgameperfect

  10. Backwardinduction • This is a convinientmethodtofigureoutthesubgameperfectNashequilibria. • Thisprincipleinvolvesrulingoutactions, ratherthanstrategiesthatplayerswould not playbecauseotheractionsgivehigher pay-offs.

  11. StrategicBehaviour • Thomas Schellinginitiatedtheformalstudy of strategicbehaviourandintroducedmanyimportantconcepts in his book “TheStrategy of Conflict” (1960). • Threats: denote a penaltyto be imposed on a rivalifshetakessomeaction. • Promises: involve a rewardto be conferred on a rivalifshetakessomeaction.

  12. A keyissue is whetherthesethreatsandpromisesarecredible. • The role of a strategicmove is toconvert a threator a promiseinto a commitment. 4 elementsarerequiredfor a moveto be strategic: • Sequentialmoves • Communication • Affectincentives • Rationalexpectations

  13. StackelbergGame • TheStackelberggame is identicaltotheCournutgame in thatfirmscompeteoverquantities. But it differs in thetiming of productiondecisions. • IntheStackelberggameoutput is chosensequentially. The “leader” movesfirstandchoosesquantitiy. The “follower” firmobservetheleader’smoveandmakesitsownquantitychoice. • Theleadertakesintoaccountthefollower’s optimal response(rationalexpectations).

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