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8/29/2014
1
MAE 301 T2 2014MICROECONOMIC THEORY AND POLICY
LECTURE 6
GAME THEORY:Dynamic Games
(sequential and repeated)
Learning Objectives Analyze sequentially played games
Extensive-form games Subgame Perfect Nash Equilibrium
Discuss the implications of repeated interactions
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Dynamic Games
In dynamic games: players move either sequentially or repeatedly players are assumed to have complete
information about payoff functions at each move, players are assumed to have
perfect information about previous moves of all players
Extensive Form
Dynamic games are analyzed in their extensive form, which specifies the n players the sequence of their moves the actions they can take at each move the information each player has about players
previous moves the payoff function over all possible strategies
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Actions v Strategies
Consider a single period two-stage game: First stage: player 1 moves Second stage: player 2 moves
In games where players move sequentially, we distinguish between an action and a strategy An action is a move that a player makes at a
specified point in the game tree A strategy is a battle plan that specifies the action a
player will make condition on information available at each move
Return to the Airline Game to demonstrate these concepts Assume American chooses its output before United
does
The Airline Game
This game treeshows decision nodes:
indicates which players turn it is
branches:indicates all possible actions available
subgames: subsequent decisions available given previous actions
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SPNE
To predict the outcome of the Airline game (and, more generally, the outcome of any sequential game with a finite number of nodes), we use a strong version of Nash equilibrium, the Subgame Perfect Nash Equilibrium (SPNE)
SPNE requires players to believe that their opponents will act optimally (in their own best interest) at any decision node
A set of strategies forms a subgame perfect Nash equilibrium if the players strategies are a Nash equilibrium in every subgame The Airline game has four subgames; three subgames
at second stage where United makes a decision and an additional subgame at the time of the first-stage decision
Backward induction We can solve for the subgame perfect Nash equilibrium
using backward induction Once game is understood through backward induction,
players play it forward To apply backward induction, first determine optimal
actions at last decision nodes that result in terminal nodes
Then determine optimal actions at next-to-last decision nodes, assuming that optimal actions will follow at next decision nodes
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Backward induction Continue backward process until root node is reached Backward induction implicitly assumes that a players
strategy will consist of optimal actions at every node in the game tree
Principle of sequential rationality: At any point in the game tree, players strategy should consist of optimal actions from that point on given other players strategies
Backward induction Backward induction is where we determine:
the best response by the last player to move the best response for the player who made the
next-to-last move repeat the process until we reach the beginning of
the game Airline Game
If American chooses 48, United selects 64, Americans profit=3.8
If American chooses 64, United selects 64, Americans profit=4.1
If American chooses 96, United selects 48, Americans profit=4.6
Thus, American chooses 96 in the first stage
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How reasonable is backward induction as a behavioral principle?
May work to explain actual outcomes in simple games, with few players and moves
More difficult to use in complex sequential move games such as Chess We cant draw out the game tree because there are too
many possible moves, estimated to be on the order of 10120
Need a rule for assigning payoffs to non-terminal nodes an intermediate valuation function
May not always predict behavior if players are unduly concerned with fair behavior by other players and do not act so as to maximize their own payoff, e.g., they choose to punish unfair behavior
Sequential Battle-of-the-Sexes
As an example of a sequential game, consider the Battle-of-the-Sexes game
Husband prefers going to fights and wife prefers opera
However, they both prefer spending their leisure time together
In the simultaneously played game, there exist two pure-strategy Nash equilibria (both going to the opera or both to the fights)
Assume the husband take the lead
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Game tree for Battle-of-the- Sexes
Sequential Battle-of-the-Sexes
Suppose the husband picks opera If the wife picks opera as well, the husband will end up
with a payoff of 2 and the wife with a payoff of 5 If the husband picks fights, it will be optimal for the
wife to pick fights (resulting payoffs are 5 for husband and 2 for wife)
For husband (first player), 5 is greater than 2. Both going to the fights is not only an overall equilibrium, but also an equilibrium in each of the subgames, that is, a subgame perfect Nash equilibrium
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The Simultaneous-Move Bargaining Game Management and a union are negotiating, among other
things, a wage increase Strategies are wage offers & wage demands Successful negotiations lead to $600 million in surplus,
which must be split among the parties Failure to reach an agreement results in a loss to the
firm of $100 million and a union loss of $3 million An agreement is reached ONLY if wage offer=wage
demand (rationale: higher wage offered at worse conditions, such as less job security, or longer shifts)
Otherwise you can assume that an agreement is reached if (all other things being equal) wage offer is higher or equal to wage demand
Only one-shot in making a deal
The Simultaneous-Move Bargaining Game
Union
Man
agem
ent Strategy W = AU$10 W = AU$5 W = AU$1
W = AU$10 100, 500 -100, -3 -100, -3W=AU$5 -100, -3 300, 300 -100, -3W=AU$1 -100, -3 -100, -3 500, 100
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The Simultaneous-Move Bargaining Game
Union
Man
agem
ent
3 Nash Equilibria!
Strategy W = AU$10 W = AU$5 W = AU$1W = AU$10 100, 500 -100, -3 -100, -3
W=AU$5 -100, -3 300, 300 -100, -3W=AU$1 -100, -3 -100, -3 500, 100
The Sequential-Move Bargaining Game
Now suppose that the management moves first by making the union a take-it-or-leave-it offer
Firm
10
5
1
Union
Union
Union
Accept
Accept
Accept
Reject
Reject
Reject
100, 500
-100, -3
300, 300
-100, -3
500, 100
-100, -3
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The Sequential-Move Bargaining Game
Identify the firms feasible strategies: the management has three feasible strategies: (1) Offer AU$10; (2) Offer AU$5; (3) Offer AU$1
Identify the unions feasible strategies: the union has eight feasible strategies (23): (1) Accept AU$10, Accept AU$5, Accept AU$1; (2) Accept AU$10, Accept AU$5, Reject AU$1; (3) Accept AU$10, Reject AU$5, Accept AU$1; (4) Accept AU$10, Reject AU$5, Reject AU$1; (5) Reject AU$10, Accept AU$5, Accept AU$1; (6) Reject AU$10, Accept AU$5, Reject AU$1; (7) Reject AU$10, Reject AU$5, Accept AU$1; (8) Reject AU$10, Reject AU$5, Reject AU$1
The Sequential-Move Bargaining Game
Identify Nash Equilibrium Outcomes: outcomes such that neither the firm nor the union has an incentive to change its strategy, given the strategy of the other
Find the Sub-game Perfect Nash Equilibrium Outcomes (SPNE): outcomes where no player has an incentive to change its strategy, given the strategy of the rival, and the outcomes are based on credible actions, that is, they are not the result of empty threats by the rival
For a firms strategy to be a credible threat rivals must believe that the firms strategy is rational (it is in the firms best interest to use it)
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The Sequential-Move Bargaining Game
Accept $10, Accept $5, Accept $1Accept $10, Accept $5, Reject $1Accept $10, Reject $5, Accept $1Reject $10, Accept $5, Accept $1Accept $10, Reject $5, Reject $1Reject $10, Accept $5, Reject $1Reject $10, Reject $5, Accept $1Reject $10, Reject $5, Reject $1
Union's Strategy Firm's Best Response
Mutual Best Response?
$1 Yes$5$1$1
$10
YesYesYesYes
$5 Yes$1
NoYes
$10, $5, $1
Nash and Credible Nash Only Neither Nash Nor Credible
The Sequential-Move Bargaining Game
We have identified many (7) combinations of strategies that are mutual best response - Nash equilibrium strategies
In all but one the firm does something that isnt in its self interest (and thus entail threats that are not credible)
Only one Nash equilibrium outcome (out of 3 Nash equilibrium outcomes) is based on credible actions/threats
Empty threats will not succeed in inducing a player to select some action
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The Sequential-Move Bargaining Game
Firm
10
5
1
Union
Union
Union
Accept
Accept
Accept
Reject
Reject
Reject
100, 500
-100, -3
300, 300
-100, -3
500, 100
-100, -3
There are 3 Nash Equilibrium Outcomes!
The Sequential-Move Bargaining Game
Firm
10
5
1
Union
Union
Union
Accept
Accept
Accept
Reject
Reject
Reject
100, 500
-100, -3
300, 300
-100, -3
500, 100
-100, -3
Only 1 Subgame-Perfect Nash Equilibrium Outcome!
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The Sequential-Move Bargaining Game
In take-it-or-leave-it bargaining, there is a first-mover advantage
Management can gain by making a take-it-or-leave-it offer to the union. But...
Management should be careful; real world evidence suggests that people sometimes reject offers on the basis of principle instead of cash considerations
What would happen if the union moved first?
The Ultimatum Game
Player 1 moves first and proposes a division of $1.00. Suppose there are just 3 possible discrete divisions, limited to $0.25 increments: Player 1 can propose x=$0.25, x=$0.50, or x=$0.75
for himself, with the remainder, 1-x going to Player 2
Player 2 moves second and can either acceptor reject Player 1s proposal
If Player 2 accepts, the proposal is implemented
If Player 2 rejects, both players get $0 each. The $1.00 gains from trade vanish
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The Ultimatum Game
Problems with Take-it-or-Leave-it
Take-it-or-leave-it games are too trivial; there is no back-and-forth bargaining
Another problem is the credibility of take-it-or leave-it proposals If player 2 rejects player 1s offer, is it really
believable that both players walk away even though there are potential gains from trade?
Or do they continue bargaining? What about fairness? Is it really likely that Player
1 will keep as much of M as possible for himself?
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The Centipede Game
RepeatedGames Wenowexaminetheeffectofrepetition onstrategic
behavioringameswithperfectinformation. Ifagameisplayedrepeatedly,withthesameplayers,the
playersmaybehaveverydifferentlythanifthegameisplayedjustonce(a oneshotgame).
Twotypesofrepeatedgames: Finitelyrepeated:thegameisplayedforafiniteandknownnumberofrounds,forexample,2rounds/repetitions.
Infinitely orIndefinitelyrepeated:thegamehasnopredeterminedlength;playersactasthoughitwillbeplayedindefinitely,oritendsonlywithsomeprobability.
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Infinitely RepeatedGames Finitelyrepeatedgamesareinteresting,butrelativelyrare;
howoftendowereallyknowforcertainwhenagameweareplayingwillend?(Sometimes,butnotoften).
Someofthepredictionsforfinitelyrepeatedgamesdonotholdupwellinexperimentaltests: Theuniquesubgameperfectequilibriuminthefinitelyrepeated
ultimatumgameorprisonersdilemmagame(alwaysconfess)arenotusuallyobservedinallroundsoffinitelyrepeatedgames.
Ontheotherhand,weroutinelyplaymanygamesthatareindefinitelyrepeated (noknownend).Wecallsuchgamesinfinitelyrepeatedgames,andwenowconsiderhowtofindsubgameperfectequilibriainthesegames.
DiscountinginInfinitelyRepeatedGames Recallfromourearlieranalysisofbargaining,thatplayersmay
discount payoffsreceivedinthefutureusingaconstantdiscountfactor,d=1/(1+r),where0
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DiscountinginInfinitelyRepeatedGames,Cont. Theinfinitesum, convergesto
Simpleproof:Letx=Noticethatx=solve
Hence,thepresentdiscountedvalue ofreceiving$pineveryfutureroundis$p[d/(1d)]or $pd/(1d)
Notefurtherthatusingthedefinition,d=1/(1+r), d/(1d)=[1/(1+r)]/[11/(1+r)]=1/r,sothepresentvalueoftheinfinitesumcanalsobewrittenas$p/r.
Thatis,$pd/(1d)=$p/r,sincebydefinition,d=1/(1+r).
1
...32
...32 x ...)( 32
1 ;)1(:for xxxxx
ThePrisonersDilemmaGame Considerathefollowingprisonersdilemmagame,where
higherpayoffs arenowpreferredtolowerpayoffs.
Tomakethisaprisonersdilemma,wemusthave:b>c>d>a.Wewillusethisexampleinwhatfollows.
C D
C
D
c,c a,b
b,a d,d
C=cooperate,(dontconfess)D=defect(confess)
C D
C
D
4, 4 0, 6
6, 0 2, 2
Supposethepayoffsnumbersareindollars
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SustainingCooperationintheInfinitelyRepeatedPrisonersDilemmaGame
TheoutcomeC,Cforever,yieldingpayoffs(4,4)canbeasubgameperfectequilibriumoftheinfinitelyrepeatedprisonersdilemmagame,providedthat1)thediscountfactorthatbothplayersuseissufficientlylargeand2)eachplayerusessomekindofcontingent ortriggerstrategy.Forexample,thegrimtriggerstrategy: Firstround:PlayC. Secondandlaterrounds:solongasthehistoryofplayhasbeen(C,C)in
everyround,playC.OtherwiseplayDunconditionallyandforever. Proof:Consideraplayerwhofollowsadifferentstrategy,playing
CforawhileandthenplayingDagainstaplayerwhoadherestothegrimtriggerstrategy.
CooperationintheInfinitelyRepeatedPrisonersDilemmaGame,Continued
Considertheinfinitelyrepeatedgamestartingfromtheroundinwhichthedeviantplayerfirstdecidestodefect.Inthisroundthedeviantearns$6,or$2morethanfromC,$6$4=$2.
SincethedeviantplayerchoseD,theotherplayersgrimtriggerstrategyrequirestheotherplayertoplayDforeverafter,andsobothwillplayDforever,alossof$4$2=$2inallfuturerounds.
Thepresentdiscountedvalueofalossof$2inallfutureroundsis$2d/(1d)
Sotheplayerthinkingaboutdeviatingmustconsiderwhethertheimmediategainof2>2d/(1d),thepresentvalueofallfuturelostpayoffs,orif2(1d)>2d,or2>4d,or1/2>d.
If
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CooperationintheInfinitelyRepeatedPrisonersDilemmaGame,Continued
d* =1/2isthecriticaldiscountfactor. Thecriticaldiscountfactoristhethresholdofthediscount
factorabovewhichcollusioncanbesustainedasaSPNEoftheinfinitelyrepeatedgame.
Foranyd >d*thepresentvalueofallfuturelostpayoffsexceedstheimmediategainfromdeviating,implyingthatcollusionissustainable.
OtherSubgamePerfectEquilibriaarePossibleintheRepeatedPrisonersDilemmaGame
ThesetofsubgameperfectNashEquilibria,isthegreenarea,asdeterminedbyaveragepayoffsfromallroundsplayed(forlargeenoughdiscountfactor,d).
ColumnPlayerAvg.Payoff
Row
Player
Avg.
Payoff
TheFolktheoremofrepeatedgamessaysthatalmostanyoutcomethatonaverage yieldsthemutualdefectionpayofforbettertobothplayerscanbesustainedasasubgameperfectNashequilibriumoftheindefinitelyrepeatedPrisonersDilemmagame.
Theefficient,mutualcooperationinallroundsequilibriumoutcomeishere,at4,4.
ThesetoffeasiblepayoffsistheunionofthegreenandyellowregionsMutualdefectioninall
roundsequilibrium
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MustWeDiscountPayoffs? Answer1:Howelsecanwedistinguishbetweeninfinite
sums ofdifferentconstantpayoffamounts? Answer2:Wedonthavetoassumethatplayersdiscount
futurepayoffs.Instead,wecanassumethatthereissomeconstant,knownprobabilityq,0
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GrimTriggerStrategiesatwork:Anexample
TheAdvertisingGame: Twofirms(Kelloggs&GeneralMills)seektomaximizeprofits
Strategiesconsistofadvertisingcampaigns Simultaneousmoves Repeatedinteraction
GrimTriggerStrategiesatwork:Anexample
Strategy None Moderate HighNone 12,12 1, 20 -1, 15
Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2
GeneralMills
Kello
ggs
Nash Equilibrium of the one shot advertising game
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GrimTriggerStrategiesatwork:Anexample
Strategy None Moderate HighNone 12,12 1, 20 -1, 15
Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2
GeneralMills
Kello
ggs
Collusive Equilibrium
GrimTriggerStrategiesatwork:Anexample
Ifthegameisrepeatedafinitenumberoftimes,e.g.2times,collusioncannotwork(bybackwardsinduction)!
Inperiod2,thegameisaoneshotgame,soequilibriumentailsHighAdvertisinginthelastperiod=>period1isreallythelastperiod,sinceeveryoneknowswhatwillhappeninperiod2.
EquilibriumentailsHighAdvertisingbyeachfirminbothperiods.
Thesameholdstrueifwerepeatthegameanyknown,finitenumberoftimes.
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GrimTriggerStrategiesatwork:Anexample
Cancollusionworkiffirmsplaythegameeachyear,forever? Considerthefollowinggrimtriggerstrategy byeachfirm:
Dontadvertise,providedtherivalhasnotadvertisedinthepast.Iftherivaleveradvertises,punishitbyengaginginahighlevelofadvertisingforeverafter.
GrimTriggerStrategiesatwork:Anexample
Eachfirmagreestocooperateaslongastherivalhasntcheatedinthepast.Cheatingtriggerspunishmentinallfutureperiods.
SupposeGeneralMillsadoptsthistriggerstrategy.Kelloggsprofits?
Wehavetocomparethevalueofcooperationwiththevalueofcheating.
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GrimTriggerStrategiesatwork:Anexample
Strategy None Moderate HighNone 12,12 1, 20 -1, 15
Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2
GeneralMills
Kello
ggs
Consider a deviation by Kellogg. In the deviation period, Moderate is the best response against None.
Deviation (or Cheating) Payoff
GrimTriggerStrategiesatwork:Anexample
Cooperate =12+12+12 2+123+=12+12/(1 ) = 12/(1 )
with =1/(1+r)beingthediscountfactorCheat =20+2+2 2+2 3 +
=20+2 /(1 )Cheat Cooperate = 20+2 /(1 ) 12/(1 ) ==(8 18)/(1 )ifr=0.05=> =0.95238Cheat Cooperate =191.67
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GrimTriggerStrategiesatwork:Anexample
Ifr=5%,itdoesntpaytodeviate=>CollusionisaNashequilibriumintheinfinitelyrepeatedgame!
Benefits&CostsofCheating:ImmediateBenefit (20 12 today) versus PresentValueofFutureCost (12 2 foreverafter).
IfImmediateBenefit PresentValueofFutureCost>0=>Paystocheat;IfImmediateBenefit PresentValueofFutureCost 0=>Doesntpaytocheat.
ThecriticaldiscountfactorcanbecomputedbysolvingCooperate =12/(1 )Cheat =20+2 /(1 ).
*=4/9.Forany > *collusionissustainable.
KeyInsights CollusioncanbesustainedasaNashequilibriumwhen
thereisnocertainendtoagame. Collusionsustainabilityrequires:
Abilitytomonitoractionsofrivals Ability(andreputationfor)punishingdefectors Lowinterestrate Highprobabilityoffutureinteraction.