introduction to game theory - adam brandenburger
TRANSCRIPT
IntroductiontoGameTheory:GameTrees
Version10/29/17
10/29/178:59PM 2
ASecond“Non-Cooperative”GameModel
Thegametreeisamodelspecifyingwhichplayermovesfirstandwhatpossiblemovesareavailabletothatplayer;whichplayermovesnextandwhatmovesareavailabletothatplayer;andsoon
Alsospecifiedistheinformationavailabletoeachplayerateachofthecontingenciesunderwhichthatplayermightneedtochooseamove
Thisinformationismadeupofanyinformationtheplayerhasaboutpreviousmovesmadebyotherplayers(andbyhimself,ifwewanttoallowforlimitedmemory)andaboutpreviousmovesbyNature(i.e.,chancemoves)
10/29/178:59PM 3
Tic-Tac-ToeRevisited(SymmetrizedGameTree)
http://commons.wikimedia.org/wiki/File:Tic-tac-toe-game-tree.svg
10/29/178:59PM 4
DefinitionofaStrategy
Astrategy foraplayerspecifies,ateachnode(later:informationset)thatbelongstothatplayer,aparticularmove
http://commons.wikimedia.org/wiki/File:Tic-tac-toe-game-tree.svg
A particular strategy for the second mover
10/29/178:59PM 5
EndpointsandOutcomes
http://commons.wikimedia.org/wiki/File:Tic-tac-toe-game-tree.svg
Anendpoint
Anoutcome
Ann’spayoff Bob’spayoffWhatifpayoffsarenot‘transparent’totheplayers?Soon!
10/29/178:59PM 6
SimplifiedPoker
Amove(ormoves)byNaturecanbeusedtoexpandtheapplicabilityofthegame-treemodel
�� Ann
Naturepickseachbranchwithprobability1/4
�
Bet Drop
Aninformationset
�AA AK KKKA
¢
Ann
Bet Drop
� �
K = KingA = AceK = KingA = Ace
Bet Drop Bet Drop
� �Bob Bob
Call Fold Call Fold Call Fold Call Fold
http://slideplayer.com/slide/6274988/
10/29/178:59PM 7
CaseStudy:APolitical‘Game’
Lyndon Johnson was in Houston on February 23 [1937] … when he suddenly saw, on a park bench, a copy of the Houston Post with the banner headline: CONGRESSMAN JAMES P. BUCHANAN OF BRENHAM DIES. He knew at once, he was to recall, that “this was my chance….”
From: TheYearsofLyndonJohnson:ThePathtoPower,byRobertCaro,Vintage,1990,pp.389-399;photofromhttps://media1.britannica.com/eb-media/18/126018-004-EC84F13F.jpg
A strategy, money, an organization --- these would give this unknown candidate [Johnson] a slim chance of victory against every opponent but one. Against that one opponent, nothing could give him a chance. Nothing could offset the sentimental appeal of a vote for Old Buck’s [Buchanan’s] widow.... And it began to look as if she was going to run…. So Lyndon Johnson went … and asked his father’s advice. Sam Johnson [Lyndon’s father] didn’t even have to think before giving it. Recalls Lyndon’s brother: “Lyndon started saying he was thinking of waiting to see what she [Mrs. Buchanan] does, and Daddy says, “Goddammit, Lyndon, you never learn anything about politics.” Lyndon says, “What do you mean?”
10/29/178:59PM 8
CaseStudyCont’d:TwoGameTrees
�
�B
J
Bwins(afterafight)
Bwins
Run Don’t
Run Don’t�J
Jwins Thirdplayerwins
Run Don’t�
�J
B
Bwins(afterafight)
J wins
Run Don’t
Run Don’t�B
B wins Thirdplayerwins
Run Don’t
10/29/178:59PM 9
CaseStudyCont’d:BackwardInduction
�
�B
J
Bwins(afterafight)
Bwins
Run Don’t
Run Don’t�J
Jwins Thirdplayerwins
Run Don’t�
�J
B
Bwins(afterafight)
J wins
Run Don’t
Run Don’t�B
B wins Thirdplayerwins
Run Don’t
10/29/178:59PM 10
TheBackward-InductionAlgorithm
• Startatfinaldecisionnodes--- i.e.,atdecisionnodesthatleadonlytoterminalnodesofthetree
• Selectthebestchoicefortherelevantplayerateachofthesenodesandreplacethenodeswiththeresultingpayoffs(toalltheplayers)
• Repeatthisprocessontheprunedtree
• Theprocessendswhenitreachestherootofthetree
10/29/178:59PM 11Rosenthal,R.,“GamesofPerfectInformation,PredatoryPricing,andtheChainStore,”JournalofEconomicTheory,1980,25,92-100;McKelvey,R.,andT.Palfrey,“AnExperimentalStudyoftheCentipedeGame,”1992,Econometrica,60,803-836;http://commons.wikimedia.org/wiki/File:Stick_Figure.jpg
$2$1
Out
$1$4
Out
$4$3
Out
• • •
$17$20
Out Out
In
$20$19
$19$22
InIn
$3$6
Out
�� � � � �In In In
Ann Bob
TheCentipedeGame
Whatdoesthebackward-inductionalgorithmyieldinthistree?
10/29/178:59PM 12
SomeGameTreeTerminologyandFacts
Agametreehasperfect (respectively,imperfect)informationifeveryinformationsetis(respectively,isnot)asingleton
Thebackward-inductionalgorithmappliestoperfect-informationtrees
Agamematrixcanbeviewedasaparticular(‘simultaneous-move’)tree
(Advancedtopicwewillnotpursuehere:Everytreecanbemappedtoa(perhaps,big)matrix,so,what,atadeeperlevel,istherelationshipbetweenthesetwotypesofmodel?)
Y X
X
Ya b
Ann
Bob
α β
c
γ
d
δ�
�
�
Ann
Bob
X Y
X XY Y
dδ
cγ
bβ
aα
10/29/178:59PM 13
GameswithUncertaintyAboutPayoffs(akaBayesianorIncomplete-InformationGames)
IfNaturechoosesthebranchlabeledwithprobabilityp,thenBobgetsapayoffof0whenAnnchoosesLandhechoosesr
IfNaturechoosesthebranchlabeledwithprobability1−p,thenBobgetsapayoffof3whenAnnchoosesLandhechoosesr
ThekeyisthatwhenAnngetstomoveLorR,shedoesnotknowwhichareBob’spayoffs
�
�
Bob
22
00
11
L R
l r�
�
Bob
22
03
11
L R
l* r*
Ann
¢
(p) (1−p)
10/29/178:59PM 14
ForwardInduction
FirmA hasthepossibilityof(irrecoverably)pre-investingafractiona oftheupfrontinvestmentF neededtoenterthemarket
(Advancedtopicwewillnotpursuehere:Whatisaformalmethodofanalysisthatcapturesforward-inductionreasoning?)
Out In
In
Out 0
0
0
1
1
0
−F
−F Firm A
Firm B Out In
In
Out −aF
0
−aF
1
1
0
−F
−F Firm A
Firm B
! Firm A
No Pre-invest