SOLUTION MANUAL
Game Theory Basics 1st Edition
By Bernhard von Stengel. Chapters 1 - 12
1
,TABLE OF CONTENTS T T T
1 - Nim and Combinatorial Games
T T T T T
2 - Congestion Games
T T T
3 - Games in Strategic Form
T T T T T
4 - Game Trees with Perfect Information
T T T T T T
5 - Expected Utility
T T T
6 - Mixed Equilibrium
T T T
7 - Brouwer’s Fixed-Point Theorem
T T T T
8 - Zero-Sum Games
T T T
9 - Geometry of Equilibria in Bimatrix Games
T T T T T T T
10 - Game Trees with Imperfect Information
T T T T T T
11 - Bargaining
T T
12 - Correlated Equilibrium
T T T
2
,Game Theory Basics
T T
SolutionsT toT Exercises
©T BernhardTvonTStengelT2022
SolutionTtoTExerciseT1.1
(a) LetT≤TbeTdefinedTbyT(1.7).T ToTshowTthatT≤TisTtransitive,TconsiderTx,Ty,TzTwithTxT ≤TyTandTyT≤Tz.TIfT
xT=TyTthenTxT≤Tz,TandTifTyT=TzTthenTalsoTxT≤Tz.TSoTtheTonlyTcaseTleftTisTxT<TyTandTyT <Tz,Twhich
TimpliesT xT <T zTbecauseT<TisTtransitive,TandThenceT xT ≤Tz.
Clearly,T≤TisTreflexiveTbecauseTxT=TxTandTthereforeTxT ≤Tx.
ToTshowTthatTTTTT
≤ isTantisymmetric,TconsiderTxTandTyTwithTxTTTTTyTand
≤ TyTTTTTx.TIf
≤TweThadTxT≠TyT
thenTxT<TyTandTyT<Tx,TandTbyTtransitivityTxT<TxTwhichTcontradictsT(1.38).THenceTxT =T y,TasTreq
uired.T ThisTshowsTthatT≤TisTaTpartialTorder.
Finally,TweTshowT(1.6),TsoTweThaveTtoTshowTthatTxT<TyTimpliesTxTTTyTandT≤ xT≠TyTandTviceTversa.
TLetTxT<Ty,TwhichTimpliesTxTyTbyT(1.7).TIfTweThadTxT=TyTthenTxT<Tx,TcontradictingT(1.38),TsoTwe
≤
TalsoThaveTxT≠Ty.T Conversely,TxTTT yTandTxT≠TyTimplyTbyT(1.7)TxT <T yT orT xT =T yT whereTtheTsecondTc
≤
aseTisTexcluded,ThenceTxT <T y,TasTrequired.
(b) ConsiderTaTpartialTorderTand≤TassumeT(1.6)TasTaTdefinitionTofT<.TToTshowTthatT<TisTtransitive,
TsupposeTxT<Ty,TthatTis,TxTyTandTxT≠Ty,TandTyT<Tz,TthatTis,TyTzTandTyT≠Tz.TBecauseTTTTisTtransitive
≤ ≤
,TxTTTTz.TIfT≤weThadTxT=TzTthen≤
TxTTTTTyTandTyTTTTTxTandThenceTxT=TyTbyTantisymmetryTofTTTT,Twhic
≤ ≤
hTcontradictsT xT ≠T y,TsoTweThaveT xTTTT zT andT xT ≠T z,TthatTis,TxT <T zTbyT(1.6),TasTrequired.
≤ ≤
Also,T<TisTirreflexive,TbecauseTxT<TxTwouldTbyTdefinitionTmeanTxTTTxTandT≤ xT≠Tx,TbutTtheTlatter
TisTnotTtrue.
Finally,TweTshowT(1.7),TsoTweThaveTtoTshowTthatTxT ≤TyTimpliesTxT<TyTorTxT=TyTandTviceTversa,T
givenTthatT<TisTdefinedTbyT(1.6).TLetTxT≤Ty.TThenTifTxT=Ty,TweTareTdone,TotherwiseTxT≠TyTandTt
henTbyTdefinitionTxT<Ty.THence,TxT≤TyTimpliesTxT<TyTorTxT=Ty.TConversely,TsupposeTxT <T yTorTx
T=Ty.T IfTxT <T yTthenTxT ≤TyTbyT(1.6),TandTifTxT=TyTthenTxT ≤T yTbecauseT ≤TisTreflexive.T ThisTcomp
letesTtheTproof.
SolutionTtoTExerciseT1.2
(a) InT analysingT theT gamesT ofT threeT NimT heapsT whereT oneT heapT hasT sizeT one,T weT firstT lookTatTsom
eTexamples,TandTthenTuseTmathematicalTinductionTtoTproveTwhatTweTconjectureTtoTbeTtheTlosingTp
ositions.TATlosingTpositionTisToneTwhereTeveryTmoveTisTtoTaTwinningTposition,TbecauseTthenTt
heTopponentTwillTwin.T TheTpointTofTthisTexerciseTisTtoTformulateTaTpreciseTstatementTtoTbeTpro
ved,TandTthenTtoTproveTit.
First,TifTthereTareTonlyTtwoTheapsTrecallTthatTtheyTareTlosingTifTandTonlyTifTtheTheapsTareTofT
equalTsize.T IfTtheyTareTofTunequalTsize,TthenTtheTwinningTmoveTisTtoTreduceTtheTlargerTheapTs
oTthatTbothTheapsThaveTequalTsize.
3
, ConsiderTthreeTheapsTofTsizesT1,Tm,Tn,TwhereT1TTTTTm≤ TTTTTn.
≤TWeTobserveTtheTfollowing:T1,T1,T
mTisTwinning,TbyTmovingTtoT1,T1,T0.TSimilarly,T1,Tm,TmTisTwinning,TbyTmovingTtoT0,Tm,Tm.TNe
xt,T1,T2,T3TisTlosingT(observedTearlierTinTtheTlecture),TandThenceT1,T2,TnTforTnT4TisTwinning.T1,
T3,TnTisTwinningTforTanyTnT3TbyTmovingTtoT1,T3,T2.TForT1,T4,T5,TreducingTanyTheapTproducesT
≥ ≥
aTwinningTposition,TsoTthisTisTlosing.
TheTgeneralTpatternTforTtheTlosingTpositionsTthusTseemsTtoTbe:T1,Tm,TmT1,TforTeven
+ Tnumbers
Tm.T ThisTincludesTalsoTtheTcaseTmT=T0,TwhichTweTcanTtakeTasTtheTbaseTcaseTforTanTinduction.T
WeTnowTproceedTtoTproveTthisTformally.
FirstTweTshowTthatTifTtheTpositionsTofTtheTformT1,Tm,TnTwithTmTTTTTTnTare ≤ TlosingTwhenTmTisTe
venTandTnT=TmT1,TthenT+ theseTareTtheTonlyTlosingTpositionsTbecauseTanyTotherTpositionT1,Tm,Tn
T withTmT T nT isTwinning.T Namely,TifTmT =TnT thenTaTwinningTmoveTfromT1,Tm,TmTisTtoT0,Tm,Tm,Tso
≤
TweTcanTassumeTmT<Tn.T IfTmTisTevenTthenTnT>TmT T 1T(otherwiseTweTwouldTbeTinTtheTpositionT1,T
+
m,TmT T 1)TandTsoTtheTwinningTmoveTisTtoT1,Tm,TmT T 1.TIfTmTisToddTthenTtheTwinningTmoveTisTto
+ +
T1,Tm,TmT1,TtheTsameTasTpositionT1,TmT1,TmT(thisTwouldT alsoT beT aT winningT moveT fromT 1,Tm,TmT so
T thereT theT winningT moveT isT notT unique).
– −
Second,TweTshowTthatTanyTmoveTfromT1,Tm,TmT+T1TwithTevenTmTisTtoTaTwinningTposition,TusingTa
sTinductiveThypothesisTthatT1,TmJ,TmJT+T1TforTevenTmJTandTmJT<TmTisTaTlosingTposition.TTheTm
oveTtoT0,Tm,TmT+T1TproducesTaTwinningTpositionTwithTcounter-
moveTtoT0,Tm,Tm.TATmoveTtoT1,TmJ,TmT+T1TforTmJT<TmTisTtoTaTwinningTpositionTwithTtheTcounter-
moveTtoT1,TmJ,TmJT+T1TifTmJTisTevenTandTtoT1,TmJ,TmJT−T1TifTmJTisTodd.TATmoveTtoT1,Tm,TmTisTto
TaTwinningTpositionTwithTcounter-
moveTtoT0,Tm,Tm.TATmoveTtoT1,Tm,TmJTwithT mJT<T mTisTalsoTtoTaTwinningTpositionTwithTtheTcounte
r-
moveTtoT1,TmJT−T1,TmJTifT mJTisTodd,TandTtoT1,TmJT 1,TmJTifTmJTisTevenT(inTwhichTcaseTmJT 1T<TmTbe
causeTmTisTeven).TThis+
Tconcludes Tthe Tinduction Tproof. +
ThisTresultTisTinTagreementTwithTtheTtheoremTonTNimTheapTsizesTrepresentedTasTsumsTofTpowersT
ofT2:T 1T T mT ∗T Tn+∗ 0
TisTlosingTifTandTonlyTif,TexceptTforT2 ,TtheTpowersTofT2TmakingTupTmTandTnTcome
+∗
0
TinTpairs.TSoTtheseTmustTbeTtheTsameTpowersTofT2,TexceptTforT1T=T2 ,TwhichToccursTinTonlyTmTorT
n,TwhereTweThaveTassumedTthatTnTisTtheTlargerTnumber,TsoT1TappearsTinTtheTrepresentationTofT n:
T WeT haveT mT =T 2
aTTTTTT2bTTTTTT2c
+ + +T ·T ·T · ·T ·T·T ≥
forT aT >T bT >T cT >TTTTTTTT 1,Ts
+ + + T ·T ·T ·T + +
oT mT isT even,T and,T withT theT sameT a,Tb,Tc,T.T.T.,T nT =T 2aT T T 2bT T T 2c 1T =T mTTTT 1.T Then
1 m
∗T +T ∗ +T ∗ ≡T∗
TTTTTT TTTTT n TTTTTT 0.T The T following T isT an T example T using T the T bit T representation T where
mT =T12T(whichTdeterminesTtheTbitTpatternT1100,TwhichTofTcourseTdependsTonTm):
1 = 0001
12 = 1100
13 = 1101
Nim-sum 0 = 0000
(b) WeTuseT(a).TClearly,T1,T2,T3TisTlosingTasTshownTinT(1.2),TandTbecauseTtheTNim-
sumTofTtheTbinaryTrepresentationsT01,T10,T11TisT00.TExamplesTshowTthatTanyTotherTposition
TisTwinning.TTheTthreeTnumbersTareTn,TnT 1,TnT T 2.TIfTnTisTevenTthenTreducingTtheTheapTofTsizeTn
+ +
T2TtoT1TcreatesTtheTpositionTn,TnT 1,T1TwhichTisTlosingTasTshownTinT(a).TIfTnTisTodd,TthenTnT 1
+ +
TisTevenTandTnTTT2T=T nTTT1TTT1TsoTbyTtheTsameTargument,TaTwinningTmoveTisTtoTreduceTtheT
+ + (T +T )T+
NimTheapTofTsizeTnTtoT1T(whichTonlyTworksTifTnT >T1).
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