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
Solutions to Exercises
T T
©T BernhardTvonTStengelT2022
SolutionTtoTExerciseT1.1
(a) LetT≤TbeTdefinedTbyT(1.7).T ToTshowTthatT≤TisTtransitive,TconsiderTx,Ty,TzTwithTxT ≤TyTandTyT≤Tz.TIfTxT=T
yTthenTxT≤Tz,TandTifTyT=TzTthenTalsoTxT≤Tz.TSoTtheTonlyTcaseTleftTisTxT<TyTandT yT <T z,TwhichTimplies
T xT <T zTbecauseT <TisTtransitive,TandThenceT xT ≤Tz.
Clearly,T≤TisTreflexiveTbecauseTxT=TxTandTthereforeTxT ≤Tx.
ToTshowTthatTTTTT≤isTantisymmetric,TconsiderTxTandTyTwithTxTTTTTyTand≤TyTTTTTx.TIfTwe≤ThadTxT≠TyTthen
TxT<TyTandTyT<Tx,TandTbyTtransitivityTxT<TxTwhichTcontradictsT(1.38).THenceTxT =T y,TasTrequired.T T
hisTshowsTthatT≤TisTaTpartialTorder.
Finally,TweTshowT(1.6),TsoTweThaveTtoTshowTthatTxT<TyTimpliesTxTTTyTandTxT≠≤TyTandTviceTversa.TLet
TxT<Ty,TwhichTimpliesTxTyTbyT(1.7).TIfTweThadTxT=TyTthenTxT<Tx,TcontradictingT(1.38),TsoTweTalsoTha
≤
veTxT≠Ty.T Conversely,TxTTT yTandTxT≠TyTimplyTbyT(1.7)TxT <T yTorT xT =T yTwhere
≤ TtheTsecondTcaseTisTexclu
ded,ThenceT xT <T y,TasTrequired.
(b) ConsiderTaTpartialTorderTand≤TassumeT(1.6)TasTaTdefinitionTofT<.TToTshowTthatT<TisTtransitive,Ts
upposeTxT<Ty,TthatTis,TxTyTandTxT≠Ty,TandT≤ yT<Tz,TthatTis,TyTzTandTyT≠Tz.TBecauseTTTTis≤
Ttransitive,TxTTTT
z.TIfTweThad ≤ TxT=TzTthenTxTTTTTyTandTyTTTTTxTandThenceTxT=TyTbyTantisymmetryT ofTTTT ,TwhichTcontrad
≤ ≤ ≤
ictsT xT ≠T y,TsoTweThaveT xTTTT zTandT xT ≠T z,TthatTis,TxT <T zTbyT(1.6),TasTrequired.
≤ ≤
Also,T<TisTirreflexive,TbecauseTxT<TxTwouldTbyTdefinitionTmeanTxTTTxTandTxT≠≤Tx,TbutTtheTlatterTisT
notTtrue.
Finally,TweTshowT(1.7),TsoTweThaveTtoTshowTthatTxT ≤TyTimpliesTxT<TyTorTxT=TyTandTviceTversa,Tgiv
enTthatT<TisTdefinedTbyT(1.6).TLetTxT≤Ty.TThenTifTxT=Ty,TweTareTdone,TotherwiseTxT≠TyTandTthenTb
yTdefinitionTxT<Ty.THence,TxT≤TyTimpliesTxT<TyTorTxT=Ty.TConversely,TsupposeTxT <T yTorTxT=Ty.T IfTx
T <T yTthenTxT ≤TyTbyT(1.6),TandTifTxT=TyTthenTxT ≤T yTbecauseT ≤TisTreflexive.T ThisTcompletesTtheTpro
of.
SolutionTtoTExerciseT1.2
(a) InT analysingT theT gamesT ofT threeT NimT heapsT whereT oneT heapT hasT sizeT one,T weT firstT lookTatTsomeTe
xamples,TandTthenTuseTmathematicalTinductionTtoTproveTwhatTweTconjectureTtoTbeTtheTlosingTpositi
ons.TATlosingTpositionTisToneTwhereTeveryTmoveTisTtoTaTwinningTposition,TbecauseTthenTtheTopp
onentTwillTwin.T TheTpointTofTthisTexerciseTisTtoTformulateTaTpreciseTstatementTtoTbeTproved,Tand
TthenTtoTproveTit.
First,TifTthereTareTonlyTtwoTheapsTrecallTthatTtheyTareTlosingTifTandTonlyTifTtheTheapsTareTofTeq
ualTsize.T IfTtheyTareTofTunequalTsize,TthenTtheTwinningTmoveTisTtoTreduceTtheTlargerTheapTsoTtha
tTbothTheapsThaveTequalTsize.
3
, ConsiderTthreeTheapsTofTsizesT1,Tm,Tn,TwhereT1TTTTTmTTTTT
≤n.TWe≤ TobserveTtheTfollowing:T1,T1,TmTisT
winning,TbyTmovingTtoT1,T1,T0.TSimilarly,T1,Tm,TmTisTwinning,TbyTmovingTtoT0,Tm,Tm.TNext,T1,T2,
T3TisTlosingT(observedTearlierTinTtheTlecture),TandThenceT1,T2,TnTforTnT4TisTwinning.T1,T3,TnTisTwi
nningTforTanyTnT3TbyTmovingTtoT1,T3,T2.TForT1,T4,T5,TreducingTanyTheapTproducesTaTwinningTpo
≥ ≥
sition,TsoTthisTisTlosing.
TheTgeneralTpatternTforTtheTlosingTpositionsTthusTseemsTtoTbe:T1,Tm,TmT1,TforTeven + TnumbersTm
.T ThisTincludesTalsoTtheTcaseTmT=T0,TwhichTweTcanTtakeTasTtheTbaseTcaseTforTanTinduction.T WeTn
owTproceedTtoTproveTthisTformally.
FirstTweTshowTthatTifTtheTpositionsTofTtheTformT1,Tm,TnTwithTmTTTTTTnTare≤ TlosingTwhenTmTisTevenT
andTnT=TmT1,TthenTtheseT+areTtheTonlyTlosingTpositionsTbecauseTanyTotherTpositionT1,Tm,TnT withT
mT T nT isTwinning.T Namely,T≤ifTmT =TnT thenTaTwinningTmoveTfromT1,Tm,TmTisTtoT0,Tm,Tm,TsoTweTcanTas
sumeTmT<Tn.T IfTmTisTevenTthenTnT>TmT T 1T(otherwiseTweTwouldTbeTinTtheTpositionT1,Tm,TmT T 1)Tand
+
TsoTtheTwinningTmoveTisTtoT1,Tm,T mT T 1.TIfTmTisToddTthenTtheTwinningTmoveTisTtoT1,Tm,T mT 1,TtheTsa
+ +
meTasTpositionT1,TmT1,TmT(thisTwouldT alsoT beT aT winningT moveT fromT 1,Tm,TmT soT thereT theT winningT
moveT isT notT unique). – −
Second,TweTshowTthatTanyTmoveTfromT1,Tm,TmT+T1TwithTevenTmTisTtoTaTwinningTposition,TusingTasTi
nductiveThypothesisTthatT1,TmJ,TmJT+T1TforTevenTmJTandTmJT<TmTisTaTlosingTposition.TTheTmoveTt
oT0,Tm,TmT+T1TproducesTaTwinningTpositionTwithTcounter-
moveTtoT0,Tm,Tm.TATmoveTtoT1,TmJ,TmT+T1TforTmJT<TmTisTtoTaTwinningTpositionTwithTtheTcounter-
moveTtoT1,TmJ,TmJT+T1TifTmJTisTevenTandTtoT1,TmJ,TmJT−T1TifTmJTisTodd.TATmoveTtoT1,Tm,TmTisTtoTaTwi
nningTpositionTwithTcounter-
moveTtoT0,Tm,Tm.TATmoveTtoT1,Tm,TmJTwithT mJT<T mTisTalsoTtoTaTwinningTpositionTwithTtheTcounter-
moveTtoT1,TmJT−T1,TmJTifT mJTisTodd,TandTtoT1,TmJT 1,TmJTifTmJTisTevenT(inTwhichTcaseTmJT 1T<TmTbecau
seTmTisTeven).TThisTconcludesTtheTinductionTproof.
+ +
ThisTresultTisTinTagreementTwithTtheTtheoremTonTNimTheapTsizesTrepresentedTasTsumsTofTpowersTofT
2:T 1T T mT T nT∗isTTlosing 0
TifTandTonlyTif,TexceptTforT2 ,TtheTpowersTofT2TmakingTupTmTandTnTcomeTinTpa
+∗ +∗
irs.TSoTtheseTmustTbeTtheTsameTpowersTofT2,TexceptTforT1T=T20,TwhichToccursTinTonlyTmTorTn,TwhereT
weThaveTassumedTthatTnTisTtheTlargerTnumber,TsoT1TappearsTinT theTrepresentationT ofT n:T WeThaveT
mT =T 2aTTTTTT2bTTTTTT2c
+ + +T ·T ·T · ·T ·T ·T ≥
forT aT >T bT >T cT >TTTTTTTT 1,TsoT
+ + +T ·T ·T ·T + +
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 nTTTTTT 0.T The T followingT isT anT example T using T theT bitT 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.TExamplesTshowTthatTanyTotherTpositionTisT
winning.TTheTthreeTnumbersTareTn,TnT 1,TnT+T 2.TIfT+
nTisTevenTthenTreducingTtheTheapTofTsizeTnT2TtoT
1TcreatesTtheTpositionTn,TnT 1,T1TwhichTisTlosingTasTshownTinT(a).TIfTnTisTodd,TthenTnT 1TisTevenT
+ +
andTnTTT2T=T nTTT1TTT1TsoTbyTtheTsameTargument,TaTwinningTmoveTisTtoTreduceTtheTNimTheapTof
+ + (T +T )T+
TsizeTnTtoT1T(whichTonlyTworksTifTnT >T1).
4
