SOLUTION MANUAL
Game Theory Basics 1st Edition
By Bernhard von Stengel. Chapters 1 - 12
1
,TABLE OF CONTENTS s s s
1 - Nim and Combinatorial Games
s s s s s
2 - Congestion Games
s s s
3 - Games in Strategic Form
s s s s s
4 - Game Trees with Perfect Information
s s s s s s
5 - Expected Utility
s s s
6 - Mixed Equilibrium
s s s
7 - Brouwer’s Fixed-Point Theorem
s s s s
8 - Zero-Sum Games
s s s
9 - Geometry of Equilibria in Bimatrix Games
s s s s s s s
10 - Game Trees with Imperfect Information
s s s s s s
11 - Bargaining
s s
12 - Correlated Equilibrium
s s s
2
,Game Theory Basics
s s
Solutions to Exercises
s s
©s BernhardsvonsStengels2022
SolutionstosExercises1.1
(a) Lets≤sbesdefinedsbys(1.7).s Tosshowsthats≤sisstransitive,sconsidersx,sy,szswithsxs ≤sysandsys≤sz.sIfsxs=systhensxs
≤sz,sandsifsys=szsthensalsosxs≤sz.sSosthesonlyscasesleftsissxs<sysands ys <s z,swhichsimpliessxs <s zsbecauses<sisst
ransitive,sandshencesxs ≤sz.
Clearly,s≤sissreflexivesbecausesxs=sxsandsthereforesxs ≤sx.
Tosshowsthatsssssis≤santisymmetric,sconsidersxsandsyswithsxsssssysandsysssssx.sIf≤sweshadsxs≠sy≤
sthensxs<sysandsys
<sx,sandsbystransitivitysxs<sxswhichscontradictss(1.38).sHencesxs =s y,sassrequired.s Thissshowssthats≤siss
aspartialsorder.
Finally,swesshows(1.6),ssosweshavestosshowsthatsxs<sysimpliessxsssysandsxs≠sysandsvice ≤ sversa.sLetsxs<sy,swh
ichsimpliessxsysbys(1.7).sIfsweshadsxs=systhensxs<sx,scontradicting
≤ s(1.38),ssoswesalsoshavesxs≠sy.s Converse
ly,sxsss ysandsxs≠sysimplysbys(1.7)sxs <s ysors xs =s yswheresthessecondscasesissexcluded,
≤ shences xs <s y,sassrequire
d.
(b) Considersaspartialsordersandsassume ≤ s(1.6)sassasdefinitionsofs<.sTosshowsthats<sisstransitive,ssuppose
sxs<sy,sthatsis,sxsysandsxs≠sy,sandsys<sz,sthatsis,syszsandsys≠sz.sBecausessssisstransitive,sxssssz.sIfsweshadsxs=szsthe
≤ ≤
nsxsssssysands≤
ysssssxsandshencesxs=sys≤bysantisymmetrysofssss,swhichscontradicts
≤ s xs ≠s y,ssosweshaves xssss zs ands xs
≤
≠s z,sthatsis,sxs <s zsbys(1.6),sassrequired.
≤ ≤
Also,s<sissirreflexive,sbecausesxs<sxswouldsbysdefinitionsmeansxsssxsandsxs≠sx,sbut≤stheslattersissnotstrue.
Finally,swesshows(1.7),ssosweshavestosshowsthatsxs ≤sysimpliessxs<sysorsxs=sysandsvicesversa,sgivensthats<s
issdefinedsbys(1.6).sLetsxs≤sy.sThensifsxs=sy,swesaresdone,sotherwisesxs≠sysandsthensbysdefinitionsxs<sy.s
Hence,sxs≤sysimpliessxs<sysorsxs=sy.sConversely,ssupposesxs <s ysorsxs=sy.s Ifsxs <s ysthensxs ≤sysbys(1.6),san
dsifsxs=systhensxs ≤sysbecauses≤sissreflexive.s Thisscompletessthesproof.
SolutionstosExercises1.2
(a) Ins analysings thes gamess ofs threes Nims heapss wheres ones heaps hass sizes one,s wes firsts looksatssomesexampl
es,sandsthensusesmathematicalsinductionstosproveswhatswesconjecturestosbestheslosingspositions.sAslosin
gspositionsissoneswhereseverysmovesisstosaswinningsposition,sbecausesthensthesopponentswillswin.s T
hespointsofsthissexercisesisstosformulatesasprecisesstatementstosbesproved,sandsthenstosprovesit.
First,sifstheresaresonlystwosheapssrecallsthatstheysareslosingsifsandsonlysifsthesheapssaresofsequalssize.s
Ifstheysaresofsunequalssize,sthenstheswinningsmovesisstosreducestheslargersheapssosthatsbothsheapssh
avesequalssize.
3
, Considersthreesheapssofssizess1,sm,sn,swheres1sssssmsssssn.sWe
≤sobserve
≤ sthesfollowing:s1,s1,smsisswinning,s
bysmovingstos1,s1,s0.sSimilarly,s1,sm,smsisswinning,sbysmovingstos0,sm,sm.sNext,s1,s2,s3sisslosings(obser
vedsearliersinstheslecture),sandshences1,s2,snsforsns4sisswinning.s1,s3,snsisswinningsforsanysns3sbysmovi
ngstos1,s3,s2.sFors1,s4,s5,sreducingsanysheapsproducessaswinningsposition,ssosthississlosing.
≥ ≥
Thesgeneralspatternsforstheslosingspositionssthussseemsstosbe:s1,sm,sms1,sforsevensnumbers + sm.s Thiss
includessalsosthescasesms=s0,swhichswescanstakesassthesbasescasesforsansinduction.s Wesnowsproceedst
osprovesthissformally.
Firstswesshowsthatsifsthespositionssofsthesforms1,sm,snswithsmssssssnsareslosingswhen ≤ smsissevensandsns=s
ms1,sthensthesesaresthesonly + slosingspositionssbecausesanysotherspositions1,sm,sns withsms s ns isswinnin
g.s Namely,sifsms =sns thensaswinning ≤ smovesfroms1,sm,smsissto s0,sm,sm,ssoswescan sassumesms<sn.s Ifsmsissevens
thensns>sms s 1s(otherwisesweswouldsbesinsthespositions1,sm,sms s 1)sandssostheswinningsmovesisstos1,sm,s
+
ms s 1.sIfsmsissoddsthenstheswinningsmovesisstos1,sm,sms1,sthessamesasspositions1,sms1,sms(thisswoulds alsos b
+ +
es as winnings moves froms 1,sm,sms sos theres thes winnings moves iss nots unique).
– −
Second,swesshowsthatsanysmovesfroms1,sm,sms+s1swithsevensmsisstosaswinningsposition,susingsassinductiv
eshypothesissthats1,smJ,smJs+s1sforsevensmJsandsmJs<smsissaslosingsposition.sThesmovestos0,sm,sms+s1spro
ducessaswinningspositionswithscounter-
movestos0,sm,sm.sAsmovestos1,smJ,sms+s1sforsmJs<smsisstosaswinningspositionswithsthescounter-
movestos1,smJ,smJs+s1sifsmJsissevensandstos1,smJ,smJs−s1sifsmJsissodd.sAsmovestos1,sm,smsisstosaswinningsposi
tionswithscounter-
movestos0,sm,sm.sAsmovestos1,sm,smJswiths mJs<s msissalsostosaswinningspositionswithsthescounter-
movestos1,smJs−s1,smJsifs mJsissodd,sandstos1,smJs 1,smJsifsmJsissevens(inswhichscasesmJs 1s<smsbecausesmsisseve
n).sThissconcludessthesinductionsproof.
+ +
ThissresultsissinsagreementswithsthestheoremsonsNimsheapssizessrepresentedsasssumssofspowerssofs2:s 1s
0
s ms s nsisslosingsifsandsonlysif,sexceptsfors2 ,sthespowerssofs2smakingsupsmsandsnscomesinspairs.sSosthesesmu
∗s +∗ +∗
stsbesthessamespowerssofs2,sexceptsfors1s=s20,swhichsoccurssinsonlysmsorsn,swheresweshavesassumedsthatsns
isstheslargersnumber,ssos1sappearssins thes representations ofs n:s Wes haves ms =s 2assssss2bssssss2c
fors as >s bs >s cs >ssssssss 1,ssos ms iss e
a s s s bs s s c + + +s ·s ·s · ·s ·s·s ≥
ven,s and,s withs thes sames a,sb,sc,s.s.s.,s ns =s 2 2 2 1s =s mssss 1.s Then
+ + +s ·s ·s ·s + +
∗1s ssssss
+s ∗msssss n+ssssss
s∗
0.s≡The
s∗
s followings iss ans examples usings thes bits representations where
ms =s12s(whichsdeterminessthesbitspatterns1100,swhichsofscoursesdependssonsm):
1 = 0001
12 = 1100
13 = 1101
Nim-sum 0 = 0000
(b) Wesuses(a).sClearly,s1,s2,s3sisslosingsassshownsins(1.2),sandsbecausesthesNim-
sumsofsthesbinarysrepresentationss01,s10,s11siss00.sExamplessshowsthatsanysotherspositionsisswinni
ng.sThesthreesnumberssaresn,sns 1,sns s 2.sIfsnsis+
sevensthensreducingsthesheapsofssizesns2stos1screatessthes
+
positionsn,sns 1,s1swhichsisslosingsassshownsins(a).sIfsnsissodd,sthensns 1sissevensandsnsss2s=s nsss1sss1ssosb
+ +
ysthessamesargument,saswinningsmovesisstosreducesthesNimsheapsofssizesnstos1s(whichsonlysworkssifs
+ + (s +s )s+
ns >s1).
4
Game Theory Basics 1st Edition
By Bernhard von Stengel. Chapters 1 - 12
1
,TABLE OF CONTENTS s s s
1 - Nim and Combinatorial Games
s s s s s
2 - Congestion Games
s s s
3 - Games in Strategic Form
s s s s s
4 - Game Trees with Perfect Information
s s s s s s
5 - Expected Utility
s s s
6 - Mixed Equilibrium
s s s
7 - Brouwer’s Fixed-Point Theorem
s s s s
8 - Zero-Sum Games
s s s
9 - Geometry of Equilibria in Bimatrix Games
s s s s s s s
10 - Game Trees with Imperfect Information
s s s s s s
11 - Bargaining
s s
12 - Correlated Equilibrium
s s s
2
,Game Theory Basics
s s
Solutions to Exercises
s s
©s BernhardsvonsStengels2022
SolutionstosExercises1.1
(a) Lets≤sbesdefinedsbys(1.7).s Tosshowsthats≤sisstransitive,sconsidersx,sy,szswithsxs ≤sysandsys≤sz.sIfsxs=systhensxs
≤sz,sandsifsys=szsthensalsosxs≤sz.sSosthesonlyscasesleftsissxs<sysands ys <s z,swhichsimpliessxs <s zsbecauses<sisst
ransitive,sandshencesxs ≤sz.
Clearly,s≤sissreflexivesbecausesxs=sxsandsthereforesxs ≤sx.
Tosshowsthatsssssis≤santisymmetric,sconsidersxsandsyswithsxsssssysandsysssssx.sIf≤sweshadsxs≠sy≤
sthensxs<sysandsys
<sx,sandsbystransitivitysxs<sxswhichscontradictss(1.38).sHencesxs =s y,sassrequired.s Thissshowssthats≤siss
aspartialsorder.
Finally,swesshows(1.6),ssosweshavestosshowsthatsxs<sysimpliessxsssysandsxs≠sysandsvice ≤ sversa.sLetsxs<sy,swh
ichsimpliessxsysbys(1.7).sIfsweshadsxs=systhensxs<sx,scontradicting
≤ s(1.38),ssoswesalsoshavesxs≠sy.s Converse
ly,sxsss ysandsxs≠sysimplysbys(1.7)sxs <s ysors xs =s yswheresthessecondscasesissexcluded,
≤ shences xs <s y,sassrequire
d.
(b) Considersaspartialsordersandsassume ≤ s(1.6)sassasdefinitionsofs<.sTosshowsthats<sisstransitive,ssuppose
sxs<sy,sthatsis,sxsysandsxs≠sy,sandsys<sz,sthatsis,syszsandsys≠sz.sBecausessssisstransitive,sxssssz.sIfsweshadsxs=szsthe
≤ ≤
nsxsssssysands≤
ysssssxsandshencesxs=sys≤bysantisymmetrysofssss,swhichscontradicts
≤ s xs ≠s y,ssosweshaves xssss zs ands xs
≤
≠s z,sthatsis,sxs <s zsbys(1.6),sassrequired.
≤ ≤
Also,s<sissirreflexive,sbecausesxs<sxswouldsbysdefinitionsmeansxsssxsandsxs≠sx,sbut≤stheslattersissnotstrue.
Finally,swesshows(1.7),ssosweshavestosshowsthatsxs ≤sysimpliessxs<sysorsxs=sysandsvicesversa,sgivensthats<s
issdefinedsbys(1.6).sLetsxs≤sy.sThensifsxs=sy,swesaresdone,sotherwisesxs≠sysandsthensbysdefinitionsxs<sy.s
Hence,sxs≤sysimpliessxs<sysorsxs=sy.sConversely,ssupposesxs <s ysorsxs=sy.s Ifsxs <s ysthensxs ≤sysbys(1.6),san
dsifsxs=systhensxs ≤sysbecauses≤sissreflexive.s Thisscompletessthesproof.
SolutionstosExercises1.2
(a) Ins analysings thes gamess ofs threes Nims heapss wheres ones heaps hass sizes one,s wes firsts looksatssomesexampl
es,sandsthensusesmathematicalsinductionstosproveswhatswesconjecturestosbestheslosingspositions.sAslosin
gspositionsissoneswhereseverysmovesisstosaswinningsposition,sbecausesthensthesopponentswillswin.s T
hespointsofsthissexercisesisstosformulatesasprecisesstatementstosbesproved,sandsthenstosprovesit.
First,sifstheresaresonlystwosheapssrecallsthatstheysareslosingsifsandsonlysifsthesheapssaresofsequalssize.s
Ifstheysaresofsunequalssize,sthenstheswinningsmovesisstosreducestheslargersheapssosthatsbothsheapssh
avesequalssize.
3
, Considersthreesheapssofssizess1,sm,sn,swheres1sssssmsssssn.sWe
≤sobserve
≤ sthesfollowing:s1,s1,smsisswinning,s
bysmovingstos1,s1,s0.sSimilarly,s1,sm,smsisswinning,sbysmovingstos0,sm,sm.sNext,s1,s2,s3sisslosings(obser
vedsearliersinstheslecture),sandshences1,s2,snsforsns4sisswinning.s1,s3,snsisswinningsforsanysns3sbysmovi
ngstos1,s3,s2.sFors1,s4,s5,sreducingsanysheapsproducessaswinningsposition,ssosthississlosing.
≥ ≥
Thesgeneralspatternsforstheslosingspositionssthussseemsstosbe:s1,sm,sms1,sforsevensnumbers + sm.s Thiss
includessalsosthescasesms=s0,swhichswescanstakesassthesbasescasesforsansinduction.s Wesnowsproceedst
osprovesthissformally.
Firstswesshowsthatsifsthespositionssofsthesforms1,sm,snswithsmssssssnsareslosingswhen ≤ smsissevensandsns=s
ms1,sthensthesesaresthesonly + slosingspositionssbecausesanysotherspositions1,sm,sns withsms s ns isswinnin
g.s Namely,sifsms =sns thensaswinning ≤ smovesfroms1,sm,smsissto s0,sm,sm,ssoswescan sassumesms<sn.s Ifsmsissevens
thensns>sms s 1s(otherwisesweswouldsbesinsthespositions1,sm,sms s 1)sandssostheswinningsmovesisstos1,sm,s
+
ms s 1.sIfsmsissoddsthenstheswinningsmovesisstos1,sm,sms1,sthessamesasspositions1,sms1,sms(thisswoulds alsos b
+ +
es as winnings moves froms 1,sm,sms sos theres thes winnings moves iss nots unique).
– −
Second,swesshowsthatsanysmovesfroms1,sm,sms+s1swithsevensmsisstosaswinningsposition,susingsassinductiv
eshypothesissthats1,smJ,smJs+s1sforsevensmJsandsmJs<smsissaslosingsposition.sThesmovestos0,sm,sms+s1spro
ducessaswinningspositionswithscounter-
movestos0,sm,sm.sAsmovestos1,smJ,sms+s1sforsmJs<smsisstosaswinningspositionswithsthescounter-
movestos1,smJ,smJs+s1sifsmJsissevensandstos1,smJ,smJs−s1sifsmJsissodd.sAsmovestos1,sm,smsisstosaswinningsposi
tionswithscounter-
movestos0,sm,sm.sAsmovestos1,sm,smJswiths mJs<s msissalsostosaswinningspositionswithsthescounter-
movestos1,smJs−s1,smJsifs mJsissodd,sandstos1,smJs 1,smJsifsmJsissevens(inswhichscasesmJs 1s<smsbecausesmsisseve
n).sThissconcludessthesinductionsproof.
+ +
ThissresultsissinsagreementswithsthestheoremsonsNimsheapssizessrepresentedsasssumssofspowerssofs2:s 1s
0
s ms s nsisslosingsifsandsonlysif,sexceptsfors2 ,sthespowerssofs2smakingsupsmsandsnscomesinspairs.sSosthesesmu
∗s +∗ +∗
stsbesthessamespowerssofs2,sexceptsfors1s=s20,swhichsoccurssinsonlysmsorsn,swheresweshavesassumedsthatsns
isstheslargersnumber,ssos1sappearssins thes representations ofs n:s Wes haves ms =s 2assssss2bssssss2c
fors as >s bs >s cs >ssssssss 1,ssos ms iss e
a s s s bs s s c + + +s ·s ·s · ·s ·s·s ≥
ven,s and,s withs thes sames a,sb,sc,s.s.s.,s ns =s 2 2 2 1s =s mssss 1.s Then
+ + +s ·s ·s ·s + +
∗1s ssssss
+s ∗msssss n+ssssss
s∗
0.s≡The
s∗
s followings iss ans examples usings thes bits representations where
ms =s12s(whichsdeterminessthesbitspatterns1100,swhichsofscoursesdependssonsm):
1 = 0001
12 = 1100
13 = 1101
Nim-sum 0 = 0000
(b) Wesuses(a).sClearly,s1,s2,s3sisslosingsassshownsins(1.2),sandsbecausesthesNim-
sumsofsthesbinarysrepresentationss01,s10,s11siss00.sExamplessshowsthatsanysotherspositionsisswinni
ng.sThesthreesnumberssaresn,sns 1,sns s 2.sIfsnsis+
sevensthensreducingsthesheapsofssizesns2stos1screatessthes
+
positionsn,sns 1,s1swhichsisslosingsassshownsins(a).sIfsnsissodd,sthensns 1sissevensandsnsss2s=s nsss1sss1ssosb
+ +
ysthessamesargument,saswinningsmovesisstosreducesthesNimsheapsofssizesnstos1s(whichsonlysworkssifs
+ + (s +s )s+
ns >s1).
4
