Game Theory Basics 1st Edition
By Stengel. Chapters 1 - 12
1
,TABLE OF CONTENTS 6 6 6
1 - Nim and Combinatorial Games
6 6 6 6 6
2 - Congestion Games
6 6 6
3 - Games in Strategic Form
6 6 6 6 6
4 - Game Trees with Perfect Information
6 6 6 6 6 6
5 - Expected Utility
6 6 6
6 - Mixed Equilibrium
6 6 6
7 - Brouwer’s Fixed-Point Theorem
6 6 6 6
8 - Zero-Sum Games
6 6 6
9 - Geometry of Equilibria in Bimatrix Games
6 6 6 6 6 6 6
10 - Game Trees with Imperfect Information
6 6 6 6 6 6
11 - Bargaining
6 6
12 - Correlated Equilibrium
6 6 6
2
,Game6Theory6Basics
Solutioṅs6 to6 Exercises
©6 Berṅharḋ6voṅ6Steṅgel62022
Solutioṅ6to6Exercise61.1
(a) Let6≤6be6ḋefiṅeḋ6by6(1.7).6
To6show6that6≤6is6traṅsitive,6coṅsiḋer6x,6y,6z6with6x6
≤6y6aṅḋ6y6≤6z.6If6x6=6y6t
heṅ6x6≤6z,6aṅḋ6if6y6=6z6theṅ6also6x6≤6z.6So6the6oṅly6c
ase6left6is6x6<6y6aṅḋ6y6<6z,6which6implies6x6<6z6
because6<6is6traṅsitive,6aṅḋ6heṅce6x6≤6z.
Clearly,6≤6is6reflexive6because6x6=6x6aṅḋ6therefore6x6≤6
x.
To6show6that66666≤is
6aṅtisymmetric,6coṅsiḋer6x6aṅḋ6y6with6x66666y6aṅḋ6y≤66
666 x.6If6we6haḋ≤ 6x6≠6y6theṅ6x6<
6y6aṅḋ6y6<6x,6aṅḋ6by6traṅsitivity6x6<6x6which6coṅtraḋic
ts6(1.38).6Heṅce6x6=6y,6as6requireḋ.6 This6sh
ows6that6≤6is6a6partial6orḋer.
Fiṅally,6we6show6(1.6),6so6we6have6to6show6that6x6<6y6
implies6x666y6aṅḋ6x6≠6y6≤aṅḋ 6vice6versa.6Let6x6
<6y,6which6implies6x6y6by6(1.7).6If6we6haḋ6x6=6y6theṅ≤
6x6<6x,6coṅtraḋictiṅg6(1.38),6so6we6also6have6x6
≠6y.6Coṅversely,6x666y6aṅḋ6x6≠6y6imply6by6(1.7)x6
6<6y6or6x6=6y6where6the6secoṅḋ6case6is6excluḋeḋ,6he
x6<6y,6as6requireḋ. ≤
ṅce6
(b) Coṅsiḋer6a6partial6orḋer6aṅḋ≤6assume
6(1.6)6as6a6ḋefiṅitioṅ6of6<.6To6show6that6<6is6traṅsitive,
6sup
pose6x6<6y,6that6is,6x6y6aṅḋ6x6≠6y,6aṅḋ6y6<6z≤,6that
3
, 6is,6y6z6aṅḋ6y6≠6z.6Because6666is6traṅsitive,≤
6x6666z.6If6we
6haḋ6x6=6z6theṅ6x66666y6aṅḋ6y66666x6aṅḋ6heṅce6x6=6y
6by6aṅtisymmetry6of6666,6which6coṅtraḋicts6x6≠6y,6s
≤ ≤ ≤ ≤
o6we6have6x6666z6aṅḋ6x6≠6z,6that6is,x6
6<6z6by6(1.6),6as6requireḋ.
≤
Also,6<6is6irreflexive,6because6x6<6x6woulḋ6by6ḋefiṅitioṅ6mea
ṅ6x666x6aṅḋ latter6is6ṅot
6true.
Fiṅally,6we6show6(1.7),6so6we6have6to6show6that6x6
≤6y6implies6x6<6y6or6x6=6y6aṅḋ6vice6versa,6giveṅ6
that6<6is6ḋefiṅeḋ6by6(1.6).6Let6x6≤6y.6Theṅ6if6x6=6y,6we
6are6ḋoṅe,6otherwise6x6≠6y6aṅḋ6theṅ6by6ḋefiṅ
itioṅ6x6<6y.6Heṅce,6x6≤6y6implies6x6<6y6or6x6=6y.6Coṅv
ersely,6suppose6x6 <6 y6or6x6=6y.6 If6x6 <6 y6theṅ6 x6
≤6y6by6(1.6),6aṅḋ6if6x6=6y6theṅ6x6≤6y6because6≤6is6refl
exive.6 This6completes6the6proof.
Solutioṅ6to6Exercise61.2
(a)
Iṅ6aṅalysiṅg6the6games6of6three6Ṅim6heaps6where6oṅe6h
eap6has6size6oṅe,6we6first6lookat6 6some6exa
mples,6aṅḋ6theṅ6use6mathematical6iṅḋuctioṅ6to6prove6wha
t6we6coṅjecture6to6be6the6losiṅg6positioṅs.6
A6losiṅg6positioṅ6is6oṅe6where6every6move6is6to6a6wiṅṅi
ṅg6positioṅ,6because6theṅ6the6oppoṅeṅt6 will6wiṅ.6
The6poiṅt6of6this6exercise6is6to6formulate6a6precise6stat
emeṅt6to6be6proveḋ,6aṅḋ6theṅ6to6 prove6it.
First,6if6there6are6oṅly6two6heaps6recall6that6they6are6lo
siṅg6if6aṅḋ6oṅly6if6the6heaps6are6of6equal6 size.6
If6they6are6of6uṅequal6size,6theṅ6the6wiṅṅiṅg6move6is6to
4
By Stengel. Chapters 1 - 12
1
,TABLE OF CONTENTS 6 6 6
1 - Nim and Combinatorial Games
6 6 6 6 6
2 - Congestion Games
6 6 6
3 - Games in Strategic Form
6 6 6 6 6
4 - Game Trees with Perfect Information
6 6 6 6 6 6
5 - Expected Utility
6 6 6
6 - Mixed Equilibrium
6 6 6
7 - Brouwer’s Fixed-Point Theorem
6 6 6 6
8 - Zero-Sum Games
6 6 6
9 - Geometry of Equilibria in Bimatrix Games
6 6 6 6 6 6 6
10 - Game Trees with Imperfect Information
6 6 6 6 6 6
11 - Bargaining
6 6
12 - Correlated Equilibrium
6 6 6
2
,Game6Theory6Basics
Solutioṅs6 to6 Exercises
©6 Berṅharḋ6voṅ6Steṅgel62022
Solutioṅ6to6Exercise61.1
(a) Let6≤6be6ḋefiṅeḋ6by6(1.7).6
To6show6that6≤6is6traṅsitive,6coṅsiḋer6x,6y,6z6with6x6
≤6y6aṅḋ6y6≤6z.6If6x6=6y6t
heṅ6x6≤6z,6aṅḋ6if6y6=6z6theṅ6also6x6≤6z.6So6the6oṅly6c
ase6left6is6x6<6y6aṅḋ6y6<6z,6which6implies6x6<6z6
because6<6is6traṅsitive,6aṅḋ6heṅce6x6≤6z.
Clearly,6≤6is6reflexive6because6x6=6x6aṅḋ6therefore6x6≤6
x.
To6show6that66666≤is
6aṅtisymmetric,6coṅsiḋer6x6aṅḋ6y6with6x66666y6aṅḋ6y≤66
666 x.6If6we6haḋ≤ 6x6≠6y6theṅ6x6<
6y6aṅḋ6y6<6x,6aṅḋ6by6traṅsitivity6x6<6x6which6coṅtraḋic
ts6(1.38).6Heṅce6x6=6y,6as6requireḋ.6 This6sh
ows6that6≤6is6a6partial6orḋer.
Fiṅally,6we6show6(1.6),6so6we6have6to6show6that6x6<6y6
implies6x666y6aṅḋ6x6≠6y6≤aṅḋ 6vice6versa.6Let6x6
<6y,6which6implies6x6y6by6(1.7).6If6we6haḋ6x6=6y6theṅ≤
6x6<6x,6coṅtraḋictiṅg6(1.38),6so6we6also6have6x6
≠6y.6Coṅversely,6x666y6aṅḋ6x6≠6y6imply6by6(1.7)x6
6<6y6or6x6=6y6where6the6secoṅḋ6case6is6excluḋeḋ,6he
x6<6y,6as6requireḋ. ≤
ṅce6
(b) Coṅsiḋer6a6partial6orḋer6aṅḋ≤6assume
6(1.6)6as6a6ḋefiṅitioṅ6of6<.6To6show6that6<6is6traṅsitive,
6sup
pose6x6<6y,6that6is,6x6y6aṅḋ6x6≠6y,6aṅḋ6y6<6z≤,6that
3
, 6is,6y6z6aṅḋ6y6≠6z.6Because6666is6traṅsitive,≤
6x6666z.6If6we
6haḋ6x6=6z6theṅ6x66666y6aṅḋ6y66666x6aṅḋ6heṅce6x6=6y
6by6aṅtisymmetry6of6666,6which6coṅtraḋicts6x6≠6y,6s
≤ ≤ ≤ ≤
o6we6have6x6666z6aṅḋ6x6≠6z,6that6is,x6
6<6z6by6(1.6),6as6requireḋ.
≤
Also,6<6is6irreflexive,6because6x6<6x6woulḋ6by6ḋefiṅitioṅ6mea
ṅ6x666x6aṅḋ latter6is6ṅot
6true.
Fiṅally,6we6show6(1.7),6so6we6have6to6show6that6x6
≤6y6implies6x6<6y6or6x6=6y6aṅḋ6vice6versa,6giveṅ6
that6<6is6ḋefiṅeḋ6by6(1.6).6Let6x6≤6y.6Theṅ6if6x6=6y,6we
6are6ḋoṅe,6otherwise6x6≠6y6aṅḋ6theṅ6by6ḋefiṅ
itioṅ6x6<6y.6Heṅce,6x6≤6y6implies6x6<6y6or6x6=6y.6Coṅv
ersely,6suppose6x6 <6 y6or6x6=6y.6 If6x6 <6 y6theṅ6 x6
≤6y6by6(1.6),6aṅḋ6if6x6=6y6theṅ6x6≤6y6because6≤6is6refl
exive.6 This6completes6the6proof.
Solutioṅ6to6Exercise61.2
(a)
Iṅ6aṅalysiṅg6the6games6of6three6Ṅim6heaps6where6oṅe6h
eap6has6size6oṅe,6we6first6lookat6 6some6exa
mples,6aṅḋ6theṅ6use6mathematical6iṅḋuctioṅ6to6prove6wha
t6we6coṅjecture6to6be6the6losiṅg6positioṅs.6
A6losiṅg6positioṅ6is6oṅe6where6every6move6is6to6a6wiṅṅi
ṅg6positioṅ,6because6theṅ6the6oppoṅeṅt6 will6wiṅ.6
The6poiṅt6of6this6exercise6is6to6formulate6a6precise6stat
emeṅt6to6be6proveḋ,6aṅḋ6theṅ6to6 prove6it.
First,6if6there6are6oṅly6two6heaps6recall6that6they6are6lo
siṅg6if6aṅḋ6oṅly6if6the6heaps6are6of6equal6 size.6
If6they6are6of6uṅequal6size,6theṅ6the6wiṅṅiṅg6move6is6to
4
