COVER PAGE FOR A WRITTEN EXAMINATION (ON CAMPUS)
Name of subject : Games and Economic Behavior
Subject code : 35V5B2-B-6
Date of examination : June 21, 2021
Length of examination : 3 hours
Lecturers : Prof. Dr. P.E.M. Borm
Telephone of departmental secretary : 013 4662430
Students are expected to conduct themselves properly during examinations and to obey any
instructions given to them by examiners and invigilators.
Firm action will be taken in the event that academic fraud is discovered.
SPECIFICS:
Open book examination : no
Use of graphical calculator : yes
Print on both sides : yes
Scrap paper available : yes
EXAM INSTRUCTIONS:
1. This exam consists of five exercises. The maximum number of score points is 100. For
each exercise, the maximal score is indicated separately.
2. Please provide complete solutions to each exercise. Each of your answers should be
clearly motivated.
3. Please have a look through the whole exam before you start working on the exercises. In
doing the exercises you do not have to follow the order of the questions in the exam.
4. Do not forget to write down your name and student number (SNR, 7 digits) on each page
you hand in.
5. You can use scrap paper to work out the exercises. Scrap paper however will not be
graded.
6. After the examination, the exercises of the exam, final answers, and a grading scheme
will become available via Canvas.
All students need to hand in the exam and their work (including scrap paper).
Good luck!
, Exam Games and Economic Behavior (35V5B2-B-6) June 21, 2021
Exercise 1 (25 points)
a) Consider the game v ∈ T U N with N = {1, 2, 3} given by
S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
v(S) 0 0 b b 12 24 0 36
with b ∈ R.
(i) Let b = 3. Decompose v into unanimity games.
(ii) Determine all values of b such that v is S-equivalent to the game w ∈ T U N
given by
S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
w(S) 0 0 21 35 29 49 60 78
(iii) Let b = 3. Using a picture, determine all extreme points of the core C(v).
(iv) Let b = 12. Using a picture, determine all extreme points of the core C(v).
(v) Determine all values of b such that C(v) has exactly four extreme points. For
each of these values of b, explicitly provide the corresponding extreme points.
b) Consider the 4-person cooperative game v ∈ T U N with N = {1, 2, 3, 4} given by
0 if |S| = 1,
1 if |S| = 2 and S 6= {3, 4},
α if S = {3, 4},
v(S) =
4 if |S| = 3 and S 6= {2, 3, 4},
α if S = {2, 3, 4},
β if |S| = 4,
with α, β ∈ R. Determine all combinations of α and β such that v is superadditive
and (5, 1, 1, 2) ∈ C(v).
1
Name of subject : Games and Economic Behavior
Subject code : 35V5B2-B-6
Date of examination : June 21, 2021
Length of examination : 3 hours
Lecturers : Prof. Dr. P.E.M. Borm
Telephone of departmental secretary : 013 4662430
Students are expected to conduct themselves properly during examinations and to obey any
instructions given to them by examiners and invigilators.
Firm action will be taken in the event that academic fraud is discovered.
SPECIFICS:
Open book examination : no
Use of graphical calculator : yes
Print on both sides : yes
Scrap paper available : yes
EXAM INSTRUCTIONS:
1. This exam consists of five exercises. The maximum number of score points is 100. For
each exercise, the maximal score is indicated separately.
2. Please provide complete solutions to each exercise. Each of your answers should be
clearly motivated.
3. Please have a look through the whole exam before you start working on the exercises. In
doing the exercises you do not have to follow the order of the questions in the exam.
4. Do not forget to write down your name and student number (SNR, 7 digits) on each page
you hand in.
5. You can use scrap paper to work out the exercises. Scrap paper however will not be
graded.
6. After the examination, the exercises of the exam, final answers, and a grading scheme
will become available via Canvas.
All students need to hand in the exam and their work (including scrap paper).
Good luck!
, Exam Games and Economic Behavior (35V5B2-B-6) June 21, 2021
Exercise 1 (25 points)
a) Consider the game v ∈ T U N with N = {1, 2, 3} given by
S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
v(S) 0 0 b b 12 24 0 36
with b ∈ R.
(i) Let b = 3. Decompose v into unanimity games.
(ii) Determine all values of b such that v is S-equivalent to the game w ∈ T U N
given by
S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
w(S) 0 0 21 35 29 49 60 78
(iii) Let b = 3. Using a picture, determine all extreme points of the core C(v).
(iv) Let b = 12. Using a picture, determine all extreme points of the core C(v).
(v) Determine all values of b such that C(v) has exactly four extreme points. For
each of these values of b, explicitly provide the corresponding extreme points.
b) Consider the 4-person cooperative game v ∈ T U N with N = {1, 2, 3, 4} given by
0 if |S| = 1,
1 if |S| = 2 and S 6= {3, 4},
α if S = {3, 4},
v(S) =
4 if |S| = 3 and S 6= {2, 3, 4},
α if S = {2, 3, 4},
β if |S| = 4,
with α, β ∈ R. Determine all combinations of α and β such that v is superadditive
and (5, 1, 1, 2) ∈ C(v).
1
