Learning Outcomes
• Mahasiswa akan dapat menyelesaikan masalah
permainan dengan metoda grafik.
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Outline Materi:
• Konsep Dasar permainan
• Penyelesaian dgnMetoda Grafik
• Contoh kasus..
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Metoda Grafik
• Ada beberapa metode yang dapat digunakan untuk mencari
nilai permainan antara lain:
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Metoda Grafik
Metoda Analitis
Metoda Aljabar Matriks
Metoda Linear Programming..
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Aturan Permainan
Semua permainan 2 x n (yaitu, pemain baris
mempunyai dua strategi dan pemain kolom
mempunyai n strategi) dan pemain m x 2 (yaitu
pemain baris mempunyai m strategi dan pemain
kolom mempunyai dua strategi) dapat diselesaikan
secara grafik. Untuk dapat menyelesaikan permainan
strategi campuran secara grafik, dimensi pertama
matriks permainan harus 2.
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•
Jika seorang pemain dapat memilih salah satu dari dua strategi
yang mungkin,maka situasi keputusan dari pemain dapat di
gambarkan dalam grafik, Misalnya sbb:
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Strategi 3
Strategi 1
10
Pilih C
Pemain
9.5
Pemain
Pilih A
7
1.0
6
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Probabiliti (Pemilihan Strategi 1)
0.2
0.1
00.0
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• Penggambaran dapat dilakukan dengan memperhatikan
cara penggambaran pada metoda linier programming
metoda grafik.
• CONTOH.
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Glossary
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Payoff/Reward Matrix
Two Person Game
Zero-sum
Column domination
Row domination
Saddle/Equilibrium Point
Strategies
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Two-person zero-sum game with a saddle point
Player #2
Player
#1
1
2
3
Row Domination:
A
6
9
9
B
5
7
8
C
-4
-2
-3
(2) > (1)
Payoff
Matrix to
Player #1
(3) > (1)
Eliminate Row (1)
Column Domination:
A > C, B > C
Eliminate Columns A, B
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Reduced Matrix
2
3
C
-2
-3
• The best strategy for player #1 is to choose 2.
• The best strategy for player #2 is to choose C
• This results in a saddle/equilibrium point which
gives us these simple strategies for each player
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Saddle Point
1
2
3
A
6
9
9
B
5
7
8
C
-4
-2
-2
-3
The value shown is the smallest in
its row - to player #2’s advantage and the largest in its column - to
player #1’s advantage.
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Mixed Strategy
1
2
3
A
7
8
-5
B
10
9
0
C
-4
-1
6
D
2
5
8
Domination:
• Row -
none
• Column -
A<B
eliminate B
D>C
eliminate D
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Revised Matrix
1
2
3
A
7
8
-5
C
-4
-1
6
Domination
Row (2) > (1)
2
3
No further domination
eliminate row (1)
A
8
-5
C
-1
6
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Row strategies
prob.
p
1-p
2
3
A
8
-5
C
-1
6
EA = 8p + (-5)(1 - p) = -5 + 13p
EC = -1p + (6)(1 - p) = 6 - 7p
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The best strategy for
player #2 for a given p
is to choose the green
line. Knowing this,
player #1 chooses the
peek at p = 0.55
-5 + 13p = 6 - 7p
20p = 11
p = 11/20 = 0.55
V = -5 + 13(0.55) = 6 - 7(0.55) = 2.15
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Column strategies
prob.
q
1-q
2
3
A
8
-5
C
-1
6
E2 = 8q + (-1)(1 - q) = -1 + 9q
E3 = -5q + (6)(1 - q) = 6 - 11q
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The best strategy for
player #1 for a given q
is to choose the green
line. Knowing this,
player #2 chooses the
low point at q = 0.35
-1 + 9q = 6 - 11q
20q = 7
q = 7/20 = 0.35
V = -1 + 9(0.35) = 6 - 11(0.35) = 2.15
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Conclusions:
Player #1 : (1) 0% (2) 55% (3) 45%
Player #2 : (A) 35% (B) 0% (C) 65% (D) 0%
Value of the Game (to Player #1) : 2.15
Further Considerations
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Linear programming
Internet - JavaScript routine
Non-zero sum games
More than two players
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