Permainan Metoda Grafik Pertemuan 11: Mata kuliah: K0194-Pemodelan Matematika Terapan Tahun: 2008

2 Learning Outcomes Mahasiswa akan dapat menyelesaikan masalah permainan dengan metoda grafik.

3 Outline Materi: Konsep Dasar permainan Penyelesaian dgnMetoda Grafik Contoh kasus..

4 Metoda Grafik Ada beberapa metode yang dapat digunakan untuk mencari nilai permainan antara lain: –Metoda Grafik –Metoda Analitis –Metoda Aljabar Matriks –Metoda Linear Programming..

5 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.

6 Jika seorang pemain dapat memilih salah satu dari dua strategi yang mungkin,maka situasi keputusan dari pemain dapat di gambarkan dalam grafik, Misalnya sbb:

7 Penggambaran dapat dilakukan dengan memperhatikan cara penggambaran pada metoda linier programming metoda grafik. CONTOH.

8 Glossary Payoff/Reward Matrix Two Person Game Zero-sum Column domination Row domination Saddle/Equilibrium Point Strategies

9 Two-person zero-sum game with a saddle point Row Domination:(2) > (1)(3) > (1) Eliminate Row (1) Column Domination:A > C,B > C Eliminate Columns A, B Player #1 Player #2 Payoff Matrix to Player #1

10 Reduced Matrix 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

11 Saddle Point 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.

12 Mixed Strategy Domination: Row-none Column -A < Beliminate B D > Celiminate D

13 Revised Matrix Domination Row (2) > (1)eliminate row (1) No further domination

14 Row strategies prob. p 1-p E A = 8p + (-5)(1 - p) = p E C = -1p + (6)(1 - p) = 6 - 7p

p = 6 - 7p 20p = 11 p = 11/20 = 0.55 V = (0.55) = 6 - 7(0.55) = 2.15 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

16 Column strategies prob. q 1 - q E 2 = 8q + (-1)(1 - q) = q E 3 = -5q + (6)(1 - q) = q

q = q 20q = 7 q = 7/20 = 0.35 V = (0.35) = (0.35) = 2.15 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

18 Further Considerations Linear programming Internet - JavaScript routine Non-zero sum games More than two players 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

19