Game Theory History John von Neumann and Morgenstern

Game Theory

History • John von Neumann and Morgenstern wrote a book titled “Theory of Games and Economic Behavior” • Used in Business, Economics, Social Sciences, etc.

Example * * Saddle Point

Non-strictly determined games • No saddle points * *

Random Variable • Dependent: P(X|Y) = P(X, Y) / P(Y) • Independent: P(X, Y) = P(X|Y) * P(Y) P(X, Y) = P(X) * P(Y)

ตวอยาง X = 10 X = 20 X =30 P(X) 0. 2 0. 3 0. 5 P(X)*X 2 6 15 E[X] = 2+6+15 = 23 2

Non-strictly determined games robert ใชกลยทธ เลอก dime 20% เลอก quarter 80% carol ใชกลยทธ เลอก dime 70% เลอก quarter 30% Value of game = expected payoff ของ Robert? = Dime Quarter Dime 0. 2*0. 7 = 0. 14 0. 2 * 0. 3 = 0. 06 0. 2 Quarter 0. 8*0. 7 = 0. 56 0. 8 * 0. 3 = 0. 24 . 8 . 7 . 3

Dime Quarter Dime 10 -10 Quarter -25 25 Dime Quarter Dime 0. 14 * 10 0. 06 * -10 Quarter 0. 56 * -25 0. 24 * 25 Expected Pay-off = 1. 4 - 0. 6 – 14 + 6 = -7. 2

Non-strictly determined games Dime Quarter Dime 10 -10 Quarter -25 25 • หา Optimal Strategy ของทง 2 players • กรณ Robert ให P(dime) = r • P(quarter) = 1 -r • ถา Carol เลอก dime, Robert ได pay-off เฉลย = r * 10 + (1 -r) *-25 = 35 r - 25 • ถา Carol เลอก quarter, Robert ได pay-off เฉลย = r * -10 + (1 -r) * 25 = -35 r + 25 35 r – 25 = -35 r + 25 r = 50/70 = 5/7 P(Robert เลอก dime) = 5/7, P(Robert เลอก quarter) = 2/7

Non-strictly determined games Dime Quarter Dime 10 -10 Quarter -25 25 • กรณ Carol ให P(dime) = c • P(quarter) = 1 -c • ถา Robert เลอก dime, Carol ได pay-off เฉลย = c * -10 + (1 -c) *10 = -20 c + 10 • ถา Robert เลอก quarter, Carol ได pay-off เฉลย = c * 25 + (1 -c) * -25 = 50 c - 25 20 c - 10 = -50 c + 25 c = 35/70 P(Carol เลอก dime) = 1/2, P(Carol เลอก quarter) = 1/2

Dime Quarter Dime 10 -10 Quarter -25 25 Dime Quarter Dime 5/7*1/2 Quarter 2/7*1/2 Expected Payoff ��� Robert + 10 * 5/14 = + 10 - * 5/14 + 25 - * 2/14 25 * 2/14 0 =

Reduction by Dominance