Transcript Introduction to Game Theory

Introduction to Game Theory
Kusdhianto Setiawan
Department of Management
Faculty of Economics
Gadjah Mada University
Language of Games
Game
Player Y’s
Strategies
Game
Player X’s
Strategies
Y1
Y2
X1
2
7
X2
6
-4
The Figure shows:
• Each number in the
box represents player
X’s gain and Y’s loss,
regarding each player
strategies.
• The game represent a
zero sum game: one
player’s gain causing
the other player’s loss
at the same level.
Pure Strategy Games
• The best strategy for X is
strategy 1, why?
Game
Player Y’s
Strategies
Game
Player X’s
Strategies
X1
X2
Y1
Y2
3
5
1
-2
• Knowing that X will always
play strategy 1, the best
strategy for Y is strategy 2,
why?
• Regardless what strategies
that Y will play, player X will
always gain 3 points by
playing strategy 1
• Because player X will always
play strategy 1, the minimum
loss of 3 points will be
achieved by player Y if he/she
uses strategy 1 too.
Minimax Criterion: Pessimistic?
Game
Player Y’s
Strategies
Game
Player X’s
Strategies
Minimum
row number
Y1
Y2
X1
10
6
6
X2
-12
2
-12
Maximum col. Number
10
6
Upper Value
Lower Value
Objective:
Minimize Losses or
Maximize Gains for each
player.
Mix Strategy Games
X1
X2
Y1 Y2
3 5
1 -2
Q
1-Q
P
3
1
1-P
5
-2
• It is played
when there is no
saddle point
Steps for 2x2 Game:
• Q, 1-Q: fraction
of the time X
plays strategy
X1 and X2
respectively
1. To find X’s best strategy, multiply Q and 1-Q times the
appropriate game outcome numbers and solve for Q
and 1-Q by setting column 1 equal to column 2 in the
game
• P, 1-P: fraction
of the time Y
plays strategy
Y1 and Y2
respectively
2. To find Y’s best strategy, multiply P and 1-P times the
appropriate game outcome numbers and solve for P
and 1-P by setting row 1 equal to row 2 in the game
3. Value of the game: the sum of each outcome multiplied
by its P Factor and Q Factor (any shortcut?)
Dominance
Y1
Y2
X1
4
3
X2
2
20
X3
1
1
Y1
Y2
X1
4
3
X2
2
20
Dominance
Y1
Y2
Y3
Y4
X1
-5
4
6
-3
X2
-2
6
2
-20
Y1
Y4
X1
-5
-3
X2
-2
-20