Game Theory
Game theory is a mathematical theory that deals
with the general features of competitive situations.
The final outcome depends primarily upon the
combination of strategies selected by the
adversaries.
Two key Assumptions:
(a) Both players are rational
(b) Both players choose their strategies solely
to increase their own welfare.
Payoff Table
Strategy
1
Player 1 2
3
Player 2
1
2
3
1
2
4
1
0
5
0
1 -1
Each entry in the payoff table for player 1
represents the utility to player 1 (or the negative
utility to player 2) of the outcome resulting from the
corresponding strategies used by the two players.
A strategy is dominated by a second strategy
if the second strategy is always at least as
good regardless of what the opponent does.
A dominated strategy can be eliminated
immediately from further consideration.
Strategy
1
Player 1 2
3
Player 2
1
2
3
1
2
4
1
0
5
0
1 -1
For player 1,
strategy 3 can be eliminated. ( 1 > 0, 2 > 1, 4 > -1)
1
2
1
1
1
2
2
0
3
4
5
For player 2,
strategy 3 can be eliminated. ( 1 < 4, 1 < 5 )
1
2
1
1
1
2
2
0
For player 1,
strategy 2 can be eliminated. ( 1 = 1, 2 < 0 )
1
1
1
2
2
For player 2,
strategy 2 can be eliminated. ( 1 < 2 )
Consequently, both players should select their
strategy 1.
A game that has a value of 0 is said to be
a fair game.
Minimax criterion:
To minimize his maximum losses whenever
resulting choice of strategy cannot be
exploited by the opponent to then improve his
position.
Player 2
Strategy
1
2
3 Minimum
-3
1
-3
-2
6
0
2
0
2
Player 1 2
-4
3
5
-2 -4
0
6
Maximum: 5
Minimax value
Maximin value
The value of the game is 0, so this is fair game
Saddle Point:
A Saddle point is an entry that is both the
maximin and minimax.
Player 2
Strategy
1
2
3 Minimum
-3
1
-3
-2
6
0
2
0
2
Player 1 2
-4
3
5
-2 -4
0
6
Maximum: 5
Saddle point
There is no saddle point.
An unstable solution
Player 2
Strategy
1
2
3 Minimum
-2
1
0
-2
2
-3
5
4
-3
Player 1 2
-4
3
2
3
-4
4
2
Maximum: 5
Mixed Strategies
xi = probability that player 1 will use strategy i
( i = 1,2,…,m),
y j = probability that player 2 will use strategy j
( j = 1,2,…,n),
m
Expected payoff for player 1 =
n
 p x y ,
i 1 j 1
ij i
j
Minimax theorem:
If mixed strategies are allowed, the pair of
mixed strategies that is optimal according to the
minimax criterion provides a stable solution
with v  v  v (the value of the game), so that
neither player can do better by unilaterally
changing her or his strategy.
v = maximin value
v = minimax value
Graphical Solution Procedure
x2  1  x1
Player 2
Probability
Pure
Probability Strategy
1
x1
Player 1
2
1  x1
( y1 , y2 , y3 )
(1,0,0)
(0,1,0)
(0,0,1)
y2
y1
1
0
5
2
-2
4
Expected Payoff
0 x1  5(1  x1 )  5  5 x1
 2x1  4(1  x1 )  4  6x1
2 x1  3(1  x1 )  3  5 x1
y3
3
2
-3
( y1 , y2 , y3 )
(1,0,0)
(0,1,0)
(0,0,1)
Expected Payoff
0 x1  5(1  x1 )  5  5 x1
 2x1  4(1  x1 )  4  6x1
2 x1  3(1  x1 )  3  5 x1
Expected payoff for player 1 =
y1 (5  5x1 )  y2 (4  6 x1 )  y3 (3  5x1 ).
Expected payoff
Player 1 wants to maximize the minimum
expected payoff. Player 2 wants to minimize
the expected payoff.
6
5
5  5x1 Maximin point
4
3
2 4  6 x1
1
0
1
1
3
1.0
-1
4
2
4
-2
 3  5x1  4  6 x1
-3  3  5x1
7 4
 ( x1 , x2 )  ( , )
-4
11 11
x1
The optimal mixed strategy for player 1 is
7 4
 ( x1 , x2 )  ( , )
11 11
So the value of the game is
7 2
v  v  3  5  
 11  11
The optimal strategy
( y1* , y*2 , y*3 )
2
(1)
y (5  5 x1 )  y (4  6 x1 )  y (3  5 x1 )  v  v 
11
*
1
*
2
*
3
7
When player 1 is playing optimally ( x1 
),
11
this inequality will be an equality, so that
20 * 2 * 2 *
2
y1  y2  y3  v 
11
11
11
11
Because y j is a probability distribution,
y  y  y  1.
*
1
*
2
*
3
(2)
y1*  0 because y1*  0 would violate (2),
 2

 11
*
*
y2 (4  6 x1 )  y3 (3  5 x2 )
2

 11
for 0  x1  1,
7
for x1 
11
2
Because the ordinate of this line must equal
11 2
7
at x1  , and because it must never exceed
,
11
11
2
*
*
y2 (4  6 x1 )  y3 (3  5 x2 )  , for 0  x1  1.
11
*
3
*
2
To solve for y and y , select two values of
(say, 0 and 1),
2
4 y  3y  ,
11
2
*
*
 2 y 2  2 y3 
11
5 * 6
*
 y 2  , y3 
11
11
The optimal mixed strategy for player 2 is
*
2
*
3
5 6
( y , y , y )  (0, , ).
11 11
*
1
*
2
*
3
Solving by Linear Programming
m
Expected payoff for player 1 =
n
 p x y
i 1 j 1
The strategy ( x1 , x2 ,, xm ) is optimal if
m
n
 p x y
i 1 j 1
ij i
j
vv
ij i
j
For each of the strategies ( y1 , y2 ,, yn ) where
one and the rest equal 0. Substituting these
values into the inequality yields
m
p x
i 1
ij i
v
for j  1,2, , n,
Because the xi are probabilities,
x1  x2    xm  1
xi  0,
fori  1,2,, m
The two remaining difficulties are
(1) v is unknown
(2) the linear programming problem has
no objective function.
Replacing the unknown constant v by the
variable xm1 and then maximizing xm1 ,
so that xm1 automatically will equal v at the
optimal solution for the LP problem.
Maximize
x m 1,
s.t. p11x1  p 21x 2    p m1x m  x m 1  0
p12 x1  p 22 x 2    p m 2 x m  x m 1  0

p1n x1  p 2n x 2    p mn x m  x m 1  0
x1  x 2    x m  1
x i  0, for i  1,2, , m.
Minimize
y n 1,
s.t. p11y1  p12 y 2    p1n y n  y n 1  0
p 21y1  p 22 y 2    p 2n y n  y n 1  0

p m1y1  p m 2 y 2    p mn y n  y n 1  0
y1  y 2    y n  1
y j  0, for j  1,2,  , n.
Player 2
Example
y1
Probability
Pure
Probability Strategy
1
x1
Player 1
2
1  x1
Maximize
s.t.
1
0
5
y2
2
-2
4
y3
3
2
-3
x3,
5x 2  x 3  0
 2x1  4x 2  x 3  0
2 x1  3x 2  x 3  0
x1  x 2
1
x1  0, x 2  0, x 3  0.
7 4 2
(x , x , x )  ( , , )
11 11 11
*
1
*
2
*
3
Player 2
The dual
Probability
Pure
Probability Strategy
1
x1
Player 1
2
1  x1
Minimize
s.t.
y1
1
0
5
y2
2
-2
4
y4 ,
 2 y 2  2 y3  y 4  0
5y1  4 y 2  3y3  y 4  0
y1  y 2  y3
*
*
*
*
( y1 , y 2 , y3 , y 4 ) 
1
y1  0, y 2  0, y3  0, y 4  0.
5 6 2
(0, , , )
11 11 11
y3
3
2
-3
Question 1
Consider the game having the following payoff table.
Strategy
Player 1
1
2
3
1
5
2
3
Player 2
2
3
0
3
4
3
2
0
4
1
2
4
(a) Formulate the problem of finding optimal mixed
strategies according to the minimax criterion as a linear
programming problem.
(b) Use the simplex method to find these optimal mixed
strategies.