Game Theory, Part 1
Game theory applies to more than just games.
Corporations use it to influence business
decisions, and militaries use it to guide their
strategies.
In fact, game theory grew in popularity and
acceptance during WWII.
Game Theory, Part 1
In essence, the goal of game theory is to
determine the best strategy for a participant in
any sort of competition to use.
Game Theory, Part 1
How do we determine the best strategy for each
competitor?
Begin with constructing a payoff matrix.
Game Theory, Part 1
How do we determine the best strategy for each
competitor?
Begin with constructing a payoff matrix.
The payoff matrix shows the outcome for each
possible combination of strategies by each
competitor.
Game Theory, Part 1
We will consider only competitions that are
limited to two participants.
The rows of the payoff matrix will represent the
actions of one competitor, the columns will
represent the options for the other competitor.
Game Theory, Part 1
Very important:
The values in the payoff matrix represent the
payoff for the row competitor.
Game Theory, Part 1
Very important:
The values in the payoff matrix represent the
payoff for the row competitor.
Those values represent exactly the opposite
payoff for the column competitor.
Game Theory, Part 1
Example: Sol and Tina play a game where they
each choose to display either heads or tails on a
coin. They reveal their coins at the same time,
and have the following payoff matrix.
Tina
H T
H  3 1
Sol
 1 2 
T 

Game Theory, Part 1
Remember that the values in the matrix show
the payoff for Sol.
Tina
H T
H  3 1
Sol
 1 2 
T 

If Sol and Tina each play heads, Sol wins
3 pennies. This means, of course, that
Tina loses 3 pennies.
Game Theory, Part 1
Remember that the values in the matrix show
the payoff for Sol.
Tina
H T
H  3 1
Sol
 1 2 
T 

If Sol plays heads and Tina plays tails,
then Sol loses 1 penny; Tina gains 1
penny—exactly opposite of Sol.
Game Theory, Part 1
Remember that the values in the matrix show
the payoff for Sol.
Tina
H T
H  3 1
Sol
 1 2 
T 

What strategy should each player follow?
Game Theory, Part 1
One method of choosing a strategy is to try to
minimize the potential damage or loss. In other
words, choose the least of all evils.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1
Sol
 1 2 
T 

For Sol, look at the rows. If Sol plays
heads, the possible outcomes are to win 3
pennies, or to lose 1 penny.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 

If Sol plays heads, the worst that can
happen is that he will lose 1 penny. This
is the row minimum for row 1.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
In order to minimize Sol’s losses, he
should choose the maximum value of all
the row minimums.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
This value is called the maximin—the
maximum of the row minimums—which in
this case is –1.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
Now consider Tina. Remember that the
values in the payoff matrix are exactly the
opposite for her. In other words, large
positive numbers mean she loses money.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3
Because of this, we find the worst-case
scenario for her by searching for the
largest numbers in each column.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3
If Tina plays heads, the worst thing that
can happen is that she’ll lose 3 pennies,
as opposed to possibly winning 1 penny.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3 -1
If she plays tails, the worst thing that can
happen is that she’ll win 1 penny, rather
than 2.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3 -1
In order to minimize her losses, we want
to choose the minimum of the column
maximums, called the minimax.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3 -1
Notice that Sol’s maximin is the same as
Tina’s minimax. This suggests that the
outcome of the game will be the same
every time—Tina will win 1 penny.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3 -1
This is an example of a strictly determined
game, where the maximin and the
minimax are the same.
Game Theory, Part 1
Consider the worst-case scenario for each
player, for each option.
Tina
H T
H  3 1 -1
Sol
 1 2 
T 
 -2
3 -1
That value for the maximin and minimax
is called the saddle point of the game. It
shows the outcome each game for the
row player.
Game Theory, Part 1
Consider another game where each player
displays a card with the letter A, B, C, or D.
This game has the following payoff matrix:
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Sol took a close look at his options, represented in
the rows, and noticed something interesting.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Regardless of what Tina does, row A never provides
a better outcome for Sol than row D.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Row D is said to dominate row A. Because it can
never outdo row D, Sol can simply eliminate A as an
option.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Likewise, Tina can compare her options in the
columns of the payoff matrix. Remember, though,
that she wants the smallest (most negative)
numbers, as those represent more money won.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Column C always outperforms column A, regardless
of what Sol does. So Tina can eliminate column A.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1 


C  1 3 5 2 

D  5 2 4
2
Now that Sol and Tina have both eliminated A as an
option, we really only need to worry about a 3x3
payoff matrix. Evaluate the maximin and minimax.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1  -3


C  1 3 5 2  -5
 -2
D  5 2 4
2
The maximin is –2, which occurs for strategy D.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1  -3


C  1 3 5 2  -5
 -2
D  5 2 4
2
3
4 2
This suggests that Sol should play D, and Tina
should play D. In such an event, Sol will win 2
pennies.
Game Theory, Part 1
A
Sol
Tina
B
C
D
1
A  4 2 3
B  2 1 3 1  -3


C  1 3 5 2  -5
 -2
D  5 2 4
2
3
4 2
How long do you think Tina will play this game?