game theory, part 1
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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. - PowerPoint PPT PresentationTRANSCRIPT
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.
3 1
1 2
H TTina
HT
Sol
Game Theory, Part 1
Remember that the values in the matrix show the payoff for Sol.
3 1
1 2
H THT
Tina
Sol
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.
3 1
1 2
H THT
Tina
Sol
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.
3 1
1 2
H THT
Tina
Sol
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.
3 1
1 2
H THT
Tina
Sol
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.
3 1
1 2
H THT
Tina
Sol
If Sol plays heads, the worst that can happen is that he will lose 1 penny. This is the row minimum for row 1.
-1
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
In order to minimize Sol’s losses, he should choose the maximum value of all the row minimums.
-1
-2
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
This value is called the maximin—the maximum of the row minimums—which in this case is –1.
-1
-2
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
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.
-1
-2
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
Because of this, we find the worst-case scenario for her by searching for the largest numbers in each column.
-1
3-2
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
If Tina plays heads, the worst thing that can happen is that she’ll lose 3 pennies, as opposed to possibly winning 1 penny.
-1
3-2
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
If she plays tails, the worst thing that can happen is that she’ll win 1 penny, rather than 2.
-1
3-2
-1
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
In order to minimize her losses, we want to choose the minimum of the column maximums, called the minimax.
-1
3-2
-1
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
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.
-1
3-2
-1
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
This is an example of a strictly determined game, where the maximin and the minimax are the same.
-1
3-2
-1
Game Theory, Part 1
Consider the worst-case scenario for each player, for each option.
3 1
1 2
H THT
Tina
Sol
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.
-1
3-2
-1
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:
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
Game Theory, Part 1
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
Sol took a close look at his options, represented in the rows, and noticed something interesting.
Game Theory, Part 1
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
Regardless of what Tina does, row A never provides a better outcome for Sol than row D.
Game Theory, Part 1
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
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
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
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
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
Column C always outperforms column A, regardless of what Sol does. So Tina can eliminate column A.
Game Theory, Part 1
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
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
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
The maximin is –2, which occurs for strategy D.
-3
-5
-2
Game Theory, Part 1
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
This suggests that Sol should play D, and Tina should play D. In such an event, Sol will win 2 pennies.
-3
-5
-2
3 4 2
Game Theory, Part 1
4 2 3 1
2 1 3 1
1 3 5 2
5 2 4 2
A B C D
A
B
C
D
Sol
Tina
How long do you think Tina will play this game?
-3
-5
-2
3 4 2