Welcome to Game Theory!
Ever played Rock-Paper-Scissors and tried to predict your friend's next move? Or wondered how businesses decide on prices when they know their rivals are watching? That is Game Theory in action! In this chapter of Decision Mathematics 2, we look at how two "players" make the best possible decisions when their success depends on the choices of the other person. It is essentially the mathematics of strategy.
1. The Basics: Two-Person Zero-Sum Games
Before we dive into the calculations, we need to understand the "rules" of the games we study in 8FM0.
Key Terms:
- Two-person game: Quite simply, a game with exactly two players.
- Zero-sum game: A game where one player's gain is exactly equal to the other player's loss. If I win £5, you lose £5. The total "sum" of our scores is always zero.
- Pay-off Matrix: A table showing the results of the game. In your exam, this matrix is almost always written from the Row Player's point of view.
Example: Imagine a matrix where a value is \( 4 \). This means the Row Player wins 4 points and the Column Player loses 4 points. If the value is \( -3 \), the Row Player loses 3 points and the Column Player wins 3 points.
Quick Review:
Positive number = Row Player wins / Column Player loses.
Negative number = Row Player loses / Column Player wins.
Key Takeaway
In a zero-sum game, the players are in total conflict. The Row Player wants the numbers in the matrix to be as large as possible, while the Column Player wants them to be as small (or as negative) as possible.
2. Play-Safe Strategies and Stable Solutions
Most players are "pessimists" — they assume the other person is playing perfectly to beat them. Because of this, they use a play-safe strategy to guarantee a minimum result regardless of what the opponent does.
Step-by-Step: Finding the Play-Safe Strategy
For the Row Player (The "Maximin" Strategy):
1. Look at each row and find the minimum value (the worst-case scenario for that choice).
2. From those minimums, pick the maximum one. This is the Row Maximin.
Memory Aid: "Row wants the Best of the Worsts."
For the Column Player (The "Minimax" Strategy):
1. Look at each column and find the maximum value (the worst-case scenario for the Column player, as Row wins more).
2. From those maximums, pick the minimum one. This is the Column Minimax.
Memory Aid: "Column wants to Minimize their Maximum loss."
Stable Solutions (Saddle Points)
Sometimes, the Row Maximin and the Column Minimax are the same number. When this happens, we have a Stable Solution, also known as a Saddle Point.
Did you know? In a stable solution, neither player can improve their outcome by changing their strategy unilaterally. If they change, they'll just end up worse off!
Important Point: A game is "stable" if and only if:
\( \text{Row Maximin} = \text{Column Minimax} \)
Key Takeaway
Always check for a saddle point first! If the Maximin equals the Minimax, the game is solved, and you don't need to do any complex graphing.
3. Optimal Mixed Strategies (Graphical Method)
What happens if there is no saddle point? If you keep picking the same "safe" row, your opponent will eventually figure you out and beat you. To stay unpredictable, you must use a Mixed Strategy.
In a mixed strategy, you play different options with certain probabilities. For AS Level 8FM0, you only need to know how to solve this graphically for \( 2 \times n \) or \( n \times 2 \) games (where one player has only 2 options).
How to Solve a \( 2 \times n \) Game (Row Player has 2 options)
Don't worry if this seems tricky at first; the graph does most of the work for you!
1. Define Probabilities: Let Row Player choose Strategy 1 with probability \( p \). Therefore, they must choose Strategy 2 with probability \( (1 - p) \).
2. Write Expected Pay-offs: For each of the Column Player's choices, write an equation for the expected gain.
Example: If Column chooses a strategy where Row wins 3 in Option 1 and 5 in Option 2, the equation is: \( V = 3p + 5(1-p) \).
3. Draw the Graph: Draw a graph with \( p \) on the x-axis (from 0 to 1) and Expected Pay-off \( V \) on the y-axis.
4. Plot the Lines: Each of Column's strategies becomes a straight line on your graph.
5. Find the "Highest Point on the Lower Boundary": Since the Row Player wants to maximize their minimum gain, look at the very bottom "edge" of all your lines (the lower envelope). Find the highest point on that specific boundary.
6. Calculate: This point is usually where two lines intersect. Set those two equations equal to each other to find the optimal value of \( p \).
Common Mistake to Avoid
When solving an \( n \times 2 \) game (where the Column Player has 2 options), the process is similar, but you are looking for the Lowest point on the Upper Boundary. This is because the Column Player wants to minimize the maximum gain the Row player can get.
Key Takeaway
The graphical method turns a strategy problem into a visual one. Row Player looks for the "highest floor" (lower boundary), and Column Player looks for the "lowest ceiling" (upper boundary).
Quick Summary for Revision
1. Pay-off Matrix: Always read from the Row Player's perspective.
2. Play-Safe: Find Row Maximin and Column Minimax.
3. Saddle Point: If Maximin = Minimax, the game is stable.
4. Mixed Strategy: Use when no saddle point exists. Use \( p \) and \( (1-p) \) to create linear equations and solve via a graph.
5. Value of the Game: This is the expected pay-off when both players play optimally.
You've got this! Practice drawing the graphs carefully, and always remember to label your axes. Game Theory is just about finding the best path through a competitive situation.