Game theory is a branch of mathematics that deals with strategic decision-making in competitive scenarios. One of the most fundamental tools in game theory is the payoff matrix, which represents the possible outcomes of a game for each player. In this article, we will explore how to solve the game theory payoff matrix.
What Is a Payoff Matrix?
A payoff matrix is a table that shows the possible outcomes of a game for each player. The rows of the table represent the strategies available to one player, while the columns represent the strategies available to another player. Each cell in the table shows the payoffs for each player when they play their respective strategies.
For example, consider a simple game where two players can choose either to cooperate or defect. If both players cooperate, they each receive 3 points.
If both players defect, they each receive only 1 point. However, if one player cooperates and the other defects, then the defector receives 5 points while the cooperator receives nothing.
This game can be represented by a payoff matrix as follows:
| Cooperate | Defect | |
|---|---|---|
| Cooperate | (3,3) | (0,5) |
| Defect | (5,0) | (1,1) |
In this matrix, (3,3) denotes that both players receive three points if they both cooperate.
Dominant Strategies
One way to solve a payoff matrix is to look for dominant strategies. A dominant strategy is one that always provides a better outcome regardless of the other player’s strategy.
In the example above, we can see that if Player 2 chooses to cooperate, then Player 1’s best response is to defect, since they will receive 5 points instead of 3. Similarly, if Player 2 chooses to defect, then Player 1’s best response is still to defect, since they will receive 1 point instead of nothing. Therefore, Player 1 has a dominant strategy of defection.
Likewise, if Player 1 chooses to cooperate, then Player 2’s best response is to defect. And if Player 1 chooses to defect, then Player 2’s best response is also to defect. Therefore, both players have a dominant strategy of defection.
Nash Equilibrium
Another way to solve a payoff matrix is to look for Nash equilibrium. A Nash equilibrium is a set of strategies where no player can improve their outcome by changing their strategy unilaterally.
In the example above, we can see that (defect, defect) is a Nash equilibrium. If either player changes their strategy unilaterally, they will receive a worse payoff. For example, if Player 1 switches from defection to cooperation while Player 2 still defects, then Player 1’s payoff will decrease from 1 point to nothing.
Iterated Elimination of Dominated Strategies
Sometimes a game may have multiple dominant strategies or no dominant strategies at all. In such cases, we can use the iterated elimination of dominated strategies method.
This method involves repeatedly eliminating dominated strategies until only one strategy remains for each player. A dominated strategy is one that always provides a worse outcome regardless of the other player’s strategy.
For example, consider the following payoff matrix:
| Left | Middle | Right | |
|---|---|---|---|
| Up | (1,1) | (0,2) | (0,0) |
| Down | (2,0) | (1,1) | (0,2) |
In this matrix, there are no dominant strategies. However, we can eliminate the dominated strategy “Right” for Player 1.
If Player 2 chooses “Middle”, then Player 1’s best response is “Up”. And if Player 2 chooses “Down”, then Player 1’s best response is still “Up”. Therefore, we can eliminate the column for “Right”.
The resulting matrix is:
| Left | Middle | |
|---|---|---|
| Up | (1,1) | (0,2) |
| Down | (2,0) | (1,1) |
Now we can eliminate the dominated strategy “Down” for Player 2. If Player 1 chooses “Left”, then Player 2’s best response is “Up”.
And if Player 2 chooses “Middle”, then Player 2’s best response is still “Up”. Therefore, we can eliminate the row for “Down”.
| Left | Middle | |
|---|---|---|
| Up | (1,1) | (0,2) |
The final result is that Player 1’s only strategy is “Left” and Player 2’s only strategy is “Middle”. Therefore, (Left, Middle) is the unique outcome of this game.
Conclusion
In conclusion, solving a game theory payoff matrix involves finding dominant strategies, Nash equilibrium or iteratively eliminating dominated strategies. These methods help to identify the best strategies for each player in a given game. By understanding these techniques, you can gain insight into how different players may behave in competitive scenarios.