Game theory is a branch of mathematics that deals with decision-making in situations where multiple players are involved. It is a fascinating subject that has applications in fields such as economics, politics, psychology, and biology. One of the most popular games in game theory is the 3 by 3 game.
A 3 by 3 game is a two-player game where each player has three possible actions. The outcome of the game depends on the actions taken by both players. The possible outcomes are represented in a table called a payoff matrix.
To solve a 3 by 3 game, we need to find the optimal strategies for both players. An optimal strategy is one that maximizes the player’s payoff given their opponent’s strategy. There are different methods to find the optimal strategies, but one of the most popular methods is called dominance.
Dominance occurs when one action is always better than another action, regardless of what the opponent does. We can use dominance to eliminate actions that are dominated by other actions.
Let’s take an example of a 3 by 3 game:
| | A | B | C |
|—|—|—|—|
| X | 2 | 1 | 4 |
| Y | 5 | 0 | -1|
| Z |-2 | 6 |-3 |
In this game, player X has three possible actions: A, B, or C. And player Y also has three possible actions: X, Y, or Z. The numbers in each cell represent the payoff for each player.
To solve this game using dominance, we start by looking at each row and column and identifying any dominated actions. An action is dominated if there exists another action that yields a higher payoff regardless of what the opponent does.
For example, let’s look at row X. Action A yields a payoff of 2 when player Y chooses X as their action.
But if player Y chooses Y or Z, then action B yields a higher payoff of 1 or 4 respectively. Therefore, action A is dominated by action B and can be eliminated.
We can apply the same logic to other rows and columns and eliminate any dominated actions. After eliminating the dominated actions, we get a reduced game:
| | B | C |
|—|—|—|
| Y | 0 |-1 |
| Z | 6 |-3 |
In this reduced game, player X only has one action left (action C) and player Y only has two actions left (actions Y and Z).
Now, we can solve this reduced game using another method called the minimax algorithm. The minimax algorithm is a method that finds the optimal strategy for a player assuming that their opponent plays optimally as well.
To find the optimal strategy for player X, we look at each possible action (in this case only action C) and calculate the minimum payoff that player Y can guarantee by choosing their best response. In this case, if player X chooses action C, then the minimum payoff that player Y can guarantee is -1 by choosing action Y. Therefore, the optimal strategy for player X is to choose action C.
To find the optimal strategy for player Y, we look at each possible action (in this case actions Y and Z) and calculate the maximum payoff that they can achieve by choosing their best response. If player Y chooses action Y, then they can achieve a maximum payoff of 0 if player X chooses action C. But if they choose action Z instead, they can achieve a maximum payoff of 6 if player X also chooses action Z. Therefore, the optimal strategy for player Y is to choose action Z.
In conclusion, to solve a 3 by 3 game using dominance and minimax algorithm involves identifying any dominated actions in the payoff matrix and finding the optimal strategies assuming that each player plays optimally. While this example is simple, the same principles can be applied to more complex games with larger payoff matrices. Game theory is an incredibly useful tool for decision-making in various fields, and it’s exciting to see how it can be used to model real-world situations.