Game theory is a branch of mathematics that deals with the study of strategic decision-making. It has applications in various fields such as economics, political science, psychology, and sociology. One of the most important concepts in game theory is the dominant strategy. A dominant strategy is a strategy that leads to the best outcome for a player regardless of the strategy chosen by the other players.

Understanding Game Theory

Game theory is a mathematical framework that models the behavior of rational players in strategic situations. It involves analyzing the interaction between two or more players who have conflicting interests. Each player has a set of possible strategies, and the outcome of the game depends on the strategies chosen by all the players.

The goal of game theory is to predict the behavior of the players and the outcome of the game. To achieve this, game theorists use various tools such as payoff matrices, decision trees, and Nash equilibrium.

What is a Dominant Strategy?

A dominant strategy is a strategy that leads to the best outcome for a player regardless of the strategy chosen by the other players. In other words, a dominant strategy is the best response to any strategy chosen by the other players. A player who has a dominant strategy will always choose that strategy, regardless of what the other players do.

For example, consider a game where two players have to choose between two options: A and B. If Player 1 chooses A, Player 2 can either choose A or B. If Player 1 chooses B, Player 2 can either choose A or B. The payoff matrix for this game is as follows:

	Player 2 chooses A	Player 2 chooses B
Player 1 chooses A	(2,3)	(0,0)
Player 1 chooses B	(0,0)	(3,2)

In this game, the dominant strategy for Player 1 is to choose option A. This is because, regardless of what Player 2 chooses, Player 1 will get a higher payoff by choosing A. Similarly, the dominant strategy for Player 2 is to choose option A.

Finding Dominant Strategies

Finding dominant strategies can be a complex task, especially for games with more than two players or more than two strategies. However, there are some techniques that can be used to simplify the process.

Elimination of Dominated Strategies

The first technique is the elimination of dominated strategies. A dominated strategy is a strategy that always leads to a worse outcome than another strategy, regardless of the strategies chosen by the other players. If a player has a dominated strategy, that strategy can be eliminated from consideration.

For example, consider a game where two players have to choose between four options: A, B, C, and D. If Player 1 chooses A, B, or C, Player 2 can choose any option. If Player 1 chooses option D, Player 2 can only choose option A or B. The payoff matrix for this game is as follows:

	Player 2 chooses A	Player 2 chooses B	Player 2 chooses C
Player 1 chooses A	(1,1)	(0,2)	(2,0)
Player 1 chooses B	(2,0)	(1,1)	(0,2)
Player 1 chooses C	(0,2)	(2,0)	(1,1)
Player 1 chooses D	(0,0)	(0,0)	(0,0)

In this game, option D is a dominated strategy for Player 1. This is because, regardless of what Player 2 chooses, Player 1 will get a higher payoff by choosing A, B, or C. Therefore, option D can be eliminated from consideration.

After eliminating option D, the payoff matrix becomes:

	Player 2 chooses A	Player 2 chooses B	Player 2 chooses C
Player 1 chooses A	(1,1)	(0,2)	(2,0)
Player 1 chooses B	(2,0)	(1,1)	(0,2)
Player 1 chooses C	(0,2)	(2,0)	(1,1)

Now, we can check if any option is dominated for Player 2. If we eliminate options A and B, the payoff matrix becomes:

	Player 2 chooses C
Player 1 chooses A	(2,0)
Player 1 chooses B	(0,2)
Player 1 chooses C	(1,1)

Now, we can see that option C is a dominant strategy for Player 2. This is because, regardless of what Player 1 chooses, Player 2 will get a higher payoff by choosing option C. Therefore, the dominant strategy for Player 2 is to choose option C.

After eliminating options A and B, the payoff matrix becomes:

	Player 2 chooses C
Player 1 chooses A	(1,1)
Player 1 chooses B	(0,2)
Player 1 chooses C	(2,0)

Now, we can see that option A is a dominant strategy for Player 1. This is because, regardless of what Player 2 chooses, Player 1 will get a higher payoff by choosing option A. Therefore, the dominant strategy for Player 1 is to choose option A.

Nash Equilibrium

Another technique for finding dominant strategies is Nash equilibrium. Nash equilibrium is a situation where each player's strategy is the best response to the strategies chosen by the other players. In other words, no player can improve their payoff by changing their strategy, given the strategies chosen by the other players.

For example, consider a game where two players have to choose between two options: A and B. The payoff matrix for this game is as follows:

	Player 2 chooses A	Player 2 chooses B
Player 1 chooses A	(2,2)	(0,3)
Player 1 chooses B	(3,0)	(1,1)

To find the Nash equilibrium for this game, we need to find the strategies that are the best response for each player given the strategies chosen by the other player. If Player 2 chooses option A, Player 1's best response is to choose option A. If Player 2 chooses option B, Player 1's best response is to choose option B. Therefore, there are two Nash equilibria for this game: (A,A) and (B,B).

Conclusion

Finding dominant strategies is an important concept in game theory. Dominant strategies can simplify the decision-making process for players and help predict the outcome of the game. There are various techniques for finding dominant strategies, such as elimination of dominated strategies and Nash equilibrium. By using these techniques, players can make better decisions and improve their chances of winning the game.

Related video of How to Find Dominant Strategy Game Theory