Game theory is a mathematical approach to understanding the behavior of individuals and groups in competitive situations. It is used in a wide range of fields, including economics, political science, and psychology.
One of the most important concepts in game theory is the dominant strategy. A dominant strategy is a strategy that is always the best choice for a player, regardless of what the other players do.
What is a Strategy?
A strategy is a plan of action that a player chooses in a game. It is based on the player's beliefs about what the other players will do, as well as the player's own goals and preferences.
In game theory, a game is defined as a set of players, a set of actions available to each player, and a set of payoffs that each player receives depending on the actions taken by all players.
For example, consider a game between two players, A and B. Each player can choose to either cooperate or defect. If both players cooperate, they both receive a payoff of 3. If one player defects and the other cooperates, the defector receives a payoff of 5 and the cooperator receives a payoff of 1. If both players defect, they both receive a payoff of 2.
The set of strategies available to each player in this game is {cooperate, defect}. The payoffs for each player are given in the following table:
| Cooperate | Defect | |
|---|---|---|
| Cooperate | (3, 3) | (1, 5) |
| Defect | (5, 1) | (2, 2) |
In this game, each player's strategy depends on what they think the other player will do. For example, if player A thinks that player B will cooperate, then player A should defect in order to receive a higher payoff. Similarly, if player B thinks that player A will cooperate, then player B should defect.
What is a Dominant Strategy?
A dominant strategy is a strategy that is always the best choice for a player, regardless of what the other players do. In other words, a dominant strategy is the optimal strategy for a player in any situation.
For example, in the game described above, the strategy of defecting is dominant for both players. No matter what the other player does, each player will receive a higher payoff by defecting.
It is important to note that not all games have dominant strategies. In some games, there may be multiple Nash equilibria, which are situations in which no player can improve their payoff by changing their strategy, given the strategies of the other players.
Examples of Dominant Strategies
There are many examples of dominant strategies in game theory. One classic example is the prisoner's dilemma, which is a game between two suspects who have been arrested and are being interrogated separately.
In the prisoner's dilemma, each suspect can either cooperate with the other suspect and remain silent, or defect and confess. If both suspects cooperate, they both receive a relatively light sentence. If both suspects defect, they both receive a relatively heavy sentence. However, if one suspect defects and the other cooperates, the defector goes free and the cooperator receives a very heavy sentence.
In this game, the dominant strategy for each suspect is to defect, even though both would be better off if they both cooperated.
Conclusion
In summary, a dominant strategy is a strategy that is always the best choice for a player, regardless of what the other players do. It is an important concept in game theory that helps us understand how people and groups behave in competitive situations.
Understanding dominant strategies can be extremely useful in a wide range of fields, from economics and political science to psychology and sociology.
