Game theory is a branch of mathematics used to study decision-making in situations where two or more individuals or groups have conflicting interests. In game theory, a strategy is a plan of action that a player will follow in order to achieve their objectives. A dominant strategy is a strategy that is always the best choice for a player, regardless of what their opponents do.
The Prisoner's Dilemma
The most famous example of game theory is the prisoner's dilemma. In this scenario, two criminals are arrested and held in separate cells. They are both given the option to confess to the crime and betray their accomplice, or remain silent. If both confess, they will both receive a shorter sentence than if neither confesses. However, if one confesses and the other remains silent, the one who confesses will receive no sentence while the other will receive a long sentence.
The Game Theory Matrix
In game theory, a matrix is used to represent the different outcomes of a game. Each cell of the matrix represents a combination of strategies chosen by the players and the resulting outcomes. The player on the left side of the matrix is referred to as player 1, and the player on the top is referred to as player 2.
For example, in the prisoner's dilemma, the matrix would look like this:
| Confess | Remain Silent | |
| Confess | Both receive shorter sentence | One receives no sentence, other receives long sentence |
| Remain Silent | One receives no sentence, other receives long sentence | Both receive longer sentence |
From the matrix, it is clear that the dominant strategy for each player is to confess, even though both players would be better off if they both remained silent.
Nash Equilibrium
Another important concept in game theory is the Nash equilibrium. This is a state in which each player is making the best decision they can, given the decisions of the other players. In other words, no player can improve their situation by changing their strategy, assuming the other players' strategies remain unchanged.
In the prisoner's dilemma, the Nash equilibrium is for both players to confess. This is because, if one player remains silent, the other player will receive a shorter sentence by confessing. Therefore, both players will confess, even though they would be better off if they both remained silent.
Iterated Prisoner's Dilemma
The prisoner's dilemma can also be played repeatedly over a period of time, known as the iterated prisoner's dilemma. In this scenario, players can learn from their opponents' strategies and adjust their own strategies accordingly.
One strategy that has been found to be successful in the iterated prisoner's dilemma is known as tit-for-tat. This strategy involves starting with cooperation and then copying the opponent's previous move. If the opponent cooperates, the player also cooperates. If the opponent defects, the player defects.
The tit-for-tat strategy has been shown to be successful because it is forgiving, meaning it is willing to cooperate after the opponent has defected, but it is also retaliatory, meaning it will defect if the opponent defects again.
Conclusion
Dominant strategy game theory is a powerful tool for understanding decision-making in situations where conflicting interests are at play. By analyzing the different strategies available to players and the resulting outcomes, game theory can help predict how individuals or groups will behave in a given situation. The concepts of the prisoner's dilemma, the game theory matrix, Nash equilibrium, and the iterated prisoner's dilemma are all important concepts in game theory that are used to understand decision-making in a variety of settings.
