Game theory is a popular and useful tool for analyzing strategic interactions between individuals or groups. One of the most important concepts in game theory is that of normal form games.
A normal form game is a mathematical representation of a strategic situation in which players have a finite set of possible actions, and each player’s payoff depends on both their own actions and the actions of the other players. Normal form games are also known as matrix games, because they can be represented by a matrix that shows each player’s possible actions and their corresponding payoffs.
In a normal form game, each player must choose an action without knowing what the other players will do. This is called simultaneous play, since all players make their decisions at the same time. The payoffs for each player depend on the combination of actions chosen by all players.
To illustrate this concept, consider the classic example of the Prisoner’s Dilemma. In this game, two suspects are arrested and held separately.
Each suspect is offered a deal: if they confess and implicate their partner in the crime, they will receive a reduced sentence while their partner gets a harsher sentence. If both suspects confess, they both get moderate sentences. If neither confesses, they both get light sentences.
We can represent this game in normal form as follows:
| | Confess | Remain Silent |
|——–|———|—————|
| Confess | -5,-5 | -10,0 |
| Remain Silent | 0,-10 | -1,-1 |
In this matrix, the first number represents the payoff to Player 1 (the row player), while the second number represents the payoff to Player 2 (the column player). For example, if both players confess, they each receive a payoff of -5 (which means five years in prison). If Player 1 confesses but Player 2 remains silent, then Player 1 receives a payoff of -10 (ten years in prison) while Player 2 receives a payoff of 0 (no additional sentence).
Normal form games can be analyzed using a variety of techniques, including dominance, Nash equilibrium, and mixed strategies. Dominance refers to situations where one player’s action is always better than another player’s action, regardless of what the other player does.
Nash equilibrium is a concept that describes a situation where each player’s strategy is the best response to the others’ strategies. Mixed strategies involve randomizing between different actions with certain probabilities.
In conclusion, normal form game theory is an important tool for analyzing strategic interactions between individuals or groups. By representing games in matrix form, we can analyze them using various techniques and gain insights into the behavior of rational decision makers.
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