Game theory is a mathematical approach to understanding strategic interactions between individuals, corporations, or nations. It’s a way of modeling decision-making processes in situations where the outcome depends on the choices made by multiple actors.
A simple game in game theory is a scenario that involves only a few players and choices. In this article, we’ll explore what a simple game in game theory is and how it works.
What is Game Theory?
Game theory is a branch of mathematics that deals with the analysis of strategic interactions between individuals, corporations, or nations. It provides a framework for understanding how people make decisions when their outcomes depend on the actions of others.
In game theory, we use models to represent these decision-making processes in situations where two or more players interact. These models can be used to predict outcomes and determine optimal strategies for each player based on their goals and objectives.
What is a Simple Game?
A simple game is a scenario in game theory that involves only a few players and choices. Typically, these games are represented using what’s called a “normal form” or “matrix” representation.
In this representation, each player has a set of possible actions they can take, and the outcomes depend on the combination of choices made by all players. The payoffs for each player are also specified in advance.
For example, let’s consider the following simple game:
Player 1 / Player 2 | Choice A | Choice B
———— | ————- | ————-
Choice X | (1,1) | (0,0)
Choice Y | (0,0) | (2,2)
In this simple game, Player 1 can choose either X or Y while Player 2 can choose either A or B. The payoffs for each player are listed inside the parentheses next to each choice combination.
If Player 1 chooses X and Player 2 chooses A, both players receive a payoff of 1. If Player 1 chooses X and Player 2 chooses B, both players receive a payoff of 0. And so on for the other three possible combinations.
How Does a Simple Game Work?
In a simple game, each player is assumed to be rational and self-interested. That is, they will choose the action that maximizes their expected payoff given their beliefs about what the other player(s) will do.
To solve for the optimal strategy in a simple game, we use what’s called “dominance” and “Nash equilibrium.” Dominance means that one choice is always better than another regardless of what the other player(s) do. Nash equilibrium means that each player’s choice is optimal given the choices of all other players.
For example, in the simple game above, Player 1 should choose Y since it dominates X (i.e., it has a higher payoff regardless of what Player 2 does). Similarly, Player 2 should choose B since it dominates A.
The Nash equilibrium for this game is (Y,B), which means that neither player can improve their payoff by changing their choice given the other player’s choice.
In conclusion, a simple game in game theory is a scenario that involves only a few players and choices. These games are represented using a normal form or matrix representation, where each player has a set of possible actions they can take and the payoffs are specified in advance.
To solve for optimal strategies in these games, we use dominance and Nash equilibrium. Simple games are useful for understanding basic concepts in game theory before moving on to more complex scenarios with multiple rounds or imperfect information.