Normal Form Games in Game Theory

Normal Form Games

• A normal form game is defined by three elements:
  • A set of players (I)
  • A set of feasible actions for each player (Xi)
  • A profit function (Pi) for each player, which is a function of the actions taken by all players

Strategies

• A pure strategy is an action taken by a player in their set of feasible actions (Xi)
• A mixed strategy is a probability distribution over the elements of Xi
• In a normal form game, a player chooses a strategy (pure or mixed) to maximize their profit

Best Response

• A best response is a strategy that gives a player the largest profit possible given the actions of other players
• A best response correspondence occurs when there are multiple strategies that maximize a player's profit

Nash Equilibrium

• A Nash equilibrium is a strategy profile (X*) where each player is best responding to the strategies of other players
• In a Nash equilibrium, each player's strategy is in the best response correspondence to the strategies chosen by all other players

Finite Games

• A finite game is a game where the strategy set of each player has a finite number of possible actions
• In finite games, a pure strategy Nash equilibrium may not exist, but a mixed strategy Nash equilibrium is guaranteed to exist

Continuous Games

• A continuous game is a game where the strategy set is made up of real numbers
• In continuous games, if the payoff function is quasi-concave in Xi and continuous in X, a sufficient condition for the existence of a Nash equilibrium in pure strategies is fulfilled

Normal Form Games

• Defined by three elements: set of players (I), set of feasible actions for each player (Xi), and profit function (Pi) for each player.
• Profit function is a function of the actions taken by all players.

Strategies

• Pure strategy: an action taken by a player in their set of feasible actions (Xi).
• Mixed strategy: a probability distribution over the elements of Xi.
• Players choose a strategy (pure or mixed) to maximize their profit.

Best Response

• Best response: a strategy that gives a player the largest profit possible given the actions of other players.
• Best response correspondence: occurs when there are multiple strategies that maximize a player's profit.

Nash Equilibrium

• Nash equilibrium: a strategy profile (X*) where each player is best responding to the strategies of other players.
• In a Nash equilibrium, each player's strategy is in the best response correspondence to the strategies chosen by all other players.

Finite Games

• Finite game: a game where the strategy set of each player has a finite number of possible actions.
• In finite games, a pure strategy Nash equilibrium may not exist, but a mixed strategy Nash equilibrium is guaranteed to exist.

Continuous Games

• Continuous game: a game where the strategy set is made up of real numbers.
• In continuous games, a sufficient condition for the existence of a Nash equilibrium in pure strategies is fulfilled if the payoff function is quasi-concave in Xi and continuous in X.

Learn about normal form games, including the definition, elements, and strategies involved, such as pure and mixed strategies. Understand how players make decisions to maximize their profit in a game.