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.