Normal Form Games in Game Theory
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Questions and Answers

Which of the following is NOT an element in the definition of a normal form game?

  • A set of feasible actions for each player
  • A profit function for each player
  • A set of possible outcomes (correct)
  • A set of players

What is a mixed strategy in a normal form game?

  • A best response to the actions of other players
  • A strategy that maximizes a player's profit
  • A pure strategy chosen by a player
  • A probability distribution over the elements of the set of feasible actions $X_i$ (correct)

What is a characteristic of a Nash equilibrium?

  • Each player's strategy is a maximin strategy
  • Each player's strategy is a pure strategy
  • Each player's strategy is a mixed strategy
  • Each player's strategy is a best response to the strategies of other players (correct)

Which type of game is guaranteed to have a mixed strategy Nash equilibrium?

<p>Finite games (D)</p> Signup and view all the answers

What is a sufficient condition for the existence of a Nash equilibrium in pure strategies in continuous games (with $x_i$ being an action of player $i$ and $x$ is the vector of actions chosen by all players).

<p>The payoff function is quasi-concave in $x_i$ and continuous in $x$ (A)</p> Signup and view all the answers

What is a best response correspondence?

<p>A situation where there are multiple strategies that maximize a player's profit (B)</p> Signup and view all the answers

Flashcards

Normal Form Game

A game defined by players, their possible actions, and how those actions affect each player's payoff.

Pure Strategy

A specific action a player chooses.

Mixed Strategy

Randomly choosing an action, weighted by probability.

Best Response

The strategy that gives the highest payoff possible to a player given the actions of other players.

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Best Response Correspondence

Multiple strategies that can achieve a best response.

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Nash Equilibrium

When every player is choosing the best possible strategy given what the other players are choosing.

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Finite Game

A game with a limited number of possible actions for each player.

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Continuous Game

A game where actions can be any value within a range.

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Strategy Profile

Combination of strategies chosen by all players in a game.

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Profit Function

A function determining a player's payoff based on all players' actions.

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Set of Players (I)

Collection of individuals participating in the game.

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Set of Feasible Actions (Xi)

The possible choices available to each player.

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Mixed Strategy Nash Equilibrium

A Nash equilibrium where at least one player uses a mixture of strategies

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Quasi-concave

A function where increasing one variable doesn't necessarily decrease the output.

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Continuous in X

A function that doesn't abruptly change in response to a change in strategies

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Payoff Function

Determining the outcome of a specific strategy profile

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Pure Strategy Nash Equilibrium

A Nash Equilibrium where all players choose a single strategy.

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Finite Actions

Discrete possible actions that a player can take

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Real Number Actions

Numerical action from a continuous spectrum

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Strategy

A complete plan of action for a player

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Study Notes

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.

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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.

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