Game Theory Basics Quiz

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Questions and Answers

What is the best strategy for player 1 regardless of player 2's choice?

  • D
  • R
  • L
  • U (correct)

What will player 2 choose if player 1 selects U?

  • D
  • L (correct)
  • R
  • U

How are the payoffs represented in two-player strategic form games?

  • In matrix form (correct)
  • As a table of values
  • Through a vector
  • In a graph

What is player 2's payoff-maximizing strategy dependent on?

<p>Player 1's strategy choice (A)</p> Signup and view all the answers

Which strategy pair is the only sensible outcome of the game described?

<p>(U, L) (A)</p> Signup and view all the answers

What does the term 'dominant strategy' imply in the context of game theory?

<p>A strategy that is beneficial regardless of opponents' strategies (A)</p> Signup and view all the answers

What is the outcome deduced when a game is played by rational players?

<p>The equilibrium outcome based on strategies (D)</p> Signup and view all the answers

What does the notation [$−i$] signify in the context of joint pure strategies?

<p>The strategies of all players except player i (D)</p> Signup and view all the answers

Which concept indicates a more advanced understanding of strategic interaction than merely recognizing payoffs?

<p>Subgame perfection (D)</p> Signup and view all the answers

What characterizes a strategic decision rather than a non-strategic decision?

<p>It requires anticipating the actions of other players. (C)</p> Signup and view all the answers

In a strategic form game, how is the payoff for a player determined?

<p>Through a function of their own strategy and those chosen by other players. (B)</p> Signup and view all the answers

What is an essential feature of a Nash equilibrium?

<p>Each player's strategy is an optimal response to the other players' strategies. (B)</p> Signup and view all the answers

What is the role of unpredictability in strategic decision making?

<p>It arises as players try to maximize their payoffs while considering opponents' strategies. (D)</p> Signup and view all the answers

What distinguishes a Bayesian-Nash equilibrium from a regular Nash equilibrium?

<p>It involves players having beliefs about the types of other players. (C)</p> Signup and view all the answers

Which structure describes the strategic situation involving multiple players and their available strategies?

<p>Strategic form game (B)</p> Signup and view all the answers

In what way is subgame perfection stronger than Nash equilibrium?

<p>It requires rational play in all subgames. (D)</p> Signup and view all the answers

What condition defines a strictly dominant strategy for player i?

<p>It is always better than any other strategy for player i, regardless of opponent's choice. (D)</p> Signup and view all the answers

In the elimination of dominated strategies, what is a characteristic of a strategy that is strictly dominated?

<p>It is never chosen because it yields a lower payoff than another strategy. (A)</p> Signup and view all the answers

If player 1's strategy C is always outperformed by D, what can be inferred about C?

<p>C is a strictly dominated strategy for player 1. (A)</p> Signup and view all the answers

What does the notation $S_i^n$ represent?

<p>The strategies of player i remaining after n rounds of elimination. (C)</p> Signup and view all the answers

Which of the following scenarios illustrates strictly dominated strategies?

<p>Player 2's strategy R always yields a higher payoff than strategy M when Player 1 plays D. (B)</p> Signup and view all the answers

What result can be concluded when both strategies C and M are removed from the strategies set?

<p>Both strategies were strictly dominated by other strategies. (A)</p> Signup and view all the answers

What is the outcome represented by the pair (3, 0) in the context of game strategies?

<p>It indicates the best outcome for player 1 with no payoff for player 2. (B)</p> Signup and view all the answers

Which of the following best describes strategies after the nth round of elimination?

<p>They are the strategies that remain undominated and viable for player i. (A)</p> Signup and view all the answers

What characterizes a strategy that is iteratively strictly undominated?

<p>It belongs to the set of strategies for player i across all iterations. (B)</p> Signup and view all the answers

In game theory, a weakly dominated strategy is defined as one that:

<p>Has at least one comparison where it performs worse than another strategy. (D)</p> Signup and view all the answers

Which strategies are characterized as weakly dominated in the given situation?

<p>D and R are weakly dominated by U and L. (B)</p> Signup and view all the answers

What does the notation $W_i^n$ represent in game theory?

<p>The strategies that survive the nth round of elimination of strictly dominated strategies. (D)</p> Signup and view all the answers

Under what condition does a strategy qualify as iteratively weakly undominated?

<p>It is not weakly dominated in the set of strategies remaining after each elimination round. (C)</p> Signup and view all the answers

What happens to weakly dominated strategies when they are eliminated?

<p>They can make eliminating strictly dominated strategies ineffective. (C)</p> Signup and view all the answers

What is the implication of having no strictly dominated strategies present?

<p>Players have a clearer choice in selecting optimal strategies. (B)</p> Signup and view all the answers

Why is it important to identify both strictly and weakly dominated strategies in a game?

<p>It ensures players can avoid dominated strategies to maximize their payoffs. (D)</p> Signup and view all the answers

What condition must be satisfied for a situation to be considered a Nash equilibrium?

<p>No player can improve their payoff by unilaterally changing their own strategy. (B)</p> Signup and view all the answers

Which statement about pure strategy Nash equilibria is true?

<p>A game may possess more than one pure strategy Nash equilibrium. (B)</p> Signup and view all the answers

How is a mixed strategy defined in the context of Nash equilibrium?

<p>It is a probability distribution over multiple strategies. (A)</p> Signup and view all the answers

In which of the following cases can a Nash equilibrium exist?

<p>When players randomize their choices and have no predictable strategy. (A)</p> Signup and view all the answers

Which of the following describes a scenario where no pure strategy Nash equilibrium exists?

<p>Players have conflicting goals and strategies. (B)</p> Signup and view all the answers

What is the expected utility of both players in the batter-pitcher example when they use mixed strategies?

<p>$0$ (B)</p> Signup and view all the answers

What implication does knowing the opponents' strategy have in playing a Nash equilibrium?

<p>It gives no advantage as changes lead to unpredictability. (D)</p> Signup and view all the answers

Which of the following statements is incorrect regarding the characteristics of Nash equilibria?

<p>All players have complete ignorance of others' strategies. (A)</p> Signup and view all the answers

What does a mixed strategy allow players to do?

<p>Randomize their choices among strategies (C)</p> Signup and view all the answers

In the context of Nash equilibrium, what does the expected utility of player i's strategy represent?

<p>The predicted payoff based on mixed strategies (B)</p> Signup and view all the answers

How is the joint strategy represented in the context of mixed strategies?

<p>As a combination of all players' actions (C)</p> Signup and view all the answers

According to Theorem 7.1, what is one condition for a strategy to be considered a Nash equilibrium?

<p>It must yield at least the same utility as any alternative strategy (C)</p> Signup and view all the answers

What does the term 'incomplete information' refer to in game theory?

<p>Some players are uncertain about others' payoff functions (A)</p> Signup and view all the answers

What is required for a game to be considered under complete information?

<p>Each player's payoff function is known to all players (A)</p> Signup and view all the answers

What does the notation $u_i(m)$ signify in the context of expected utility?

<p>Utility based on mixed strategy outcomes (B)</p> Signup and view all the answers

What fundamentally changes when players use mixed strategies compared to pure strategies?

<p>Strategies can involve randomness (C)</p> Signup and view all the answers

What is a significant outcome of Nash's Existence Theorem?

<p>Every finite strategic form game has at least one Nash equilibrium (A)</p> Signup and view all the answers

Which of the following describes the normal-form representation of a game?

<p>Players' strategies and payoffs are clearly outlined (B)</p> Signup and view all the answers

What is an essential characteristic of a mixed strategy set, $M_i$?

<p>It can represent any pure strategy via probability distribution (C)</p> Signup and view all the answers

Which statement best describes a pure strategy within the context of game theory?

<p>A consistent choice that does not involve randomness (D)</p> Signup and view all the answers

What is the role of each player's utility function in the context of a strategic game?

<p>To determine the payoffs based on chosen strategies (C)</p> Signup and view all the answers

Flashcards

Nash Equilibrium

A situation where no player can improve their payoff by unilaterally changing their strategy, assuming other players remain unchanged.

Bayesian-Nash Equilibrium

Nash equilibrium where players' strategies depend on their private information or beliefs.

Backward Induction

A method for solving sequential games by working backwards from the end of the game to determine the optimal strategy.

Subgame Perfection

A refinement of Nash equilibrium that requires strategies to be the best response at every sub-game.

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Sequential Equilibrium

A solution concept for games with incomplete information that combines the idea of perfect Bayesian equilibrium and the concept of sequential rationality.

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Non-strategic Decision

A decision that can be made without considering the actions of others.

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Strategic Form Game

A game where the strategies available to each player and the payoffs for each combination of strategies are specified.

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Strategic Game Elements

The common elements involving players, strategies, and payoffs depending on a player's strategy and those of others.

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Dominant Strategy

A strategy that yields the best payoff for a player regardless of the strategy chosen by other players.

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Rational Players

Players who aim to maximize their own payoffs in a game.

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Joint Pure Strategies

The combinations of strategies chosen by all players in a game.

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Solving a Game

Determining the outcome of a game when played by rational players.

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Payoff Matrix

A table showing the payoffs for all combinations of strategies chosen by players in a game.

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Strategy Set

All possible choices available to a player in a game.

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Payoff Vector

A vector showing the payoffs for each player in a game given a combination of strategies chosen by all players.

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Strictly Dominant Strategy

A strategy for a player that always yields a higher payoff than any other strategy, regardless of the opponent's strategy.

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Strictly Dominated Strategy

A strategy that is always worse than another strategy, regardless of the opponent's strategy.

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Iterative Elimination of Strictly Dominated Strategies

A process to find the best possible outcome in a game by repeatedly removing strategies that are strictly dominated by other strategies.

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Payoff

The result or outcome for each strategy in a game.

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Elimination of strategies

Removal of suboptimal choices from the strategy set.

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Best Choice

The strategy that provides the highest payout for a player.

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N-th Round Elimination

Strategies that are strictly dominated in the previous round are removed. The surviving strategies become part of the new strategy set for the N-th round.

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Iteratively Strictly Undominated Strategy

A strategy for a player that survives multiple rounds of eliminating strictly dominated strategies.

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Strictly Dominated Strategies - Elimination

A strategy for a player is strictly dominated by another strategy if the other strategy always yields a higher payoff, regardless of what the other players do.

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Weakly Dominated Strategies - Elimination

A strategy for a player is weakly dominated if another strategy always yields a payoff that is at least as good, and sometimes better.

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Iteratively Weakly Undominated Strategy

A strategy for a player that survives multiple rounds of eliminating weakly dominated strategies.

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Domination vs. Elimination

Domination is the relationship between strategies, while Elimination is the process of removing strategies.

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Multiple Rounds of Elimination

Iterative elimination refers to multiple rounds of removing dominated strategies, where the new set of strategies becomes the basis for further elimination.

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Impact of Dominated Strategies

Dominated strategies can be removed because a player can always do better by switching to the dominating strategy.

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Unique Strategy Pair

After eliminating dominated strategies, sometimes a single strategy combination for all players remains as the most rational choice.

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Joint Strategy

A combination of strategies chosen by all players in a game.

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Pure Strategy Nash Equilibrium

When each player's strategy maximizes their payoff given the other players' fixed strategies, and no player can improve by changing their own.

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Mixed Strategy

A player's strategy that involves choosing different actions with certain probabilities.

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Mixed Strategy Nash Equilibrium

A situation where each player's randomized strategy maximizes their expected payoff, given the other players' randomized strategies.

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Expected Utility (in games)

The average payoff a player expects to receive by using a mixed strategy, considering the probabilities of different outcomes.

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Unilaterally Change (Strategy)

Changing your strategy without coordinating with other players.

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Pure Strategy

A specific action that a player can choose in a game. Think of it as a definite plan of action.

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Joint Mixed Strategy

A combination of mixed strategies chosen by all players in a game.

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Expected Utility

The average payoff a player expects to receive when using a mixed strategy.

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What is a simplified Nash Equilibrium test?

A method to verify if a joint strategy is a Nash Equilibrium by analyzing individual player's payoffs for each pure strategy given positive or zero weight by their mixed strategy.

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What does the Nash Existence Theorem state?

Every finite strategic form game possesses at least one Nash Equilibrium.

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What is complete information in a game?

When all players know each other's payoffs for every possible combination of strategies.

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Incomplete Information Game

A game where some players have uncertainty about other players' payoff functions.

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What defines a normal-form representation of a game?

It specifies each player's strategy options and their payoffs associated with all possible combinations of strategies chosen by all players.

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What is the Prisoners' Dilemma?

A classic example of a game where two individuals acting in their own self-interest result in a worse outcome for both compared to if they cooperated.

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What is a payoff function?

A function that determines a player's payoff based on the combination of strategies chosen by all players.

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What is a strategy space?

The set of all possible strategies available to a player in a game.

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What is a combination of strategies?

A specific choice made by each player in a game.

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What is a strategic form game?

A game where all players choose their strategies simultaneously.

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Study Notes

Lecture 1 - Strategic Form Games

  • Game theory = the systematic study of how rational agents behave in strategic situations.
  • Each agent must know the others' decisions before determining their own best choice.
  • Includes concepts like Nash equilibrium, Bayesian-Nash equilibrium, backward induction, and subgame perfection.
  • Knowing when to apply each concept is important for applied economics.

Lecture 2 - Strategic Form Games under Incomplete Information

  • Incomplete information = situations where some players are uncertain about other player's payoff functions.
  • This is different than the standard game's payoff function.
  • Includes Bayesian games.
  • Information about others' types.

Lecture 3 - Extensive Form Games

  • Extensive form = a graphical representation of a game, using a game tree.
  • The nodes represent decisions and actions.
  • The lines connecting the nodes represent a logical sequence.
  • End nodes display payoffs.

Lecture 4 - Extensive Form Games under Incomplete Information

  • Sequential equilibrium = a solution concept for extensive form games with incomplete information, building on Nash Equilibrium.
  • It addresses the issue of beliefs in cases where backward induction isn't directly applicable, due to incomplete information.

Lecture 5 - Repeated Games

  • Deals with games that are played multiple times.
  • Explores the interactions between players when they know the game will be played many times.

Lecture 6 - Mixed Strategies examples

  • Mixed strategies = players randomize their choices in a game.
  • Introduces mixed strategies' Nash Equilibria.

Additional concepts

  • Iterative elimination of strictly dominated strategies: A method for simplifying games by eliminating dominated strategies, to determine which strategies are rational for players.
  • Rationality: Players choose strategies that maximize their expected payoffs (given their beliefs)
  • Common knowledge: A property is common knowledge if everyone knows it, everyone knows that everyone knows it, and so on.
  • Nash equilibrium: A strategy profile where no player can improve their payoff by unilaterally changing their strategy.
  • Subgame perfect equilibrium: A Nash equilibrium where every subgame of the game is a Nash Equilibrium
  • Correlated Equilibrium: A refinement of Nash Equilibrium that allows for correlated strategies, where players' actions may be coordinated.
  • Bayesian Equilibrium: A refinement of Nash Equilibrium that accounts for beliefs about the other players' types or strategies; used in situations with incomplete information.
  • Maxmin (and Minmax) strategies: A method for choosing strategies that maximize the minimum possible payoff and minimize the maximum possible loss.

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