Game Theory and Nash Equilibrium Concepts
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Questions and Answers

What characterizes a Nash Equilibrium?

  • It is a strategy where players do not have any best responses.
  • It requires players to randomly select their strategies.
  • It is when all players maximize their utility given others' strategies. (correct)
  • It occurs only in games with a single strategy for each player.

    • According to Nash's existence result, which of the following statements is true?

  • Only games with mixed strategies can have a Nash equilibrium.
  • Every finite game has at least one Nash equilibrium. (correct)
  • A game with multiple players cannot have a Nash equilibrium.
  • Nash equilibrium can only be found in infinite games.

    • In game theory, a Best Response refers to:

  • The only strategy that guarantees success in all situations.
  • A random choice made by a player to confuse opponents.
  • The strategy that provides the highest possible utility against any opponent's strategy. (correct)
  • A strategy that minimizes the payoff of opponents.

    • What implication does the Nash Equilibrium have regarding player strategies?

    <p>Players react to opponents' strategies and adjust their own to achieve best responses.</p> Signup and view all the answers

    Which of the following statements about Nash Equilibria is correct?

    <p>Nash equilibria can include both pure and mixed strategies.</p> Signup and view all the answers

    What is a key feature of strategic interaction in decision problems?

    <p>Your outcome depends on the choices of other agents.</p> Signup and view all the answers

    In which situation would Nash Equilibrium be particularly relevant?

    <p>A competitive environment where players make decisions simultaneously.</p> Signup and view all the answers

    What is the implication of wanting to maximize one's payoff in strategic scenarios?

    <p>You need to consider the strategies of other agents.</p> Signup and view all the answers

    Which of the following examples demonstrates strategic interaction?

    <p>Oligopolists choosing production levels.</p> Signup and view all the answers

    What characterizes the normative aspect of Analytical Game Theory?

    <p>It examines how decisions should be made based on ideal strategies.</p> Signup and view all the answers

    In the Battle of the Sexes game, what utility does your partner receive when attending a heavy metal concert?

    <p>2</p> Signup and view all the answers

    What is the utility level for you when attending gangsta rap events in the Battle of the Sexes game?

    <p>2</p> Signup and view all the answers

    What happens if both criminals in the Prisoners' Dilemma decide to defect?

    <p>They serve 10 years each.</p> Signup and view all the answers

    In the context of the Prisoners' Dilemma, what is the outcome for both criminals if neither of them defects?

    <p>They are convicted of the minor charge.</p> Signup and view all the answers

    Which of the following correctly describes a Nash equilibrium in the context of the Battle of the Sexes?

    <p>Both partners choose to attend heavy metal concerts together.</p> Signup and view all the answers

    What is a key characteristic of the Prisoners' Dilemma?

    <p>Defecting always leads to the best individual outcome regardless of the other.</p> Signup and view all the answers

    In the Battle of the Sexes, what is the utility level for both partners if they cannot agree and go to separate concerts?

    <p>0</p> Signup and view all the answers

    What does the payoff-matrix help to reveal in a game like the Prisoners' Dilemma?

    <p>Potential payoffs for different strategy combinations.</p> Signup and view all the answers

    What is the outcome of finite repetition in game theory?

    <p>Involves non-credible threats to defect.</p> Signup and view all the answers

    In an anti-coordination game, what happens if the player meets their ex-boyfriend?

    <p>They receive negative utility.</p> Signup and view all the answers

    What characterizes a Nash equilibrium in mixed strategies?

    <p>Each player is indifferent among pure strategies given the other player's strategy.</p> Signup and view all the answers

    Which of the following statements about the anti-coordination game is true?

    <p>It is also referred to as a zero-sum game.</p> Signup and view all the answers

    What could induce efficient outcomes in infinite repetition of a game?

    <p>Patient players with high discount factors.</p> Signup and view all the answers

    In the context of the anti-coordination game, what is the preferred outcome for the player?

    <p>Receiving a utility of 1.</p> Signup and view all the answers

    What is one consequence of modifying the payoff structure in game theory?

    <p>It can induce credible, strategy-dependent fines.</p> Signup and view all the answers

    What does it mean for a strategy profile to be Pareto-efficient in the anti-coordination game?

    <p>At least one player's utility is maximized without reducing another player's utility.</p> Signup and view all the answers

    What does it mean for a strategy profile X to Pareto dominate strategy profile Y?

    <p>All players weakly prefer the outcome under X over Y.</p> Signup and view all the answers

    What characterizes a Pareto optimal strategy profile?

    <p>It is not Pareto dominated by any other strategy profile.</p> Signup and view all the answers

    In the context of the Prisoners’ Dilemma, why is the Nash equilibrium not Pareto optimal?

    <p>It results in an inefficient allocation that all rational players would prefer to avoid.</p> Signup and view all the answers

    What is one proposed method for rational players to overcome inefficiencies in the Prisoners’ Dilemma?

    <p>Engaging in pre-play communication.</p> Signup and view all the answers

    What is indicated by the term 'cheap talk' in the context of pre-play communication?

    <p>Verbal agreements that lack credibility and are often ignored.</p> Signup and view all the answers

    Why might players in the Prisoners’ Dilemma fail to achieve a Pareto optimal outcome?

    <p>Their rational decisions lead them to betray each other instead of cooperating.</p> Signup and view all the answers

    Which statement is true about the relationship between Nash equilibrium and Pareto optimality?

    <p>Some Nash equilibria can be Pareto optimal, while others are not.</p> Signup and view all the answers

    What is a consequence of players not being able to coordinate in an efficient allocation within a Nash equilibrium?

    <p>An allocation that favors individual rationality over collective benefit occurs.</p> Signup and view all the answers

    What is the primary characteristic of the resource in the Chicken Game?

    <p>It is rivalrous but non-excludable.</p> Signup and view all the answers

    In the context of the Hawk-Dove Game, what does a higher relative payoff imply for players?

    <p>Players will replicate more due to higher fitness.</p> Signup and view all the answers

    What is the only Evolutionarily Stable Strategy (ESS) in the Hawk-Dove Game?

    <p>Mixed strategy Nash equilibrium.</p> Signup and view all the answers

    In the Entry Game, if the entrant decides to enter and the incumbent does nothing, what is the incumbent's profit?

    <p>2</p> Signup and view all the answers

    If the entrant does not enter the market, what is their profit?

    <p>2</p> Signup and view all the answers

    What happens to profits if the incumbent starts a price war after the entrant enters?

    <p>Both profits fall to 0.</p> Signup and view all the answers

    In the context of the Entry Game, what is the profit outcome for the entrant if they choose to enter and the incumbent reacts by doing nothing?

    <p>4</p> Signup and view all the answers

    Which statement accurately reflects the nature of the threats in the Entry Game?

    <p>Some threats are credible while others are not.</p> Signup and view all the answers

    Study Notes

    Behavioural Economics: Strategic Interaction

    • This chapter covers strategic interaction and game theory, specifically analytical game theory (normative) and behavioural game theory (positive).
    • Real-world decision problems often have a strategic/interactive element. This means the outcome depends not only on your choice, but also on the choices of other actors.
    • If you aim to maximize your own payoff, you must consider how others will act.
    • Examples include chess, conflict, cooperation, and oligopolists deciding on production levels.
    • Game theory analyzes strategic interactions.

    Analytical Game Theory: Definitions

    • A game is a decision problem where the ultimate result depends on all actors' decisions, as well as external circumstances.
    • Agents in the game are called players.
    • A strategy is a comprehensive action plan that outlines how a player will act under different scenarios.
    • A strategy profile presents a player's strategy for each participant.
    • The payoff matrix shows the payoffs (utility levels) for different player strategy combinations.

    Exercise 5.1: Exam Situation

    • Two students miss an exam due to oversleeping and invent the excuse of a flat tire.
    • One question on their resit exam: "Which tyre was broken?"
    • The task is to examine this experience through a game's lens and determine the components (i.e. players, strategies, payoff matrix).

    Best Response and Nash Equilibrium

    • Best response: For a chosen strategy of other players, a best response strategizes to maximize your utility. Analytical game theory suggests rational players select a specific equilibrium strategy profile.
    • Nash equilibrium: A strategy profile is a Nash equilibrium if every strategy within that profile is the best response against all other strategies in the same profile. This defines mutual best responses.

    Theorem 5.1: Nash Equilibrium Existence

    • Every finite game where all players have a limited number of strategies must possess a Nash equilibrium.
    • Nash equilibria may not be limited to pure strategies. It might involve probability-based approaches.

    Exercise 5.2: Battle of the Sexes

    • Both participants desire mutual interaction (concerts), but have varying preferences on the type of concert.
    • If they cannot agree, their utility is zero.
    • Examine this situation using a payoff matrix and determine the pure strategy Nash equilibria.

    Exercise 5.3: Prisoners' Dilemma

    • Two criminals are apprehended in separate crimes.
    • The prosecutor can only convict on a minor offense.
    • The deal involved one getting a free ride if they admit to the greater/major offense with the other receiving punishment.
    • The strategy is to examine this scenario with a payoff matrix. Identify and locate the pure-strategy Nash Equilibria.

    Pareto Dominance and Optimality

    • Pareto dominance: Strategy Profile X surpasses Y when each player prefers the outcome of strategy X over strategy Y.
    • Pareto optimality: A strategy profile is Pareto optimal, meaning it can't be surpassed, if it's not dominated by a different profile.
    • Determine whether the Nash equilibria located in Exercise 5.2 and 5.3 meet the criteria of Pareto optimality.

    Prisoners' Dilemma and Pareto Optimality

    • The Nash equilibrium in a prisoners' dilemma isn't Pareto optimal in most cases.
    • Even if all are rational, they can't come to a consensus for an efficient strategy allocation. This violates Adam Smith's invisible hand concept.
    • How rational players can avoid the limitations of this dilemma. Strategies for overcoming the inefficiency of the game are discussed.

    Exercise 5.4: Anti-Coordination Game

    • Your goal is to meet your new boyfriend/girlfriend for coffee.
    • You have selected two venues (A or B).
    • Your ex wants to meet you and assumes you'll choose either A or B.
    • Create a payoff matrix for this game considering when interaction occurs. Locate the best responses. Determine the Pareto-efficient equilibrium strategy profiles, considering that both utilities are influenced by meeting/not meeting criteria.

    Definition 5.5: Nash Equilibrium in Mixed Strategies

    • Mixed strategy Nash equilibrium: Each player uses a probability distribution over pure strategies in which they're neutral to the other player's randomization strategy distribution function.

    Remarks on Theorem 5.1 and Mixed Strategies

    • Not all games are guaranteed to have a Nash equilibrium in pure strategies. Multiple Nash equilibria might be present in some games, for example, a battle of the sexes.
    • Rational player behaviour needs 'randomization' amongst multiple strategy options to ensure a mixed equilibrium. The probabilities assigned for different options rely on player utility outcomes from other player choices.

    Exercise 5.5: Chicken (Hawk-Dove) Game

    • Two drivers approaching a cliff.
    • The first person to brake is considered the 'chicken' and loses the game.
    • If both avoid braking at the same time, both suffer. Determine the payoff matrix. Identify all Nash equilibria and Pareto optimal strategy profiles.
    • Examine the implications and discussion about the 'Chicken' game as applied to conflict or political situations.

    Exercise 5.6: Entry Game

    • An established monopoly (Incumbent) faces a potential new player (Entrant).
    • The entrant decides whether to enter the market.
    • Monopoly can respond to the entry by not retaliating (no action) or by initiating a price war. Construct the payoff matrix. Identify all Nash equilibria in pure strategies. Represent this interaction using an extensive game form.
    • Discuss the likely outcomes, based on the Nash equilibriums identified.

    Definition 5.7: Subgame-Perfect Equilibrium

    • A subgame-perfect equilibrium constitutes a Nash equilibrium for all subgames within a larger game.

    Theorem 5.3: Subgame-Perfect Equilibrium Existence

    • In finite extensive-form games, a subgame-perfect Nash equilibrium exists.
    • Subgame-perfect equilibria can often be recognized using backward induction in extensive game forms.

    Example 5.2: Mutually Assured Destruction (MAD)

    • MAD is a Cold War-era military strategy.
    • This doctrine revolves around the threat of complete annihilation if one party attacks.
    • Analyze the issues that arise in strategic interactions concerning deterrence strategies like this, and the plausibility of its success.

    Recap of Concepts

    • Key concepts include best response, Nash equilibrium, coordination, anti-coordination, mixed strategies, Pareto optimality, prisoners' dilemma, subgame-perfect equilibrium.
    • Game theory has applications across various disciplines, including politics, economics, and biology. Understanding concepts can help to predict and assess strategic scenarios in the real world.

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    Description

    This quiz explores essential concepts of game theory, focusing on Nash Equilibrium and its applications. You'll encounter scenarios like the Prisoners' Dilemma and the Battle of the Sexes to understand player strategies and best responses. Test your knowledge on the implications of strategic interactions in decision-making processes.

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