Game Theory: Mixed Strategies and Nash Equilibrium
21 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the expected payoff for Bill if he plays Heads and Ben plays Heads with probability $p$?

  • $2p - 1$
  • $1 - 2p$ (correct)
  • $p(-1) + (1-p)(1)$
  • $p(1) + (1-p)(-1)$

Under what condition would Bill be indifferent between playing Heads and Tails?

  • When $p = 1$
  • When $p > 1/2$
  • When $p = 1/2$ (correct)
  • When $p = 0$

If $p < 1/2$, which strategy will Bill choose according to his expected payoffs?

  • He will play Heads (correct)
  • He will mix his strategies
  • He will play Tails
  • He will play randomly

What is the expected payoff for Ben when he plays Tails and the probability of Bill playing Heads is $q$?

<p>$1 - 2q$ (C)</p> Signup and view all the answers

If $p = 1$, what would be Bill's expected payoff for playing Heads?

<p>-1 (D)</p> Signup and view all the answers

What defines a mixed strategy in game theory?

<p>A probability distribution over a player's available strategies. (A)</p> Signup and view all the answers

In the game of Matching Pennies, what is the Nash equilibrium?

<p>Each player plays Head and Tail with a probability of one half. (A)</p> Signup and view all the answers

How does a player maximize their expected payoff using a mixed strategy?

<p>By assessing probabilities within their own strategies and those of the opponent. (D)</p> Signup and view all the answers

What is the expected payoff of an outcome that has a 50/50 chance of yielding a payoff of 10 and a 5?

<p>$7.5 (B)</p> Signup and view all the answers

Which of the following best describes a pure strategy?

<p>Playing one strategy with a probability of 1. (C)</p> Signup and view all the answers

What does Nash's Theorem state about Nash equilibrium?

<p>Every finite game will have at least one Nash equilibrium. (B)</p> Signup and view all the answers

In the context of the Left-Centre-Right game mentioned, what is a mixed strategy represented by?

<p>A probability distribution over three available pure strategies (p, q, 1-p-q). (C)</p> Signup and view all the answers

Why might players choose to employ a mixed strategy rather than a pure strategy?

<p>To avoid being easily countered by opponents. (B)</p> Signup and view all the answers

What will Ben choose if the probability q is less than 1/2?

<p>Ben will play Tails (D)</p> Signup and view all the answers

In which scenario does Mixed Strategies make sense according to game theory?

<p>In zero-sum competition games (C)</p> Signup and view all the answers

At what value of q is the Woman indifferent between attending the Opera and the Fight in the Battle of the Sexes?

<p>q = 1/3 (B)</p> Signup and view all the answers

What is the best response for Bill when he is playing against the given strategy by Ben?

<p>His best response is (1/2, 1/2) (D)</p> Signup and view all the answers

In the context of the Battle of the Sexes, what are the expected payoffs for the Man when he chooses the Opera if p is the probability he plays Heads?

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

How many Nash equilibria exist in the game scenario described?

<p>Three (C)</p> Signup and view all the answers

What is the Nash equilibrium strategy for both Ben and Bill?

<p>(1/2, 1/2) (D)</p> Signup and view all the answers

When p equals 2/3, what will the Man choose to do in the Battle of the Sexes?

<p>Go to the Opera (A)</p> Signup and view all the answers

Flashcards

Bill's belief in Ben playing Heads

The probability that Bill believes Ben will play Heads.

Ben's belief in Bill playing Heads

The probability that Ben believes Bill will play Heads.

Nash Equilibrium in Mixed Strategies

The situation where both players are indifferent between playing Heads or Tails, meaning they have no incentive to change their strategy.

Bill's Equilibrium Strategy

The probability of Bill playing Heads that makes Ben indifferent between playing Heads or Tails.

Signup and view all the flashcards

Bill's Indifference Point

The point where Bill's expected payoff from playing Heads equals his expected payoff from playing Tails.

Signup and view all the flashcards

Mixed Strategy

A strategy where a player chooses an action with a certain probability, rather than always choosing the same action.

Signup and view all the flashcards

Matching Pennies

A game where players simultaneously choose between two options, and the outcome depends on the matching or mismatching of choices.

Signup and view all the flashcards

Nash Equilibrium

A state where no player can improve their outcome by unilaterally changing their strategy, even if they know the other player's strategy.

Signup and view all the flashcards

Mixed Strategy Equilibrium

A combined strategy where each player chooses their actions with certain probabilities, maximizing their expected payoff when considering the other player's strategy.

Signup and view all the flashcards

Expected Payoff

The average payoff a player can expect to receive, considering the probabilities of different outcomes.

Signup and view all the flashcards

Identifying a Mixed Strategy Equilibrium

The process of identifying the combined strategies where no player can improve their expected payoff by changing their own strategy, given the other player's strategy.

Signup and view all the flashcards

Expected Value Calculation

The mathematical tool used to determine the best mixed strategy for a player in a game, considering the probabilities of different outcomes.

Signup and view all the flashcards

Uniform Mixed Strategy

A type of mixed strategy where the probabilities of each action are equal for all players.

Signup and view all the flashcards

Mixed strategy Nash Equilibrium

A Nash Equilibrium in a mixed strategy game where players are indifferent between their available actions.

Signup and view all the flashcards

Strategy probability

The probability with which a player chooses a specific action in a mixed strategy.

Signup and view all the flashcards

Best response function

A scenario where a player's best response depends on the probability with which the other player chooses their actions.

Signup and view all the flashcards

Zero-sum game

A game where the gain of one player is equal to the loss of the other player.

Signup and view all the flashcards

Coordination game

A type of game where players benefit from coordinating their actions to achieve a shared outcome.

Signup and view all the flashcards

Bluffing

A strategy that involves intentionally misleading an opponent about one's true intentions.

Signup and view all the flashcards

Study Notes

Mixed Strategies

  • Pure strategies studied so far can be replaced by mixed strategies in some games
  • A mixed strategy involves a probability distribution over a player's available pure strategies.

Matching Pennies

  • A game where Bill and Ben simultaneously show one face of a coin
  • Ben wins if both faces are the same, Bill if they are different.
  • A payoff matrix, with payoffs circled, shows the outcome for each action for Bill and Ben
  • A Nash equilibrium exists in every finite game

Finding a Nash Equilibrium

  • The Nash equilibrium is where both players choose their actions with a probability of 1/2
  • Two pure strategies (Head or Tail) determine mixed strategies
  • The probability distribution for a player is (p, 1-p) where p = Probability(Head), 1-p= Probability(Tail)

Mixed Strategies Examples

  • Matching Pennies: Playing Head/Tail equally often, or randomly
  • Left-Centre-Right Game: Each of the three options is chosen with a 1/3 probability

Identifying a Mixed Strategy Equilibrium

  • Players aim to maximize expected payoffs
  • Expected payoffs calculated using probabilities of both player's actions

Expected Payoffs Calculations

  • Example: Outcome Y has a 50/50 chance of yielding a payoff of 10 or 5
  • Expected payoff = (Payoff 1 x Probability1) + (Payoff2 x Probability2)
  • In this example, the expected payoff would be 7.5

Nash Equilibrium and Mixed Strategies

  • A combination of mixed strategies forms a Nash equilibrium
  • Each player's mixed strategy is optimal given the mixed strategies of the other players

Bill and Ben's Beliefs

  • Bill assumes Ben plays Heads with probability p and Tails with 1-p
  • Ben assumes Bill plays Heads with probability q and Tails with 1-q

Identifying the Nash Equilibrium in Mixed Strategies

  • Bill's expected payoffs when choosing Head and Tail are calculated
  • Ben's expected payoffs when choosing Head and Tail are calculated
  • The values to make both players indifferent between both actions lead to a Nash equilibrium

Finding the Equilibrium Strategy

  • Equating expected payoffs for Heads and Tails determines Bill's strategy, example: 1-2p = 2p-1, thus, p=0.5
  • Equating expected payoffs leads to Ben's strategy, q=0.5

Mixed Strategies in the Real World

  • Poker: Bluffing is often needed for a good strategy
  • Penalty shootouts: Goalkeepers and shooters randomize their strategies

Battle of the Sexes Revisited

  • Mixed strategies achieve a "fair" Nash equilibrium in coordination games

Battle of the Sexes Revisited - Payoff Calculations

  • Expected payoffs for the Woman are calculated, based on possible outcomes
  • Expected payoffs for the Man are calculated, based on possible outcomes

Battle of the Sexes Revisited - Finding the Equilibrium

  • The Woman is indifferent between Opera and Fight when 2q = 1 - q, leading to q =1/3
  • The Man is indifferent between Opera and Fight when p = 2 - 2p,leading to p = 2/3

Battle of the Sexes Revisited Results

  • The game has three Nash equilibria

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

Related Documents

Mixed Strategies PDF

Description

Explore the concepts of mixed strategies and Nash equilibrium in game theory. This quiz covers examples like Matching Pennies and examines how to determine strategies using probability distributions. Perfect for understanding strategic interactions in finite games.

More Like This

Use Quizgecko on...
Browser
Open
Browser
Browser