Game Theory: Mixed Strategies and Nash Equilibrium

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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 $q$?

$1 - 2q$ Signup and view all the answers

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

-1 Signup and view all the answers

What defines a mixed strategy in game theory?

A probability distribution over a player's available strategies. Signup and view all the answers

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

Each player plays Head and Tail with a probability of one half. Signup and view all the answers

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

By assessing probabilities within their own strategies and those of the opponent. 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?

$7.5 Signup and view all the answers

Which of the following best describes a pure strategy?

Playing one strategy with a probability of 1. Signup and view all the answers

What does Nash's Theorem state about Nash equilibrium?

Every finite game will have at least one Nash equilibrium. Signup and view all the answers

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

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

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

To avoid being easily countered by opponents. Signup and view all the answers

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

Ben will play Tails Signup and view all the answers

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

In zero-sum competition games 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?

q = 1/3 Signup and view all the answers

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

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

How many Nash equilibria exist in the game scenario described?

Three Signup and view all the answers

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

(1/2, 1/2) Signup and view all the answers

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

Go to the Opera Signup and view all the answers

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

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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.