Game Theory 101 (#10): Battle of the Sexes

Name: Game Theory 101 (#10): Battle of the Sexes
Uploaded: 2012-09-04T22:23:46.000Z
Duration: 16 min 53 s

Introduction to Battle of the Sexes

In this section, William Spaniel introduces the concept of the "Battle of the Sexes" game in game theory. The game involves a man and a woman who want to spend an evening together but have different preferences for entertainment options.

Understanding the Game

The man prefers going to see a fight, while the woman prefers going to see a ballet.

Both players prefer being together than being alone, even if they end up at their own preferred forms of entertainment without their partner.

The payoffs for each player depend on whether they coordinate or mismatch their choices.

Payoff Matrix Analysis

Analyzing the payoff matrix, we can identify potential Nash equilibria.

Ballet-Ballet is a Nash equilibrium because neither player has an incentive to deviate from this outcome.

Ballet-Fight and Fight-Ballet are not Nash equilibria because both players have profitable deviations available.

Fight-Fight is also a Nash equilibrium because neither player has an incentive to switch their choice.

Coordination Challenges

While coordination is beneficial for both players, it may not be easy due to conflicting preferences.

There may be situations where they end up in conflictual outcomes instead of coordinating perfectly.

Mixed Strategy Nash Equilibrium

In this section, William Spaniel discusses the possibility of mixed strategy Nash equilibria in the Battle of the Sexes game and suggests using a mixed strategy algorithm to analyze such cases.

Mixed Strategy Analysis

To determine if there are any mixed strategy Nash equilibria, we can apply a mixed strategy algorithm.

If there is a mixed strategy Nash equilibrium, it indicates that players should randomize their choices based on certain probabilities.

Conclusion

The Battle of the Sexes game involves coordination challenges and potential Nash equilibria. While pure strategy Nash equilibria exist, mixed strategy Nash equilibria may also be possible. Analyzing the game using different strategies can provide insights into decision-making and coordination in similar situations.

The transcript provided does not contain enough information to create additional sections.

Expected Utility for Player Two

This section discusses the expected utility for player two in a game, specifically focusing on the left and right strategies as a function of player one's mixed strategy.

Calculation of Player Two's Expected Utility for Left Strategy

The expected utility for player two choosing the left strategy is calculated as a function of player one's mixed strategy.

The probability of player one choosing up (Sigma up) multiplied by 2 is added to the probability of player one choosing down (1 - Sigma) multiplied by 0.

Calculation of Player Two's Expected Utility for Right Strategy

Similarly, the expected utility for player two choosing the right strategy is calculated as a function of player one's mixed strategy.

The probability of player one choosing up (Sigma up) multiplied by 0 is added to the probability of player one choosing down (1 - Sigma) multiplied by 1.

Equating Expected Utilities and Solving for Player One's Mixed Strategy

To find the mixed strategy Nash equilibrium, we set the expected utilities for left and right equal to each other and solve for Sigma.

By solving algebraically, it is determined that when player one plays ballet with a probability of 1/3 and fight with a probability of 2/3, player two becomes indifferent between choosing ballet or fight.

Mixed Strategy Equilibrium

This section explores the mixed strategy equilibrium in the game, where both players have probabilities assigned to their strategies.

Calculation of Player Two's Mixed Strategy

To determine player two's mixed strategy, we compare the expected utilities for player one when playing up and down.

The expected utility for playing up involves multiplying the probability that player two plays left by 1 and adding it to 1 minus Sigma left (the probability that she plays right) multiplied by 0.

The expected utility for playing down involves multiplying the probability that player two plays left by 0 and adding it to 1 minus Sigma left (the probability that she plays right) multiplied by 2.

Solving for Player Two's Mixed Strategy

By setting the expected utilities for up and down equal to each other, we can solve for Sigma left.

Through algebraic manipulation, it is found that Sigma left is equal to 2/3.

This results in a mixed strategy equilibrium where player two mixes with a probability of 2/3 on ballet and 1/3 on fight.

Comparing Equilibria

This section discusses the need to compare the mixed strategy Nash equilibrium with pure strategy Nash equilibria in terms of welfare or payoffs.

Lack of Payoff Information for Mixed Strategy Nash Equilibrium

Currently, there is no obvious payoff information available for the mixed strategy Nash equilibrium.

Only the actions chosen by players are known, but not their welfare when playing this equilibrium.

Importance of Calculating Payoffs

In the next video, payoffs will be calculated to allow for a comparison of welfare between the mixed strategy Nash equilibrium and pure strategy Nash equilibria.

It is crucial to consider both types of equilibria when analyzing games.

Conclusion

The transcript covers the concept of expected utility in game theory and explores both pure strategy and mixed strategy Nash equilibria. It highlights how expected utilities are calculated as functions of players' strategies and demonstrates how to solve for mixed strategies. The importance of comparing different equilibria in terms of welfare or payoffs is emphasized.