Game Theory 101 (#65): Solving for Bayesian Nash Equilibrium

Summary

In this Game Theory 101 lecture, William Spaniel explores the concept of Bejan Nash equilibrium (BE) in incomplete information games. He compares it to Nash equilibrium, highlighting the added complexity when players face uncertainty about each other's types. Through a detailed example involving a prisoners dilemma type and a stag hunt type, Spaniel illustrates how to construct a single payoff matrix and find the BE by eliminating strictly dominated strategies. The lecture concludes with insights on mixed strategies, emphasizing how uncertainty can create the illusion of mixing from an external perspective, and sets the stage for future discussions on solving other incomplete information games.

Takeaways

• 😀 The Béla Nash equilibrium is a more complex solution concept than the standard Nash equilibrium, as it requires strategies for each type of player in games with incomplete information.
• 🤔 Incomplete information settings introduce uncertainty about players' types, impacting their strategies and outcomes.
• 🔄 We can use Nash equilibrium to find the Béla Nash equilibrium by converting matrices from the original game into a new configuration.
• 🎲 Player one faces uncertainty regarding player two's type, which can either be a 'prisoners dilemma' type or a 'stag hunt' type.
• 🧩 The payoff matrices are critical for determining the expected outcomes for each player based on their types and strategies.
• ⚖️ Player two, when identified as the prisoners dilemma type, will opt for the defect strategy, which strictly dominates the cooperative option.
• 📊 Calculating expected payoffs accurately is essential, as mistakes in arithmetic can lead to incorrect equilibria.
• ✂️ Iterated elimination of strictly dominated strategies can simplify the game and make finding equilibria easier.
• 🔄 A mixed-strategy Béla Nash equilibrium can occur when one player mixes their strategies while the other plays a pure strategy.
• 🔍 Uncertainty about player types can create the illusion of mixing strategies from the perspective of a player, even when multiple types are playing pure strategies.

Q & A

What is the main topic of the lecture?

The lecture focuses on finding the Bayesian Nash Equilibrium in incomplete information games.

How does Bayesian Nash Equilibrium differ from Nash Equilibrium?

Bayesian Nash Equilibrium accounts for different types of players and their strategies under uncertainty, making it more complex than traditional Nash Equilibrium, which assumes full information.

What types of players are discussed in the example?

The example discusses two types of Player 2: the Prisoners Dilemma type and the Stag Hunt type.

What is the significance of Player 1's Stag Hunt preferences?

Player 1's Stag Hunt preferences indicate that he aims for cooperative outcomes, while facing uncertainty about Player 2's type, which complicates his strategic decisions.

How are the payoff matrices structured for the game?

The payoff matrices are structured to reflect the outcomes based on different strategy combinations of Player 1 and Player 2, with each type of Player 2 having distinct payoffs.

What mathematical method is used to find the Bayesian Nash Equilibrium?

The method of iterated elimination of strictly dominated strategies is used to simplify the game and identify the equilibrium.

What role does probability play in calculating expected payoffs?

Probability is used to weight the payoffs from different outcomes based on the likelihood of Player 2 being a specific type, thus calculating expected payoffs for Player 1.

What happens when strategies are strictly dominated?

When strategies are strictly dominated, they can be eliminated from consideration, simplifying the analysis of the game and helping to identify viable strategies.

What are the final results of the Nash equilibria calculations?

The calculations yield two Nash equilibria: one where Player 1 chooses 'up' and Player 2's strategy depends on her type, and a mixed strategy equilibrium for Player 1 and Player 2.

What future topics will be addressed in the upcoming lectures?

Future lectures will explore more complex incomplete information games and methods for finding Bayesian Nash Equilibrium beyond simple matrix representations.