GT Bayes Nash equilibrium

Bayesian Nash Equilibrium Explained

Introduction to Bayesian Nash Equilibrium

• The lecture introduces the concept of Bayesian Nash equilibrium, extending the traditional Nash equilibrium to scenarios involving private information among players.

• A motivating example is presented with two firms in a market where Firm 1 knows its marginal cost but lacks knowledge about Firm 2's marginal cost, illustrating incomplete information.

Complete vs. Incomplete Information

• In complete information scenarios, players are aware of all strategy sets and payoffs; however, in incomplete information situations, they lack knowledge about certain aspects such as opponents' costs or strategies.

• The approach involves transforming games of incomplete information into those of perfect or imperfect information by modeling uncertainty through a state space with a common probability distribution.

Example: The Gift Game

• The gift game serves as a simpler illustration where Player 1 can be either a friend or an enemy and must decide whether to give a gift (G) or not (N).

• Nature selects whether Player 1 is a friend or enemy with probabilities p and (1-p), affecting Player 2's decision to accept or reject the gift based on Player 1's type.

Payoff Structure in the Gift Game

• Player 2 prefers accepting gifts from friends over enemies; thus, the nature of the gift influences their acceptance decision after observing it.

• The game is represented in Bayesian normal form by calculating expected payoffs for each strategy profile based on different types of Player 1.

Calculating Expected Payoffs

• Each player has multiple strategies leading to various outcomes; for instance, if Player 1 gives a gift when being friendly and Player 2 accepts it, specific payoffs are calculated based on probabilities.

• An example calculation shows that if both players follow certain strategies under given probabilities, their expected payoffs can be derived systematically through multiplication and addition of outcomes.

Understanding Bayesian Nash Equilibria in Game Theory

Introduction to Payoff Matrices

• The discussion begins with the simplicity of payoff matrices where certain strategies yield consistent outcomes (zero payoffs), emphasizing the importance of practicing these standard questions.

Finding Nash Equilibria

• The focus shifts to identifying Nash equilibria within a Bayesian normal form, leading to the concept of Bayes Nash or Bayesian Nash equilibria.

• It is noted that the value of P (probability) significantly influences player two's payoffs, particularly when P is greater than 0.5.

Example: Corner Duopoly

• A motivating example involving a corner duopoly is introduced, where two firms independently choose quantities q_1 and q_2.

• The inverse demand curve is defined as p = 1 - q_1 - q_2, establishing how each firm's revenue depends on their chosen quantity.

Marginal Revenue and Market Dynamics

• For those unfamiliar with calculus, marginal revenue is provided as 1 - 2Q_i - Q_j, aiding in understanding firm behavior.

• An assumption is made that Firm One has zero marginal cost, illustrating a competitive market scenario akin to farmers' markets or street vendors.

Cost Realizations and Bayesian Games

• Firm Two's marginal cost can either be low (zero cost with probability 0.5) or high (quarter probability), introducing variability into the game.

• This simple model serves as an introduction to incomplete information scenarios where one firm lacks knowledge about another's costs.

Payoff Functions and Best Responses

• Firm One’s payoff function incorporates both high-cost and low-cost scenarios for Firm Two, reflecting its uncertainty regarding costs.

• The average output (Q_2) combines both cost types since firms are equally likely to be high or low-cost types.

Conclusion on Payoff Structures

• The final expressions for Firm Two’s payoffs are clarified, ensuring accurate representation of their decision-making process based on known costs while highlighting the need for corrections in notation.

Understanding Marginal Cost and Best Response Functions

Marginal Cost and Firm Strategies

• The concept of marginal cost is introduced, emphasizing that firm one operates under expectations regarding firm two's costs. It uses an average of high and low quantities (Q2H and Q2L).

• For firm one, the marginal revenue equation is established as 1 - Q_2 - Q_1, which is set equal to zero to find the optimal quantity for firm one based on firm two's output.

• Firm two's profit maximization involves setting its marginal revenue equal to its marginal cost, leading to a similar response function as firm one but adjusted for type H’s higher costs.

Graphing Best Response Functions

• A graphical representation of best response functions shows how each firm's output decisions depend on the other's production levels, with specific points indicating optimal strategies.

• The graph illustrates that when firm two produces a quantity of 1, firm one optimally produces 0. Conversely, if firm two produces 0, then firm one would produce half.

Expected Values in Bayesian Games

• The differences in responses between types are highlighted; type H has a lower output due to higher marginal costs. Firm one's strategy relies on expected values since it lacks visibility into firm two's cost structure.

• Firm one's best response considers the average expected value of Q2 due to equal probabilities assigned to both high and low-cost types for firm two.

Finding Equilibrium Quantities

• To determine equilibrium quantities, adjustments are made using the best response functions derived from previous calculations involving both types of firms.

• Substituting expressions for Q2 from both types leads to complex equations that ultimately yield specific equilibrium outputs: Q_1^* = 3/8, Q_2H = 3/16, and Q_2L = 5/16.

Key Takeaways on Bayesian Nash Equilibrium

• Understanding Bayesian games requires familiarity with matrix representations and continuous action spaces. Graphing best response functions can aid comprehension but may not always be necessary unless specified in questions.

• The concept of Bayesian Nash equilibrium extends traditional Nash equilibrium principles into scenarios where players have incomplete information about others' costs or strategies.