🎲Game Theory and Business Decisions Unit 2 – Nash Equilibrium in Simultaneous Games

Key Concepts

• Nash equilibrium represents a stable state in a game where no player has an incentive to unilaterally change their strategy
• Simultaneous games involve players making decisions independently and simultaneously without knowledge of the other players' choices
• Payoff matrices summarize the outcomes and payoffs for each combination of strategies chosen by the players
• Dominant and dominated strategies can simplify the process of finding Nash equilibria by eliminating certain strategy profiles
• Mixed strategies allow players to randomize their actions based on a probability distribution over their available strategies
• Pareto optimality refers to a situation where no player can be made better off without making another player worse off
• Zero-sum games have the property that the sum of all players' payoffs is always zero, meaning one player's gain is another player's loss

Nash Equilibrium Defined

• Nash equilibrium is a key concept in game theory that describes a stable state in a game where each player is using their best response to the strategies of the other players
• In a Nash equilibrium, no player can improve their payoff by unilaterally changing their strategy while the other players keep their strategies unchanged
• Nash equilibrium can be pure (players choose a single strategy) or mixed (players randomize over multiple strategies based on a probability distribution)
• The existence of a Nash equilibrium does not guarantee that it is unique; some games may have multiple Nash equilibria
• Nash equilibrium is named after mathematician and economist John Nash, who introduced the concept in his 1950 paper "Equilibrium Points in n-Person Games"
• Finding Nash equilibria involves analyzing the payoff matrix and identifying strategy profiles where each player's strategy is a best response to the other players' strategies
• The concept of Nash equilibrium has been widely applied in various fields, including economics, political science, and psychology, to model and predict strategic interactions

Types of Games

• Simultaneous games, also known as static games, involve players making decisions independently and simultaneously without knowledge of the other players' choices
• Sequential games, in contrast, involve players making decisions in a specific order, with each player aware of the previous players' actions
• Cooperative games allow players to form binding agreements and make joint decisions to maximize their collective payoffs
• Non-cooperative games do not allow for binding agreements, and players make decisions independently to maximize their individual payoffs
• Zero-sum games have the property that the sum of all players' payoffs is always zero, meaning one player's gain is another player's loss (Matching Pennies)
• Non-zero-sum games have payoffs that do not always sum to zero, allowing for outcomes where all players can benefit or all players can lose (Prisoner's Dilemma)
• Symmetric games have identical strategy sets and payoff functions for all players, while asymmetric games have different strategy sets or payoff functions for different players

Finding Nash Equilibria

• To find Nash equilibria, start by constructing a payoff matrix that summarizes the outcomes and payoffs for each combination of strategies chosen by the players
• Identify any dominant strategies, which are strategies that provide a higher payoff for a player regardless of the other players' choices
• Eliminate any dominated strategies, which are strategies that provide a lower payoff for a player regardless of the other players' choices
• Look for pure strategy Nash equilibria by finding strategy profiles where each player's strategy is a best response to the other players' strategies
• If no pure strategy Nash equilibria exist, consider mixed strategies where players randomize over their available strategies based on a probability distribution
• To find mixed strategy Nash equilibria, set up indifference conditions that ensure each player is indifferent between their available strategies given the other players' mixed strategies
• Solve the system of equations resulting from the indifference conditions to determine the probability distribution for each player's mixed strategy
• Verify that the identified strategy profiles are indeed Nash equilibria by checking that no player can unilaterally improve their payoff by deviating from their equilibrium strategy

Real-World Applications

• Nash equilibrium has been applied to various real-world situations, including market competition, political campaigns, and international relations
• In the classic "Prisoner's Dilemma" game, the Nash equilibrium is for both players to defect, even though mutual cooperation would yield a better outcome for both
• In the "Tragedy of the Commons" game, the Nash equilibrium is for all players to overexploit a shared resource, leading to its depletion, even though restraint would be better for everyone in the long run
• In the "Chicken" game, which models a situation where two players engage in a dangerous confrontation (two drivers heading towards each other), the Nash equilibria involve one player yielding while the other player stays the course
• In the "Stag Hunt" game, which represents a coordination problem, there are two Nash equilibria: both players cooperating to hunt a stag, or both players defecting to hunt rabbits individually
• In the "Battle of the Sexes" game, which models a coordination problem with conflicting preferences, there are two pure strategy Nash equilibria and one mixed strategy Nash equilibrium
• In the "Cournot Duopoly" game, which models competition between two firms choosing their production quantities, the Nash equilibrium involves each firm producing a specific quantity based on the other firm's expected output

Limitations and Criticisms

• Nash equilibrium assumes that players are rational and have complete information about the game structure and payoffs, which may not always hold in real-world situations
• The concept of Nash equilibrium does not account for the possibility of players engaging in communication, bargaining, or coalition formation, which can alter the game's outcomes
• In some games, the Nash equilibrium may not be Pareto optimal, meaning there could be other outcomes that make all players better off without making any player worse off
• The existence of multiple Nash equilibria in some games can make it difficult to predict which equilibrium will actually occur in practice
• Nash equilibrium is a static concept and does not capture the dynamic aspects of real-world interactions, such as learning, adaptation, and evolution
• Some critics argue that the focus on individual rationality in Nash equilibrium neglects the importance of social norms, emotions, and bounded rationality in shaping human behavior
• The assumption of common knowledge of rationality, which underlies the concept of Nash equilibrium, has been challenged by some researchers as being too strong and unrealistic

Practice Problems

1. Consider the following payoff matrix for a simultaneous game:

   	Player 2: A	Player 2: B
   Player 1: A	(2, 1)	(0, 0)
   Player 1: B	(0, 0)	(1, 2)

   Find all pure strategy Nash equilibria in this game.

2. In the "Matching Pennies" game, two players simultaneously choose either Heads (H) or Tails (T). Player 1 wins if the choices match, while Player 2 wins if the choices differ. The payoff matrix is as follows:

   	Player 2: H	Player 2: T
   Player 1: H	(1, -1)	(-1, 1)
   Player 1: T	(-1, 1)	(1, -1)

   Determine whether this game has any pure strategy Nash equilibria or mixed strategy Nash equilibria.

3. Consider the following payoff matrix for a simultaneous game:

   	Player 2: A	Player 2: B
   Player 1: A	(3, 3)	(1, 4)
   Player 1: B	(4, 1)	(2, 2)

   Find all pure strategy Nash equilibria and determine if any of them are Pareto optimal.

4. In the "Prisoner's Dilemma" game, two suspects are being interrogated separately. Each suspect can either Confess (C) or Deny (D) the crime. The payoff matrix is as follows:

   	Suspect 2: C	Suspect 2: D
   Suspect 1: C	(-5, -5)	(-10, 0)
   Suspect 1: D	(0, -10)	(-1, -1)

   Find the Nash equilibrium and explain why it is not Pareto optimal.

5. Consider the following payoff matrix for a simultaneous game:

   	Player 2: A	Player 2: B
   Player 1: A	(2, 2)	(0, 3)
   Player 1: B	(3, 0)	(1, 1)

   Find all pure strategy Nash equilibria and mixed strategy Nash equilibria in this game.

Further Reading

• "A Beautiful Mind" by Sylvia Nasar, a biography of John Nash that covers the development of the Nash equilibrium concept and its impact on game theory
• "The Strategy of Conflict" by Thomas Schelling, a seminal work that explores the application of game theory to international relations and nuclear deterrence
• "Game Theory: Analysis of Conflict" by Roger B. Myerson, a comprehensive textbook that covers the foundations of game theory and its various applications
• "The Evolution of Cooperation" by Robert Axelrod, a book that examines the conditions under which cooperation can emerge in a world of self-interested individuals, using the Prisoner's Dilemma game as a central example
• "Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life" by Avinash Dixit and Barry Nalebuff, a non-technical introduction to game theory and its applications in various real-world contexts
• "Game Theory 101: The Complete Textbook" by William Spaniel, an accessible and engaging textbook that covers the basics of game theory and includes numerous examples and exercises
• "Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction" by Herbert Gintis, a textbook that presents game theory from an evolutionary perspective and emphasizes its interdisciplinary applications