Nash Equilibrium in Location Choices

Assumptions and Scenario Definition

• The context revolves around two individuals who want to set up a lemonade stand (or any similar drink selling venture) along a stretch of beach.
• Both individuals sell the same product, hence they are competitors.

Key Question: Where Should They Be Located?

• The main question posed is about determining their optimal locations on the beach to maximize customers. This leads to an exploration of the Nash equilibrium concept in the context of their choices.

Potential Locations

• Example scenario presented: Individual A opens shop at location A and Individual B opens shop at location B.

Evaluating Nash Equilibrium

• The first proposed arrangement (A and B) is initially considered as a potential Nash equilibrium.
• A discussion ensues on whether, knowing the other's location, either individual has an incentive to change their location. If one is located at A and the other at B:
  • A customer will walk to the stand closer to them since both are selling identical products.
  • Individuals will continuously change their positions to gain a competitive advantage, thus demonstrating that A and B cannot constitute a Nash equilibrium.

Definition of Nash Equilibrium

• A Nash equilibrium occurs when neither individual has the incentive to unilaterally change their position given the other’s choice.
• For example, if both set up at the center, they will effectively split customers equally between them. This scenario prevents either from gaining an advantage by changing locations.

Real-World Examples of Nash Equilibrium

• Observations in the real world suggest similar behavior in business ventures:
  • Gas stations along a highway often cluster at intersections, maximizing accessibility for motorists, which can be partly explained by Nash equilibrium.

Payoff Matrices and Nash Equilibria

• Mention of using payoff matrices to evaluate choices.
• The matrices can vary (e.g., 2x2, 3x3, 4x4), and the goal is to arrive at the Nash equilibrium by identifying overlapping best responses given the options available.

Process of Finding Nash Equilibrium in Payoff Matrices

• The participant is instructed to analyze a given payoff matrix:
  • Identify best responses for each player given the other's choice.
  • Example provided: If Player 2 plays left, Player 1 chooses their best option from that column, and so forth.
• Intersection of best responses indicates the Nash equilibrium, ensuring both players have no incentives to change their strategies.

Mixed Strategies in Nash Equilibrium

Definition of Mixed Strategy

• A mixed strategy involves a player randomizing over available strategies rather than choosing one explicitly.
• Example: Player 2 can randomly choose:
  • Left with probability 0.3
  • Center with probability 0.3
  • Right with probability 0.4

Determining Best Response to Mixed Strategies

• To compute the best response for Player 1 when Player 2 has a mixed strategy, one must calculate expected payoffs for each potential decision (up, middle, down).
• The player will choose the strategy that maximizes expected value given the probabilities assigned to Player 2’s choices.
• If two strategies yield equal expected payoffs, a player could choose either or a mix thereof.

Expectation Calculation Example

• The calculation of expected values involves using assessments of probabilities from Player 2's mixed strategy.
• If Player 1's responses are evaluated against Player 2's choices, one can calculate individual expected values for up, middle, and down.
• The results will allow us to derive which choice maximizes Player 1's expected payoff:
  • Given probabilities and resulting payoffs, expected values should be computed and compared.

Best Responses and Continuous vs. Discrete Strategies

Switching from Mixed Strategies to Continuous Variables

• Notion that in more complex games, players might vary their quantities or price options infinitely as opposed to selecting distinct categorical choices.
• The next focus includes applications such as the Krunal model where two competing firms choose quantities simultaneously (duopoly context).

Implications of Duopoly in Quantity Decisions

• Emphasis on firm behaviors, the relationship between competing firms, and simultaneous decision-making leading to potential Nash equilibria.
• Reaction functions guide what each firm should produce based on the output of its competitor, assisting in finding equilibrium points visually or numerically by solving equations.

Examples and Further Exploration of Strategic Models

Krunal Model Details

• Theoretical framework highlights that firm interactions must account for each other's potential outputs.
• Marginal revenues for both firms lead to established reaction curves, with intersections defining the equilibrium points.

Bertrand Competition Context

• Exploration of simultaneous pricing decisions in an oligopoly (Bertrand model).
• Firms must set prices without knowledge of their competitor’s choice, with the competitive implication being a tendency to lower prices to marginal cost due to the demand for identical goods.

Collusion versus Competition Outcomes

• Theoretical exploration of collusion in profits maximization compared to competition, where firms would benefit by acting as a monopolist, thereby maximizing collective profits at the expense of consumer welfare (theory of antitrust laws).

Summary of Equilibrium Concepts

• Pure versus Mixed Strategy: Importance of understanding the structure of responses whether discrete (like matrix entries) or continuous based (like choice of quantity).

• Each scenario serves to illustrate deeper strategic interactions and practical implications of decision-making within economic models involving competition.

• Understanding Nash Equilibrium serves as a crucial element to analyze behavior in strategic environments, whether that be for location choices, quantity production in duopolies or competitive pricing strategies.