Game Theory

Introduction to Game Theory

1. Introduction

Game Theory studies interactions among rational agents where one agent's actions impact the payoffs of others.Key elements of games include:

  • Players: Individuals or entities participating in the game.

  • Strategies: Choices available to the players.

  • Payoffs: Outcomes received based on the chosen strategies.

Graphical Representations of Games

  1. Matrices for simultaneous moves:

    • A matrix presents the strategies of players along the rows and columns, with each cell representing the payoff outcomes for all players based on these stra

    • egies.

    • Example:

      Player B: Strategy 1

      Player B: Strategy 2

      Player A: Strategy 1

      (3,2)

      (0,4)

      Player A: Strategy 2

      (4,1)

      (2,3)

      • In this table, (x,y) represents the payoff for Player A and Player B respectively.

  2. Game Trees for sequential moves:

    • A game tree starts with a single node (the initial state), from which branches represent the possible actions available to players. Each decision point (node) can lead to further decision points or terminal nodes that represent outcomes.

    • Example:

      Player A  
  /   \  
 A1    A2  
 / \     \

    B1 B2 B3

    - In this tree, Player A has two choices (A1 and A2), and depending on Player A's choice, Player B can choose among different strategies (B1, B2, B3).

1.2 What Is Game Theory?

Examines how individual actions (e.g., pricing strategies by firms, contributions to public goods) influence others.Real-world examples:

  • A firm lowering prices impacting competitors’ sales.

  • Personal decisions during a pandemic (masks, vaccinations) affecting community safety.

1.3 Definition

Game Theory Definition: Studies the interaction between rational agents behaving strategically.

  • Rationality: Players aim to optimize their payoffs with common knowledge of rational behavior.

  • Strategic Behavior: Necessitates consideration of others' reactions and actions.

  • Example from pop culture illustrating common knowledge: "The Princess Bride"'s Vizzini scene demonstrates strategic decision-making under uncertainty.

1.3 Main Elements in a Game

1.3.1 Players

  • Definition: Set of Agents (individuals, firms, or countries).

  • Number of players, N: At least two players are required in a game (N ≥ 2).

1.3.2 Strategies

  • Definition: A complete plan detailing a player’s actions based on possible situations.

  • Types of Strategies:

    • Discrete Strategies: Limited choices (e.g., select from finite outputs like {6, 10}).

    • Continuous Strategies: Unlimited choices within a range (e.g., any real number in [0, 10]).

  • Strategy Sets: Notation examples:

    • sA for player A's strategy.

1.3.3 Payoffs

  • Definition: A mapping of strategy profiles to real numbers representing player returns.

  • Payoff Function Example: uA(sA, sB) = 8 means player A scores 8 with strategies sA and sB chosen.

1.4 Two Graphical Approaches

1.4.1 Matrices

  • Represent simultaneous-move games.

1.4.2 Game Trees

  • Illustrate sequential-move games, showing turns and responses.

1.5 Imperfect Information in Game Trees

  • Information Set: Connects nodes in which a player cannot distinguish prior moves.

1.6 Identifying Equilibrium Behavior

1.6.1 Existence of Equilibria

  • A game must have at least one strategy profile as an equilibrium.

1.6.2 Uniqueness

  • Equilibria may not be unique; multiple equilibria can arise in some games.

1.6.3 Robustness to Payoff Changes

  • Equilibria should remain stable under small changes to payoffs.

1.6.4 Social Optimality

  • Equilibria may not align with socially optimal outcomes; discussed in the context of the Prisoner's Dilemma.

2 Equilibrium Dominance

2.1 Introduction

  • Solutions focus on excluding strategies that a rational actor wouldn't choose.

2.2 Strictly Dominated Strategies

  • Delete strategies yielding lower payoffs across all scenarios.

2.3 Iterated Deletion of Strictly Dominated Strategies (IDSDS)

  • Remove dominated strategies sequentially until no further iterations are possible.

2.4 Applying IDSDS

  • Evaluating games like the Prisoner's Dilemma, coordinating strategies, etc.

3 Nash Equilibrium

3.1 Introduction

  • The Nash Equilibrium is a major refinement over IDSDS, indicating mutual best response strategies.

3.2 Finding Best Responses

  • Illustrating through payoff matrices the responses of players to each other's strategies.

3.3 Evaluating the Nash Equilibrium

  • Comparing mutual best responses and classifying them into equilibria.

3.4 Applications of Rationalizability

  • Practical scenarios such as the Beauty Contest and Cournot competition illustrates applications in various economic settings to determine equilibria.