GAME THEORY

GAME THEORY

Instructor: NGUYEN BICH DIEP, PhD

CONCEPTS

  • Game Theory:

    • Definition: The study of strategic interactions among economic agents.

    • Strategic Interactions: A scenario where the payoff of one agent relies on the choices made by others.

ESSENTIAL ELEMENTS OF A GAME

  • Players: The individuals or entities participating in the game.

  • Strategies: The actions or plans of actions available to players.

  • Payoffs: The outcomes resulting from players' strategies.

  • Objective: Each player's goal is to maximize their individual payoff.

TYPES OF GAMES

Simultaneous vs. Sequential Games

  • Simultaneous Games:

    • Players make decisions simultaneously without knowing opponents' moves.

    • Example: Rock, paper, scissors.

  • Sequential Games:

    • Players take turns making moves, with each player observing the previous player's move.

    • Example: Chess.

Single-Play vs. Repeated Games

  • Single-Play Games:

    • Played once, resulting in a single outcome.

    • Example: A one-time rock, paper, scissors game.

  • Repeated Games:

    • Played multiple times by the same players.

    • Example: A series of rock, paper, scissors, where the first to win five times is declared the winner.

REPRESENTATION OF GAMES

Normal Form

  • Example Table:

    • Players: A and B

    • Strategies (A: Sports, Comedy; B: Sports, Comedy)

    • Payoff Matrix:
      | | Player B: Sports | Player B: Comedy |
      |----------|------------------|------------------|
      | Player A: Sports | (3,2) | (1,1) |
      | Player A: Comedy | (0,0) | (2,3) |

Extensive Form

  • Illustrates the sequence of moves with nodes (decision points).

  • Same example as normal form, detailing decisions:

    • Player A makes a choice, then Player B follows, leading to outcomes with designated payoffs:

    • Payoffs for decisions lead to pairs like (3,2), (1,1), etc.

DOMINANT STRATEGIES

Key Concepts

  • Prisoners' Dilemma Example:

    • Payoff Table:
      | Player B | Confess | Deny |
      |--------------|---------|-------|
      | Player A: Confess | (-3,-3) | (0,-6) |
      | Player A: Deny | (-6,0) | (-1,-1)|

  • Dominant Strategy: A strategy that results in higher payoffs than any other strategy regardless of what the opponent does.

  • Dominated Strategy: A strategy that always results in lower payoffs than any other strategy, regardless of the opponent’s actions.

  • Equilibrium in Dominant Strategies: The outcome where players do their best given the opponent's actions and each plays their dominant strategy, leading to a stable outcome.

Outcome Analysis

  • In the Prisoners' Dilemma:

    • Dominant strategies result in both players choosing to confess, reaching the equilibrium of (Confess, Confess).

    • If both chose deny instead, their outcomes would have been better, indicating a non-Pareto efficient equilibrium.

  • Pareto Efficiency: No alternative exists that can improve one player’s payoff without harming another.

Oligopolies and Dominant Strategies

  • In an oligopoly scenario modeled as a prisoners' dilemma:

    • Firm B Payoffs Table:
      | Firm B | High Production | Low Production |
      |--------------|------------------|------------------|
      | Firm A: High | (16,16) | (20,15) |
      | Firm A: Low | (15,20) | (18,18) |

Case of One Dominant Strategy

  • Payoff Table:

    • Firm B: Advertise vs. Don’t Advertise
      | Firm B | Advertise | Don’t Advertise |
      |-----------------|-----------|-----------------|
      | Firm A: Advertise | (10,5) | (15,0) |
      | Firm A: Don’t Advertise | (6,8) | (20,2) |

NASH EQUILIBRIUM

  • Nash Equilibrium: The situation where each player's strategy is optimal given the strategy of the other player(s).

    • At equilibrium, players have no incentive to change their strategy unilaterally.

  • Pure Coordination Game Example:

    • Payoff Table:
      | Player B | Left | Right |
      |--------------|-------|---------|
      | Player A: Left | (10,10) | (0,0) |
      | Player A: Right | (0,0) | (10,10) |

EXERCISE

  • Product Choice Problem Example:

    • Firm A vs. Firm B:
      | Firm B | Crispy | Sweet |
      |-----------|---------|--------|
      | Firm A: Crispy | (-5,-5) | (10,10) |
      | Firm A: Sweet | (10,10) | (-5,-5) |

NASH EQUILIBRIUM IN VARIOUS CONTEXTS

Battles of the Sexes Example

  • Payoff Table:

    • Player B: Football vs. Shopping
      | Player B | Football | Shopping |
      |--------------|-----------|-----------|
      | Player A: Football | (10,5) | (0,0) |
      | Player A: Shopping | (0,0) | (5,10) |

Stag Hunt Example

  • Payoff Table:

    • Player B: Stag vs. Hare
      | Player B | Stag | Hare |
      |--------------|-------|-------|
      | Player A: Stag | (8,8) | (0,7) |
      | Player A: Hare | (7,0) | (5,5) |

MIXED STRATEGIES

  • Pure Strategy: Strategy involves choosing a specific action.

  • Mixed Strategy: A strategy involving random choices among two or more actions, with specific probabilities assigned.

  • Nash Equilibria in Mixed Strategies: Every game has at least one Nash equilibrium when mixed strategies are allowed.

Example: Battles of the Sexes (General Case)


  • Payoff Table:

    Player B

    Football

    Shopping


    Player A: Football

    (10,5)

    (0,0)


    Player A: Shopping

    (0,0)

    (5,10)

    Mixed Strategy Equilibrium Calculations

    • For Player A: 10q + 0(1 - q) = 0q + 5(1 - q)

      • Solving gives: q=rac13q = rac{1}{3}

    • For Player B: 5p + 0(1 - p) = 0p + 10(1 - p)

      • Solving gives: p=rac23p = rac{2}{3}

    • Mixed-Strategy Nash Equilibrium:

      • Recognized as ext{[(2/3, 1/3), (1/3, 2/3)]}.

    SEQUENTIAL GAMES

    • Backward Induction: A method for analyzing sequential games by solving them backwards from the end of the game.

    • Example representation details the flow from player A to B, with payoffs linked to strategic choices.