NATS1595 13 - Mixed Strategies

NATS1595

  • Course Title: Iterated Games and Mixed Strategies

Introduction to the Iterated Prisoner’s Dilemma

  • Background: Focus on understanding cooperation through experiments.

  • Key Question: What sorts of experiments have people conducted to explore cooperation?

Axelrod’s Tournament

  • Competition Overview: In 1980, Professor Robert Axelrod held a competition to analyze strategies in the Iterated Prisoner’s Dilemma.

  • Game Mechanics:

    • Iterative play: Players repeatedly engage in the game.

    • Simultaneous moves: Both players decide whether to Cooperate (C) or Defect (D) at the same time.

    • Strategy influenced by previous moves.

Example of Rounds in the Iterated Game

  • Round Description:

    • Round 1: C vs C -> (3,3)

    • Round 2: D vs C -> (8,1)

    • Round 3: C vs D -> (1,8)

    • Round 4: C vs C -> (6,6)

  • Final Score Observations: Reveals the outcomes based on chosen strategies and consequences based on player actions.

Winning Strategy Identified

  • Best Strategy: Tit-for-Tat (TFT)

    • Mechanism:

      • Start with Cooperation on the first move.

      • In subsequent moves, mimic the previous action of the opponent (Cooperate or Defect).

    • Character of the Strategy: Considered a "nice" strategy.

Cooperation Findings

  • Emerging Conclusion: Cooperation may lead to better outcomes in certain situations.

Mixed Strategy Nash Equilibrium

  • Concept Introduction: What if there is no dominant strategy?

    • Implication of playing the game iteratively suggests mixing strategies prevents exploitation.

  • Strategy Consideration: How often and when to alternate between different strategies?

Understanding Expected Value

  • Definition: The expected value is the average outcome based on probabilities of different possible results.

  • Formula: E(X) = x1 * p1 + x2 * p2 + ...

  • Example Using a Die:

    • If repeatedly rolling a 6-sided die, consider what the average value will be over many rolls.

Practical Example of Expected Value with a Spinner

  • Spinner Average Moves: 1.75 moves per turn based on given outcomes.

Probabilities in Simple Experiments

  • Shirt Color Example:

    • Probability of wearing blue shirts: 0.4

    • Probability of wearing red shirts: 1 - 0.4 = 0.6

Detailed Mixed Strategy Nash Equilibrium

  • Condition: A 2x2 payoff matrix lacking a pure strategy leads to a mixed strategy Nash equilibrium.

  • Strategy Probabilities:

    • Player 1 selects strategy A with probability q, and B with 1-q.

    • Player 2 selects strategy A with p, and B with 1-p.

Example Payoff Calculations

  • Dynamic Payoff Insights:

    • If Player 2 plays A with probability p and B with 1-p, predict expected payoffs based on Player 1's strategy.

    • Equalize payoffs to determine mixed strategies for optimal play.

Hawk and Dove Concept

  • Behavioral Strategy:

    • Hawks: Aggressive, escalate conflicts resulting in possible injuries.

    • Doves: Avoid conflict, seldom lead to injury.

Strategy Outcomes in Hawk and Dove Scenarios

  • Common Resource Competition: Outcomes depend on behavior (Hawk versus Dove) and expected payoffs (Benefit B vs Cost C).

  • Dominant Strategy Dynamics: If B > C, both players behaving as Hawks leads to dominant strategy identified through Nash equilibrium.

Population Dynamics of Hawks and Doves

  • Probability of Encounters: Awareness of how prevalent each behavior is in a population.

    • Varies between 100% probability to lower percentages in mixed populations.

Implications of Increasing Hawk Population**

  • Assess the impact on strategy as the prevalence of Hawks in the population rises.