2.4 Converting between normal and extensive form representations

Game theory offers different ways to represent strategic interactions. Normal form and extensive form are two key representations, each with its strengths. Understanding how to convert between these forms is crucial for analyzing games effectively.

Converting between normal and extensive form can reveal hidden insights or simplify complex games. However, it's important to note that some information may be lost in the process, particularly when converting from extensive to normal form.

Game Representations

Strategy Profiles and Reduced Normal Form

• A strategy profile specifies the strategies chosen by each player in a game
  • Consists of a set of strategies, one for each player
  • Represents a possible outcome or solution to the game
• The reduced normal form of a game eliminates strategies that are never best responses for any player
  • Simplifies the game by removing dominated strategies (always worse than another strategy)
  • Helps identify the essential strategic choices available to players
• Key terms: strategy profile, reduced normal form

Behavioral Strategies

• In extensive form games, a behavioral strategy assigns a probability distribution over actions at each information set
  • Allows for randomization of choices at decision points (rolling a die to decide)
  • Provides flexibility in modeling player behavior in dynamic games
• Behavioral strategies can be used to represent mixed strategies in extensive form games
  • Mixed strategies assign probabilities to entire strategies
  • Behavioral strategies assign probabilities to actions at each information set
• Key terms: behavioral strategy

Equivalence and Transformations

Strategic Equivalence and Game Transformation

• Two games are strategically equivalent if they have the same reduced normal form
  • Players have the same strategic choices and payoffs in equivalent games
  • Allows for comparing games with different representations (normal form vs. extensive form)
• Game transformations are operations that modify a game while preserving strategic equivalence
  • Examples: interchanging players, relabeling strategies, adding/removing duplicate strategies
  • Useful for simplifying games and identifying essential features
• Key terms: strategic equivalence, game transformation

Information Loss in Conversion

• Converting an extensive form game to normal form can result in information loss
  • Normal form does not capture the dynamic structure and information sets of the extensive form
  • Some strategically relevant aspects may be lost in the conversion (timing of moves, information available at decision points)
• Information sets in extensive form games represent the knowledge available to players at each decision point
  • Players may have different information sets at different stages of the game
  • Converting to normal form treats all strategies as simultaneous choices, losing the sequential nature
• Key terms: information loss, information set