2.4 Converting between normal and extensive form representations
2 min read•august 7, 2024
Game theory offers different ways to represent strategic interactions. and 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
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- A 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 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 assigns a probability distribution over actions at each
- 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
- Examples: interchanging players, relabeling strategies, adding/removing duplicate strategies
- Useful for simplifying games and identifying essential features
- Key terms: strategic equivalence,
Information Loss in Conversion
- Converting an extensive form game to normal form can result in
- 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
Key Terms to Review (22)
Backward induction: Backward induction is a method used in game theory to determine optimal strategies by analyzing a game from the end to the beginning. It involves looking at the last possible moves of players and determining their best responses, then moving sequentially backward through the game tree to deduce the optimal actions of earlier moves. This technique is particularly relevant in analyzing strategic interactions in sequential games and helps in identifying subgame perfect equilibria.
Battle of the Sexes: The Battle of the Sexes is a classic game in game theory that illustrates a coordination problem between two players who prefer different outcomes but must reach an agreement on one. The game typically represents a situation where two players (often depicted as a male and female) want to go out together but have different preferences on where to go, such as one preferring to attend a football game while the other prefers going to the ballet. This scenario highlights the challenges of achieving a mutually beneficial outcome while dealing with conflicting interests, and it connects with various concepts in game theory such as normal and extensive form representations, dominant strategies, Nash equilibria, and rationalizability.
Behavioral Strategy: A behavioral strategy is a plan of action that specifies the choices a player will make at each decision point in a game, typically incorporating randomization or mixed strategies to respond to opponents' actions. This approach allows players to adapt their strategies based on the history of previous moves and can lead to more nuanced decision-making in complex scenarios. By using behavioral strategies, players can potentially enhance their chances of achieving favorable outcomes through unpredictability and responsiveness.
Dominant Strategy: A dominant strategy is a course of action that yields the highest payoff for a player, regardless of the strategies chosen by other players. This concept is key in understanding how individuals or firms make decisions in strategic situations where their outcomes depend on the choices of others.
Extensive Form: Extensive form is a way of representing games that shows the sequential nature of decisions made by players, typically using a tree diagram. It highlights the order of moves and the choices available at each decision point, making it easier to analyze dynamic interactions among players. This representation is especially useful for understanding complex strategies and outcomes that depend on the timing of decisions and players' responses.
Game Transformation: Game transformation refers to the process of converting a game from one representation format to another, typically between normal form and extensive form. This transformation allows for different insights and strategies to be analyzed, as each form has its unique strengths in representing player actions, payoffs, and the sequence of moves. Understanding how to switch between these forms enhances strategic analysis and decision-making in game theory.
Game Tree: A game tree is a graphical representation of a strategic game that illustrates the possible moves and outcomes from each player's decisions over time. It organizes the game's structure by showing the sequence of actions, including branching points for choices made by players, which leads to various terminal nodes representing outcomes. Game trees are essential for understanding how players can strategize in extensive form games and are particularly useful in applying concepts like backward induction to identify optimal strategies.
Information Loss: Information loss refers to the reduction or omission of crucial data when converting from one representation of a game to another, such as from extensive form to normal form. This occurs because the detailed sequential decisions and strategies represented in extensive form may be compressed into a simpler matrix in normal form, potentially obscuring critical strategic nuances. Understanding information loss is essential, as it influences how players perceive their options and affects the overall analysis of strategic interactions.
Information Set: An information set is a collection of decision nodes in a game where a player cannot distinguish between them, meaning the player does not know which node they are at when making a decision. This concept is crucial in understanding how players make strategic choices under uncertainty and helps to illustrate the differences between normal and extensive form representations of games, sequential decision-making processes, and the formation of beliefs in dynamic environments.
John Nash: John Nash was an influential mathematician and economist best known for his groundbreaking work in game theory, particularly the concept of Nash equilibrium. His theories have fundamentally shaped our understanding of strategic interactions among rational decision-makers, making them essential for analyzing competitive behaviors in various fields, including economics, political science, and biology.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and polymath, widely regarded as one of the founders of game theory. His groundbreaking work laid the foundation for analyzing strategic interactions among rational decision-makers, influencing fields such as economics, computer science, and social sciences.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where no player can benefit by unilaterally changing their strategy if the strategies of the other players remain unchanged. This means that each player's strategy is optimal given the strategies of all other players, resulting in a stable outcome where players have no incentive to deviate from their chosen strategies.
Normal Form: Normal form is a representation of a game that summarizes the strategies and payoffs of players in a matrix format. This format allows for a clear visual comparison of the possible outcomes based on the chosen strategies by each player. It highlights the simultaneous nature of decisions made by players and is essential for analyzing strategic interactions in different types of games.
Payoff Matrix: A payoff matrix is a table that shows the payoffs or outcomes for each player in a game, given all possible combinations of strategies chosen by the players. It visually represents the choices available to players and their potential results, making it essential for analyzing strategic interactions in various types of games.
Perfect Information: Perfect information refers to a situation in a game where all players are fully aware of all the actions that have taken place prior to their turn. This means every player knows the history of moves and choices made by others, allowing them to make informed decisions. This concept is essential for understanding the structure of extensive form games, where game trees represent the sequential nature of decision-making, and it plays a significant role in analyzing strategies through backward induction.
Prisoner's dilemma: The prisoner's dilemma is a standard example of a game in which two players must choose between cooperation and betrayal, with the outcome for each dependent not only on their own choice but also on the choice made by the other player. This scenario highlights the conflict between individual rationality and collective benefit, demonstrating how two rational individuals may not cooperate even if it appears that it is in their best interest.
Reduced Normal Form: Reduced normal form is a simplified representation of a game in normal form, where dominated strategies have been eliminated from consideration. This approach allows players to focus on the strategies that remain relevant and likely to influence the outcome of the game, making it easier to analyze strategic interactions. By reducing the normal form, players can identify equilibrium points more efficiently, leading to better strategic decision-making.
Strategic Equivalence: Strategic equivalence refers to a situation in game theory where two different strategic forms, such as normal and extensive form representations, lead to the same outcomes for players. This concept highlights that different ways of presenting a game can still capture the same strategic interactions and results. Understanding strategic equivalence is crucial when converting between these forms, as it helps to simplify complex games while ensuring that the essence of the strategies and payoffs remains intact.
Strategic Moves: Strategic moves are actions taken by players in a game to influence the outcome and the decisions of others, often aimed at gaining a competitive advantage. These moves can involve commitments, threats, or promises that shape the strategies available to other players and can alter the payoff structure in their favor. Understanding strategic moves is crucial when converting between different representations of games, as they can reveal insights into player motivations and potential equilibrium outcomes.
Strategy Profile: A strategy profile is a complete description of the strategies chosen by each player in a game, detailing what action each player will take in every possible situation they may encounter. This concept is crucial as it helps analyze the overall outcome of strategic interactions and assists in determining the equilibrium points within various game formats. Understanding strategy profiles is essential for evaluating decision-making processes, especially when converting between different game representations or analyzing equilibria in games with incomplete information.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games where players make decisions at various stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that the strategies are credible and optimal, even if the game is played out from any point along the decision path.
Transformation Rules: Transformation rules refer to the systematic procedures used to convert game representations from one form to another, specifically between normal form and extensive form. These rules help in understanding the strategic aspects of games by translating the same scenario into different formats, making it easier to analyze player decisions, payoffs, and strategies.