3.1 Extensive Form Games and Decision Trees
2 min read•july 22, 2024
use to map out sequential decision-making in strategic interactions. These trees show players, moves, and , allowing for analysis of complex scenarios where timing and information matter.
give players full knowledge of past moves, while introduce uncertainty. Understanding these differences is crucial for solving games through techniques like and predicting optimal .
Extensive Form Games
Game trees for sequential moves
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- Extensive form games use game trees to represent sequential-move games
- depict decision points or outcomes in the game
- illustrate the actions or moves available to players
- The initial node serves as the root or starting point of the game tree
- indicate the end of the game and specify the payoffs for each player
- Subgames are smaller extensive form games embedded within the larger game tree
- Originate at decision nodes and encompass all subsequent nodes and edges
Components of extensive form games
- Players are the individuals or entities making decisions in the game
- Identified by labels or symbols at each decision node (Player 1, Player 2)
- Strategies refer to the complete set of actions available to a player at each decision point
- Determined by the edges extending from a player's decision nodes (Cooperate, Defect)
- Payoffs represent the outcomes or rewards for each player at the terminal nodes
- Expressed as a tuple for multi-player games (3, 5) for a two-player game
Perfect vs imperfect information
- Perfect information games:
- Players have complete knowledge of all prior moves and the current game state
- Represented by single decision nodes for each player (Chess, Go)
- Imperfect information games:
- Players may lack complete knowledge of previous moves or the current state
- Represented by information sets connecting decision nodes with dashed lines or ovals
- Players cannot differentiate between nodes within the same information set (Poker, Battleship)
Move order and player knowledge
- The order of moves follows the sequence of nodes in the game tree
- Earlier moves appear at the top, with later moves below
- The information available to players varies based on the type of game:
- Perfect information games:
- Players have complete knowledge at each decision node
- Imperfect information games:
- Players may have incomplete knowledge at certain decision nodes
- Information sets signify the player's uncertainty about the current state
- Perfect information games:
- Backward induction solves extensive form games by working backwards from terminal nodes
- At each decision node, the player selects the action leading to their best payoff
- Assumes optimal play by all players in subsequent moves (Minimax algorithm)
Key Terms to Review (24)
Backward induction: Backward induction is a method used in game theory to analyze decision-making processes by reasoning backwards from the end of a problem to determine optimal strategies. This technique involves considering the final outcomes of a game and working back through each player's possible choices to identify the best decisions at each step. It is crucial for understanding how players can make rational choices in extensive form games, evaluate subgame perfect equilibria, and establish credible threats and promises.
Cox Model: The Cox model, also known as the Cox proportional hazards model, is a statistical technique used to explore the relationship between the survival time of subjects and one or more predictor variables. This model is particularly useful in the context of survival analysis, allowing researchers to understand how different factors affect the risk of an event occurring over time. It’s essential for estimating hazard ratios, which help in comparing the risk between different groups while accounting for censored data.
Discounting Future Payoffs: Discounting future payoffs refers to the process of determining the present value of future cash flows or benefits, reflecting the principle that money available today is worth more than the same amount in the future due to its potential earning capacity. This concept is crucial for making strategic decisions in scenarios where outcomes unfold over time, influencing how players evaluate options in extensive form games and decision trees.
Dominant strategy: A dominant strategy is a strategy that yields a higher payoff for a player, regardless of what the other players choose. This concept is central to understanding decision-making in strategic interactions, where players assess their options based on the potential responses of others, leading to predictable outcomes in competitive environments.
Edges: In the context of extensive form games and decision trees, edges represent the connections between nodes that illustrate the possible moves or decisions available to players at various points in the game. Each edge signifies a choice made by a player leading to a different outcome or state in the game, helping to map out the flow of decisions and their consequences. This structure allows for a clear visualization of strategic interactions among players over time.
Expected Utility Theory: Expected utility theory is a decision-making framework that helps individuals and organizations evaluate risky choices by assigning values to different outcomes based on their probabilities and the utilities associated with those outcomes. This theory assumes that people choose options that maximize their expected utility, guiding their decisions in uncertain situations. It serves as a fundamental concept in understanding how rational agents approach risk and uncertainty in various contexts, including extensive form games and decision trees.
Extensive Form Games: Extensive form games are a representation of strategic interactions among players that illustrate the sequence of actions and decisions made over time. They use decision trees to depict how players make choices at different points, taking into account the possible moves and outcomes. This format allows for a clearer understanding of the game's dynamics, including information sets, where players may not know what others have chosen at earlier stages.
Game Trees: Game trees are graphical representations used to illustrate the possible moves in a strategic game, showing how players can make decisions at various points. They help visualize the sequential nature of extensive form games, where players choose actions one after another, allowing for the analysis of potential outcomes and strategies based on earlier choices.
Hirschleifer's Model: Hirschleifer's Model is a framework used in game theory to analyze decision-making under uncertainty and the strategic interactions of individuals when facing risk. This model emphasizes how players make choices based on the expected outcomes of their actions, particularly in contexts where information is incomplete or asymmetric. It connects to extensive form games and decision trees by illustrating how players map out their options and potential payoffs over time, considering the possible moves of other players.
Imperfect Information Games: Imperfect information games are strategic interactions where players do not have complete knowledge of all relevant aspects of the game, particularly the actions or types of other players. This lack of complete information can lead to uncertainty in decision-making and can significantly affect the strategies employed by players. In such games, players may need to make educated guesses based on available information, which can complicate the analysis of the game and the outcomes derived from it.
John Nash: John Nash was an influential mathematician and economist best known for his contributions to game theory, particularly for developing the concept of Nash equilibrium. His work transformed how we understand strategic decision-making in competitive environments, laying the groundwork for numerous applications in economics, politics, and business.
John von Neumann: John von Neumann was a Hungarian-American mathematician and polymath who made foundational contributions to various fields, most notably game theory. His work established the mathematical framework for analyzing strategic interactions, which is vital for understanding decision-making processes in competitive environments, including extensive form games, repeated games, and mixed strategies.
Mixed strategy: A mixed strategy is a decision-making approach where a player chooses between different strategies randomly, assigning a probability to each possible action rather than sticking to a single strategy. This adds an element of unpredictability to the player's choices, which can be crucial in competitive situations. It connects closely to game elements such as players, their available strategies, and the associated payoffs, allowing for a more complex analysis in various game forms.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where players, knowing the strategies of their opponents, choose their optimal strategies resulting in a situation where no player has anything to gain by changing their own strategy unilaterally. This balance occurs when each player's strategy is the best response to the strategies chosen by others, highlighting the interdependence of player decisions and strategic decision-making.
Nodes: In the context of extensive form games and decision trees, nodes represent the points of decision or action within a game. Each node can indicate a player's turn to act, a chance event, or a terminal outcome, helping to map out the structure of the game and the possible choices available at each step. Nodes are crucial for visualizing strategies and understanding how decisions lead to different outcomes.
Pareto Efficiency: Pareto efficiency refers to a state where resources are allocated in a way that no individual's situation can be improved without worsening someone else's situation. This concept highlights the importance of mutual benefit in various strategic interactions and economic environments, emphasizing that an optimal allocation exists when it is impossible to make any participant better off without making at least one other participant worse off.
Payoffs: Payoffs refer to the outcomes or rewards that players receive in a game based on the strategies they choose. They represent the utility or benefit associated with each possible combination of strategies, allowing players to assess their choices and make informed decisions. Understanding payoffs is crucial for analyzing how players interact, predict behavior, and determine the best strategies to maximize their outcomes.
Perfect Information Games: Perfect information games are strategic interactions where all players have complete knowledge of the game's structure, including the actions taken by other players at all times. This means that there are no hidden moves or incomplete information, allowing players to make fully informed decisions based on the current state of the game. These games can be represented using extensive form, often illustrated through decision trees, which display the sequential nature of the decisions made by players.
Risk Assessment: Risk assessment is the process of identifying, evaluating, and prioritizing potential risks that could impact a decision-making scenario. It involves analyzing the likelihood and consequences of various uncertainties, enabling individuals and organizations to make informed choices. This practice is essential in extensive form games and decision trees, where players need to anticipate the potential outcomes of their decisions based on the risks involved.
Sequential games: Sequential games are a type of game in which players make decisions one after another, rather than simultaneously. This structure allows players to observe the actions of others before making their own choices, which can significantly influence the outcome of the game. The analysis of these games often involves constructing extensive form representations to understand the strategic interactions and potential payoffs involved.
Strategies: In game theory, strategies refer to the complete plans of action that a player can take in a game, outlining how they will respond to different scenarios and the actions of other players. These strategies can be either pure, where a player consistently chooses one action, or mixed, where they randomize over possible actions. Understanding strategies is crucial as they shape the decisions made by players in both normal and extensive form games, influencing the outcomes based on the interactions among participants.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium used in dynamic games, where players make decisions at different stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that players' strategies are optimal even when the game reaches any point in the future. This concept helps analyze decision-making processes in extensive form games and supports the evaluation of credible threats and promises in strategic interactions.
Terminal Nodes: Terminal nodes are the endpoints in a decision tree or extensive form game, where no further actions or decisions occur. They represent final outcomes that can lead to payoffs for players involved in the game, and understanding these nodes is crucial for analyzing strategies, outcomes, and the overall structure of the game. They help in visualizing the end results of various strategic paths and play a key role in evaluating the effectiveness of decisions made along the way.
Utility Maximization: Utility maximization is the principle that individuals or firms make choices to achieve the highest level of satisfaction or benefit possible, given their preferences and constraints. This concept is central to understanding how decisions are made in various situations, particularly in strategic interactions where outcomes depend on the actions of multiple players. It underscores the importance of analyzing preferences, payoffs, and potential strategies to achieve the best possible result in different scenarios.