Sequential games model strategic interactions where players make decisions in a specific order. They capture the dynamic nature of real-world economic scenarios, providing insights into decision-making processes in various contexts.
Understanding sequential games is crucial for analyzing market entry strategies, policy implementation, and other economic interactions. Key concepts include game trees, , and , which help predict outcomes in complex scenarios.
Definition of sequential games
Sequential games form a crucial component of game theory in mathematical economics, modeling strategic interactions where players make decisions in a specific order
These games capture the dynamic nature of many real-world economic scenarios, allowing for analysis of strategic behavior over time
Understanding sequential games provides insights into decision-making processes in various economic contexts, from market entry strategies to policy implementation
Key characteristics
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Union-firm wage negotiations often involve multi-stage bargaining
International trade agreements modeled as sequential bargaining between countries
Debt restructuring negotiations analyzed using sequential bargaining frameworks
Political bargaining models examine coalition formation and policy-making processes
Limitations and extensions
While sequential games provide valuable insights, they also have limitations in modeling certain economic scenarios
Recognizing these constraints and exploring extensions allows for more comprehensive analysis of complex strategic interactions
Understanding limitations and extensions helps economists choose appropriate models for specific economic problems
Imperfect information
Many real-world scenarios involve incomplete knowledge about previous actions
Imperfect information complicates the application of backward induction
Requires the use of more advanced solution concepts (perfect Bayesian equilibrium)
Can lead to multiple equilibria, making predictions more challenging
Modeling imperfect information often involves probability distributions over information sets
Repeated games
Extend sequential games to multiple periods or infinite horizons
Allow for the analysis of long-term relationships and reputation effects
Introduce concepts like trigger strategies and folk theorems
Can lead to cooperation in scenarios where one-shot games predict conflict
Require consideration of discounting and long-term payoffs in strategy formulation
Mathematical formalization
Mathematical formalization of sequential games provides a rigorous framework for analysis in economics
This approach allows for precise definition of game elements and solution concepts
Formalization enables the application of mathematical techniques to derive equilibria and analyze strategic behavior
Strategies and payoffs
Strategy si for player i maps each information set to an action
Strategy profile s=(s1,...,sn) represents a complete description of all players' strategies
Payoff function ui(s) assigns a real-valued payoff to player i for each strategy profile
Expected payoffs calculated using probability distributions over chance nodes
Utility functions may incorporate risk preferences in games with uncertainty
Equilibrium conditions
Subgame perfect equilibrium s∗ satisfies ui(si∗,s−i∗)≥ui(si,s−i∗) for all si and all subgames
Nash equilibrium condition: ui(si∗,s−i∗)≥ui(si,s−i∗) for all si and all players i
Equilibrium in behavioral strategies may be necessary for games with imperfect information
Mixed strategy equilibria involve probability distributions over pure strategies
Existence of equilibrium often proved using fixed-point theorems (Kakutani's theorem)
Examples of sequential games
Examining specific examples of sequential games helps illustrate key concepts and their applications in economics
These examples demonstrate how sequential game analysis can provide insights into real-world economic scenarios
Understanding these examples aids in developing intuition for solving and interpreting more complex sequential games
Entry deterrence
Incumbent firm decides on capacity investment before potential entrant's decision
Entrant observes incumbent's choice and decides whether to enter the market
Backward induction reveals whether entry deterrence is a credible strategy
Outcomes depend on the relative costs of capacity expansion and market entry
Illustrates the importance of credible commitments in strategic interactions
Stackelberg competition
Sequential move game in an oligopolistic market with a leader and follower(s)
Leader chooses output first, followed by the follower(s)
Backward induction solves for the subgame perfect equilibrium
Typically results in higher profits for the leader compared to simultaneous-move Cournot competition
Demonstrates the first-mover advantage in certain market structures
Behavioral considerations
Behavioral aspects of decision-making introduce additional complexities to sequential game analysis in economics
Incorporating behavioral factors can lead to more realistic models of economic interactions
Understanding these considerations helps explain observed deviations from standard game-theoretic predictions
Time inconsistency
Refers to situations where optimal plans made in the present become suboptimal in the future
Can lead to deviations from subgame perfect equilibrium predictions
Often modeled using hyperbolic discounting or present-biased preferences
Relevant in areas such as savings decisions, addiction, and policy-making
May require commitment devices or institutional arrangements to mitigate
Credible vs non-credible threats
Credible threats influence opponent's behavior and are rational to carry out if tested
Non-credible threats lack in execution and should be disregarded by opponents
Subgame perfect equilibrium eliminates non-credible threats through backward induction
Reputation effects in repeated games can make otherwise non-credible threats credible
Understanding threat credibility is crucial for analyzing bargaining and negotiation processes
Key Terms to Review (16)
Backward induction: Backward induction is a method used in game theory and decision-making that involves reasoning backward from the end of a problem or game to determine optimal strategies. This approach helps identify the best course of action by analyzing potential future outcomes and choices, ultimately leading to a well-informed decision. It is particularly useful in scenarios where decisions are made sequentially, allowing players to anticipate the actions of others and optimize their own strategies accordingly.
Branches: In the context of sequential games, branches refer to the different paths or decision points that can arise as players make choices over time. Each branch represents a possible action that a player can take at a given stage of the game, leading to different outcomes and subsequent decisions. This structure is crucial for analyzing the strategic interactions between players, as it allows us to visualize and evaluate how choices evolve throughout the game.
Centipede Game: The centipede game is a strategic form of game theory involving two players who alternately decide whether to take a larger share of a growing pot of resources or pass it to the other player. The game highlights the tension between cooperation and self-interest in sequential decision-making scenarios, revealing how players can choose different strategies based on their expectations of the other's behavior.
Common Knowledge: Common knowledge refers to information that is known by all parties involved in a situation, meaning that everyone is aware of the same facts and understands that others are also aware of them. This concept plays a crucial role in strategic decision-making, especially in scenarios where players' actions are influenced by their expectations of other players' knowledge and behaviors.
Complete information: Complete information refers to a situation in game theory where all players have access to all relevant information about the game, including the strategies available to each player and the payoffs associated with each possible outcome. This concept is crucial for analyzing sequential games, as it allows players to make informed decisions based on a shared understanding of the game's structure and potential results.
Decision trees: Decision trees are graphical representations used to model decisions and their possible consequences, including chance event outcomes, resource costs, and utility. They help to visualize the sequential nature of decisions, making it easier to analyze different strategies and outcomes. Decision trees are particularly useful in understanding complex problems where uncertainty exists, allowing decision-makers to evaluate potential actions and their impacts.
Extensive form game: An extensive form game is a representation of a strategic situation where players make decisions at different points in time, often depicted using a tree-like structure. This format allows for the modeling of sequential moves, showing how players' choices lead to different outcomes and enabling the analysis of strategies over time. It is particularly useful for understanding dynamic interactions among players, taking into account their potential actions and reactions at each stage of the game.
Game Matrices: Game matrices are structured tools used to represent the strategic interactions between players in a game, detailing their available strategies and the corresponding payoffs. They provide a visual representation that helps to analyze the decisions of players based on the potential outcomes of their choices. By organizing the information clearly, game matrices facilitate the understanding of complex interactions, particularly in sequential games where players make decisions one after another.
Imperfect information: Imperfect information refers to situations where all parties involved do not have complete knowledge about the relevant factors or actions of others, leading to uncertainty and potentially suboptimal decision-making. This concept is crucial in understanding how individuals and firms make choices when they lack full awareness of outcomes, strategies, or payoffs, which can influence strategic interactions and decision processes.
Nash equilibrium: Nash equilibrium is a concept in game theory where no player can benefit by changing their strategy while the other players keep theirs unchanged. This idea highlights a state of mutual best responses, making it essential in analyzing strategic interactions among rational decision-makers. Understanding Nash equilibrium helps to explore various scenarios, including competitive markets, sequential games, and different strategic approaches, thus providing a foundation for equilibrium analysis and the existence of stable outcomes.
Nodes: In game theory, nodes represent decision points in a game tree where players make choices. Each node corresponds to a specific state of the game, allowing players to analyze their strategies based on possible outcomes that stem from their decisions. Nodes are essential in visualizing the sequential nature of games and help in applying concepts like backward induction to determine optimal strategies.
Payoff structure: The payoff structure refers to the outcomes or rewards that players receive from different strategies in a game, particularly in sequential games where decisions are made one after another. Understanding the payoff structure is essential as it influences players' choices, strategies, and the overall equilibrium of the game. It determines how each player's actions impact not only their own payoffs but also those of their opponents, making it a critical component in analyzing decision-making processes.
Perfect Bayesian Equilibrium: Perfect Bayesian Equilibrium (PBE) is a refinement of Nash Equilibrium used in sequential games, where players' strategies are based on their beliefs about other players' actions. It incorporates the concept of beliefs that update based on the observed actions in a game and ensures that players choose optimal strategies at every stage, given these beliefs. This equilibrium concept balances the notions of rationality and belief consistency in dynamic settings, making it crucial for analyzing strategies in sequential games.
Rationality: Rationality refers to the behavior of individuals making decisions based on logical reasoning and consistent preferences, aiming to maximize their utility or payoffs in strategic situations. This concept is central to understanding how players interact in games, as it influences their choices and strategies, leading to outcomes that depend on the actions of others. Rationality assumes that players have clear preferences and will choose the best possible option available to them based on their beliefs about other players' actions.
Stackelberg competition: Stackelberg competition is a strategic model of competition where firms make production decisions sequentially rather than simultaneously. In this framework, one firm, known as the leader, sets its output level first, and the other firm, called the follower, makes its output decision based on the leader's choice. This structure reflects the dynamics of real-world markets where firms often take turns in making strategic decisions.
Subgame perfect equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium used in dynamic games, where players' strategies constitute a Nash equilibrium in every subgame of the original game. This concept ensures that players' strategies remain optimal not only for the overall game but also for every possible scenario that may arise as the game unfolds. By applying backward induction, players can determine their best responses at every stage of the game, leading to a more robust understanding of strategic interactions.