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, backward induction, and subgame perfect equilibrium, 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

• Players take turns making decisions, with each move observed by subsequent players
• Order of play significantly impacts strategic choices and outcomes
• Incorporates the concept of information sets, representing what players know at each decision point
• Allows for the analysis of credible threats and promises in economic interactions
• Often represented using game trees to visualize the sequence of decisions and potential outcomes

Comparison to simultaneous games

• Sequential games differ from simultaneous games in the timing of decisions
• Players in sequential games can condition their strategies on observed actions of previous players
• Equilibrium concepts like subgame perfect equilibrium become relevant in sequential games
• Sequential games often lead to different outcomes compared to their simultaneous counterparts (Stackelberg vs Cournot competition)
• Modeling sequential games requires consideration of information flow and strategic foresight

Game tree representation

• Game trees serve as a visual tool in mathematical economics to depict the structure and flow of sequential games
• This representation aids in analyzing strategic interactions, identifying optimal strategies, and predicting outcomes in complex economic scenarios
• Game trees form the foundation for applying concepts like backward induction and subgame perfect equilibrium

Nodes and branches

• Nodes represent decision points or chance events in the game
• Decision nodes indicate where players make choices, often depicted as circles or squares
• Terminal nodes (leaf nodes) represent the end of the game, showing final payoffs
• Branches connect nodes, representing possible actions or moves available to players
• Each path from the root to a terminal node represents a possible sequence of decisions in the game

Information sets

• Group decision nodes where a player cannot distinguish between positions in the game
• Represented by dashed lines connecting indistinguishable nodes
• Crucial for modeling games with imperfect information
• Affect strategy formulation as players must consider multiple possible game states
• Can lead to more complex equilibrium concepts (perfect Bayesian equilibrium)

Backward induction process

• Backward induction forms a fundamental technique in solving sequential games within mathematical economics
• This process allows economists to analyze strategic decision-making in multi-stage interactions
• Backward induction provides insights into optimal strategies and equilibrium outcomes in various economic scenarios

Starting from terminal nodes

• Begin analysis at the end of the game tree, examining final payoffs
• Identify optimal choices for the last player at each terminal decision node
• Consider all possible outcomes resulting from the final player's decisions
• Assign these optimal choices to the corresponding decision nodes
• Prepare to move up the game tree to earlier decision points

Working backwards

• Move to the penultimate player's decision nodes, considering the last player's optimal responses
• Determine the best choice for the penultimate player given the subsequent optimal play
• Continue this process, moving up the game tree to earlier decision points
• At each stage, players choose the action leading to their preferred outcome
• Reach the initial node, determining the optimal strategy for the first player

Subgame perfect equilibrium

• Subgame perfect equilibrium (SPE) represents a refinement of Nash equilibrium in sequential games
• This concept plays a crucial role in analyzing strategic behavior in dynamic economic interactions
• SPE helps economists predict outcomes in scenarios involving multi-stage decision-making processes

Definition and properties

• Strategy profile constituting Nash equilibrium in every subgame of the original game
• Eliminates non-credible threats by requiring optimal play in all contingencies
• Ensures that players' strategies remain optimal even off the equilibrium path
• Can be found using backward induction in games of perfect information
• May not exist in games with imperfect information or continuous action spaces

Comparison to Nash equilibrium

• SPE is a stronger solution concept than Nash equilibrium in sequential games
• All subgame perfect equilibria are Nash equilibria, but not vice versa
• SPE eliminates Nash equilibria that rely on non-credible threats
• Provides more accurate predictions of behavior in multi-stage economic interactions
• Becomes equivalent to Nash equilibrium in simultaneous-move games

Solving sequential games

• Solving sequential games forms a critical aspect of game-theoretic analysis in mathematical economics
• This process involves identifying optimal strategies and equilibrium outcomes in dynamic strategic interactions
• Understanding solution methods enables economists to analyze complex economic scenarios and predict behavior

Extensive form vs normal form

• Extensive form represents the game as a tree, showing the sequence of moves
• Normal form displays the game as a matrix of payoffs for all possible strategy combinations
• Extensive form better captures the dynamic nature of sequential games
• Normal form can be derived from the extensive form but may lose information about the order of play
• Solution methods may differ depending on the chosen representation (backward induction for extensive form)

Backward induction algorithm

• Start at the terminal nodes and work backwards to the initial node
• At each decision node, determine the optimal choice for the acting player
• Assume rational play by all subsequent players when evaluating choices
• Eliminate dominated strategies at each stage of the analysis
• Continue until reaching the initial node, yielding the subgame perfect equilibrium

Applications in economics

• Sequential games find widespread applications across various fields of economics
• These models help analyze strategic interactions in markets, firms, and policy-making
• Understanding sequential games enables economists to predict outcomes and design effective strategies in dynamic economic environments

Industrial organization

• Entry deterrence models analyze incumbent firms' strategies to prevent new competitors
• Stackelberg competition examines sequential output decisions in oligopolistic markets
• Research and development races model firms' sequential investment decisions
• Vertical integration decisions consider sequential choices in supply chain management
• Pricing strategies in multi-period settings analyze dynamic pricing behavior

Bargaining models

• Alternating offer models (Rubinstein bargaining) analyze sequential negotiation processes
• 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 $s_i$ for player $i$ maps each information set to an action
• Strategy profile $s = (s_1, ..., s_n)$ represents a complete description of all players' strategies
• Payoff function $u_i(s)$ 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 $u_i(s^*_i, s^*_{-i}) \geq u_i(s_i, s^*_{-i}) for all $s_i$ and all subgames
• Nash equilibrium condition: u_i(s^*_i, s^*_{-i}) \geq u_i(s_i, s^_{-i}) for all $s_i$ 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 rationality 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