Extensive Form Games

Extensive Form Games

• The simplified Stackelberg model is represented in extensive form, where firms make sequential decisions.
• Firm 1 (Leader) has a single decision node with two actions: (q_L) (Low) and (q_H) (High).
• Firm 2 (Follower) has two decision nodes, leading to four possible strategies based on the leader's choice.
• Strategies are contingent plans specifying decisions at each node, essential for analyzing outcomes in dynamic environments.

Backward Induction

• Backward induction is a method for finding Nash equilibria in games of perfect information.
• Steps include identifying Nash equilibria in final subgames and progressively reducing the game until terminal nodes are reached.
• This method is useful for finite-stage games but ineffective for infinitely repeated games, which require different analysis.

Bayesian Nash Equilibrium

• Involves players with private, payoff-relevant information, leading to games of incomplete information.
• It requires that the strategy profile forms a Nash equilibrium where expected payoffs account for uncertainties about other players' types.
• It involves random variables determined by Nature, representing the probability distribution of types players can have.

Perfect Bayesian Nash Equilibrium (PBNE)

• Refines Nash equilibrium in games where players observe predecessors' actions to update their beliefs about types.
• Requires that strategies are sequentially rational given the belief system, enhancing the robustness of decision-making processes.
• A belief system assigns probabilities to decision nodes ensuring consistency with observed events.

Types of Perfect Bayesian Equilibria

• Separating Equilibrium: Informed players signal different types through distinct actions, allowing uninformed players to infer types.
• Pooling Equilibrium: All informed players take the same action; uninformed players cannot deduce type information, leading to retained prior beliefs.

Key Mathematical Notations

• (X_1) and (X_2) are strategy sets for firms, while (θ_i) denotes the type variable associated with each player, chosen by Nature.
• Payoff functions are defined in terms of strategies and types, thereby incorporating the uncertainty present in the players' decisions.

References for Further Reading

• Introductory texts: Gibbons (1992), Mas-Colell et al. (1995) cover foundational concepts of game theory.
• Advanced study: Fudenberg and Tirole (1991) provide comprehensive insights into dynamic games and strategic interactions.

This quiz covers two critical concepts in game theory: the extensive form representation of games and backward induction.