Question

Consider an extensive-form game in which player 1 is one of two types: A and B. Suppose that types A and B have exactly the same preferences; the difference between these types has something to do with the payoff of another player. Is it possible for such a game to have a separating PBE, where \(\mathrm{A}\) and \(\mathrm{B}\) behave differently?

Short answer

No, since A and B have identical preferences, they cannot behave differently, making a separating PBE impossible.

Step-by-step solution

Understand the Separating PBE Concept

In a separating Perfect Bayesian Equilibrium (PBE), players of different types choose different actions, allowing other players or observers to infer the type based on the action taken. Thus, each type of player selects an action that distinguishes them from the other types.

Consider Player Type Preferences

Review the problem statement: Types A and B have identical preferences. This implies that their decision-making process, based solely on their preferences, would not differ. Hence, if a separating equilibrium existed, it can't be due to their own payoffs preferences.

Analyze Game's Impact on Other Player's Payoff

Although types A and B have the same preferences, their presence can impact another player's payoff differently. Examine how this external party's response could create divergence in the behavior of A and B if that player's response affects A and B's strategies differently.

Evaluate Feasibility of Separating PBE

To achieve a separating equilibrium, the variation in the impact on another player's payoff should incentivize A and B to choose different actions. However, since A and B have identical preferences, even with their different impacts on another player, if that does not change their own payoffs or perceptions, their behavior is likely to be the same.

Conclude on the Possibility of Separating PBE

Given that A and B have identical preferences, any differences in the other player's payoff do not directly change the decision environment of A and B. Therefore, despite potential external differences, A and B will choose the same actions, precluding a separating equilibrium.

Key concepts

Extensive-form games

Extensive-form games are a way to represent games that showcase every possible move and decision point within the game. They are like a roadmap of choices for the players involved. Imagine having a game tree that branches out, showing various paths and strategies that a player could take. In this format, you see not only the choices but also the outcomes and payoffs associated with each decision. This setup is particularly useful in games where timing and sequencing of moves matter.

• Representation: Extensive-form games are shown as trees with nodes (decision points) and branches (actions).
• Information Set: It shows what a player knows when making a decision.
• Moves: Can be simultaneous or sequential, affecting the game’s strategy.

Understanding these elements is crucial because it helps analyze how decisions are influenced by earlier actions and the possible payoffs at the end.

Perfect Bayesian Equilibrium

A Perfect Bayesian Equilibrium (PBE) is a solution concept in game theory that merges beliefs with strategies. It helps predict how rational players will behave in games where some information is missing or imperfect. It is especially useful when players have private information about their types or intentions.

In a PBE, every player's strategy maximizes their expected utility, given their beliefs about other players’ strategies and the types they may encounter. Beliefs are updated using Bayes' rule wherever possible, ensuring rational decision-making throughout the game.

• Credibility: Players use past play to infer others' types, influencing future strategies.
• Strategy Consistency: Strategies should make sense with the given beliefs.
• Equilibrium Components: Includes strategies and beliefs that satisfy Bayesian updating.

By focusing on PBE, players can refine their strategies in the face of uncertainty, leading to more predictable outcomes.

Player types

Player types in game theory refer to the different kinds of players that might exist within a game, distinguished by their preferences or characteristics. In the given exercise, player 1 can be either type A or Type B. Even though both types have the same preferences, their existence might influence another player's payoff differently.

Understanding player types is essential because it allows us to explore how every type's behavior or strategy can affect the game’s outcome. Here are some key points about player types:

• Preference Similarities: Types A and B have identical preferences, meaning their desired outcomes are aligned.
• Impact on Payoffs: While A and B sharing preferences might limit separating strategies, their influence on others' payoffs can lead to strategic differences.
• Role in PBE: Differentiating player types can help in understanding potential PBE and separating equilibria.

Grasping the concept of player types can shape our perception of strategic decision-making in games.

Payoffs

Payoffs in game theory are the rewards players receive as a result of their chosen strategies. They are fundamental to understanding a player's incentives and ultimately their decision-making process. In this exercise, although players A and B have the same preferences, their actions can still impact another player's payoffs.

The crucial aspect of analyzing payoffs is to determine how each player's actions directly affect their own and other players' payoffs. Understanding payoffs helps predict which strategies players are likely to choose in order to maximize their returns.

• Direct Payoffs: The immediate reward a player receives from their actions.
• Indirect Payoffs: Rewards based on the interdependence of actions between players.
• Influence on Strategy: Payoffs significantly guide a player’s strategic choices and evolution within the game.

Mastering payoffs is integral because it shapes strategic choices and gears players toward optimizing their outcomes.

Equilibrium analysis

Equilibrium analysis in game theory involves determining the set of strategies where players have no incentive to deviate. At equilibrium, each player's strategy is optimal, given the strategies of the other players in the game. The focus here in the exercise is on analyzing whether a separating equilibrium is feasible.

To conduct equilibrium analysis, we examine each player's potential strategies and evaluate if any player can benefit by unilaterally changing their strategy. Several factors influence this analysis:

• Identifying Equilibria: Finding strategies where no player can gain by changing their strategy unilaterally.
• Strategy Evaluation: Analyzing which combinations of players' strategies form a stable outcome.
• Separating Equilibrium: Checking if players of different types behave differently, allowing others to infer their types.

Through thorough equilibrium analysis, we determine if players A and B can indeed produce different strategies despite having identical preferences, ensuring strategic stability.