Question

Are the subgames of a sequential game visible when the entire game is presented in strategic form? Explain.

Short answer

No, subgames are not visible in strategic form, as it abstracts sequence details essential for identifying subgames.

Step-by-step solution

Understanding Sequential and Strategic Forms

Sequential games are represented through extensive form, focusing on the order of moves and information sets. Strategic form, on the other hand, captures simultaneous actions without displaying the sequence of moves. To determine if subgames are visible, we must first understand if strategic form provides all necessary structural elements of extensive form.

Examining Subgames

A subgame is a portion of an extensive form game that starts from a single node and contains all its subsequent gameplay. This structure shows the progression and branching of decisions starting from a unique history point.

Comparing Game Representations

In strategic form, each player’s strategies are listed as if they are making decisions simultaneously, omitting explicit order, which are crucial in determining subgames. Without the sequence and nodes shown, strategic form fails to specify where one subgame ends and another begins.

Realizing Subgame Visibility

Given a strategic form lacks the explicit node-based configuration, it does not display inherent branching structures necessary to isolate subgames. Subgames, deeply reliant on sequential decisions, depend on extensive form for visibility, which isn't provided in strategic form.

Key concepts

Sequential Games

Sequential games are a fascinating aspect of game theory where players make decisions one after another. Unlike simultaneous games where all decisions occur at once, sequential games emphasize the order in which decisions are made. This order is crucial because players can observe the actions of others before making their move. As a result, they can develop strategies based on the previous actions of their opponents.

Key elements of sequential games include sequences of moves and decision nodes. These decision nodes are points where players must choose an action based on the information available from previous moves.

• Players await the actions of others before deciding.
• Information sets help determine the knowledge each player has.
• Decisions depend heavily on visible actions from earlier in the game.

Sequential games are often represented in extensive form to capture this dynamic nature.

Strategic Form

Strategic form, also known as normal form, is a way to represent games where players make decisions simultaneously or are unaware of other players' actions while deciding. Typically shown in a matrix format, strategic form allows us to see all possible strategies and their corresponding payoffs for every combination of decisions.

However, this form does not capture the sequence of events or dynamic nature of games with multiple stages. Instead, it provides a snapshot of strategic interactions where:

• Players choose their strategies without knowledge of others' choices.
• Every player determines their best response given potential strategies of their opponents.
• Payoffs are assessed based on the combination of strategies selected by all players.

While strategic form is potent for analyzing certain games, it doesn't showcase the decision-making process over time.

Subgames

Subgames are like mini-games within a larger game, specifically found in extensive form representations. Each subgame begins at a single decision node and includes all future actions and decisions that derive from that point. They help in simplifying complex games by breaking them down into tractable parts.

In essence, subgames are valuable because:

• They allow for the analysis of smaller, more manageable sections of the game.
• Each subgame must resemble a complete game starting from a new decision node.
• This concept aids in applying backward induction to find optimal strategies.

However, in the strategic form, subgames lose their visibility because the form lacks nodes and sequence, which are crucial for identifying subgame boundaries.

Extensive Form

Extensive form is the go-to representation for understanding the structure of sequential games. It displays games as a decision tree or graph, capturing the order of moves, decision nodes, possible actions, and player information throughout the game. This form is vital for showing how the game unfolds over time.

Extensive form provides clarity on several fronts:

• Identifies the exact sequence of moves by showcasing decision nodes.
• Exposes the information available to players at different stages.
• Highlights opportunities for making strategic moves based on prior actions.

Unlike strategic form, it is ideal for highlighting subgames and structuring backward induction analysis. Essentially, extensive form reveals the true dynamics of gameplay that strategic form cannot, due to its linear, matrix-bound perspective.