proper subgame

Understanding Proper Subgame in Game Theory

Proper subgame is a fundamental concept in the field of game theory, especially in the study of dynamic games and extensive form representations. It plays a crucial role in the analysis of strategic decision-making over multiple stages or periods. By understanding what constitutes a proper subgame and how it functions within the larger context of a game, strategists and economists can better predict outcomes, analyze strategic stability, and design mechanisms that lead to desired equilibria. This article aims to provide a comprehensive overview of proper subgames, their characteristics, significance, and applications in game theory.

Fundamentals of Game Theory and Subgames

What is a Game?

A game in game theory is a formal model that represents strategic interactions among rational decision-makers, known as players. The game specifies the set of players, strategies available to each player, and payoffs associated with each strategy profile. Games can be static or dynamic; the latter involve sequential moves, information sets, and potential observations of previous actions.

Extensive Form Representation

Many dynamic games are represented in extensive form, which visualizes the game as a tree-like structure. This form explicitly shows the order of moves, possible decisions at each node, and information available at each decision point. It aids in analyzing sequential strategies and the concept of subgames.

Defining a Subgame

What is a Subgame?

A subgame is a subset of the original game that can be considered a game in its own right. More formally, a subgame is a part of the game tree that starts at a single decision node and includes all subsequent nodes and branches that follow from it, satisfying specific criteria. Subgames are used to analyze parts of the game independently, especially when applying backward induction or subgame perfect equilibrium concepts.

Criteria for a Subgame

For a subset of the game tree to qualify as a subgame, it must meet the following conditions:

• Start at a single decision node: The subgame begins at a particular node within the larger game tree.
• Contain all successor nodes: All decision nodes that follow the initial node must be included.
• Be closed under the information structure: If the game involves imperfect information, the subgame must include all nodes that are indistinguishable to the players at the initial node.
• Include all actions and payoffs: The subgame must contain all the relevant strategies, moves, and payoffs that follow from the initial node.

Proper Subgame: Definition and Significance

What is a Proper Subgame?

A proper subgame is a subgame that is strictly smaller than the entire game and satisfies the criteria for being a subgame. The term "proper" emphasizes that it is neither the entire game nor an improper subset that does not adhere to the rules of subgame formation. In essence, a proper subgame is a sub-part of the game that can be analyzed independently and is suitable for backward induction and equilibrium refinement. Proper subgames are essential because they allow the decomposition of complex, multi-stage games into manageable segments.

Importance of Proper Subgames

Proper subgames serve several vital roles in game theory:
• Enabling backward induction: They allow the analysis of subgames from the end to the beginning, facilitating the derivation of subgame perfect equilibria.
• Refinement of equilibria: Proper subgames help eliminate non-credible threats by focusing on credible strategies within each subgame.
• Modular analysis: They allow the division of extensive form games into smaller, analyzable components, simplifying complex strategic interactions.
• Application in mechanism design: Proper subgames are used to design credible commitments and strategies within specific parts of a larger game.

Characteristics of Proper Subgames

Key Features

Proper subgames exhibit several distinctive features:
• Independence: They can be analyzed independently of the entire game due to their self-contained nature.
• Start at a decision node: Each proper subgame begins at a single node and includes all subsequent nodes.
• Closure under information sets: If the game involves imperfect information, the subgame must include entire information sets, not just individual nodes.
• Strict subset: They are strictly smaller than the entire game, ensuring that the analysis focuses on a portion rather than the whole.

Difference from Improper Subgames

An improper subgame might violate one or more of the criteria above, such as:
• Starting at multiple nodes (not a single node).
• Excluding parts of an information set.
• Not being closed under the information structure.
• Including the entire game, which essentially makes it the whole.
Proper subgames are distinguished precisely because they adhere strictly to the rules, enabling meaningful strategic analysis.

Examples of Proper Subgames

Simple Example: Sequential Game with Two Stages

Consider a sequential game where Player 1 chooses to "Invest" or "Not Invest," and, if Player 1 invests, Player 2 then chooses "Cooperate" or "Defect." The game tree starts with Player 1's decision node, followed by Player 2's decision node if Player 1 invests.
• The subgame starting at Player 2's decision node (after Player 1 invests) is a proper subgame. It contains:
• Player 2's decision node.
• All subsequent nodes, including their payoffs.
• The information set for Player 2 (assuming perfect information).
• The initial decision node of Player 1 is not a subgame, but the part starting from Player 2's node qualifies as a proper subgame.

Complex Example: Repeated Games

In repeated or multi-stage games, proper subgames can be identified at each stage:
• The subgame starting at any decision node where a player makes a move, along with all subsequent nodes, can be a proper subgame.
• For instance, in a repeated Prisoner’s Dilemma, each period's decision node can be viewed as a proper subgame if analyzed separately.

Applications of Proper Subgames in Game Theory

Subgame Perfect Equilibrium (SPE)

One of the primary applications of proper subgames is in the derivation of subgame perfect equilibrium, a refinement of Nash equilibrium applicable to dynamic games.
• Definition: A strategy profile constitutes a subgame perfect equilibrium if it induces a Nash equilibrium in every proper subgame.
• Methodology: Backward induction involves solving the game starting from the smallest proper subgames, ensuring strategies are credible at every stage.
• Significance: This eliminates non-credible threats and promises, leading to more realistic and robust predictions.

Mechanism Design and Credibility

Proper subgames are instrumental in mechanism design where the goal is to create strategies or rules ensuring certain outcomes:
• By focusing on subgames, designers can ensure that strategies are credible and enforceable at each stage.
• Proper subgames help identify where commitments are necessary and whether threats or promises are credible.

Analyzing Multi-stage and Repeated Games

Proper subgames facilitate the decomposition of complex strategic interactions:
• They enable players to analyze their strategies within each subgame independently.
• This approach simplifies the process of finding equilibria in complex, multi-stage settings.

Challenges and Limitations of Proper Subgames

Information Set Considerations

In games with imperfect information, identifying proper subgames can be complicated:
• When players have incomplete information, subgames must include entire information sets.
• This restriction may limit the applicability of subgame analysis.

Non-Existence of Proper Subgames

Some extensive form games may not contain proper subgames, especially if:
• The game has a single decision node.
• The structure does not satisfy the closure under information sets.

Complexity in Large Games

In large, multi-stage games, identifying all proper subgames can be computationally intensive:
• The process involves examining numerous nodes and information sets.
• Automated tools or algorithms are often employed to facilitate analysis.

Conclusion

Proper subgames are a cornerstone concept in game theory, providing a framework for analyzing complex, dynamic strategic interactions. They enable the application of backward induction, the refinement of equilibria, and the design of credible strategies. Understanding their characteristics, how to identify them, and their role in solution concepts such as subgame perfect equilibrium is essential for both theorists and practitioners. While they offer powerful analytical tools, their application can be limited by information structures and complexity considerations. Nonetheless, proper subgames remain a vital concept for advancing the analysis of strategic behavior in multi-stage and extensive form games. By mastering the concept of proper subgames, analysts can better understand the strategic nuances of sequential decision-making, improve the robustness of their predictions, and contribute to the development of more effective strategic mechanisms.

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