Subgame

Published Sep 8, 2024

Definition of Subgame

Subgame is a concept in game theory that refers to a part of an extensive game which constitutes a game in itself. A subgame starts at a decision node (not at the beginning of the game) and includes all the subsequent decisions and payoffs from that node to the end of the game. For a subgame to be properly defined, it must include a unique initial decision node and all continuations from that node.

Example

Consider a business negotiation between two firms, Firm A and Firm B, about a potential merger. Initially, Firm A proposes a merger deal. Firm B then decides whether to accept or reject the proposal. If Firm B accepts, the game ends with specific payoffs for both firms. If Firm B rejects, a new negotiation phase begins where Firm A might alter its proposal, leading to a new series of decisions and outcomes. The initial decision by Firm B (accepting or rejecting the proposal) and all the subsequent negotiations can be viewed as a subgame.

To visualize this, imagine a decision tree that starts with Firm A’s proposal. From there, Firm B’s decision to accept or reject leads to different branches, each containing further decision points and outcomes. The subgame encompasses the decision tree starting from Firm B’s first decision node and includes all the subsequent nodes and payoffs.

Why Subgames Matter

Subgames are crucial for the concept of subgame-perfect equilibrium, which is a refinement of Nash equilibrium. In essence, the subgame-perfect equilibrium requires that players’ strategies constitute a Nash equilibrium in every subgame of the original game. This is important because it ensures that the strategies form a consistent and credible plan of action, even when considering any possible deviations within the game.

Subgames provide a framework to analyze dynamic and sequential strategic interactions where decisions at different stages need to be evaluated. By breaking down a strategic scenario into subgames, analysts can better understand the structure of the game, anticipate future actions, and predict outcomes more accurately.

Frequently Asked Questions (FAQ)

What is a subgame-perfect equilibrium?

A subgame-perfect equilibrium (SPE) is an extension of Nash equilibrium used in extensive-form games. It ensures that players’ strategies form a Nash equilibrium within every subgame of the original game. This concept eliminates non-credible threats by requiring optimal strategies in each subgame, leading to more realistic predictions of behavior in dynamic strategic situations.

How can identifying subgames improve strategic decision-making?

Identifying subgames helps players understand the structure of a game in a detailed manner, allowing them to anticipate potential moves by other players and adjust their strategies accordingly. This leads to more informed and strategic decision-making. By analyzing subgames, players can foresee future scenarios and make choices that are optimal at every stage of the game, thereby increasing their chances of achieving desired outcomes.

Are there limitations to using subgames in game theory analysis?

While subgames provide a detailed view of sequential decision-making, they can become complex and difficult to manage in large or highly intricate games with numerous decision nodes and outcomes. Additionally, identifying subgames and calculating subgame-perfect equilibria can be computationally intensive. Despite these challenges, the insights gained from analyzing subgames can significantly enhance understanding and strategic planning in many scenarios.

Can subgames be applied to real-world scenarios outside of business negotiations?

Yes, subgames are applicable to various real-world scenarios beyond business negotiations. They can be used in political strategy, where different stages of electoral processes or legislative actions can be treated as subgames. Legal disputes, military tactics, and even everyday decision-making processes like purchasing a home or planning a vacation can involve sequential decisions that are best analyzed using subgames. By breaking down complex decisions into smaller, manageable parts, subgames provide a structured approach to understanding and optimizing strategic behavior in numerous fields.