Subgame

3 key takeaways

• A subgame is a smaller component of a larger game, beginning at a particular decision point and including all subsequent moves and outcomes.
• Subgame perfection is a refinement of Nash equilibrium, ensuring that strategies are optimal for every subgame, promoting consistency in strategic decisions.
• Understanding subgames helps in analyzing complex strategic interactions and improving decision-making in multi-stage games and dynamic settings.

What is a subgame?

In game theory, a subgame is a portion of a larger game that begins at a specific decision node and includes all subsequent decisions and outcomes from that point forward. A subgame must satisfy the property that every move within it is part of the original game, preserving the structure and rules of the broader game. Subgames are used to analyze the strategies and outcomes of specific parts of the game, providing insights into the overall strategic interactions.

Subgames are particularly useful in extensive-form games, which represent games with a sequential structure, where players make decisions at different stages. Analyzing subgames helps in understanding how players’ strategies evolve over time and ensures that decisions are consistent and optimal at every stage of the game.

How does a subgame work?

• Identification of subgames: In an extensive-form game, subgames are identified by selecting decision nodes that serve as starting points. From these nodes, the subgame includes all subsequent nodes, decisions, and outcomes.
• Subgame perfection: A strategy is subgame perfect if it constitutes a Nash equilibrium in every subgame of the original game. Subgame perfection ensures that strategies are not only optimal for the entire game but also for every possible scenario that could arise within the game.
• Backward induction: One common method for finding subgame-perfect equilibria is backward induction, where players start from the end of the game and work backward to determine optimal strategies at each decision point. This ensures that decisions are consistent and rational throughout the game.

Examples of subgames

• Ultimatum game: In the ultimatum game, one player proposes a division of a sum of money, and the second player can accept or reject the proposal. If rejected, both players receive nothing. The game can be broken into subgames starting from each possible proposal, analyzing the second player’s decisions.
• Repeated games: In a repeated game, players engage in the same game multiple times. Each repetition can be considered a subgame, and strategies can be analyzed for consistency and optimality across the repetitions.
• Chess: In chess, each move by a player creates a new subgame, where the remaining game is played out from the current board position. Analyzing subgames helps players anticipate future moves and plan strategies accordingly.

Importance of subgames

• Strategic consistency: Analyzing subgames ensures that players’ strategies are consistent and rational at every stage of the game, leading to more robust and credible equilibria.
• Dynamic decision-making: Subgames provide insights into dynamic strategic interactions, where decisions made at one stage influence future decisions and outcomes. This helps players anticipate and plan for future contingencies.
• Complex game analysis: In complex games with multiple stages and decision points, breaking the game into subgames simplifies analysis and helps in identifying optimal strategies and potential equilibria.

Understanding subgames is crucial for analyzing strategic interactions in dynamic and multi-stage games. For further exploration, consider studying the principles of extensive-form games, the concept of Nash equilibrium and subgame perfection, and methods like backward induction for solving subgames. Additionally, examining real-world applications of subgame analysis in economics, politics, and competitive strategy can provide practical insights into the value of this approach.