Understanding Subgames in Game Theory: A Comprehensive Guide
What is a subgame in game theory?
In game theory, a subgame is a part of a larger game that can be analyzed independently. More formally, a subgame is a subset of the game that starts at a particular decision node and includes all subsequent decision nodes and outcomes that can be reached from that point. This concept is crucial in understanding strategic interactions within dynamic or sequential games.
Key Characteristics of Subgames
Starting Point: A subgame begins at one of the decision nodes of the original game. Complete Information: A subgame must include all the information relevant to the players at that decision node, meaning it should contain all the strategies and payoffs available from that point onward. Self-Contained: The subgame should be a game in itself, meaning it can be analyzed using the same principles as the original game, including the determination of optimal strategies and equilibria. Nash Equilibrium: The concept of Nash equilibrium can also apply to subgames. A strategy profile can be a Nash equilibrium in the broader game if it is also a Nash equilibrium in every subgame that it affects.Examples of Subgames
Consider a simple extensive form game represented by a game tree. If one player makes a decision that branches into different outcomes, any of those branches that lead to further decisions can be viewed as subgames. For instance, if Player A can choose between actions X and Y, and choosing X leads to further decisions by Player B, the game starting from the decision node where Player B makes their choice is a subgame.
Importance of Subgames:
Subgames are crucial for analyzing strategic interactions in games, especially in dynamic or sequential games where players make decisions at various points in time. For example, if B's strategy is to always be unkind, which means B is unkind if A moves left and unkind if B moves right, this strategy would be a Nash equilibrium as both players would have nothing to gain from changing strategies. However, it is not a subgame perfect equilibrium because if the subgame were limited to where A moves right, B's strategy of always being unkind would yield a lower payoff of 0 instead of 1 if they were kind.
Subgame Perfect Equilibrium
A subgame perfect equilibrium is a refinement of Nash equilibrium that takes into account the sequential nature of the game. This is a strategy profile where players’ strategies are optimal within every subgame of the original game. For an example, Player A always goes right, and Player B is unkind if A goes left but kind if A goes right. In every subgame, the players' actions make sense and are in Nash equilibrium.
Formalizing Subgames in Game Theory
When modeling games as trees, where each link is a possible move, every subtree corresponds to a subgame. This formalization helps in breaking down complex games into simpler, more manageable components. It allows for a more accurate analysis of strategic decisions at various points in the game, leading to better predictions and strategies for players.
Conclusion
Understanding subgames in game theory is essential for analyzing strategic interactions in dynamic and sequential games. The concept of subgame perfect equilibrium provides a more refined way of looking at Nash equilibria in the context of these games. By breaking down games into their constituent subgames, players can better strategize and make more informed decisions.