How to Find Subgame Perfect Nash Equilibrium
Subgame perfect Nash equilibrium (SPNE) is a concept in game theory that represents a solution concept for extensive-form games, which are games with a sequence of moves and decisions. It is a refinement of the Nash equilibrium, which is a solution concept for strategic-form games. In this article, we will discuss how to find subgame perfect Nash equilibrium in a step-by-step manner.
Firstly, it is essential to understand the basic components of an extensive-form game. An extensive-form game consists of players, information sets, actions, payoffs, and a sequence of moves. Players make decisions at different nodes (or information sets) in the game tree, and each decision has a set of possible actions.
To find the subgame perfect Nash equilibrium, follow these steps:
1. Identify the subgames: A subgame is a part of the original game that starts at a node and includes all the subsequent nodes and decisions. In other words, a subgame is a smaller game that can be played independently of the rest of the game. Identify all the subgames within the extensive-form game.
2. Solve the subgames: For each subgame, find the Nash equilibrium. This can be done by backward induction, which is a method of solving extensive-form games by working from the end of the game tree to the beginning. Start by finding the equilibrium of the last subgame, and then move backward to find the equilibrium of the previous subgames.
3. Check for consistency: Once you have found the Nash equilibrium for each subgame, check if the equilibria are consistent across all subgames. This means that the strategies chosen in each subgame should be compatible with the strategies chosen in other subgames.
4. Identify the subgame perfect Nash equilibrium: If the equilibria are consistent across all subgames, then you have found the subgame perfect Nash equilibrium. This equilibrium represents a set of strategies where no player can unilaterally change their strategy to improve their payoff, given the strategies of the other players.
In summary, finding the subgame perfect Nash equilibrium involves identifying subgames, solving the equilibria for each subgame using backward induction, checking for consistency across subgames, and finally identifying the consistent equilibrium as the subgame perfect Nash equilibrium.
It is important to note that finding the subgame perfect Nash equilibrium can be challenging, especially for complex games with many players and decisions. However, by following these steps and understanding the underlying principles, you can effectively find the subgame perfect Nash equilibrium for a wide range of extensive-form games.
