Subgame perfect Nash equilibrium (SPNE) is a solution concept in game theory that refines the Nash equilibrium by considering the incentives of players in all possible subgames of a game. SPNE is closely related to four key concepts in game theory: Nash equilibrium, subgames, perfect Bayesian equilibrium, and extensive form games. A Nash equilibrium is a set of strategies for all players in a game where no player can improve their payoff by unilaterally deviating from their strategy, given the strategies of the other players. A subgame is a game that results from the actions of players at some point in the original game. Perfect Bayesian equilibrium is a Nash equilibrium that takes into account the players’ beliefs about the strategies of other players, which are updated based on the information available to them. Extensive form games are games that are represented as trees, where each node represents a decision point for a player and each branch represents an action that a player can take.
Structure of Subgame Perfect Nash Equilibrium
A subgame perfect Nash equilibrium (SPNE) is a solution concept in game theory that refines the Nash equilibrium concept by requiring that players’ strategies be optimal in all subgames of the game. A subgame is a game that results from the actions of players at a particular decision node in the original game.
To find the SPNE of a game, we can use the following steps:
1. Identify all the subgames of the game.
2. For each subgame, find the Nash equilibrium.
3. Check if the Nash equilibria of all the subgames are consistent with each other.
If the Nash equilibria of all the subgames are consistent with each other, then the original game has a SPNE. If not, then the original game does not have a SPNE.
Here is a more detailed explanation of the steps:
1. Identify all the subgames of the game.
A subgame is a game that results from the actions of players at a particular decision node in the original game. To identify all the subgames of a game, we can use the following trick:
- Start at the end of the game and work backwards.
- For each decision node, consider all the possible actions that could be taken by the players.
- For each combination of actions, create a new game that starts at the decision node and ends at the end of the game.
The new games that we create are the subgames of the original game.
2. For each subgame, find the Nash equilibrium.
Once we have identified all the subgames of the game, we can find the Nash equilibrium of each subgame. A Nash equilibrium is a set of strategies, one for each player, such that no player can improve their payoff by changing their strategy while the other players’ strategies remain the same.
To find the Nash equilibrium of a subgame, we can use any of the standard methods for finding Nash equilibria, such as the best response algorithm or the minimax algorithm.
3. Check if the Nash equilibria of all the subgames are consistent with each other.
Once we have found the Nash equilibrium of each subgame, we need to check if the Nash equilibria are consistent with each other. The Nash equilibria of all the subgames are consistent with each other if the following two conditions hold:
- The strategies that players use in the Nash equilibrium of each subgame are the same as the strategies that they use in the Nash equilibrium of the original game.
- The payoffs that players receive in the Nash equilibrium of each subgame are the same as the payoffs that they receive in the Nash equilibrium of the original game.
If the Nash equilibria of all the subgames are consistent with each other, then the original game has a SPNE. If not, then the original game does not have a SPNE.
Example
Consider the following game:
Player 1 Player 2
A B C
D 1,1 -1,2 0,0
E 0,0 2,-1 1,1
The subgames of this game are:
- Subgame 1: The game that starts after Player 1 chooses A and Player 2 chooses D.
- Subgame 2: The game that starts after Player 1 chooses A and Player 2 chooses E.
- Subgame 3: The game that starts after Player 1 chooses B and Player 2 chooses D.
- Subgame 4: The game that starts after Player 1 chooses B and Player 2 chooses E.
- Subgame 5: The game that starts after Player 1 chooses C and Player 2 chooses D.
- Subgame 6: The game that starts after Player 1 chooses C and Player 2 chooses E.
The Nash equilibrium of each subgame is:
- Subgame 1: Player 1 chooses A and Player 2 chooses D.
- Subgame 2: Player 1 chooses A and Player 2 chooses E.
- Subgame 3: Player 1 chooses B and Player 2 chooses D.
- Subgame 4: Player 1 chooses B and Player 2 chooses E.
- Subgame 5: Player 1 chooses C and Player 2 chooses D.
- Subgame 6: Player 1 chooses C and Player 2 chooses E.
The Nash equilibria of all the subgames are consistent with each other. Therefore, the original game has a SPNE. The SPNE is that Player 1 chooses A and Player 2 chooses D.
Question 1:
What is the key concept behind a subgame perfect Nash equilibrium?
Answer:
A subgame perfect Nash equilibrium is a Nash equilibrium that is also a subgame equilibrium, meaning that it is an equilibrium in every subgame of the original game.
Question 2:
How does the concept of subgame perfection differ from the concept of Nash equilibrium?
Answer:
Nash equilibrium considers only the original game, while subgame perfection additionally considers all subgames and ensures that the equilibrium is stable in all of them.
Question 3:
In what types of games is the concept of subgame perfection particularly relevant?
Answer:
Subgame perfection is particularly relevant in sequential games, where players make decisions in a sequence and their decisions in later stages may depend on the outcome of earlier stages.
Whew! That was a lot of brainwork, wasn’t it? But hey, now you’re a pro at understanding subgame perfect Nash equilibrium. Just remember, it’s all about thinking strategically and looking ahead to see what your best move is. Thanks for sticking with me through this little journey into game theory. If you have any more questions or just want to geek out about games some more, come back and visit me again soon!