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Subgame Perfect Equilibrium is a critical concept in game theory, referring to a strategy profile that constitutes a Nash Equilibrium in every subgame of the original game, ensuring optimal decision-making at every point. This equilibrium refines the Nash Equilibrium by eliminating non-credible threats, leading to logically consistent outcomes in extensive-form games. Understanding subgame perfect equilibria helps students analyze strategic scenarios in various fields such as economics, political science, and evolutionary biology.
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Sign up for freeWhat is a Subgame Perfect Equilibrium?
Why might a Nash Equilibrium not be Subgame Perfect?
In the given game example, why does Player 1 choose Y?
How does backward induction contribute to finding Subgame Perfect Equilibria?
How does Subgame Perfect Equilibrium differ from Nash Equilibrium?
How is Bayesian induction used in finding Subgame Perfect Equilibrium?
Why is backward induction crucial for Subgame Perfect Equilibrium?
What is the main role of Subgame Perfect Nash Equilibrium (SPNE)?
What is a Subgame Perfect Nash Equilibrium?
What is a Subgame Perfect Nash Equilibrium?
What method is used to find Subgame Perfect Nash Equilibria?
What is a Subgame Perfect Equilibrium?
Why might a Nash Equilibrium not be Subgame Perfect?
In the given game example, why does Player 1 choose Y?
How does backward induction contribute to finding Subgame Perfect Equilibria?
How does Subgame Perfect Equilibrium differ from Nash Equilibrium?
How is Bayesian induction used in finding Subgame Perfect Equilibrium?
Why is backward induction crucial for Subgame Perfect Equilibrium?
What is the main role of Subgame Perfect Nash Equilibrium (SPNE)?
What is a Subgame Perfect Nash Equilibrium?
What is a Subgame Perfect Nash Equilibrium?
What method is used to find Subgame Perfect Nash Equilibria?
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Subgame Perfect Equilibrium is a key concept in game theory and economics. It ensures that the strategies chosen by players are optimal for every possible subgame within a larger game. Unlike the Nash Equilibrium, which might include non-credible threats, Subgame Perfect Equilibrium focuses on strategies that are credible in every subgame.Understanding how this equilibrium works is crucial for analyzing strategies in games with sequential actions, where decisions at one stage can significantly impact the outcomes at later stages.
A Subgame Perfect Equilibrium is an equilibrium where players' strategies constitute a Nash Equilibrium in every subgame of the original game.
Consider a simple extensive-form game where Player 1 chooses A or B, and then Player 2 decides to play C or D if Player 1 chooses A. If Player 1 chooses B, Player 2 gets no move. Payoffs are as follows: choosing (A, C) gives (2, 1), (A, D) gives (0, 3), and (B) results in (1, 2). By using backward induction:- If Player 1 chooses A, Player 2 will choose D because 3 > 1.- Hence, if Player 1 chooses A, the outcome will be (A, D).- Player 1 compares the payoff from (A, D) (which is 0 for Player 1) with choosing B (which gives 1).- As a result, Player 1 chooses B.The Subgame Perfect Equilibrium for this scenario is (B).
The difference between Nash Equilibrium and Subgame Perfect Equilibrium is central to many game theory applications:
Hint: Remember, not all Nash Equilibria are Subgame Perfect, but every Subgame Perfect Equilibrium is a Nash Equilibrium!
The Subgame Perfect Nash Equilibrium is a pivotal concept in game theory that extends the Nash Equilibrium to better accommodate sequential games. This type of equilibrium ensures players’ strategies are optimal across every possible subgame, thus providing a more comprehensive understanding of strategic decision-making.
A Subgame Perfect Nash Equilibrium is a strategy profile in extensive-form games whereby the players' strategies constitute a Nash Equilibrium in every subgame, ensuring credibility in every step of the game.
In extensive-form games, you represent decisions using a game tree where each node reflects a decision point. The concept of Subgame Perfect Equilibrium allows us to refine Nash Equilibria by requiring that strategies form Nash Equilibria not only for the whole game but for each subgame too.This refinement is often achieved through backward induction, a method of solving sequential games by analyzing from the end towards the start. By doing so, each player's strategy is proven credible and optimal at every possible decision node.
Imagine you are playing a game where Player 1 can choose to Invest (I) or Not Invest (NI), and Player 2 can choose to Expand (E) or Stay (S) after observing Player 1's decision. The payoffs are:
| I, E | I, S | NI, E | NI, S | |
| Player 1 | 5 | 2 | 1 | 3 |
| Player 2 | 4 | 3 | 2 | 1 |
In the context of extensive-form games, understanding why not every Nash Equilibrium is Subgame Perfect is crucial. While Nash Equilibrium considers players' strategies as optimal given the strategies of others, it does not necessarily restrain players from using non-credible threats within the game sequence. This limitation of Nash is rectified by Subgame Perfect Equilibrium:
Subgame Perfect Nash Equilibrium is a refinement of Nash Equilibrium applicable in extensive-form games. Finding it involves strategies suitable for every possible subgame, ensuring all moves are credible and optimal.To identify this equilibrium, you'll often use backward induction, a method that analyzes a game's structure from the endpoints back to the start. This involves dissecting the game into its constituent subgames and determining the optimal strategies at each decision point.
Backward induction is a systematic way of solving sequential games, essential for determining Subgame Perfect Equilibria. Start by analyzing the final decision in the game to ensure the chosen action is optimal. Then, move backward step-by-step to the initial decision nodes:1. Identify the end node payoffs.2. Determine the best decision for the player at these nodes.3. Work backwards to update the payoffs for preceding decision nodes based on the new optimal strategies.4. Repeat until reaching the initial move.
Consider a two-stage game where Player 1 decides between L and R. If L, Player 2 chooses A or B. Payoffs are:
| L, A | L, B | R | |
| Player 1 | 3 | 1 | 2 |
| Player 2 | 2 | 4 | - |
The use of backward induction in finding Subgame Perfect Nash Equilibria plays a vital role in strategic decision-making where the sequence of actions is crucial. Sequential games often involve planning for future contingencies, and backward induction works by:
Subgame Perfect Nash Equilibrium (SPNE) is vital for analyzing strategies in games with sequential moves. It refines Nash Equilibrium by considering only the credible strategies within subgames.
Consider a game with two players in sequential decisions. Player 1 can choose Move X or Move Y. If X, Player 2 can choose between C and D. The payoffs are displayed in the following table:
| X, C | X, D | Y | |
| Player 1 | 3 | 1 | 4 |
| Player 2 | 2 | 5 | - |
Remember, each subgame in the sequence requires optimal and credible strategies to ensure the overall game's equilibrium outcome.
In-depth analysis with mathematical models demonstrates that Subgame Perfect Equilibria are robust under various game structures. Consider a general sequential decision model:1. **Initial State**: Represented by the initial node (root) where decisions unfold.2. **Transition Function**: \( T(s, a) \), which denotes the transition from state \( s \) when action \( a \) is taken.3. **Payoff Function**: \( U_i(s, a) \), expresses the utility or reward obtained by player \( i \) for state \( s \) with action \( a \).Verification in SPNE involves computing backward from the end states (leaves) for each subgame:
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