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Subgame Perfect Equilibrium Explained
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.
Components of Subgame Perfect Equilibrium
- Extensive-Form Representation: This involves representing the game as a tree with nodes indicating decision points. The root node represents the initial state, and the branches depict possible actions.
- Subgames: A subgame is a section of the larger game that represents a game in itself, starting at a decision node. It must include all future decisions that follow and resemble the structure of the original game.
- Backward Induction: A method used to solve for the Subgame Perfect Equilibrium by analyzing the game from the end to the beginning, ensuring optimal strategies at each step.
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:
- Nash Equilibrium applies to simultaneous games where players select strategies independently and simultaneously. It is concerned with mixed strategies where players have no incentive to deviate unilaterally.
- Subgame Perfect Equilibrium refines Nash by accounting for credibility in sequential games. It ensures that expected payoffs are consistent through all subgames, making strategies robust against deviations.
Hint: Remember, not all Nash Equilibria are Subgame Perfect, but every Subgame Perfect Equilibrium is a Nash Equilibrium!
Definition of Subgame Perfect 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 |
- If Player 1 Invests, Player 2 chooses Expand, because 4 > 3.
- The payoff (I, E) results in (5, 4).
- If Player 1 chooses Not Invest, Player 2 also chooses Expand, achieving (1, 2).
- Player 1 compares (5, 4) and (1, 2), preferring to Invest.
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:
- Ensures that all threats and promises are credible by being enforceable at any subgame level.
- Requires for sequential rationality, meaning that players’ strategies are optimal for every decision point.
How to Find Subgame Perfect Nash 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 Subgame Perfect Equilibrium
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 | - |
- Subgame 1: If Player 2 faces the choice between A and B, Player 2 chooses B for a payoff of 4 over 2.
- Initial Decision: Player 1 considers their payoff if choosing L, which is 1 (since Player 2 chooses B), and 2 if choosing R.
- Therefore, Player 1 selects R.
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:
- Ensuring credibility: Strategies that are not sustainable at any future decision point are systematically eliminated.
- Simplifying complex decisions: By reducing the need to consider all possible future contingencies upfront, it narrows the strategic focus to pivotal decision points.
- Highlighting time-consistency: Decisions are harmonized over time, reflecting a coherent long-term strategic vision.
Subgame Perfect Nash Equilibrium Example
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 | - |
- In the subgame, if Player 1 chooses X, Player 2 picks D because 5 > 2.
- Now, Player 1 anticipates this and compares the outcome of choosing X (1) vs Y (4).
- Therefore, Player 1 chooses Y for a payoff of 4.
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:
- Evaluate terminal payoffs: quickly establish end-state utilities.
- Optimize decision nodes: recursively determine optimal actions, ensuring \(a^*\) maximizes expected payoff given \( E[U(s)] \).
- Iterate backwards: continue until the start state, forming a strategy profile \((s, a^*)\) ensuring SPNE.
subgame perfect equilibrium - Key takeaways
- Subgame Perfect Equilibrium: An equilibrium where players' strategies form a Nash Equilibrium in every subgame of the original game, ensuring credibility in sequential games.
- Difference with Nash Equilibrium: Subgame Perfect Equilibrium focuses on credible strategies in sequential games, refining Nash Equilibria by considering each subgame.
- Backward Induction: A method for finding Subgame Perfect Equilibrium by solving the game from the last decision backward to the first.
- Finding Subgame Perfect Nash Equilibrium: Use backward induction to analyze optimal strategies in each subgame, considering all decision points sequentially.
- Subgame Perfect Nash Equilibrium Example: A game where decision process by backward induction leads to a strategy profile that is credible and optimal in every subgame.
- Applications: Useful in extensive-form games to ensure strategies are credible by being enforceable at each subgame level, preventing non-credible threats.
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