Answer First
A subgame perfect Nash equilibrium is a strategy profile that forms a Nash equilibrium in every subgame. To find it, use backward induction: start at the final decision nodes, determine optimal actions, and work backward until you reach the initial node.
Problem Setup
Consider a sequential game with players 1 and 2. Key definitions:
- Subgame: a part of the game beginning at a decision node and containing all its successors.
- SPNE: strategies that form a Nash equilibrium in every subgame.
- Backward induction: solving the game from the end to the beginning.
General approach: \[ \text{SPNE} = \text{strategies chosen via backward induction} \]
Step-by-Step Explanation
1. Identify all subgames
Every decision node that is not inside an information set begins a subgame.
2. Start at the terminal nodes
Determine which action gives the highest payoff for the player moving at each final decision point.
3. Replace each subgame with its optimal action
This collapses the game tree step by step.
4. Move backward to earlier nodes
At each earlier node, choose the action that maximizes the player’s payoff given the optimal continuation strategies.
5. Write the full strategy profile
SPNE requires complete strategies:
- Player 1: what they do at every node where they could move
- Player 2: what they do at every node where they could move
6. Verify subgame perfection
Check that the strategies form a Nash equilibrium in every subgame.
Intuition
Backward induction eliminates non‑credible threats. A player cannot threaten to take an action that would hurt themselves later. SPNE keeps only strategies that are optimal at every stage of the game.
Common Exam Mistakes
- Writing actions instead of full strategies.
- Forgetting to specify off‑path behavior.
- Not identifying all subgames.
- Using Nash equilibrium instead of SPNE.
Final Summary
To find a subgame perfect Nash equilibrium, identify subgames, apply backward induction from the terminal nodes, determine optimal actions at each stage, and write the full strategy profile. SPNE eliminates non‑credible threats and ensures optimal play in every subgame.
This explanation belongs to the broader Economics Tutoring pillar.
If you want help working through these ideas for coursework or exams, you can talk directly to a tutor, not a marketer.
Call/Text: 510-398-0006
Email: tutor@californiagraduatetutor.com