Subgame Perfect Equilibrium
Also known as: Backward Induction, Sequential Equilibrium, Extensive-Form Equilibrium
Subgame Perfect Equilibrium (SPE) is a refinement of Nash Equilibrium for sequential games, introduced by Reinhard Selten in 1965. It requires that strategy profiles constitute a Nash Equilibrium in every subgame, eliminating non-credible threats and incredible promises. Backward induction is the primary computational method for finding SPE in finite games.
Key highlights
- Eliminates implausible equilibria based on non-credible threats or promises
- Computationally tractable for finite games using backward induction
- Uniquely applicable to sequential games where move order and timing are critical
- Provides a clear decision procedure via explicit algorithm, enabling direct implementation
Intuition
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How it works
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When to use it
Apply SPE analysis to sequential games with incomplete information resolved over time: negotiations, auctions with multiple rounds, entry and exit decisions, or sequential moves in competition. Essential when understanding the order of moves and credibility of commitments matters. Works best for games with perfect or complete information; imperfect information requires refinements such as Perfect Bayesian Equilibrium.
Strengths & limitations
- Eliminates implausible equilibria based on non-credible threats or promises
- Computationally tractable for finite games using backward induction
- Uniquely applicable to sequential games where move order and timing are critical
- Provides a clear decision procedure via explicit algorithm, enabling direct implementation
- Restricted to finite games with perfect recall (no information asymmetries across decision nodes)
- Exponential complexity as game tree size grows, making large sequential games intractable
- May yield multiple equilibria in games with simultaneous moves or imperfect information
- Backward induction is sensitive to game tree representation; different extensive forms can yield different equilibria
Common pitfalls
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Applications
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Frequently asked
What is the difference between a Nash Equilibrium and a Subgame Perfect Equilibrium?
Every SPE is a Nash Equilibrium, but not vice versa. SPE requires Nash equilibrium play at every subgame, eliminating equilibria sustained by non-credible out-of-equilibrium threats that would never be carried out.
Why does backward induction sometimes seem to predict unrealistic behavior?
Backward induction assumes common knowledge of rationality and perfect information. Real players may have limited foresight, bounded rationality, or imperfect information, making SPE predictions less reliable in practice.
How does SPE handle games with imperfect information?
SPE alone does not directly apply; instead, Perfect Bayesian Equilibrium extends SPE by requiring players to update beliefs about others' types using Bayes' rule and then play optimally given those beliefs.
Sources
- 1.Selten, R. (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft, 121, 301-324.
- 2.von Stackelberg, H. (1934). Marktform und Gleichgewicht. Julius Springer.
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Cite this page
ScholarGate. (2026, June 3). Subgame Perfect Equilibrium. ScholarGate. https://scholargate.app/game-theory/subgame-perfect-equilibrium