Skip to content

Subgame Perfect Equilibrium Identification

Why It Matters

Standard Nash equilibrium in sequential games can include incredible threats - commitments to actions that aren't optimal when you'd actually have to execute them. Opponents who predict your future actions correctly won't be deterred by incredible threats. SPE forces consistency: your strategy must be optimal not just at the start but at every point where you might have to act. This eliminates bluffing equilibria where threats lack commitment, reveals which first-mover advantages are real versus illusory, and identifies credible bargaining positions.

The Key Move

  1. Map the sequential game as a tree with decision nodes and subgames (subtrees starting from each node).
  2. Find all Nash equilibria using standard methods.
  3. For each Nash equilibrium, check every decision node: if play reaches this node, would the player whose turn it is actually want to take the action specified by the strategy?
  4. Check by applying backward induction from that node forward - what's optimal in the subgame starting from here?
  5. If the strategy specifies an action that isn't optimal in any subgame, eliminate that Nash equilibrium - it's not subgame perfect.
  6. Only strategies that are optimal at every node (including off-equilibrium nodes) are SPE.

Classic Application

Chain store paradox (Selten). Incumbent chain faces potential entry in 20 cities sequentially. In each city, if entrant enters, incumbent chooses "fight" (price war, both lose money) or "accommodate" (share market, both profit modestly). Entrant chooses "enter" or "stay out." Backward induction: in city 20 (last city), incumbent won't fight (no future cities to deter entry, fighting is costly). Knowing this, entrant enters city 20. But then in city 19, same logic applies - no benefit to fighting in 19 since entry in 20 is inevitable. Unraveling backward, entrant should enter all cities and incumbent should accommodate all. The "fight to build reputation" strategy isn't subgame perfect - when you reach the decision to fight, it's not optimal.

Surprising Application

Parenting and discipline threats. Parent threatens punishment if child misbehaves. Nash equilibrium could include: "If child misbehaves, parent imposes severe punishment." But when misbehavior actually occurs, is severe punishment optimal for the parent? If punishment is costly to enforce (parent suffers from child's distress, creates relationship damage), the parent won't want to follow through. Child predicts this, making the threat non-credible. SPE analysis: only punishment levels the parent will actually enforce are credible. This explains why consistent moderate consequences work better than threatened severe consequences - moderate punishment is subgame perfect (parent will actually do it), making it credible and therefore deterring. Severe threats aren't SPE and get ignored. Non-obvious because it distinguishes stated strategy from credible strategy.

Failure Modes

Computational complexity - finding all subgames and checking equilibrium in each can be intractable in complex games. Assuming common knowledge of rationality - SPE requires all players to predict all future optimal play, which requires strong rationality assumptions. Missing trembles and mistakes - perfect equilibrium further refines SPE by considering small probability mistakes. Not identifying all subgames - off-equilibrium nodes still define subgames that must be checked. Applying to simultaneous games - SPE is specifically for sequential games with proper subgames.

Go Deeper

Selten, "The Chain Store Paradox," Theory and Decision 9(2), 1978; Fudenberg & Tirole, Game Theory, Chapter 3