Extensive-Form Games and Backward Induction

Draw the entry game as a tree. The order of moves is visible at a glance.

Solve it by reading backwards from the end. First fix the last mover's best answer. Knowing that answer settles the earlier player's choice too. A threat to fight is not credible, because fighting hurts the threatener once entry happens. So that branch gets pruned away.

For example, say a neighborhood has one snack 🏪 shop and a new one wants to open. The old owner threatens a price war. But once the new shop opens, fighting costs the old owner too. The newcomer knows this and enters anyway.

SPE: optimal in every subgame\text{SPE: optimal in every subgame}

Part of these learning paths

Examples & Exercises

Find the subgame-perfect equilibrium of the entry game by backward induction.

Out→(0,2),Enter→Incumbent: {Accommodate→(1,1),  Fight→(−1,−1)}\text{Out} \to (0, 2), \quad \text{Enter} \to \text{Incumbent: } \{\text{Accommodate} \to (1,1), \; \text{Fight} \to (-1,-1)\}
Backward induction complete — the bold path is the SPE.
Show proof

  1. Start at the incumbent's node: after entry, accommodating gives 11 versus −1-1 from fighting, so accommodate is optimal. The threat to fight is empty — it would not be carried out once the node is reached.

  2. The entrant anticipates this: entering yields 11 (the incumbent will accommodate), staying out yields 00.

    Since 1>01 > 0, enter.

    The subgame-perfect equilibrium is (Enter, Accommodate) with payoffs (1,1)(1, 1).

Show that the profile (Stay Out, Fight-if-enter) is a Nash equilibrium but not subgame perfect.

The empty threat — Fight is not a credible commitment.
Show proof

  1. Why it is Nash: with the entrant staying out, the incumbent's Fight is off-path and payoff-irrelevant; believing the incumbent would fight, the entrant prefers Stay Out (00) over Enter (−1-1, getting fought). No unilateral deviation pays.

  2. But it is not subgame perfect: in the subgame after entry, Fight is not a best response (−1<1-1 < 1). Nash equilibrium never asks whether off-path behavior is "sensible," but subgame perfection demands best responses in every subgame — so this equilibrium is thrown out.

8. Game Theory