Backward Induction
What is Backward Induction?
Backward induction solves a sequential game by reasoning from the last decision backward, choosing each player's best move at every stage.
In a game tree with perfect information, you start at the final decisions, pick the optimal action there, then fold those payoffs back to earlier nodes and repeat to the start. The resulting strategy profile is the subgame-perfect equilibrium because it is optimal in every subgame, ruling out non-credible threats. It is the standard method for solving sequential/Stackelberg-type games.
Backward Induction: a worked example
Take a two-stage entry game. A challenger chooses Enter or Stay Out; if it enters, the incumbent chooses Fight or Accommodate. Payoffs are written (challenger, incumbent): Stay Out gives (0, 10), Enter then Fight gives (-2, 2), Enter then Accommodate gives (3, 5). Start at the incumbent's node, the last decision in the tree. Accommodating pays 5 against 2 for fighting, so the incumbent accommodates. Fold that back one stage: the challenger compares 3 from entering against 0 from staying out, and enters. The subgame-perfect equilibrium is Enter, Accommodate, worth (3, 5). The threat to fight was never credible, since carrying it out would cost the incumbent 3.
The mistake students make with backward induction
Students routinely count every Nash equilibrium as a backward-induction solution. In the entry game above, Stay Out paired with Fight is a Nash equilibrium: facing the threat, staying out beats entering, and the incumbent never has to act on it. Backward induction discards it, because at the node where fighting would actually happen, fighting pays 2 and accommodating pays 5. The payoff matrix is tempting precisely because it hides the order of moves, and a threat that is never tested looks free.
Backward Induction questions
What is the difference between backward induction and nash equilibrium?
Backward induction is a solving procedure that works from the final nodes of a game tree back to the start, while Nash equilibrium is a property a strategy profile either has or lacks. Every backward-induction solution is a Nash equilibrium, but not every Nash equilibrium survives backward induction. The ones it eliminates rest on non-credible threats, actions a player would refuse to take if that node were actually reached.
Does backward induction work in every game?
Backward induction needs a finite game of perfect information, meaning each player observes every move made before their own and the tree has a final stage. It fails when players move simultaneously, since neither observes the other's choice and there is no single last decision to work back from. It also cannot start in an infinitely repeated game, which has no final round. Ties in payoffs at a node leave the procedure without one prescribed action.
Why is backward induction called subgame perfect?
Backward induction yields a subgame-perfect equilibrium because the action it selects at every node is optimal in the subgame beginning there, not merely along the path players expect to travel. A subgame is the entire tree hanging below a single node. Checking optimality in all of them, including branches nobody reaches in equilibrium, is exactly what strips out threats a player would abandon if called.
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