backward induction
/ BAK-wurd in-DUK-shun /
To plan a sequence of moves wisely, the trick is to start from the finish line and work backward. A chess player asks 'if I make this move, what is my opponent's best reply, and then my best reply to that?' — reasoning from the endgame back to the present. Backward induction is that idea made precise: to solve a sequential game, you figure out what the last player to move will do, then use that to figure out the second-to-last player's best move, and keep stepping backward until you reach the very first decision.
On a game tree, you start at the final decisions (the ones just before the leaves) and, for each, mark the move the player there would rationally choose — the one with the highest payoff for them. You then replace that whole subtree with its resulting payoffs and move up one level, repeating. Step by step you 'fold up' the tree from the leaves to the root, and the path you trace out is the predicted play. Example: in the market-entry game, you first ask what the incumbent does if the startup enters. If fighting a price war would cost the incumbent more than tolerating the rival, the incumbent will accommodate — so the threat to fight is empty, and the startup, foreseeing this, enters.
Backward induction is the standard method for solving sequential games and yields what is called a subgame-perfect equilibrium, which automatically discards non-credible threats. It powers analysis of bargaining, deterrence, and timing of investment. Its honest limit is that it assumes everyone is rational, knows everyone else is rational, and can carry the reasoning all the way back — assumptions that strain in long games. In experiments, real people often deviate (as in the famous centipede game), because trust, fairness, and limited foresight intervene.
Splitting a cake by 'I cut, you choose': reasoning backward, the cutter foresees that whoever chooses will grab the bigger piece, so the cutter's best move is to cut as evenly as possible. By thinking about the last step first, the cutter is led to a fair split.
'I cut, you choose': backward induction leads the cutter to divide the cake evenly.
Backward induction assumes flawless rationality all the way down the tree. In long games like the centipede game, real players cooperate far more than the theory predicts — a reminder that the method is a benchmark, not a law of behavior.