Backward induction
Solve a sequential game by reasoning from the last move back to the first.
What it means
A method for finding the rational outcome of a sequential game by starting at the final decisions, determining the optimal action there, and working backward to the opening move. In finite games of perfect information it yields a subgame-perfect equilibrium and strips out threats a player would never actually carry out, and it is the engine behind solving bargaining, deterrence, and entry games; games with imperfect information or no last move require the wider machinery of subgame perfection and sequential equilibrium instead. Its predictions can be sharp yet behaviorally fragile: in games like the centipede game, backward induction prescribes taking at the first opportunity, but real players cooperate for many rounds, exposing how heavily the result leans on common knowledge of rationality rather than on rationality alone. It matters as the core technique of dynamic game theory and a frequent target for behavioral critique.
How the fold-back works
Draw the game as a tree and start where it ends. At a final decision the player has no future to weigh, so the best action is whichever payoff is larger — provided no two tie, since indifference at a node leaves the fold-back with more than one answer. Replace that node with its payoff, and the second-to-last decision becomes a final decision in a smaller tree. Repeat until you reach the opening move. The procedure needs a finite horizon and perfect information: everyone sees the full history at every turn. What it buys is credibility. A threat to price below cost for a year if a rival enters can survive as a Nash equilibrium, but fold-back kills it: at the node where it would be carried out, doing so hurts the threatener. Only threats someone would actually execute survive.
What the evidence shows
McKelvey and Palfrey ran the centipede game in 1992 and found the theory's sharpest prediction almost never happened: rather than take at the first node, most pairs passed for several rounds. Three decades of replication have not rescued it, and a 2020 comparison across centipede variants found the deviations too heterogeneous for one model: level-k fits some players, noisy best response others. The most instructive episode is a contradiction between two field studies of expert chess players. Palacios-Huerta and Volij reported in 2009 that about 69 percent of their chess players took at the first node, rising to every Grandmaster against a known chess opponent. Levitt, List and Sadoff repeated the exercise in 2011 and got the reverse: none of their sixteen Grandmasters stopped there. The discrepancy has never been fully settled.
Why it breaks down
The centipede result is often read as proof that people cannot reason backward. It is not. Levitt, List and Sadoff gave their players Race to 100, a two-player race whose winning strategy needs no assumption about the opponent's rationality, and in the version where players pick numbers from 1 to 9 nearly 60 percent solved it. Yet of the players who backward inducted perfectly, not one took at the first centipede node, and the best inductors passed around 84 percent of the time. Ability and behavior came apart. Fold-back needs more than your own rationality: you must believe your opponent is rational, believe they believe you are, and so on. Aumann showed the outcome follows from common knowledge of rationality, precisely what a real opponent does not supply. Binmore and Reny pressed the objection that the premise undercuts itself: when they pass at a node theory said you would never reach, the belief that drove your fold-back has just been refuted — a challenge Aumann rejected.
Using it in practice
Treat backward induction as a benchmark rather than a forecast. It tells you what happens if everyone reasons perfectly and knows everyone else does, which makes the gap between it and observed behavior the quantity worth measuring. Johnson, Camerer, Sen and Rymon tracked how people inspect a three-round bargaining problem and found many never opened the later-round payoffs in the order or for the time the method requires; once taught it, offers against a robot opponent known to play subgame-perfectly fell to $1.22 against an equilibrium of $1.25 — but trained bargainers facing untrained ones landed halfway between equilibrium and the untrained $2.11. If your counterparty may not have done the arithmetic, the equilibrium is not the number to plan around. And when you need them to see the endgame, walking them through the horizon can be worth more than arguing over the split.
Examples
To decide your first move in a multi-stage negotiation, you first work out what each side would rationally do in the final stage.
A chess player works out the forced mate three moves from the end, then chooses today's move so the board arrives exactly there; the last move decides the first.
Before promising a child one more story at bedtime, a parent thinks through what happens when that story ends, realises the plea will simply repeat, and declines the first one.
A firm designing a two-year retention grant folds back from the vesting date: an employee who has already banked the payout has nothing left to stay for, so the final month's hold is worth nothing, which tells the firm what each earlier month of the grant is actually buying.
A retailer planning a six-week clearance works out the final week's markdown first, sees that shoppers who expect it will simply wait, and starts discounting shallower and earlier.
First described in Game theory; Zermelo (1913); Selten (1965).
Key references
- García-Pola, B., Iriberri, N., & Kovářík, J. (2020). Non-equilibrium play in centipede games. Games and Economic Behavior, 120, 391-433. doi.org/10.1016/j.geb.2020.01.007
- Levitt, S. D., List, J. A., & Sadoff, S. E. (2011). Checkmate: Exploring backward induction among chess players. American Economic Review, 101(2), 975-990. doi.org/10.1257/aer.101.2.975
- Palacios-Huerta, I., & Volij, O. (2009). Field centipedes. American Economic Review, 99(4), 1619-1635. doi.org/10.1257/aer.99.4.1619
- Johnson, E. J., Camerer, C., Sen, S., & Rymon, T. (2002). Detecting failures of backward induction: Monitoring information search in sequential bargaining. Journal of Economic Theory, 104(1), 16-47. doi.org/10.1006/jeth.2001.2850
- Aumann, R. J. (1995). Backward induction and common knowledge of rationality. Games and Economic Behavior, 8(1), 6-19. doi.org/10.1016/S0899-8256(05)80015-6
- McKelvey, R. D., & Palfrey, T. R. (1992). An experimental study of the centipede game. Econometrica, 60(4), 803-836. doi.org/10.2307/2951567