Sequential Games and Backward Induction

Why thinking through a decision by working backward from the ending often reveals the smartest first move.

Most of the games covered earlier in this module - the prisoner’s dilemma, auctions, coordination problems - involve players choosing at essentially the same moment, without seeing what the other side picked first. Plenty of real strategic situations don’t work that way at all. In a sequential game, one player moves, the other player observes that move and then responds, and the order matters enormously to the outcome.

Thinking backward to move forward

The standard tool for solving a sequential game is called backward induction: instead of starting at the first move and guessing forward, you start at the very last decision in the game and work backward, step by step, figuring out what each player would rationally do at each stage. Since every player understands that later players will act rationally too, the earliest mover can predict the whole chain of responses and pick their opening move accordingly.

Here’s the intuition. Imagine a simple two-step game: a company can either enter a market currently dominated by one competitor, or stay out. If it enters, the dominant competitor can either fight (a price war that hurts both companies) or accommodate (share the market peacefully). Working backward, the new entrant asks: once I’ve entered, what will the incumbent actually do? If a price war is more costly to the incumbent than simply sharing the market, the incumbent will accommodate rather than fight - so the entrant, correctly predicting that, can enter with real confidence. Backward induction shows that the incumbent’s threat to fight, unless it’s genuinely cheaper than accommodating, isn’t credible, and a purely rational entrant should see through it.

The last slice of pizza

Two roommates are splitting a pizza. One cuts it into two pieces of their choosing; the other picks which piece to take first. Working backward: whatever the cutter does, the picker will simply take the bigger piece. Knowing that in advance, the cutter's genuinely smartest move at step one is to cut the pizza as evenly as possible, since any lopsided cut just hands the bigger half to the other person. The fair outcome doesn't happen because either roommate is being generous - it falls out of both players reasoning backward from the last move.

Credible threats versus empty ones

Backward induction is especially useful for separating a credible threat from an empty one. A threat is only credible if it would genuinely still make sense for the threatening player to carry it out once the moment actually arrives - not just as a warning beforehand. A parent who threatens to cancel a long-planned, already-paid-for vacation over a minor chore left undone is making a threat that’s expensive for them too; a child who reasons this through may correctly predict it won’t happen. Businesses face the same test: a company that threatens to slash prices to punish any competitor who enters its market needs that price war to still be its best move after entry actually happens, or a sharp-eyed competitor will call the bluff.

Subgame perfect equilibrium

Economists call the outcome that survives backward induction a subgame perfect equilibrium - a plan of action that remains rational at every single stage of the game, not merely as an opening move. It’s a stronger standard than an ordinary Nash equilibrium, discussed earlier in this module, because it also rules out strategies that only look sensible if a later, non-credible threat is bluffed successfully.

Believing a threat just because it was stated firmly

It's easy to assume a confidently delivered threat must be credible. Backward induction asks a more useful question: once the moment to act actually arrives, would carrying out that threat still serve the threatening player's own interests? If not, a sufficiently informed opponent has good reason to ignore it - and calling the bluff is often the rational move.

Key takeaways

A sequential game is one where players move in a defined order and later players observe earlier moves.
Backward induction solves these games by starting at the last decision and reasoning backward to the first.
A threat is only credible if it would still make sense to carry out once the moment to act actually arrives.
Backward induction often reveals that seemingly aggressive threats are empty, and rational opponents can see through them.
The outcome that survives this process is called a subgame perfect equilibrium, a stricter standard than an ordinary Nash equilibrium.