Backward Induction in Sequential Games

Some games aren't simultaneous — players move one after another, each seeing what happened before. A negotiation with alternating offers, a multi-round auction, or a sequence of quote-and-respond trades all have this shape. Solving them by imagining "what would I do first" gets the causality backward: the earlier player's best move depends entirely on how the later player will respond, so you have to know the ending before you can correctly choose the beginning.

The idea, stated simply

Backward induction solves a sequential game by starting at the final decision point, determining the rational choice there (since there's nothing left to anticipate), then treating that choice as a known, fixed outcome when solving the second-to-last decision, and continuing backward to the very first move. Every player, reasoning this way, effectively plays the whole game "from the end," because a rational player only makes a move that looks good given what a rational opponent will actually do next — not what they hope the opponent does.

Worked example

Two traders alternately propose how to split a fixed $100 profit pool from a deal that both need to agree to. Trader A proposes first; if Trader B rejects, the pool shrinks by half (to $50) before Trader B gets to counter-propose; if Trader A rejects that, the pool shrinks by half again (to $25) and Trader A must then simply accept whatever split is left, or both walk away with nothing. Solve backward. Last round: the pool is $25, and Trader A must accept any nonzero offer rather than get $0 — so Trader B, moving last, offers Trader A the smallest possible positive amount, say $1, keeping $24 for themself. Second round: Trader B, deciding whether to reject Trader A's second-round offer, compares walking into the last round (where B knows they'd get $24) against accepting whatever A offers now — so B will accept anything giving at least $24. Knowing this, Trader A's best second-round offer is $26 for themself, $24 for B (the least A can give while still clearing B's bar), out of a $50 pool. First round: Trader A, deciding whether to reject B's hypothetical opening resistance, compares getting $26 in round two against whatever's offered now — so Trader A will accept any first-round offer giving at least $26. Trader B's optimal first offer is $74 for themself, $26 for Trader A, out of the full $100 pool. The whole first move — a 74/26 split favoring the first proposer — was only computable by first solving the round-three endgame.

What this means in practice

Backward induction is exactly how a rational negotiator sizes up any deadline-driven negotiation, and it's why the party who effectively moves "first" in a shrinking-pool setup often captures a disproportionate share — not through aggression, but because they can see the whole tree and know the other side's true walk-away point at every future stage. It's also the logic behind why credible deadlines and diminishing outside options shift bargaining power well before any actual offer is made.