A pot sits between two players and grows every time it is passed. On your turn you can take the bigger share and end the game, or pass and let it grow.
Pass long enough and both of you end up with more than anyone could take at the start. But whoever passes last gives the other player the chance to take.
You are Player 1
Play down the tree. The other player decides at random here, like a coin toss.
Solve it from the end
Start at the last move and ask what the player there would do, then step back one move at a time. That is backward induction.
What it teaches
Backward induction can unravel a long and mutually profitable sequence at its very first move, and the longer the sequence the more obviously wrong the prediction looks.
The usual mistakeAccepting the backward induction and then concluding that the people who keep passing in experiments are simply bad at the game, rather than asking what assumption the argument needs beyond each player being rational.
The checkThe induction needs rationality to be common knowledge all the way down. The moment you think the other player might pass, passing can be your best reply, and the chain breaks at its first link.
What people actually do
Nobody takes at the first node
The theory says
In a finite centipede game with complete information and common knowledge of rationality, backward induction from the last node unravels the whole tree, so the first mover takes immediately and the players forgo almost all of the available surplus.
People actually
Taking at the first node happens in a small minority of plays, usually under a tenth, and most pairs pass for several nodes before someone takes. The rate of taking rises steeply towards the end of the tree, so the shape of the prediction is visible even though its starting point is not.
Why is still argued over.
In the real world
Fiction: Death Note
Light and L inside the task force
Light joins the investigation that is hunting him and works alongside the detective who suspects him, each of them aware of what the arrangement probably is. Every day it continues hands both of them information they could not get otherwise, and either could end it whenever he chose. Both keep passing for far longer than an immediate strike would suggest, because at each point the next exchange is worth more than stopping, and the sequence runs on until one of them finally takes.
The moves alternate, either player can end the game outright, and the prize grows the longer both decline to, so the immediate strike that backward induction recommends at the very first node is exactly what neither does, which is the centipede's lesson and not the simultaneous choice of a dilemma. The question is whether to continue at all rather than when to fire, which is what separates this from reading the same pair as a duel.
Real worldContested
Staged reciprocity in the Iran nuclear agreement
The agreement reached in 2015 was built as a sequence: verified Iranian steps on enrichment and inspection matched by staged relief from sanctions, with nothing about it enforceable against either side. It ran for roughly three years, and what had accumulated by then, inspections, trade and relief, was worth more to both parties than anything either had at the start. The United States withdrew in 2018 and reimposed sanctions, and Iran then reduced its own compliance in announced stages. The withdrawing administration's revealed preference looks like the value of stopping the sequence outright rather than the value of the next exchange in it, which is the move the model says is always available and always tempting.
The moves alternate and either side's defection ends the whole sequence rather than opening a fresh round, while the value of continuing grows at every step and the immediate gain from stopping first never goes away, which is the centipede rather than a dilemma played repeatedly.
Next door on the wall
DuelWaiting here is costly through a rising risk rather than a growing pot. Not on the wall yet.