A pot sits between two players and grows every time somebody leaves it alone. At each turn you can take it and end things with the larger share, or pass and hand the decision over. A short, valid argument says take it immediately and walk away with almost nothing. Almost nobody does, and the people who ignore it end up far richer.
Look at the last node. Whoever moves there takes, because taking beats letting the game end with the smaller share. Now look at the node before it. That player knows what happens next, so they take. Now the one before that. The reasoning unwinds all the way to the opening move.
Every step is valid. The conclusion is that a rational player, facing another rational player, grabs a tiny pot at the first opportunity while an enormous one sits one move away. Rosenthal (1981) built the game to make that as vivid as possible.
What is disputed is not the arithmetic. It is whether the reasoning survives contact with the situation it describes. Aumann proved that common knowledge of rationality entails backward induction. Binmore's reply is that if your opponent has already passed, you have evidence against the premise, so the theorem tells you what would follow in a situation you are no longer in.
The centipede is not a puzzle about greed. It is a stress test for the idea that rationality is something you can have common knowledge of.
Aumann (1995) proved it: in a game of perfect information, common knowledge of rationality entails the backward induction outcome. The proof is not in dispute. If both players are rational, and both know it, and both know that they both know it, all the way up, then the first player takes at once.
Binmore's reply is about what happens when the premise fails. Suppose your opponent passes. Backward induction says they should not have. So either they are not rational, or they do not think you are, or the common knowledge assumption was never true. Whichever it is, you are now in a situation the theorem does not describe, and continuing to reason as though it did is itself a strange thing to do.
The argument tells you what to do in a situation that your opponent's last move proved you are not in.
Pettit and Sugden (1989) pushed the same point: the reasoning requires you to hold fixed an assumption about your opponent while simultaneously working through what happens when they violate it. It is not obvious that is coherent.
Nothing here reasons at all. There is no common knowledge to have, no beliefs about anyone's rationality, and no backward induction. Plans that pay better become more common, and that is the whole mechanism.
What comes out is that the population passes well past the first node most of the time, and that the length of the chain barely matters. What matters is the ratio between what the pot gains by being passed and what you lose by being taken from. When passing pays well, populations pass. When it does not, they unravel exactly as predicted. The argument's conclusion turns out to track a payoff ratio rather than a chain of reasoning, which is a strange result for an argument whose whole content is the chain.
Real subjects do not play a fixed plan; they respond to what happened earlier in the game, and McKelvey and Palfrey's (1992) explanation turns on a small chance that your opponent is an altruist, which makes it worth building a reputation for passing. None of that is here. This shows only that a population with no beliefs at all lands somewhere the belief-based argument says it should not.