Reported at Citadel, Jane Street, Hudson River Trading
The puzzle
Four pounds sit on the table, three on Ann's side and one on Ben's. Ann can take her larger share and end the game, or pass. Each pass doubles the pot and hands the choice to the other player, who can take the larger share or pass again. After six rounds the game ends with a pot of 256. What does game theory predict?
Short answer: Backward induction unravels every pass: the last mover would take, so the one before takes first, all the way back, and Ann takes the 3 on move one.
Step 1: Start at the final decision
At the last decision, the player to move chooses between taking the larger share now and passing, which ends the game with the smaller share. The larger share is more money, so a player who cares only about money takes.
Step back one round. The player there knows that if she passes, the next player will take and leave her the smaller share of a doubled pot. Compare that smaller share with the larger share available right now. In the standard payoffs the larger share now is worth more, so she takes too.
Step 2: The whole ladder collapses
Each rung falls the same way. Whoever is about to move knows the next player will take, so passing hands away the larger share. The only subgame-perfect outcome is that Ann takes immediately and pockets 3 pounds.
The cost is enormous: 253 pounds of surplus that both players could have shared is never created. Compare it with the prisoner's dilemma, where the loss comes from a dominant strategy. Here no strategy dominates; the loss comes from a chain of reasoning six steps long.
Step 3: What the unravelling depends on
The argument needs common knowledge of rationality. Ann must know Ben is rational, know that Ben knows she is, and so on six levels deep. Remove even a little of that certainty and the logic weakens. If Ann thinks there is a small chance Ben is the cooperative type who will pass, passing early becomes a sensible gamble, because the pot is growing exponentially.
In experiments, people pass for several rounds and usually take somewhere in the middle. Doubt about the other player, or a willingness to gamble on cooperation, is enough to keep the pot growing for a while.
Step 4: Why trading firms like this one
It is a test of whether you treat a model as a tool or as a fact. The confident candidate states the unravelling argument cleanly, then explains which assumption real players violate and how much money that violation is worth. That is exactly how a trader should treat any pricing model.
Why interviewers ask it
State the induction with conviction, then say what breaks it. The follow-up is often: how much uncertainty about the other player would make passing on move one rational?
Knowing the proof is half of it. Work Centipede Game interactively, then take it under a clock with the rest of the Outcry simulators.