Solution
The Centipede Game
Show the problem again
Two players alternate over 100 rounds. Each round, the active player either takes the larger share of a growing pot (ending the game) or passes, which doubles the pot. Passing when the opponent will take next round leaves you worse off than taking now. In the unique subgame perfect equilibrium, on which round does the game end?
Worked solution
The game ends immediately: the first player takes at the very first node. At the final node, the last mover takes. Anticipating that, the previous mover prefers taking one round earlier, and this logic propagates all the way back: every node's mover takes. The unique subgame perfect equilibrium ends the game immediately with tiny payoffs, despite enormous gains available from mutual cooperation: the canonical illustration of backward induction's counterintuitive force.
Source: Introduced by Robert Rosenthal (1981). Statement written for AxiomIQ.