 Many of the issues raised by the fixed-length IPD can be raised in even starker form by a somewhat simpler game. Consider a PD in which they punishment payoff is zero. Now iterate the asynchronous version of this game a fixed number times. Imagine that both players are restricted to highly “punitive” strategies according to which, they must always defect against a player who has ever defected. (One important strategy of this variety is discussed below under the label GRIM.) The result is a centipede game. A particularly nice realization is given by Sobel 2005. A stack of \(n\) one-dollar bills lies on a table. Players take turns taking money from the stack, one or two bills per turn. The game ends when the stack runs out or one of the players takes two bills (whichever comes first). Both players keep what they have taken to that point. The extensive form of the game with for \(n=4\) is pictured below.