An infinite-horizon perfect-information "centipede" game is studied. The unique subgame-perfect Nash equilibrium (SPNE) in pure strategies has each player choose, at each opportunity, to terminate the game. In contrast, mixed strategies can yield equilibrium cooperation described as follows: for all but finitely many periods players randomize between terminating and continuing the game. These continuation probabilities are explicitly calculated. If both players' payoff functions are unbounded, then there exist SPNE exhibiting such cooperation. Conditions are also derived under which such cooperation does not arise.