# The Penny Game

Two robots are playing a game. Between them is a pile of coins. Each robot, on its turn, can take either one or two coins from the pile. So long as each elects to take one coin, play continues until the pile is exhausted. If either elects to take two, the remaining coins vanish and the game ends.

One might think that the best plan would be always to take a single coin, but if both players are rational and know it, the first player will immediately take two pennies and end the game.

He reasons thus: If there were only two pennies in the pile, I’d benefit most by taking both of them rather than just one. Now suppose there were three pennies. If I took only one, then I would leave my opponent in the position I just imagined, and being rational he’d take both remaining pennies. Therefore I should take two of the three.

And so on backward, up to any arbitrary number of pennies. Paradoxically, it seems, improvident greed is more rational than constructive cooperation. Adapted from Hollis, Martin and Sugden, Robert (1993) “Rationality in action.” Mind 103:1-35, referenced in R.M. Sainsbury, Paradoxes, 2009.

See Tug of War.