The penny partition paradox

The penny partition paradox, a perplexing parable of profit—the penny players parlay perfectly precise provable principles to a peculiarly puzzling paradoxical payout.

Allow me kindly to relate the parable of the penny partition paradox. In an earlier era, the days of penny candy and pennies saved and pennies earned, days when pennies were held in a somewhat higher regard than now, our protagonist pair Percival and Penelope enter the Penny Pavilion as distinguished invited participants in the grand penny partition. They are seated before a mountainous pile of pennies overflowing the small table now buried beneath the hoard. There are exactly one million dollars worth of pennies, which Percival and Penelope shall divide amongst themselves according to the rigid rules of the long-established penny-partition procedure. The money is available to them, theirs for the taking more or less equally to within a penny or two if they follow the rules. Each of them might hope to take home half a million dollars, a vast sum.

According to the penny-partition procedure, the penny players will alternately take turns, choosing on each turn either (1) to take one penny and continue the game; or (2) to take two pennies and end the game. If either player elects for two pennies on their turn, the process stops completely and the game is finished—no more pennies will be dispensed to either player.

Let us imagine that Percival and Penelope have what is in their day an ordinary attitude toward pennies, namely, a penny is not worth very much—just a penny—but definitely not nothing. Each extra penny is valued to some small but nonzero extent, and in any given situation, all else being equal, they'd rather have one more penny, if possible, than none. Naturally, they will each be happy to see the other person become rich along with themselves, but they aren't especially concerned about how many pennies the other person will or will not get; rather, they each just want to maximize their own take. And neither are they prone to envy; it will be fine with either of them to get fewer pennies than their partner, as long as they have maximized what is possible for themselves. In any case, because of the nature of the process, the difference between them will be at most two pennies, and so regardless of the outcome there will not be much reason for envy. In keeping with the “game” aspect of the process, there is no special concern to divide the pennies exactly equally—they will in any case get approximately the same number of pennies. Let me also state categorically that both Percival and Penelope are perfectly logical and that all this is fully known to them both in a state of common knowledge.

The penny partition process begins, with Penelope having the first move. What do you expect will happen?

Interlude