One of my favorite game theoretic insights is that rational players should not play chess: The outcome is already determined at the beginning of the game. This is what’s usually called Zermelo’s Theorem, but it is a common rant among game theorists that this is a misattribute. Zamir, Maschler and Solan, for example, in their recent game theory textbook credit the theorem to von-Neumann, citing his minimax paper. Following a conversation with David Schmeidler, who suggested that Zermelo might have used this theorem but didn’t make a fuss over it because it’s so trivial, I decided to look at Zermelo’s paper myself. This is possible thanks to translation from German by Ulrich Schwalbe and Paul Walker.

A game in Zermelo’s setup is given by a finite set {P} of positions. A position in chess includes the configuration of the board, whether White or Black has to move, whether there was castling, etc.. A history (Zermelo called it an endgame) is a sequence of positions that can follow each other in accordance with the rule of the games. An history can be finite, ending with a position which is win or loss to one of the players, but can also be infinite, in which case the outcome of the game is a draw. (In a game like chess there is also the possibility of a finite history that ends with a stalemate. This issue has no independent interest so, as mathematicians will do, we ignore it.)

What is usually called Zermelo’s Theorem in the literature now becomes:

Theorem 1 Let {q} be a position. If there is a number {N} such that the length of every history starting with {q} is smaller than {N}, then one and only one of the following holds:

  • White can force a win for White.
  • Black can force a win for Black.

  • Both players can force at least a draw.

    • I (and Zermelo) do not explicitly define strategies, but `force’ means what you think it means: That the player has a strategy such that every history which is consistent with this strategy achieves the desired outcome.

    Zermelo’s theorem is true, though I hasten to add that the assumption about the existence of such a number {N}  is unnecessary. However, this theorem is not the subject matter of Zermelo’s paper. The question that Zermelo asks is this:

    for some natural number {r}, what properties does a position q has such that White, independently of how the Black plays, can force a win in at most r moves?

    (Apparently Zermelo was motivated by the kind of `chess problem’ which bound the number of moves until a checkmate)

    So, Zermelo clearly did not set out to prove his eponymous theorem. But was he aware of it ? I guess the answer depends on what you mean by being aware of a mathematical statement (were you aware of the fact that any even number greater than 2 can be written as a sum of a prime number and an even number before you read this sentence ?). But I do believe that some of the logical implications in Zermelo’s paper only make sense if you already assume his theorem. Here is one such implication:

    The numbers {s} for which {\bar V_s(q)\neq\emptyset} are either infinite or {\leq \tau}, since the opponent, if he can win at all, must be able to force a win in at most {\tau} moves

    Here {\bar V_s(q)} are some sets histories of length {s} and Zermelo shows that {\bar V_s(q)\neq\emptyset} iff White can hold out until the {s}-th move without losing. The argument makes sense only if you assume that {\bar V_s(q) = \emptyset} (which, according to what Zermelo when White cannot postpone losing in {s} moves) implies Black can force winning in {s} moves, which is the essence of the theorem.

    My conclusion ? I believe that Zermelo and his readers were taking what is now called Zermelo’s Theorem for granted, though they didn’t recognize its significance yet.