On Zermelo's theorem

A famous result in game theory known as Zermelo's theorem says that"in chess either White can force a win, or Black can force a win, or both sides can force at least a draw". The present paper extends this result to the class of all finite-stage two-player games of complete information with alternating moves. It is shown that in any such game either the first player has a winning strategy, or the second player has a winning strategy, or both have unbeatable strategies.


1.
In this note we generalize the following proposition usually referred to in the modern game-theoretic literature as Zermelo's theorem (see the paper [6] by Zermelo and its discussion and further references in [5]). Theorem 1. In chess either White has a winning strategy, or Black has a winning strategy, or both have strategies guaranteeing at least a draw.
The main result of this paper is as follows. Theorem 2. In any finite-stage two-player game with alternating moves, either (i) player 1 has a winning strategy, or (ii) player 2 has a winning strategy, or (iii) both have unbeatable strategies.
We emphasize that in this theorem we speak of any finite-stage twoplayer game with alternating moves, with any action sets and any real-valued payoff functions, so that the formulation of the result has a maximum level of generality.
The notions introduced are regarded in this work as primitive game solution concepts. We do not try to reduce them to the conventional ones: saddle point, Nash equilibrium, dominant strategy or their versions. This approach, contrasting with the common game-theoretic methodology, constitutes the main element of novelty in this paper. It enables us to obtain the above result in its most general and natural form.
2. In a finite-stage game with alternating moves, players 1 and 2 make moves (take actions) sequentially by selecting elements from two given action sets A and B. At stage 0 player 1 makes a move a 0 ; then player 2, having observed the player 1's move a 0 , makes a move b 0 = y 0 (a 0 ). At stage 1 player 1 makes a move a 1 = x 1 (a 0 , b 0 ) depending on the previous moves a 0 and b 0 ; then player 2 makes a move b 1 = y 1 (a 0 , b 0 , a 1 ), and so on. At stage t (t ≤ N) player 1 makes a move a t = x t (a t−1 , b t−1 ) depending on the sequences of the previous actions up to time t − 1, and then player 2 makes a move y t (a t , b t−1 ) depending on a t = (a 0 , a 1 , a 2 , ..., a t ) and b t−1 = (b 0 , b 1 , b 2 , ..., b t−1 ). The game terminates at stage N, where N is some given natural number.
A strategy of player 1 is a sequence should be made at stage t given the history (a t−1 , b t−1 ) of the previous moves of the players. To specify a strategy of player 2 one has to specify a sequence of functions should be made at stage t given the history (a t , b t−1 ) of the previous moves of the players.
The outcome of the game h(ξ, η) resulting from the application of the strategies ξ and η is described by the whole history of play N , b N ) = (a 0 , a 1 , a 2 , ..., a N , b 0 , b 1 , b 2 , ..., b N ).
In the course of the proof of Theorem 2, we will establish the existence of winning and unbeatable strategies having a special structure: basic strategies. We call a strategy basic if moves of the player using this strategy depend only on the previous moves of the rival (there is no need to memorize your own moves). Thus a basic strategy of player 1 is a sequence ξ = {x 0 , x 1 (b 0 ), x 2 (b 1 ), ..., x N (b N −1 )} and a basic strategy of player 2 is a sequence η = {y 0 (a 0 ), y 1 (a 1 ), y 2 (a 2 ), ..., y N (a N )}.
3. In chess, possible actions/moves of players 1 and 2 (White and Black) can be identified with positions on the board. When selecting a move, the player selects a new position. The payoffs, depending on the history of play, are defined as follows. If White wins then White gets 1 and Black 0; if Black wins then Black gets 1 and White 0; in case of a draw, both get 1/2. Illegitimate moves (or sequences of moves) lead, by definition, to a zero payoff for the corresponding player. Winning strategies defined above in the general context correspond to winning strategies in chess, and unbeatable ones to those strategies in chess which guarantee at least a draw.
It is assumed that chess is a finite-stage game, which is justified by the following argument. There is a finite number of chess-pieces and a finite number of squares on the board, hence there is a finite number of possible positions. The game automatically terminates as a draw if the same position occurs at least three times, with the same player having to go. Therefore the game cannot last more than N stages, where N is a sufficiently large number.
Proof. The first assertion follows from the fact that for every strategy profile (ξ, η) the outcome h(ξ, η) of the game coincides with h(ξ, b N ) where b N = (b 0 , ..., b N ) is the sequence of moves of player 2 generated by the strategy profile (ξ, η). The second assertion is a consequence of the fact that for every strategy profile (ξ, η) the outcome h(ξ, η) of the game coincides with h(a N , η) where a N = (a 0 , ..., a N ) is the sequence of moves of player 1 generated by the strategy profile (ξ, η).
Proposition 2. One of the assertions (i) -(iii) listed in Theorem 2 holds if and only if the following two conditions are satisfied: (I) Either player 1 has a winning strategy, or player 2 has an unbeatable strategy.
(II) Either player 2 has a winning strategy, or player 1 has an unbeatable strategy.
Proof. "If". Suppose (I) and (II) hold. Let us show that one of the assertions (i)-(iii) holds. If (i) is valid, the assertion is proved. Suppose (i) is not valid. Then by virtue of (I), player 2 has an unbeatable strategy. If (ii) holds, the assertion is proved. Suppose not only (i) but also (ii) fails to hold. Then by virtue of (I) and (II), (iii) holds.