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The game must satisfy the following criteria: there are two players in the game; the game is of perfect information; the board game is finite; the two players can take alternate turns; and there is no chance element present. Zermelo has stated that there are many games of this type; however his theorem has been applied mostly to the game chess.
Constant sum: A game is a constant sum game if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.
A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly.This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory or computer assistance.
Chess initial position. The game of chess is commonly divided into three phases: the opening, middlegame, and endgame. [1] There is a large body of theory regarding how the game should be played in each of these phases, especially the opening and endgame.
A variant first described by Claude Shannon provides an argument about the game-theoretic value of chess: he proposes allowing the move of “pass”. In this variant, it is provable with a strategy stealing argument that the first player has at least a draw thus: if the first player has a winning move in the initial position, let him play it, else pass.
Sequential games are a type of dynamic game, a broader category where decisions occur over time (e.g., differential games), but they specifically emphasize a clear order of moves with known prior actions. Because later players know what earlier players did, the order of moves shapes strategy through information rather than timing alone.
An example of a retrograde analysis problem is shown on the left. The solver must deduce White's last move. It is not immediately apparent how the white king could have moved, since every adjacent square puts White in a seemingly impossible double check; on further examination it becomes apparent that if the white king moved from f5, then Black could have delivered the double check by playing ...
Loyd had a friend who was willing to wager that he could always find the piece which delivered the principal mate of a chess problem. Loyd composed this problem as a joke and bet his friend dinner that he could not pick a piece that didn't give mate in the main line (his friend immediately identified the pawn on b2 as being the least likely to deliver mate), and when the problem was published ...