Equiprobable and attacked-square chess entropies by phases, levels of play and game outcomes

Abstract

In any legal chess position, we define an attacked-square entropy [Formula: see text] for either Black or White pieces in terms of the square occupation probability [Formula: see text], where [Formula: see text] is the number of all possible movements to square [Formula: see text] (free or occupied by an opponent’s piece) and [Formula: see text] is the total mobility defined as the sum of all possible movements. Thus, each attacked square contributes to the entropy according to its received “firepower” concentration. A simpler nonsquare-dependant equiprobable entropy [Formula: see text] in terms of equal probabilities [Formula: see text] always yields [Formula: see text]. On average, the difference [Formula: see text] is very large in the Opening phase and [Formula: see text] decreases faster for lower ranked players after move 25. A major cause of the reduction of [Formula: see text] during a game is material loss, which is an irreversible process. By game outcome, gaps in average [Formula: see text] among winners, draws and losers are larger for Amateur players than for Elite players, both in the Middlegame and Endgame. Statistically, Elite players exhibit narrower dispersions in [Formula: see text]. Also, the entropy rates of the Elite level fluctuate much less than the entropies of other levels. Density of attacks in the four-square central zone is very high in the Opening, specially for Elite players.