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

In any legal chess position, we define an attacked-square entropy S for either Black or White pieces in terms of the square occupation probability pi=mi∕M, where m is the number of all possible movements to square i (free or occupied by an opponent’s piece) and M 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 Se in terms of equal probabilities pe=1∕M always yields Se−S≥0. On average, the difference Se−S is very large in the Opening phase and S decreases faster for lower ranked players after move 25. A major cause of the reduction of S during a game is material loss, which is an irreversible process. By game outcome, gaps in average 〈S〉 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 S. 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.