A two dimensional measure of chess performance
The proposed two dimensional measure of chess performance has more precision and discrimination power than the scalar measure because it contains more information by virtue of the fact that it is true to the trinomial nature of chess game outcomes. Scalar measures such as the Elo rating system contain less information because they rely on scoring to reduce chess game outcomes from a trinomial to a binomial process. This simplification is achieved at a cost because scoring causes some chess game outcome information to be lost. The information loss occurs at the point of scoring. No amount of mathematical wizardry downstream can recover the lost information.
The reduction of chess game outcomes to a binomial variable by the use of scoring was likely a necessary sacrifice of information for the sake of computational simplicity at a time when computational machinery were costly and scarce. It is likely that this relationship between the value of information and the cost of computation no longer applies because of advances in computer technology.
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