A General Measure for the Relative Efficiency of Any Two Scoring Systems

Abstract

Miles (1984) developed a very elegant theory for the relative efficiency of different scoring systems at correctly identifying the better player, assuming points were independent. This earlier work was limited to those situations in which the underlying probability structures of the game being modelled had certain restrictive characteristics. Using those underlying characteristics it was possible to use interpolation methods to derive efficiency measures in a restricted number of practical situations. The major objective of this research was to investigate whether Miles’ work on the efficiency of scoring systems could be extended to more general situations. Games that do not possess the restrictive probability structures noted above have been considered, and it has been shown that an extrapolation method for deriving efficiency measures can be developed and applied. In doing so the efficiency of nested scoring systems has been studied. It turns out that this extrapolation method can be used in any scoring system situation, even where the outcome is win/draw/loss rather than win/loss. It produces exactly the same efficiency formula as that produced by the interpolated method. Thus, the method for measuring efficiency has been extended to a wider range of practical scoring systems situations.