Ranking whist players

Abstract

We present methods for ranking players in whist tournaments and consider desirable properties of these methods. This work is analogous to the study of rankings for players in round-robin tournaments.

The semialgebraic geometry of saturated optimal designs for the Bradley–Terry model

“…Remark 1. When all probabilities 1) , the Bradley-Terry model can be described by algebraic equations. This means that all values of the p i j that arise for different values of π satisfy certain algebraic equations and, among the probability vectors, they are the only solutions to these equations.…”

“…It is customary to use ξ to refer to a design with weights w i j and slightly abuse notation with expressions like ξ ∈ ( m 2 ) . The information gained from one observation of Y (i, j) is encoded in the information matrix 1) , where…”

“…This implies that the designξ with equal weights 1 m−2 on E \ {1, 2} is optimal for the reduced model. Sinceλ i j = λ i j , we only have to show…”

“…We find these first. According to [23, Lemma 5.1.3], a saturated design has three positive weights whose values are all 1 3 , the remaining weights being zero. There are 6 3 = 20 possible saturated designs.…”

“…If one of the weights equals 1 3 , then (5-3) implies that another weight is zero, i.e., the design is saturated. Since the saturated cases have been dealt with in Theorem 12, we only look for solutions whose weights all lie in the open interval 0, 1 3 . There are solutions of (5-3) that satisfy this, for example, if the weights and corresponding intensities are equal.…”