pairwise function

Pairwise comparisons

Function pairwise() takes a matrix of pairwise comparisons and returns a hyper2 likelihood function. Function zermelo() gives a standard iterative procedure for likelihood maximization of pairwise Bradley-Terry likelihoods (such as those produced by function pairwise()).

Function home_away() takes two matrices, one for home wins and one for away wins. It returns a hyper2 support function that includes a home advantage ghost. Function home_away3() is the same, but returns a hyper3 object. A complex matrix is interpreted as real parts being the home wins and imaginary parts away wins.

Function white_draw3() returns a hyper3 likelihood function for pairwise comparisons, one of whom has a home team-type advantage (white player in the case of chess). It is designed to work with an array of dimensions $n*n*3$, where $n$ is the number of players. It is used in inst/kka.Rmd to create chess3 likelihood function.

pairwise(M)
zermelo(M, maxit = 100, start, tol = 1e-10, give = FALSE)
home_away(home_games_won, away_games_won)
home_away3(home_games_won, away_games_won,lambda)
white_draw3(A,lambda,D)

Arguments

• M: Matrix of pairwise comparison results

• maxit: Maximum number of iterations

• start: Starting value for iteration; if missing, use equalp()

• tol: Numerical tolerance for stopping criterion

• give: Boolean with default FALSE meaning to return the evaluate and TRUE meaning to return all iterations

• home_games_won, away_games_won: Matrices showing home games won and away games won

• lambda: The home ground advantage (or white advantage in chess)

• D: Weight of draw

• A: Array of dimension n*n*3, with A[,,i]

  corresponding to white wins, white draws, and white losses for i=1,2,3. The canonical example would be kka_array, see inst/kka.Rmd for details

Details

In function zermelo(), the diagonal is disregarded.

If home_games_won is complex, then the real parts of the entries are interpreted as home games won, and the imaginary parts as away games won.

References

• D. R. Hunter 2004. MM algorithms for generalized Bradley-Terry models . The Annals of Statistics, volume 32, number 1, pages 384--406
• S. Borozki and others 2016. An application of incomplete pairwise comparison matrices for rankingtop tennis players . arXiv:1611.00538v1 10.1016/j.ejor.2015.06.069
• R. R. Davidson and R. J. Beaver 1977. On extending the Bradley-Terry model to incorporate within-pair ordereffects . Biometrics, 33:693--702

Author(s)

Robin K. S. Hankin

Note

An extended discussion of pairwise() is given in inst/zermelo.Rmd and also inst/karate.Rmd. Functions home_away() and home_away3() are described and used in inst/home_advantage.Rmd; see Davidson and Beaver 1977.

See Also

maxp

Examples

#Data is the top 5 players from Borozki's table 1

M <- matrix(c(
0,10,0, 2,5,
4, 0,0, 6,6,
0, 0,0,15,0,
0, 8,0, 0,7,
1 ,0,3, 0,0
),5,5,byrow=TRUE) 
players <-  c("Agassi","Becker","Borg","Connors","Courier")
dimnames(M) <- list(winner=players,loser=players)
M
# e.g. Agassi beats Becker 10 times and loses 4 times
pairwise(M)
zermelo(M)
# maxp(pairwise(M))  # should be identical (takes ~10s to run)

M2 <- matrix(c(NA,19+2i,17,11+2i,16+5i,NA,12+4i,12+6i,12+2i,19+10i,
NA,12+4i,11+2i,16+2i,11+7i,NA),4,4)
teams <- LETTERS[1:4]
dimnames(M2) <- list("@home" = teams,"@away"=teams)
home_away(M2)
# home_away3(M2,lambda=1.2)  # works but takes too long (~3s)
home_away3(M2[1:3,1:3],lambda=1.2) 

M <- kka_array[,,1] + 1i*kka_array[,,3] # ignore draws
home_away(M)
# home_away3(M,lambda=1.3)  # works but takes too long (~3s)

white_draw3(kka_array,1.88,1.11)