pairwise: Pairwise comparisons

In hyper2: The Hyperdirichlet Distribution, Mark 2

Pairwise comparisons

Description

Convenience functions to generate likelihoods for pairwise comparisons.

Usage

pairwise(M)
zermelo(M, maxit = 100, start, tol = 0, give = FALSE)
home_away(home_games_won, away_games_won, monster="home")
home_away3(home_games_won, away_games_won, lambda)
home_away_table(a, give=FALSE, teams)
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	In zermelo(), Boolean with default FALSE meaning to return the evaluate and TRUE meaning to return all iterations; in home_away_table() governs output form
home_games_won, away_games_won	Matrices showing home games won and away games won
monster	The monster, with default home representing the home ground advantage
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
a, teams	In function home_away_table(), argument a is a data frame (typically of football results), give a boolean governing output form, and teams a list of football teams. See details

Details

Function pairwise(M) takes a M, a matrix of pairwise comparisons and returns a hyper2 likelihood function. If there are only two competitors, so M is 2-by-2, it might be better to use dirichlet() or race() with 2 players, or one of the pick() quartet.

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 monster [or indeed any systematic advantage]. Set the monster to NULL if the home advantage is not present. 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 home_away_table() takes a dataframe of results (each row being a single match) and returns a table amenable to analysis by home_away() or home_away3(). If give takes its default value of FALSE, draws are discarded and a complex matrix of wins and losses is returned. Files inst/monster_vs_lambda.Rmd and inst/home_advantage.Rmd show some use-cases. Argument teams is a character vector that specifies the teams to be tabulated (useful if one wishes to change the default ordering of the teams).

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\times n\times 3, where n is the number of players. It is used in inst/kka.Rmd to create chess3 likelihood function.

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.

Function zermelo() implements a standard iterative procedure for maximization of pairwise Bradley-Terry likelihoods (such as those produced by function pairwise()). It takes a matrix of pairwise comparisons; the diagonal is disregarded. The process keeps going until the difference between successive estimates does not exceed argumet tol; note that tol=0 is acceptable. If give=TRUE, a matrix with rows corresponding to successive approximations is returned; if so, argument maxit corresponds to the maximum number of rows.

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.

Experimental function pair3() is now removed as dirichlet3() is more general and has nicer idiom; pair3(a=4, b=3, lambda=1.88) and dirichlet3(c(a=4, b=3), 1.88) give identical output.

Author(s)

Robin K. S. Hankin

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 ranking top 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 order effects”. Biometrics, 33:693–702

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_away(M2, monster = NULL) # nullify the home ground monster
# 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)

hyper2 documentation built on June 23, 2026, 5:07 p.m.