Package {TheOrdinals}

TheOrdinals: Aggregation and Consensus Methods for Preference-Approvals

Description

Tools for aggregating ordinal preference data into a group consensus. The current release implements DIVA (Divide and Conquer for Preference-Approvals), a distance-based aggregation method for preference-approvals. The package is organised so that further methods for ordinal preference data can be added in future releases.

Preference-approvals

A preference-approval is a pair (\pi, A) where \pi is a (weak) ranking of n alternatives and A is the subset of approved alternatives, subject to a consistency condition linking the two components (Definition 1 of Albano and Romano, 2026). Throughout the package a set of m preference-approvals is stored as a numeric matrix with 2n columns: the first n columns hold the ranking (positions, with ties allowed) and the last n columns hold the approval indicators (1 approved, 0 not approved).

Main functions

diva

Compute the DIVA consensus preference-approval.

diva_sensitivity

Average consensus distance over a grid of the weighting parameter lambda.

pref_dist

Distance between preference-approvals.

find_approval

Admissible approvals of a ranking.

pa_universe

Universe of preference-approvals on n alternatives.

is_consistent

Check the preference-approval consistency condition.

Author(s)

Maintainer: Maurizio Romano romano.maurizio@unica.it

Authors:

• Alessandro Albano alessandro.albano@unipa.it

References

Albano, A. and Romano, M. (2026). A distance-based aggregation method for finding consensus in preference-approvals. Advances in Data Analysis and Classification. doi:10.1007/s11634-025-00663-4

Erdamar, B., Garcia-Lapresta, J. L., Perez-Roman, D. and Sanver, M. R. (2014). Measuring consensus in a preference-approval context. Mathematical Social Sciences, 39-46. doi:10.1016/j.mathsocsci.2013.10.005

DIVA consensus for preference-approvals

Description

Computes the DIVA (Divide and Conquer for Preference-Approvals) consensus of a set of preference-approvals. The consensus is the preference-approval that minimises the average distance pref_dist to the set of voters, and it is guaranteed to be admissible (its ranking and approval components are consistent).

Usage

diva(x, search = FALSE, algorithm = "quick", lambda = 0.5, verbose = FALSE)

Arguments

Details

The algorithm first looks for a direct solution in which the consensus ranking and the consensus approval are already compatible. If none exists, it explores two branches: from the consensus ranking it derives admissible approvals (find_approval), and from the consensus approval it derives admissible rankings by solving the consensus separately within approved and unapproved alternatives. The branch yielding the smaller average distance is returned.

Value

An object of class "diva": a list with components

consensus

Data frame with the consensus preference-approval(s); a ranking component followed by an approval component. More than one row is returned when several solutions are tied.

d_lambda

The minimised average distance.

elapsed

A difftime with the computing time.

lambda, algorithm, search

The arguments used.

References

Albano, A. and Romano, M. (2026). A distance-based aggregation method for finding consensus in preference-approvals. Advances in Data Analysis and Classification. doi:10.1007/s11634-025-00663-4

See Also

diva_sensitivity, pref_dist

Examples

x <- rbind(
  c(1, 2, 3, 4, 1, 1, 0, 0),
  c(2, 1, 3, 4, 1, 0, 0, 0),
  c(1, 2, 4, 3, 1, 1, 0, 0),
  c(1, 3, 2, 4, 1, 1, 1, 0)
)
res <- diva(x, algorithm = "quick")
res
res$d_lambda

Sensitivity of the DIVA consensus to the weighting parameter

Description

Computes the minimised average distance of the DIVA consensus over a grid of values of the weighting parameter lambda, which is useful to assess how the relative weight of the ranking and approval components affects the consensus. Unlike diva, only the achieved distance is returned, not the consensus preference-approval itself.

Usage

diva_sensitivity(
  x,
  search = FALSE,
  algorithm = "quick",
  lambda = seq(0, 1, by = 0.1),
  verbose = FALSE
)

Arguments

Value

A list with components d_lambda (named numeric vector of the minimised average distances, one per value of lambda) and lambda (the grid used).

See Also

diva

Examples

x <- rbind(
  c(1, 2, 3, 4, 1, 1, 0, 0),
  c(2, 1, 3, 4, 1, 0, 0, 0),
  c(1, 2, 4, 3, 1, 1, 0, 0),
  c(1, 3, 2, 4, 1, 1, 1, 0)
)
s <- diva_sensitivity(x)
s$d_lambda

Admissible approvals of a ranking

Description

Given a (weak) ranking, enumerates the approval vectors that are consistent with it, i.e. the approval boundaries that can be placed between consecutive buckets of the ranking. The result combines the ranking with each admissible approval, so that every row is a valid preference-approval.

Usage

find_approval(ranking)

Arguments

Value

A data frame with 2n columns and (buckets + 1) rows, where buckets is the number of distinct positions in ranking. The first n columns repeat the ranking and the last n columns hold the admissible approval vectors (from "approve none" to "approve all").

See Also

pa_universe, is_consistent

Examples

find_approval(c(1, 2, 2, 3))

Formula 1 World Championship (1950) preference-approvals

Description

Preference-approval data built from the 1950 Formula 1 World Championship: the classification of each of the 7 Grands Prix is read as a ranking of the 81 drivers, and the top five finishers of each race are taken as approved. Used as a real-data application in Albano and Romano (2026).

Usage

formula1_1950

Format

A numeric matrix with 7 rows (Grands Prix) and 162 columns. The first 81 columns are the ranking component and the last 81 columns the approval component (1 approved, 0 not). Both blocks are named after the drivers.

Source

PrefLib (https://preflib.github.io/PrefLib-Jekyll/); processed in the replication material of Albano and Romano (2026).

References

Albano, A. and Romano, M. (2026). A distance-based aggregation method for finding consensus in preference-approvals. Advances in Data Analysis and Classification. doi:10.1007/s11634-025-00663-4

Examples

data(formula1_1950)
dim(formula1_1950)

French Presidential Election (2002) preference-approvals

Description

Preference-approval data on the 15 candidates of the first round of the 2002 French presidential election, used as a real-data application in Albano and Romano (2026). Each voter provides a ranking of the candidates together with an approval of a subset of them. Only the admissible preference-approvals (those whose ranking and approval are concordant) are retained.

Usage

french_election_2002

Format

A numeric matrix with one row per voter and 30 columns. The first 15 columns are the ranking component (positions, ties allowed) and the last 15 columns the approval component (1 approved, 0 not). Both blocks of columns are named after the candidates' surnames.

Source

Erdamar et al. (2014) and the replication material of Albano and Romano (2026); original survey data file 00029-00000001.dat.

References

Albano, A. and Romano, M. (2026). A distance-based aggregation method for finding consensus in preference-approvals. Advances in Data Analysis and Classification. doi:10.1007/s11634-025-00663-4

Examples

data(french_election_2002)
dim(french_election_2002)

diva(french_election_2002, algorithm = "quick")

Preference-approval consistency check

Description

Checks whether a ranking and an approval vector form an admissible preference-approval, i.e. whether they satisfy the consistency condition of Definition 1 in Albano and Romano (2026): approved alternatives must be ranked above unapproved ones, and tied alternatives must share the same approval status.

Usage

is_consistent(ranking, approval)

Arguments

Value

A single logical value: TRUE if the pair is a valid preference-approval, FALSE otherwise.

References

Albano, A. and Romano, M. (2026). A distance-based aggregation method for finding consensus in preference-approvals. Advances in Data Analysis and Classification. doi:10.1007/s11634-025-00663-4

Examples

is_consistent(c(1, 2, 3, 4), c(1, 1, 0, 0))
is_consistent(c(1, 2, 3, 4), c(0, 1, 0, 0))

Small synthetic universe of preference-approvals (n = 4)

Description

The complete universe of preference-approvals on four alternatives, generated with pa_universe. Provided as a small, fast example for the documentation and the tests.

Usage

pa_small

Format

A numeric matrix with 8 columns. The first 4 columns are the ranking component and the last 4 columns the approval component.

See Also

pa_universe

Examples

data(pa_small)
head(pa_small)

Universe of preference-approvals

Description

Generates the full set (universe) of preference-approvals on n alternatives, by enumerating every weak order on n items (via univranks) and, for each, all the admissible approval boundaries (via find_approval).

Usage

pa_universe(n, verbose = FALSE)

Arguments

Value

A data frame with 2n columns, one row per preference-approval. The first n columns are the ranking component and the last n the approval component.

See Also

find_approval, diva

Examples

u <- pa_universe(3)
head(u)
nrow(u)

Distance between preference-approvals

Description

Computes the family of distances d_\lambda for preference-approvals introduced by Erdamar et al. (2014), a convex combination of the (normalised) Kemeny distance on the ranking component and the (normalised) Hamming distance on the approval component.

Usage

pref_dist(x, y = NULL, lambda = 0.5)

Arguments

Value

A numeric matrix of distances. When y is NULL it is the nrow(x) by nrow(x) matrix of pairwise distances; otherwise it is the nrow(x) by nrow(y) matrix of distances between the rows of x and the rows of y.

References

Erdamar, B., Garcia-Lapresta, J. L., Perez-Roman, D. and Sanver, M. R. (2014). Measuring consensus in a preference-approval context. Mathematical Social Sciences, 39-46. doi:10.1016/j.mathsocsci.2013.10.005

See Also

diva

Examples

x <- rbind(
  c(1, 2, 3, 4, 1, 1, 0, 0),
  c(2, 1, 3, 4, 1, 0, 0, 0),
  c(1, 2, 4, 3, 1, 1, 0, 0)
)
pref_dist(x)
pref_dist(x, lambda = 0.8)