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#' @title Lorenz-dominance relation
#' @description This function checks whether or not the awards assigned by two rules to a claims problem are Lorenz-comparable.
#' @param E The endowment.
#' @param d The vector of claims.
#' @param Rules The two rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
#' @param Info A logical value.
#' @return If Info = FALSE, the Lorenz-dominance relation between the awards vectors selected by both rules.
#' If both awards vectors are equal then cod = 2. If the awards vectors are not Lorenz-comparable then cod = 0.
#' If the awards vector selected by the first rule Lorenz-dominates the awards vector selected by the second rule then cod = 1; otherwise cod = -1.
#' If Info = TRUE, it also gives the corresponding cumulative sums.
#' @details Let \eqn{E\ge 0} be the endowment to be divided and \eqn{d\in \mathcal{R}^n}{d} the vector of claims
#' with \eqn{d\ge 0} and such that \eqn{\sum_{i=1}^{n} d_i\ge E,\;}{} the sum of claims exceeds the endowment.
#'
#' A vector \eqn{x=(x_1,\dots,x_n)}{x=(x1,...,xn)} is an awards vector for the claims problem \eqn{(E,d)} if \eqn{0\le x \le d}
#' and satisfies the balance requirement, that is, \eqn{\sum_{i=1}^{n}x_i=E}{x1+\dots+xn=E} the sum of its coordinates is equal to \eqn{E}.
#' Let \eqn{X(E,d)} be the set of awards vectors for \eqn{(E,d)}.
#'
#' Given a claims problem \eqn{(E,d)}, in order to compare a pair of awards vectors \eqn{x,y\in X(E,d)}{x,y in X(E,d)} with the Lorenz criterion,
#' first one has to rearrange the coordinates of each allocation in a non-decreasing order. Then we say that \eqn{x} Lorenz-dominates \eqn{y} (or, that \eqn{y} is Lorenz-dominated by \eqn{x})
#' if all the cumulative sums of the rearranged coordinates are greater with \eqn{x} than with \eqn{y}. That is,
#' \eqn{x} Lorenz-dominates \eqn{y} if for each \eqn{k=1,\dots,n-1} we have that
#' \deqn{\sum_{j=1}^{k}x_j \geq \sum_{j=1}^{k}y_j}{x1+\dots+xk \ge y1+\dots+yk.}
#'
#' Let \eqn{R} and \eqn{R'} be two rules. We say that \eqn{R} Lorenz-dominates \eqn{R'} if \eqn{R(E,d)} Lorenz-dominates \eqn{R'(E,d)} for all \eqn{(E,d)}.
#'
#' @seealso \link{cumawardscurve}, \link{deviationindex}, \link{indexgpath}, \link{lorenzcurve}, \link{giniindex}.
#' @examples
#' E=10
#' d=c(2,4,7,8)
#' Rules=c(AA,CEA)
#' lorenzdominance(E,d,Rules)
#' @references Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association, 9(70), 209-219.
#' @references Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. \doi{10.1007/s10058-022-00300-y}
#' @references Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.
#' @export
lorenzdominance = function(E, d, Rules, Info = FALSE) {
########################################
# Required: (E,d) must be a claims problem, i.e., E >=0, d >=0, E <= sum(d)
########################################
n = length (d)
D = sum(d) #The number of claims and the total claim
if (E < 0 || sum((d < 0)) > 0 || E > D)
stop('(E,d) is not a claims problem.',call.=F)
if(length(Rules)>2)
warning('We check if the first two rules are Lorenz-comparable. \n',call.=F,immediate.=T)
#####################
do = sort(d)
if (sum(do == d) < n){
message('The result is computed for the rearranged vector of claims.\n')
}
Rule1char = Rules[[1]](0, 0, name=TRUE)
Rule2char = Rules[[2]](0, 0, name=TRUE)
rule1 = Rules[[1]](E, do)
rule2 = Rules[[2]](E, do) #Award of Rule 1 and Rule2
sum1 = cumsum(rule1)
sum2 = cumsum(rule2)
dom1 = sum(sum1 >= sum2)
dom2 = sum(sum2 >= sum1)
if (dom1 == n & dom2 < n) {
cod=1
if (Info == TRUE){
message('The awards vector selected by the ',
Rule1char,
' rule Lorenz dominates the awards vector selected by the ',
Rule2char,
' rule.\n')
}
} else if (dom2 == n & dom1 < n) {
cod=-1
if (Info == TRUE){
message('The awards vector selected by the ',
Rule1char,
' rule is Lorenz-dominated by the awards vector selected by the ',
Rule2char,
' rule. \n')
}
} else if (dom1 == n & dom2 == n) {
cod=2
if (Info == TRUE){
message('The awards vectors selected by the ',
Rule1char,
' and the ',
Rule2char,
' rules are equal.\n')
}
}else{
cod=0
if (Info == TRUE){
message('The awards vectors selected by the',
Rule1char,
'and the',
Rule2char,
'rules are not Lorenz-comparable.\n')
}
}
if (Info == TRUE){
message('The cumulative sums for both awards vectors are:\n')
message(paste(
format(Rule1char, width = 17, justify = c("right")),
format(Rule2char, width = 17, justify = c("right")),
'\n'
))
for (i in 1:n) {
message(paste(
format(
round(sum1[i], 3),
nsmall = 3,
width = 17,
justify = c("right")
),
format(
round(sum2[i], 3),
nsmall = 3,
width = 17,
justify = c("right")
),
'\n'
))
}
}
return(list(cod=cod))
}
```

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