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#' @title Constrained egalitarian rule
#' @description This function returns the awards vector assigned by the constrained egalitarian rule (CE) rule to a claims problem.
#' @param E The endowment.
#' @param d The vector of claims.
#' @param name A logical value.
#' @return The awards vector selected by the CE rule. If name = TRUE, the name of the function (CE) as a character string.
#' @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.
#'
#' Assume that the claims are ordered from small to large, \eqn{0 \le d_1 \le...\le d_n}{%
#' 0 \le d1 \le...\le dn}.
#' The constrained egalitarian rule coincides with the constrained equal awards rule (CEA) applied to
#' the problem \eqn{(E, d/2)} if the endowment is less or equal than the half-sum of the claims \eqn{D/2}.
#' Otherwise, any additional unit is assigned to claimant \eqn{1} until she/he receives the minimum
#' of the claim and half of \eqn{d_2}{%
#' d2}. If this minimun is \eqn{d_1}{%
#' d1}, she/he stops there. If it is not, the next increment is
#' divided equally between claimants \eqn{1} and \eqn{2} until claimant \eqn{1} receives \eqn{d_1}{%
#' d1} (in this case she drops out) or they reach \eqn{d_3/2}{%
#' d3/2}.
#' If claimant \eqn{1} leaves, claimant \eqn{2} receives any aditional increment until she/he reaches \eqn{d_2}{%
#' d2} or \eqn{d_3/2}{%
#' d3/2}. In the case that claimant \eqn{1} and \eqn{2} reach \eqn{d_3/2}{%
#' d3/2}, any additional unit is divided between claimants \eqn{1}, \eqn{2}, and \eqn{3} until the first one receives \eqn{d_1}{%
#' d1} or they reach \eqn{d_4/2}{%
#' d4/2}, and so on.
#'
#' Therefore:
#'
#' If \eqn{E \le D/2} then \eqn{CE(E,d) = CEA(E,d/2)=(\min\{\frac{d_i}{2},\lambda\})_{i\in N}}{CE(E,d)=CEA(E,d/2)=(min\{di/2,\lambda\})} where \eqn{\lambda \ge 0} is chosen so as to achieve balance.
#'
#' If \eqn{E \ge D/2} then the CE rule assigns to claimant \eqn{i} the maximum of two quantities: the half-claim and the minimum of the claim and a value \eqn{\lambda \ge 0}
#' chosen so as to achieve balance.
#' \deqn{CE_i(E,d)=\max\{\frac{d_i}{2},\min\{d_i,\lambda\}\},\ i=1,\dots,n, \ where \ \sum_{i=1}^{n} CE_i(E,d)=E.}{CE(E,d) = (max\{ di/2 , min\{di,\lambda\} \}).}
#' @seealso \link{allrules}, \link{CEA}, \link{Talmud}, \link{PIN}
#' @examples
#' E=10
#' d=c(2,4,7,8)
#' CE(E,d)
#' @references Chun, Y., Schummer, J., Thomson, W. (2001). Constrained egalitarianism: a new solution for claims problems. Seoul J. Economics 14, 269–297.
#' @references Thomson, W. (2019). How to divide when there isn't enough. From Aristotle, the Talmud, and Maimonides to the axiomatics of resource allocation. Cambridge University Press.
#' @export
CE = function(E, d, name = FALSE) {
if (name == TRUE) {
rule = "CE"
return(rule)
}
########################################
# 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)
###################
# Claims in ascending order
do = sort(d, index.return = T)$x
ordenI = sort(d, index.return = T)$ix
orden = sort(ordenI, index.return = T)$ix
dnull = sum(do == 0)
##################
# Trivial cases:
if (E == 0) {
#Null endowment
rule = rep(0, length(d))
return(rule)
} else if (E == D) {
# Endowment equal to the sum of the claims
rule = d
ruleNull = rep(0, dnull)
return(rule)
} else if (dnull > 0) {
#Some claims (but not all) are zero
do = do[(dnull + 1):n]
ruleNull = rep(0, dnull)
} else{
ruleNull = c()
}
############ THE CE RULE #########
n = length(do)
if (E <= D / 2) {
#Lower-half domain
rule = CEA(E, do / 2)
} else{
# Higher-half domain
rule = do / 2
# Each claimant gets at least di/2
Ns = 0
while (sum(rule) < E) {
R = E - sum(rule)#Extra units
pendientes1 = which(rule != do)#The claimants with award is not equal to the claim
pendientes = which(rule[pendientes1] == min(rule[pendientes1])) +
Ns #The first claimant
#with award is not equal to the claim
Requitativo = R / (length(pendientes))
#The equitable quantity to give to each claimant
for (j in (sort(pendientes, decreasing = TRUE))) {
if (j < n) {
rule[j] = min(min(do[j], rule[j] + Requitativo), rule[j + 1])#We give to each claimant the equitative award
} else {
rule[j] = min(do[j], rule[j] + Requitativo)
}
}
saturados = as.numeric(do == rule)
Ns = sum(saturados)
}
}
#######ADDING the null claimants and REORDERING #####
rule = c(ruleNull, rule)
#Adding the null claimants
rule = rule[orden]#Reordering the claimants
return(rule)
}
```

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