Nothing
R/axioms.R
In ClaimsProblems: Analysis of Conflicting Claims
Defines functions
axioms
Documented in
axioms
#' @title Properties of the rules
#' @description This function shows which of the main properties the rules satisfy.
#' @param Rules The rules: AA, APRO, CE, CEA, CEL, AV, MO, PIN, PRO, RA, Talmud. By default, \code{Rules = "All"}.
#' @param Properties The properties listed in the description section. By default, \code{Properties = "All"}.
#' @return A table with the rules and the properties. If a rule satisfies a property it returns 1, and 0 otherwise. By default, it returns a table with all rules and properties.
#' @details Let \eqn{\mathcal{N}} be the set of all finite nonempty subsets of the natural numbers \eqn{\mathbb{N}}. Let \eqn{N=\{1,\ldots,n\}} be the set of claimants, \eqn{E\ge 0} the endowment to be divided and \eqn{d\in \mathbb{R}_+^N} the vector of claims
#' such that \eqn{D=\sum_{i \in N} d_i\ge E}. We denote the class of claims problems with set of
#' claimants \eqn{N} by \eqn{C^N}. Given \eqn{z\in\mathbb{R}^N}, \eqn{S\in 2^N} and \eqn{N'\subset N\in\mathcal{N}}, let \eqn{z(S)=\sum_{j\in S}z_j}, \eqn{z_{N'}=(z_i)_{i\in N'}} and denote \eqn{z_{-i}=z_{N\backslash \{i\}}\in \mathbb{R}^{N\backslash \{i\}}}. For simplicity, we will write \eqn{z=(z_{-i},z)}.
#'
#' The minimal right of claimant \eqn{i\in N} in \eqn{(E,d)} is defined by \eqn{m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\}.}
#' Let \eqn{m(E,d)=(m_1(E,d),\dots,m_n(E,d))} be the vector of minimal rights.
#'
#' The truncated claim of claimant \eqn{i\in N} in \eqn{(E,d)} is defined by
#' \eqn{t_i(E,d)=\min\{d_i,E\}.} Let \eqn{t(E,d)=(t_1(E,d),\dots,t_n(E,d))} be the vector of truncated claims.
#'
#' Given \eqn{(E,d)\in C^N} and \eqn{k\in\mathbb{N}}, we say that \eqn{(E',d')} is a \eqn{k}-replica of \eqn{(E,d)} if \eqn{E'=k E}, \eqn{N'\supset N}, \eqn{|N'|=k|N|}, and there is a partition \eqn{(N^i)_{i\in N}} of \eqn{N'} such that for each \eqn{i\in N} and each \eqn{j\in N^i}, \eqn{|N^i|=k} and \eqn{d'_j=d_i}.
#'
#' A vector \eqn{x=(x_1,\dots,x_n)} is an awards vector for the claims problem \eqn{(E,d)\in C^N} if \eqn{0\le x \le d}
#' and satisfies the balance requirement, that is, \eqn{x(N)=E}. Let \eqn{X(E,d)} be the set of awards vectors for the problem \eqn{(E,d)}.
#' A rule is a function that assigns to each claims problem \eqn{(E,d)} an awards vector.
#'
#' A rule \eqn{\mathcal{R}} satisfies:
#'
#' 1) \bold{Anonymity}, if for each \eqn{(E,d)\in C^N}, each bijection \eqn{f} from \eqn{N} into itself, and each \eqn{i\in N}, we have that \eqn{\mathcal{R}_i(E,d)=\mathcal{R}_{f(i)}(E,(d_{f(i)})_{i\in N})}.
#'
#' 2) \bold{Continuity}, if for each sequence \eqn{\{(E^{\ell},d^{\ell})\}} of elements of \eqn{C^N} and each \eqn{(E,d)\in C^N}, if \eqn{(E^{\ell},d^{\ell})\rightarrow (E,d)} then \eqn{\mathcal{R}(E^{\ell},d^{\ell})\rightarrow \mathcal{R}(E,d)}.
#'
#' 3) \bold{Homogeneity}, if for each \eqn{(E,d)\in C^N} and each \eqn{\rho>0}, then \eqn{\mathcal{R}(\rho E,\rho d)=\rho \mathcal{R}(E,d)}.
#'
#' 4) \eqn{\mathbf{\tfrac{1}{|N|}}}\bold{-truncated-claims lower bounds on awards}, if for each \eqn{(E,d)\in C^N}, then \eqn{\mathcal{R}(E,d) \geq \frac{1}{|N|} t(E,d)}.
#'
#' 5) \eqn{\mathbf{\tfrac{1}{|N|}}}\bold{-min-of-claims-and-deficit lower bounds on losses}, if for each \eqn{(E,d)\in C^N}, then \eqn{d-\mathcal{R}(E,d) \geq \frac{1}{|N|} t(D-E,d)}.
#'
#' 6) \bold{Equal treatment of equals}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\}\in N}, if \eqn{d_i=d_j}, then \eqn{\mathcal{R}_i(E,d)=\mathcal{R}_j(E,d)}.
#'
#' 7) \bold{Equal treatment of equal groups}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{N',N''\}} of subsets of \eqn{N}, if \eqn{d(N')=d(N'')}, then \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N''}\mathcal{R}_i(E,d)}.
#'
#' 8) \bold{Order preservation}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\} \subset N}, if \eqn{d_i \leq d_j}, then \eqn{\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d)} (in awards) and \eqn{d_i-\mathcal{R}_i(E,d)\leq d_j-\mathcal{R}_j(E,d)} (in losses).
#'
#' 9) \bold{Group order preservation}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{N',N''\}} of subsets of \eqn{N}, if \eqn{d(N') \leq d(N'')}, then \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)\leq \sum_{i\in N''}\mathcal{R}_i(E,d)} (in awards) and \eqn{\sum_{i\in N'}(d_i-\mathcal{R}_i(E,d))\leq \sum_{i\in N''}(d_i-\mathcal{R}_i(E,d))} (in losses).
#'
#' 10) \bold{Conditional null compensation}, if for each \eqn{(E,d)\in C^N} and each \eqn{i\in N}, if \eqn{\sum_{j\in N}\max\{0,d_j-d_i\}\geq E}, then \eqn{\mathcal{R}_i(E,d)=0}.
#'
#' 11) \bold{Conditional full compensation}, if for each \eqn{(E,d)\in C^N} and each \eqn{i\in N}, if \eqn{\sum_{j\in N}\min\{d_j,d_i\} \leq E}, then \eqn{\mathcal{R}_i(E,d)=d_i}.
#'
#' 12) \bold{Linked claim-endowment monotonicity}, if for each \eqn{(E,d)\in C^N}, each \eqn{i\in N}, and each \eqn{\delta>0}, we have \eqn{\mathcal{R}_i(E+\delta,(d_{-i},d_i+\delta))-\mathcal{R}_i(E,d)\leq \delta}.
#'
#' 13) \bold{Claim monotonicity}, if for each \eqn{(E,d)\in C^N}, each \eqn{i \in N}, and each \eqn{d'_i \geq d_i}, \eqn{\mathcal{R}_i(E,(d_{-i}, d'_i))\geq \mathcal{R}_i(E,d)}.
#'
#' 14) \bold{Population monotonicity}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, we have \eqn{\mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E,d_{N'})}.
#'
#' 15) \bold{Linked endowment-population monotonicity}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, if \eqn{d(N')\geq E}, we have \eqn{\mathcal{R}(E,d_{N'})\leq \mathcal{R}_{N'}(E+d(N\backslash N'),d)}.
#'
#' 16) \bold{Progressivity}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, we have \eqn{\frac{\mathcal{R}_i(E,d)}{d_i}\leq \frac{\mathcal{R}_j(E,d)}{d_j}}.
#'
#' 17) \bold{Regressivity}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, we have \eqn{\frac{\mathcal{R}_i(E,d)}{d_i}\geq \frac{\mathcal{R}_j(E,d)}{d_j}}.
#'
#' 18) \bold{No transfer paradox}, if for each \eqn{(E,d)\in C^N}, each pair \eqn{\{i,j\}\subset N}, each \eqn{d'_i>d_i}, and each \eqn{0\leq d'_j<d_j}, if \eqn{d'_i+d'_j=d_i+d_j}, then \eqn{\mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}})) \geq \mathcal{R}_i(E,d)} and \eqn{\mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq \mathcal{R}_j(E,d)}.
#'
#' 19) \bold{Bounded impact of claims transfer}, if for each \eqn{(E,d)\in C^N}, each pair \eqn{\{i,j\}\subset N}, each \eqn{d'_i>d_i}, and each \eqn{0\leq d'_j<d_j}, if \eqn{d'_i+d'_j=d_i+d_j}, then \eqn{\mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))- \mathcal{R}_i(E,d)\leq d'_i-d_i} and \eqn{\mathcal{R}_j(E,d)-\mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq d_j-d'_j}.
#'
#' 20) \bold{Concavity}, if for each \eqn{(E,d)\in C^N}, each triple \eqn{\{E,E',E''\}\subset \mathbb{R}^+} such that \eqn{0<E<E'<E''\leq D}, and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, then \eqn{\frac{\mathcal{R}_j(E',d)-\mathcal{R}_j(E,d)}{\mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)}\geq \frac{\mathcal{R}_j(E'',d)-\mathcal{R}_j(E',d)}{\mathcal{R}_i(E'',d)-\mathcal{R}_i(E',d)}}, if these ratios are well defined.
#'
#' 21) \bold{Convexity}, if for each \eqn{(E,d)\in C^N}, each triple \eqn{\{E,E',E''\}\subset \mathbb{R}^+} such that \eqn{0<E<E'<E''\leq D}, and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, then \eqn{\frac{\mathcal{R}_j(E',d)-\mathcal{R}_j(E,d)}{\mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)}\leq \frac{\mathcal{R}_j(E'',d)-\mathcal{R}_j(E',d)}{\mathcal{R}_i(E'',d)-\mathcal{R}_i(E',d)}}, if these ratios are well defined.
#'
#' 22) \bold{Endowment monotonicity}, if for each \eqn{(E,d)\in C^N} and each \eqn{E' >E}, if \eqn{D \geq E'} then \eqn{\mathcal{R}(E',d) \geq \mathcal{R}(E,d)}.
#'
#' 23) \bold{Order preservation under endowment variations}, if for each \eqn{(E,d)\in C^N}, each pair \eqn{\{i,j\}\subset N} and each \eqn{E'>E}, if \eqn{D\geq E'} and \eqn{d_i\leq d_j}, then \eqn{\mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E',d)-\mathcal{R}_j(E,d)}.
#'
#' 24) \bold{Order preservation under claims variations}, if for each \eqn{(E,d)\in C^N} with \eqn{|N|\geq 3}, each \eqn{i\in N}, each \eqn{d'_i>d_i}, and each pair \eqn{\{j,k\}\subset N\backslash\{i\}}, if \eqn{d_j\leq d_k}, then \eqn{\mathcal{R}_j(E,d)-\mathcal{R}_j(E,(d_{-i},d'_i))\leq \mathcal{R}_k(E,d)-\mathcal{R}_k(E,(d_{-i},d'_i))}.
#'
#' 25) \bold{Minimal rights first}, if for each \eqn{(E,d)\in C^N}, \eqn{\mathcal{R}(E,d) =m(E,d)+\mathcal{R}\bigl( E-\underset{i\in N}{\sum} m_i(E,d),d-m(E,d) \bigr)}.
#'
#' 26) \bold{Claims truncation invariance}, if for each \eqn{(E,d)\in C^N}, \eqn{\mathcal{R}(E,d) =\mathcal{R}( E, t(E,d) )}.
#'
#' 27) \bold{Composition down}, if for each \eqn{(E,d)\in C^N} and each \eqn{E'<E}, we have \eqn{\mathcal{R}(E',d)=\mathcal{R}(E',\mathcal{R}(E,d))}.
#'
#' 28) \bold{Composition up}, if for each \eqn{(E,d)\in C^N} and each \eqn{E'>E} such that \eqn{D \geq E'}, we have \eqn{\mathcal{R}(E',d)=\mathcal{R}(E,d)+\mathcal{R}(E'-E,d-\mathcal{R}(E,d))}.
#'
#' 29) \bold{Midpoint property}, if for each \eqn{(E,d)\in C^N} such that \eqn{E=\frac{1}{2}D}, then \eqn{\mathcal{R}(E,d)=\frac{d}{2}}.
#'
#' 30) \bold{Self-duality}, if for each \eqn{(E,d)\in C^N}, \eqn{\mathcal{R}(E,d) =d- \mathcal{R}\bigl(D-E,d\bigr)}.
#'
#' 31) \bold{Claims separability}, if for each pair \eqn{(E,d),(E',d')\in C^N} and each \eqn{N'\subset N}, if \eqn{d_{N'}=d'_{N'}}, \eqn{E=E'}, and \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d')}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d')}.
#'
#' 32) \bold{Claims-and-endowment separability}, if for each pair \eqn{(E,d),(E',d')\in C^N} and each \eqn{N'\subset N}, if \eqn{d_{N'}=d'_{N'}} and \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d')}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d')}.
#'
#' 33) \bold{Endowment convexity}, if for each \eqn{d\in\mathbb{R}^N_+}, each pair \eqn{\{E,E'\}\subset \mathbb{R}_+} such that \eqn{D\geq \max\{E,E'\}}, and each \eqn{\lambda\in[0,1]}, \eqn{\mathcal{R}(\lambda E+(1-\lambda)E',d)=\lambda\mathcal{R}(E,d)+(1-\lambda)\mathcal{R}(E',d)}.
#'
#' 34) \bold{Claims-and-endowment uniformity}, if for each pair \eqn{(E,d),(E',d')\in C^N}, and each \eqn{N'\subset N}, if \eqn{d_{N'}=d'_{N'}}, then either \eqn{\mathcal{R}_{N'}(E,d)\geq \mathcal{R}_{N'}(E',d')} or \eqn{\mathcal{R}_{N'}(E,d)\leq \mathcal{R}_{N'}(E',d')}.
#'
#' 35) \bold{Endowment-and-population uniformity}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each \eqn{(E',d')\in C^{N'}}, if \eqn{d_{N'}=d'}, then either \eqn{\mathcal{R}_{N'}(E,d)\geq \mathcal{R}(E',d')} or \eqn{\mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E',d')}.
#'
#' 36) \bold{No advantageous transfer}, if for each \eqn{(E,d)\in C^N}, each \eqn{N'\subset N}, and each \eqn{(d'_i)_{i\in N'}}, if \eqn{d'(N')=d(N')}, then \eqn{\sum_{i\in N'}\mathcal{R}_i(E,((d'_i)_{i\in N'},d_{N\backslash N'}))=\sum_{i\in N'}\mathcal{R}_i(E,d)}.
#'
#' 37) \bold{Summation independence}, if for each \eqn{(E,d)\in C^N}, each \eqn{i\in N}, and \eqn{N'=N\backslash \{i\}}, for each \eqn{(d'_j)_{j\in N'}\in\mathbb{R}^{N'}_+}, if \eqn{d'(N')=d(N')}, then \eqn{\mathcal{R}_i(E,(d_i,(d'_j)_{j\in N'}))=\mathcal{R}_i(E,d)}.
#'
#' 38) \bold{Consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N} if \eqn{x=\mathcal{R}(E,d)}, then \eqn{x_{N'}=\mathcal{R}(x(N'),d_{N'})}.
#'
#' 39) \bold{Bilateral consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N} and \eqn{|N'|=2}, and each \eqn{(E,d)\in C^N} if \eqn{x=\mathcal{R}(E,d)}, then \eqn{x_{N'}=\mathcal{R}(x(N'),d_{N'})}.
#'
#' 40) \bold{Converse consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N} and \eqn{|N'|=2}, each \eqn{(E,d)\in C^N}, and each \eqn{x\in X(E,d)}, we have \eqn{x_{N'}=\mathcal{R}(x(N'),d_{N'})}, then \eqn{x=\mathcal{R}(E,d)}.
#'
#' 41) \bold{Null claims consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N} and each \eqn{(E,d)\in C^N}, if \eqn{d_{N\backslash N'}=0}, then \eqn{\mathcal{R}_{N'}(E,d)= \mathcal{R}(E,d_{N'})}.
#'
#' 42) \bold{Null compensation consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, if \eqn{\mathcal{R}_{N\backslash N'}(E,d)=0}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}(E,d_{N'})}.
#'
#' 43) \bold{Full compensation consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, if \eqn{\mathcal{R}_{N\backslash N'}(E,d)=d_{N\backslash N'}}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}(E-d(N\backslash N'),d_{N'})}.
#'
#' 44) \bold{No advantageous merging}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each \eqn{d'\in \mathbb{R}^{N'}_+}, if there is \eqn{i\in N'} such that \eqn{d'_i=d_i+d(N\backslash N')} and for each \eqn{k\in N'\backslash\{i\}}, \eqn{d'_k=d_k}, then \eqn{\mathcal{R}_i(E,d')\leq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d)}.
#'
#' 45) \bold{No advantageous splitting}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each \eqn{d'\in \mathbb{R}^{N'}_+}, if there is \eqn{i\in N'} such that \eqn{d'_i=d_i+d(N\backslash N')} and for each \eqn{k\in N'\backslash\{i\}}, \eqn{d'_k=d_k}, then \eqn{\mathcal{R}_i(E,d')\geq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d)}.
#'
#' 46) \bold{Order preservation under population variations}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each pair \eqn{\{i,j\}\subset N'}, if \eqn{d(N')\geq E} and \eqn{d_i\leq d_j}, then \eqn{\mathcal{R}_i(E,d_{N'})-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d_{N'})-\mathcal{R}_j(E,d)}.
#'
#' 47) \bold{Division invariance}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\supset N}, each \eqn{(E,d)\in C^N} and each \eqn{k\in \mathbb{N}}. Let \eqn{(E',d')} be a \eqn{k}-replica of \eqn{(E,d)} with associated partition \eqn{(N^i)_{i\in N}}, and \eqn{\mathcal{R}(E',d')} be a \eqn{k}-replica of some awards vector \eqn{x\in X(E,d)} associated with the same partition, we have \eqn{\mathcal{R}(E,d)=x}.
#'
#' 48) \bold{Replication invariance}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\supset N}, each \eqn{(E,d)\in C^N} and each \eqn{k\in \mathbb{N}}. Let \eqn{(E',d')} be a \eqn{k}-replica of \eqn{(E,d)} with associated partition \eqn{(N^i)_{i\in N}}, then for each \eqn{i\in N} and each \eqn{j\in N^i} we have \eqn{\mathcal{R}_j(E',d')=\mathcal{R}_i(E,d)}.
#'
#' @examples
#' Rules=c(AA,Talmud)
#' Properties=c(1:10)
#' axioms(Rules,Properties)
#' axioms() #Table with all the rules and properties implemented.
#' # The minimal overlap rule does not satisfy linked endowment-population
#' # monotonicity (Mirás Calvo et al. (2024)):
#' E=1;d=c(1,2,9,10); d3= d[-3]
#' MOR=MO(E+d[3],d)
#' MOR3=MO(E,d3)
#' MOR3[1]>MOR[1]
#' # The adjusted proportional rule does not satisfy order preservation under
#' # population variations (Mirás Calvo et al. (2023)):
#' E=17; d=c(1,2,3,8,10); d3=d[-1]
#' APR=APRO(E,d)
#' APR3=APRO(E,d3)
#' APR3[1]-APR[1]>APR3[2]-APR[2]
#' @references Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2023). Refining the Lorenz‐ranking of rules for
#' claims problems on restricted domains. International Journal of Economic Theory 19(3), 526-558.
#' @references Mirás Calvo, M.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2024). An algorithm to compute the average-of-awards rule for claims problems with an application to
#' the allocation of CO\eqn{_2} emissions. Annals of Operations Research 336, 1435-1459.
#' @references Mirás Calvo, M.A., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2022). The average-of-awards rule for claims problems. Social Choice and Welfare 59, 863-888.
#' @references Mirás Calvo,M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez,E. (2025). On how the rules that extend the concede-and-divide principle
#' differ for pairs of claimants. Preprint.
#' @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
axioms=function(Rules="All",Properties="All"){
if(Rules[1]=="All")
{
Rules=c(AA, APRO, CE, CEA, CEL, AV, MO, PIN, PRO, RA, Talmud)
}
if(Properties[1]=="All")
{
Properties=c(1:48)
}
R=c()
for(i in 1:length(Rules)){
R[i]=Rules[[i]](0,0,1)
}
if(sum(R=="CD")>0)
{
warning("The concede-and-divide rule is not implemented in this function.",call.=F)
R=R[-which(R=="CD")]
}
if(sum(R=="DT")>0)
{
warning("The Dominguez-Thomson rule is not implemented in this function.",call.=F)
R=R[-which(R=="DT")]
}
if(sum(R=="RTalmud")>0)
{
warning("The reverse Talmud rule is not implemented in this function.",call.=F)
R=R[-which(R=="DT")]
}
if(length(R)==0)
{
stop("The rules introduced are not implemented in the function.",call.=F)
}
#We have added de following properties.
AllProp=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Equal treatment of equal groups", "Order preservation",
"Group order preservation", "Conditional null compensation", "Conditional full compensation",
"Linked claim-endowment monotonicity","Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "Progressivity",
"Regressivity", "No transfer paradox", "Bounded impact of claims transfers", "Concavity",
"Convexity", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Composition down", "Composition up", "Midpoint property", "Self-duality",
"Claims separability", "Claim-endowment separability", "Endowment convexity",
"Claim-endowment uniformity", "Endowment-population uniformity", "No advantageous transfers",
"Summation independence", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous merging", "No advantageous splitting","Order preservation under population variations",
"Division invariance", "Replication invariance")
Properties=AllProp[Properties]
#The properties that are satisfied by each rule.
pro=c("Anonymity", "Continuity", "Homogeneity",
"Equal treatment of equals", "Equal treatment of equal groups", "Order preservation",
"Group order preservation",
"Linked claim-endowment monotonicity","Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "Progressivity",
"Regressivity", "No transfer paradox", "Bounded impact of claims transfers", "Concavity",
"Convexity", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations",
"Composition down", "Composition up", "Midpoint property", "Self-duality",
"Claims separability", "Claim-endowment separability", "Endowment convexity",
"Claim-endowment uniformity", "Endowment-population uniformity", "No advantageous transfers",
"Summation independence", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous merging", "No advantageous splitting","Order preservation under population variations",
"Division invariance", "Replication invariance")
cea=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals", "Order preservation",
"Conditional full compensation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity",
"Regressivity", "No transfer paradox", "Bounded impact of claims transfers",
"Convexity", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Claims truncation invariance",
"Composition down", "Composition up",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity",
"Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous merging","Order preservation under population variations",
"Division invariance", "Replication invariance")
cel=c("Anonymity", "Continuity", "Homogeneity",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Conditional null compensation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "Progressivity",
"No transfer paradox", "Bounded impact of claims transfers", "Concavity",
"Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first",
"Composition down", "Composition up",
"Claims separability",
"Claim-endowment uniformity", "Endowment-population uniformity",
"Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous splitting","Order preservation under population variations",
"Division invariance", "Replication invariance")
av=c("Anonymity", "Continuity", "Homogeneity",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity",
"No transfer paradox", "Bounded impact of claims transfers",
"Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations",
"Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations","Midpoint property", "Self-duality")
ce=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers",
"Endowment monotonicity",
"Midpoint property",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Division invariance", "Replication invariance")
talmud=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity",
"No transfer paradox", "Bounded impact of claims transfers","Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations","Division invariance", "Replication invariance")
pin=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity","Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers",
"Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations",
"Midpoint property",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations","Division invariance", "Replication invariance")
ra=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations")
mo=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals","Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations")
apro=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity","Claim monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Null claims consistency")
aa=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations")
#Table with rules and properties.
Properties=Properties[Properties!=0]
P=matrix(0,length(Properties),length(R))
colnames(P)=R
rownames(P)=Properties
for(i in 1:length(R)){
if(R[i]=="PRO"){#Properties satisfied by the proportional rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==pro)>0){P[j,i]=1 }
}
}
if(R[i]=="APRO"){#Properties satisfied by the adjusted proportional rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==apro)>0){P[j,i]=1}
}
}
if(R[i]=="MO"){#Properties satisfied by the minimal overlap rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==mo)>0){P[j,i]=1}
}
}
if(R[i]=="CEA"){#Properties satisfied by the CEA rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==cea)>0){P[j,i]=1}
}
}
if(R[i]=="CEL"){#Properties satisfied by the CEL rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==cel)>0){P[j,i]=1}
}
}
if(R[i]=="AV"){#Properties satisfied by the average rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==av)>0){P[j,i]=1}
}
}
if(R[i]=="CE"){#Properties satisfied by the CE rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==ce)>0){P[j,i]=1}
}
}
if(R[i]=="PIN"){#Properties satisfied by the Piniles' rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==pin)>0){P[j,i]=1}
}
}
if(R[i]=="Talmud"){#Properties satisfied by the Talmud rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==talmud)>0){P[j,i]=1}
}
}
if(R[i]=="RA"){#Properties satisfied by the random arrival rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==ra)>0){P[j,i]=1}
}
}
if(R[i]=="AA"){#Properties satisfied by the average-of-awards rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==aa)>0) {P[j,i]=1}
}
}
}
return(P)
}
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Defines functions axioms
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#' @title Properties of the rules
#' @description This function shows which of the main properties the rules satisfy.
#' @param Rules The rules: AA, APRO, CE, CEA, CEL, AV, MO, PIN, PRO, RA, Talmud. By default, \code{Rules = "All"}.
#' @param Properties The properties listed in the description section. By default, \code{Properties = "All"}.
#' @return A table with the rules and the properties. If a rule satisfies a property it returns 1, and 0 otherwise. By default, it returns a table with all rules and properties.
#' @details Let \eqn{\mathcal{N}} be the set of all finite nonempty subsets of the natural numbers \eqn{\mathbb{N}}. Let \eqn{N=\{1,\ldots,n\}} be the set of claimants, \eqn{E\ge 0} the endowment to be divided and \eqn{d\in \mathbb{R}_+^N} the vector of claims
#' such that \eqn{D=\sum_{i \in N} d_i\ge E}. We denote the class of claims problems with set of
#' claimants \eqn{N} by \eqn{C^N}. Given \eqn{z\in\mathbb{R}^N}, \eqn{S\in 2^N} and \eqn{N'\subset N\in\mathcal{N}}, let \eqn{z(S)=\sum_{j\in S}z_j}, \eqn{z_{N'}=(z_i)_{i\in N'}} and denote \eqn{z_{-i}=z_{N\backslash \{i\}}\in \mathbb{R}^{N\backslash \{i\}}}. For simplicity, we will write \eqn{z=(z_{-i},z)}.
#'
#' The minimal right of claimant \eqn{i\in N} in \eqn{(E,d)} is defined by \eqn{m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\}.}
#' Let \eqn{m(E,d)=(m_1(E,d),\dots,m_n(E,d))} be the vector of minimal rights.
#'
#' The truncated claim of claimant \eqn{i\in N} in \eqn{(E,d)} is defined by
#' \eqn{t_i(E,d)=\min\{d_i,E\}.} Let \eqn{t(E,d)=(t_1(E,d),\dots,t_n(E,d))} be the vector of truncated claims.
#'
#' Given \eqn{(E,d)\in C^N} and \eqn{k\in\mathbb{N}}, we say that \eqn{(E',d')} is a \eqn{k}-replica of \eqn{(E,d)} if \eqn{E'=k E}, \eqn{N'\supset N}, \eqn{|N'|=k|N|}, and there is a partition \eqn{(N^i)_{i\in N}} of \eqn{N'} such that for each \eqn{i\in N} and each \eqn{j\in N^i}, \eqn{|N^i|=k} and \eqn{d'_j=d_i}.
#'
#' A vector \eqn{x=(x_1,\dots,x_n)} is an awards vector for the claims problem \eqn{(E,d)\in C^N} if \eqn{0\le x \le d}
#' and satisfies the balance requirement, that is, \eqn{x(N)=E}. Let \eqn{X(E,d)} be the set of awards vectors for the problem \eqn{(E,d)}.
#' A rule is a function that assigns to each claims problem \eqn{(E,d)} an awards vector.
#'
#' A rule \eqn{\mathcal{R}} satisfies:
#'
#' 1) \bold{Anonymity}, if for each \eqn{(E,d)\in C^N}, each bijection \eqn{f} from \eqn{N} into itself, and each \eqn{i\in N}, we have that \eqn{\mathcal{R}_i(E,d)=\mathcal{R}_{f(i)}(E,(d_{f(i)})_{i\in N})}.
#'
#' 2) \bold{Continuity}, if for each sequence \eqn{\{(E^{\ell},d^{\ell})\}} of elements of \eqn{C^N} and each \eqn{(E,d)\in C^N}, if \eqn{(E^{\ell},d^{\ell})\rightarrow (E,d)} then \eqn{\mathcal{R}(E^{\ell},d^{\ell})\rightarrow \mathcal{R}(E,d)}.
#'
#' 3) \bold{Homogeneity}, if for each \eqn{(E,d)\in C^N} and each \eqn{\rho>0}, then \eqn{\mathcal{R}(\rho E,\rho d)=\rho \mathcal{R}(E,d)}.
#'
#' 4) \eqn{\mathbf{\tfrac{1}{|N|}}}\bold{-truncated-claims lower bounds on awards}, if for each \eqn{(E,d)\in C^N}, then \eqn{\mathcal{R}(E,d) \geq \frac{1}{|N|} t(E,d)}.
#'
#' 5) \eqn{\mathbf{\tfrac{1}{|N|}}}\bold{-min-of-claims-and-deficit lower bounds on losses}, if for each \eqn{(E,d)\in C^N}, then \eqn{d-\mathcal{R}(E,d) \geq \frac{1}{|N|} t(D-E,d)}.
#'
#' 6) \bold{Equal treatment of equals}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\}\in N}, if \eqn{d_i=d_j}, then \eqn{\mathcal{R}_i(E,d)=\mathcal{R}_j(E,d)}.
#'
#' 7) \bold{Equal treatment of equal groups}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{N',N''\}} of subsets of \eqn{N}, if \eqn{d(N')=d(N'')}, then \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N''}\mathcal{R}_i(E,d)}.
#'
#' 8) \bold{Order preservation}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\} \subset N}, if \eqn{d_i \leq d_j}, then \eqn{\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d)} (in awards) and \eqn{d_i-\mathcal{R}_i(E,d)\leq d_j-\mathcal{R}_j(E,d)} (in losses).
#'
#' 9) \bold{Group order preservation}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{N',N''\}} of subsets of \eqn{N}, if \eqn{d(N') \leq d(N'')}, then \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)\leq \sum_{i\in N''}\mathcal{R}_i(E,d)} (in awards) and \eqn{\sum_{i\in N'}(d_i-\mathcal{R}_i(E,d))\leq \sum_{i\in N''}(d_i-\mathcal{R}_i(E,d))} (in losses).
#'
#' 10) \bold{Conditional null compensation}, if for each \eqn{(E,d)\in C^N} and each \eqn{i\in N}, if \eqn{\sum_{j\in N}\max\{0,d_j-d_i\}\geq E}, then \eqn{\mathcal{R}_i(E,d)=0}.
#'
#' 11) \bold{Conditional full compensation}, if for each \eqn{(E,d)\in C^N} and each \eqn{i\in N}, if \eqn{\sum_{j\in N}\min\{d_j,d_i\} \leq E}, then \eqn{\mathcal{R}_i(E,d)=d_i}.
#'
#' 12) \bold{Linked claim-endowment monotonicity}, if for each \eqn{(E,d)\in C^N}, each \eqn{i\in N}, and each \eqn{\delta>0}, we have \eqn{\mathcal{R}_i(E+\delta,(d_{-i},d_i+\delta))-\mathcal{R}_i(E,d)\leq \delta}.
#'
#' 13) \bold{Claim monotonicity}, if for each \eqn{(E,d)\in C^N}, each \eqn{i \in N}, and each \eqn{d'_i \geq d_i}, \eqn{\mathcal{R}_i(E,(d_{-i}, d'_i))\geq \mathcal{R}_i(E,d)}.
#'
#' 14) \bold{Population monotonicity}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, we have \eqn{\mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E,d_{N'})}.
#'
#' 15) \bold{Linked endowment-population monotonicity}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, if \eqn{d(N')\geq E}, we have \eqn{\mathcal{R}(E,d_{N'})\leq \mathcal{R}_{N'}(E+d(N\backslash N'),d)}.
#'
#' 16) \bold{Progressivity}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, we have \eqn{\frac{\mathcal{R}_i(E,d)}{d_i}\leq \frac{\mathcal{R}_j(E,d)}{d_j}}.
#'
#' 17) \bold{Regressivity}, if for each \eqn{(E,d)\in C^N} and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, we have \eqn{\frac{\mathcal{R}_i(E,d)}{d_i}\geq \frac{\mathcal{R}_j(E,d)}{d_j}}.
#'
#' 18) \bold{No transfer paradox}, if for each \eqn{(E,d)\in C^N}, each pair \eqn{\{i,j\}\subset N}, each \eqn{d'_i>d_i}, and each \eqn{0\leq d'_j<d_j}, if \eqn{d'_i+d'_j=d_i+d_j}, then \eqn{\mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}})) \geq \mathcal{R}_i(E,d)} and \eqn{\mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq \mathcal{R}_j(E,d)}.
#'
#' 19) \bold{Bounded impact of claims transfer}, if for each \eqn{(E,d)\in C^N}, each pair \eqn{\{i,j\}\subset N}, each \eqn{d'_i>d_i}, and each \eqn{0\leq d'_j<d_j}, if \eqn{d'_i+d'_j=d_i+d_j}, then \eqn{\mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))- \mathcal{R}_i(E,d)\leq d'_i-d_i} and \eqn{\mathcal{R}_j(E,d)-\mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq d_j-d'_j}.
#'
#' 20) \bold{Concavity}, if for each \eqn{(E,d)\in C^N}, each triple \eqn{\{E,E',E''\}\subset \mathbb{R}^+} such that \eqn{0<E<E'<E''\leq D}, and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, then \eqn{\frac{\mathcal{R}_j(E',d)-\mathcal{R}_j(E,d)}{\mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)}\geq \frac{\mathcal{R}_j(E'',d)-\mathcal{R}_j(E',d)}{\mathcal{R}_i(E'',d)-\mathcal{R}_i(E',d)}}, if these ratios are well defined.
#'
#' 21) \bold{Convexity}, if for each \eqn{(E,d)\in C^N}, each triple \eqn{\{E,E',E''\}\subset \mathbb{R}^+} such that \eqn{0<E<E'<E''\leq D}, and each pair \eqn{\{i,j\}\subset N}, if \eqn{0<d_i\leq d_j}, then \eqn{\frac{\mathcal{R}_j(E',d)-\mathcal{R}_j(E,d)}{\mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)}\leq \frac{\mathcal{R}_j(E'',d)-\mathcal{R}_j(E',d)}{\mathcal{R}_i(E'',d)-\mathcal{R}_i(E',d)}}, if these ratios are well defined.
#'
#' 22) \bold{Endowment monotonicity}, if for each \eqn{(E,d)\in C^N} and each \eqn{E' >E}, if \eqn{D \geq E'} then \eqn{\mathcal{R}(E',d) \geq \mathcal{R}(E,d)}.
#'
#' 23) \bold{Order preservation under endowment variations}, if for each \eqn{(E,d)\in C^N}, each pair \eqn{\{i,j\}\subset N} and each \eqn{E'>E}, if \eqn{D\geq E'} and \eqn{d_i\leq d_j}, then \eqn{\mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E',d)-\mathcal{R}_j(E,d)}.
#'
#' 24) \bold{Order preservation under claims variations}, if for each \eqn{(E,d)\in C^N} with \eqn{|N|\geq 3}, each \eqn{i\in N}, each \eqn{d'_i>d_i}, and each pair \eqn{\{j,k\}\subset N\backslash\{i\}}, if \eqn{d_j\leq d_k}, then \eqn{\mathcal{R}_j(E,d)-\mathcal{R}_j(E,(d_{-i},d'_i))\leq \mathcal{R}_k(E,d)-\mathcal{R}_k(E,(d_{-i},d'_i))}.
#'
#' 25) \bold{Minimal rights first}, if for each \eqn{(E,d)\in C^N}, \eqn{\mathcal{R}(E,d) =m(E,d)+\mathcal{R}\bigl( E-\underset{i\in N}{\sum} m_i(E,d),d-m(E,d) \bigr)}.
#'
#' 26) \bold{Claims truncation invariance}, if for each \eqn{(E,d)\in C^N}, \eqn{\mathcal{R}(E,d) =\mathcal{R}( E, t(E,d) )}.
#'
#' 27) \bold{Composition down}, if for each \eqn{(E,d)\in C^N} and each \eqn{E'<E}, we have \eqn{\mathcal{R}(E',d)=\mathcal{R}(E',\mathcal{R}(E,d))}.
#'
#' 28) \bold{Composition up}, if for each \eqn{(E,d)\in C^N} and each \eqn{E'>E} such that \eqn{D \geq E'}, we have \eqn{\mathcal{R}(E',d)=\mathcal{R}(E,d)+\mathcal{R}(E'-E,d-\mathcal{R}(E,d))}.
#'
#' 29) \bold{Midpoint property}, if for each \eqn{(E,d)\in C^N} such that \eqn{E=\frac{1}{2}D}, then \eqn{\mathcal{R}(E,d)=\frac{d}{2}}.
#'
#' 30) \bold{Self-duality}, if for each \eqn{(E,d)\in C^N}, \eqn{\mathcal{R}(E,d) =d- \mathcal{R}\bigl(D-E,d\bigr)}.
#'
#' 31) \bold{Claims separability}, if for each pair \eqn{(E,d),(E',d')\in C^N} and each \eqn{N'\subset N}, if \eqn{d_{N'}=d'_{N'}}, \eqn{E=E'}, and \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d')}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d')}.
#'
#' 32) \bold{Claims-and-endowment separability}, if for each pair \eqn{(E,d),(E',d')\in C^N} and each \eqn{N'\subset N}, if \eqn{d_{N'}=d'_{N'}} and \eqn{\sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d')}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d')}.
#'
#' 33) \bold{Endowment convexity}, if for each \eqn{d\in\mathbb{R}^N_+}, each pair \eqn{\{E,E'\}\subset \mathbb{R}_+} such that \eqn{D\geq \max\{E,E'\}}, and each \eqn{\lambda\in[0,1]}, \eqn{\mathcal{R}(\lambda E+(1-\lambda)E',d)=\lambda\mathcal{R}(E,d)+(1-\lambda)\mathcal{R}(E',d)}.
#'
#' 34) \bold{Claims-and-endowment uniformity}, if for each pair \eqn{(E,d),(E',d')\in C^N}, and each \eqn{N'\subset N}, if \eqn{d_{N'}=d'_{N'}}, then either \eqn{\mathcal{R}_{N'}(E,d)\geq \mathcal{R}_{N'}(E',d')} or \eqn{\mathcal{R}_{N'}(E,d)\leq \mathcal{R}_{N'}(E',d')}.
#'
#' 35) \bold{Endowment-and-population uniformity}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each \eqn{(E',d')\in C^{N'}}, if \eqn{d_{N'}=d'}, then either \eqn{\mathcal{R}_{N'}(E,d)\geq \mathcal{R}(E',d')} or \eqn{\mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E',d')}.
#'
#' 36) \bold{No advantageous transfer}, if for each \eqn{(E,d)\in C^N}, each \eqn{N'\subset N}, and each \eqn{(d'_i)_{i\in N'}}, if \eqn{d'(N')=d(N')}, then \eqn{\sum_{i\in N'}\mathcal{R}_i(E,((d'_i)_{i\in N'},d_{N\backslash N'}))=\sum_{i\in N'}\mathcal{R}_i(E,d)}.
#'
#' 37) \bold{Summation independence}, if for each \eqn{(E,d)\in C^N}, each \eqn{i\in N}, and \eqn{N'=N\backslash \{i\}}, for each \eqn{(d'_j)_{j\in N'}\in\mathbb{R}^{N'}_+}, if \eqn{d'(N')=d(N')}, then \eqn{\mathcal{R}_i(E,(d_i,(d'_j)_{j\in N'}))=\mathcal{R}_i(E,d)}.
#'
#' 38) \bold{Consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N} if \eqn{x=\mathcal{R}(E,d)}, then \eqn{x_{N'}=\mathcal{R}(x(N'),d_{N'})}.
#'
#' 39) \bold{Bilateral consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N} and \eqn{|N'|=2}, and each \eqn{(E,d)\in C^N} if \eqn{x=\mathcal{R}(E,d)}, then \eqn{x_{N'}=\mathcal{R}(x(N'),d_{N'})}.
#'
#' 40) \bold{Converse consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N} and \eqn{|N'|=2}, each \eqn{(E,d)\in C^N}, and each \eqn{x\in X(E,d)}, we have \eqn{x_{N'}=\mathcal{R}(x(N'),d_{N'})}, then \eqn{x=\mathcal{R}(E,d)}.
#'
#' 41) \bold{Null claims consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N} and each \eqn{(E,d)\in C^N}, if \eqn{d_{N\backslash N'}=0}, then \eqn{\mathcal{R}_{N'}(E,d)= \mathcal{R}(E,d_{N'})}.
#'
#' 42) \bold{Null compensation consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, if \eqn{\mathcal{R}_{N\backslash N'}(E,d)=0}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}(E,d_{N'})}.
#'
#' 43) \bold{Full compensation consistency}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, and each \eqn{(E,d)\in C^N}, if \eqn{\mathcal{R}_{N\backslash N'}(E,d)=d_{N\backslash N'}}, then \eqn{\mathcal{R}_{N'}(E,d)=\mathcal{R}(E-d(N\backslash N'),d_{N'})}.
#'
#' 44) \bold{No advantageous merging}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each \eqn{d'\in \mathbb{R}^{N'}_+}, if there is \eqn{i\in N'} such that \eqn{d'_i=d_i+d(N\backslash N')} and for each \eqn{k\in N'\backslash\{i\}}, \eqn{d'_k=d_k}, then \eqn{\mathcal{R}_i(E,d')\leq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d)}.
#'
#' 45) \bold{No advantageous splitting}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each \eqn{d'\in \mathbb{R}^{N'}_+}, if there is \eqn{i\in N'} such that \eqn{d'_i=d_i+d(N\backslash N')} and for each \eqn{k\in N'\backslash\{i\}}, \eqn{d'_k=d_k}, then \eqn{\mathcal{R}_i(E,d')\geq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d)}.
#'
#' 46) \bold{Order preservation under population variations}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\subset N}, each \eqn{(E,d)\in C^N}, and each pair \eqn{\{i,j\}\subset N'}, if \eqn{d(N')\geq E} and \eqn{d_i\leq d_j}, then \eqn{\mathcal{R}_i(E,d_{N'})-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d_{N'})-\mathcal{R}_j(E,d)}.
#'
#' 47) \bold{Division invariance}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\supset N}, each \eqn{(E,d)\in C^N} and each \eqn{k\in \mathbb{N}}. Let \eqn{(E',d')} be a \eqn{k}-replica of \eqn{(E,d)} with associated partition \eqn{(N^i)_{i\in N}}, and \eqn{\mathcal{R}(E',d')} be a \eqn{k}-replica of some awards vector \eqn{x\in X(E,d)} associated with the same partition, we have \eqn{\mathcal{R}(E,d)=x}.
#'
#' 48) \bold{Replication invariance}, if for each pair \eqn{N,N'} of subsets of \eqn{\mathcal{N}} such that \eqn{N'\supset N}, each \eqn{(E,d)\in C^N} and each \eqn{k\in \mathbb{N}}. Let \eqn{(E',d')} be a \eqn{k}-replica of \eqn{(E,d)} with associated partition \eqn{(N^i)_{i\in N}}, then for each \eqn{i\in N} and each \eqn{j\in N^i} we have \eqn{\mathcal{R}_j(E',d')=\mathcal{R}_i(E,d)}.
#'
#' @examples
#' Rules=c(AA,Talmud)
#' Properties=c(1:10)
#' axioms(Rules,Properties)
#' axioms() #Table with all the rules and properties implemented.
#' # The minimal overlap rule does not satisfy linked endowment-population
#' # monotonicity (Mirás Calvo et al. (2024)):
#' E=1;d=c(1,2,9,10); d3= d[-3]
#' MOR=MO(E+d[3],d)
#' MOR3=MO(E,d3)
#' MOR3[1]>MOR[1]
#' # The adjusted proportional rule does not satisfy order preservation under
#' # population variations (Mirás Calvo et al. (2023)):
#' E=17; d=c(1,2,3,8,10); d3=d[-1]
#' APR=APRO(E,d)
#' APR3=APRO(E,d3)
#' APR3[1]-APR[1]>APR3[2]-APR[2]
#' @references Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2023). Refining the Lorenz‐ranking of rules for
#' claims problems on restricted domains. International Journal of Economic Theory 19(3), 526-558.
#' @references Mirás Calvo, M.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2024). An algorithm to compute the average-of-awards rule for claims problems with an application to
#' the allocation of CO\eqn{_2} emissions. Annals of Operations Research 336, 1435-1459.
#' @references Mirás Calvo, M.A., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2022). The average-of-awards rule for claims problems. Social Choice and Welfare 59, 863-888.
#' @references Mirás Calvo,M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez,E. (2025). On how the rules that extend the concede-and-divide principle
#' differ for pairs of claimants. Preprint.
#' @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
axioms=function(Rules="All",Properties="All"){
if(Rules[1]=="All")
{
Rules=c(AA, APRO, CE, CEA, CEL, AV, MO, PIN, PRO, RA, Talmud)
}
if(Properties[1]=="All")
{
Properties=c(1:48)
}
R=c()
for(i in 1:length(Rules)){
R[i]=Rules[[i]](0,0,1)
}
if(sum(R=="CD")>0)
{
warning("The concede-and-divide rule is not implemented in this function.",call.=F)
R=R[-which(R=="CD")]
}
if(sum(R=="DT")>0)
{
warning("The Dominguez-Thomson rule is not implemented in this function.",call.=F)
R=R[-which(R=="DT")]
}
if(sum(R=="RTalmud")>0)
{
warning("The reverse Talmud rule is not implemented in this function.",call.=F)
R=R[-which(R=="DT")]
}
if(length(R)==0)
{
stop("The rules introduced are not implemented in the function.",call.=F)
}
#We have added de following properties.
AllProp=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Equal treatment of equal groups", "Order preservation",
"Group order preservation", "Conditional null compensation", "Conditional full compensation",
"Linked claim-endowment monotonicity","Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "Progressivity",
"Regressivity", "No transfer paradox", "Bounded impact of claims transfers", "Concavity",
"Convexity", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Composition down", "Composition up", "Midpoint property", "Self-duality",
"Claims separability", "Claim-endowment separability", "Endowment convexity",
"Claim-endowment uniformity", "Endowment-population uniformity", "No advantageous transfers",
"Summation independence", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous merging", "No advantageous splitting","Order preservation under population variations",
"Division invariance", "Replication invariance")
Properties=AllProp[Properties]
#The properties that are satisfied by each rule.
pro=c("Anonymity", "Continuity", "Homogeneity",
"Equal treatment of equals", "Equal treatment of equal groups", "Order preservation",
"Group order preservation",
"Linked claim-endowment monotonicity","Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "Progressivity",
"Regressivity", "No transfer paradox", "Bounded impact of claims transfers", "Concavity",
"Convexity", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations",
"Composition down", "Composition up", "Midpoint property", "Self-duality",
"Claims separability", "Claim-endowment separability", "Endowment convexity",
"Claim-endowment uniformity", "Endowment-population uniformity", "No advantageous transfers",
"Summation independence", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous merging", "No advantageous splitting","Order preservation under population variations",
"Division invariance", "Replication invariance")
cea=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals", "Order preservation",
"Conditional full compensation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity",
"Regressivity", "No transfer paradox", "Bounded impact of claims transfers",
"Convexity", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Claims truncation invariance",
"Composition down", "Composition up",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity",
"Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous merging","Order preservation under population variations",
"Division invariance", "Replication invariance")
cel=c("Anonymity", "Continuity", "Homogeneity",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Conditional null compensation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "Progressivity",
"No transfer paradox", "Bounded impact of claims transfers", "Concavity",
"Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first",
"Composition down", "Composition up",
"Claims separability",
"Claim-endowment uniformity", "Endowment-population uniformity",
"Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"No advantageous splitting","Order preservation under population variations",
"Division invariance", "Replication invariance")
av=c("Anonymity", "Continuity", "Homogeneity",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity",
"No transfer paradox", "Bounded impact of claims transfers",
"Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations",
"Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations","Midpoint property", "Self-duality")
ce=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers",
"Endowment monotonicity",
"Midpoint property",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Division invariance", "Replication invariance")
talmud=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity",
"No transfer paradox", "Bounded impact of claims transfers","Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations","Division invariance", "Replication invariance")
pin=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity","Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers",
"Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations",
"Midpoint property",
"Claims separability", "Claim-endowment separability",
"Claim-endowment uniformity", "Endowment-population uniformity", "Consistency", "Bilateral consistency", "Converse consistency",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations","Division invariance", "Replication invariance")
ra=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations")
mo=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"Equal treatment of equals","Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations")
apro=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity","Claim monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Null claims consistency")
aa=c("Anonymity", "Continuity", "Homogeneity", "1/|N|-Truncated-claims lower bounds on awards",
"1/|N|-min-of-claims-and-deficit lower bounds on losses",
"Equal treatment of equals", "Order preservation",
"Linked claim-endowment monotonicity", "Claim monotonicity",
"Population monotonicity", "Linked-endowment-population monotonicity", "No transfer paradox", "Bounded impact of claims transfers", "Endowment monotonicity", "Order preservation under endowment variations",
"Order preservation under claims variations", "Minimal rights first", "Claims truncation invariance",
"Midpoint property", "Self-duality",
"Null claims consistency", "Null compensation consistency", "Full compensation consistency",
"Order preservation under population variations")
#Table with rules and properties.
Properties=Properties[Properties!=0]
P=matrix(0,length(Properties),length(R))
colnames(P)=R
rownames(P)=Properties
for(i in 1:length(R)){
if(R[i]=="PRO"){#Properties satisfied by the proportional rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==pro)>0){P[j,i]=1 }
}
}
if(R[i]=="APRO"){#Properties satisfied by the adjusted proportional rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==apro)>0){P[j,i]=1}
}
}
if(R[i]=="MO"){#Properties satisfied by the minimal overlap rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==mo)>0){P[j,i]=1}
}
}
if(R[i]=="CEA"){#Properties satisfied by the CEA rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==cea)>0){P[j,i]=1}
}
}
if(R[i]=="CEL"){#Properties satisfied by the CEL rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==cel)>0){P[j,i]=1}
}
}
if(R[i]=="AV"){#Properties satisfied by the average rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==av)>0){P[j,i]=1}
}
}
if(R[i]=="CE"){#Properties satisfied by the CE rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==ce)>0){P[j,i]=1}
}
}
if(R[i]=="PIN"){#Properties satisfied by the Piniles' rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==pin)>0){P[j,i]=1}
}
}
if(R[i]=="Talmud"){#Properties satisfied by the Talmud rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==talmud)>0){P[j,i]=1}
}
}
if(R[i]=="RA"){#Properties satisfied by the random arrival rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==ra)>0){P[j,i]=1}
}
}
if(R[i]=="AA"){#Properties satisfied by the average-of-awards rule.
for(j in 1:length(Properties)){
if(sum(Properties[j]==aa)>0) {P[j,i]=1}
}
}
}
return(P)
}
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