axioms: Properties of the rules
In ClaimsProblems: Analysis of Conflicting Claims
axioms R Documentation
Properties of the rules
Description
This function shows which of the main properties the rules satisfy.
Usage
axioms(Rules = "All", Properties = "All")
Arguments
Rules
The rules: AA, APRO, CE, CEA, CEL, AV, MO, PIN, PRO, RA, Talmud. By default, Rules = "All".
Properties
The properties listed in the description section. By default, Properties = "All".
Details
Let \mathcal{N} be the set of all finite nonempty subsets of the natural numbers \mathbb{N}. Let N=\{1,\ldots,n\} be the set of claimants, E\ge 0 the endowment to be divided and d\in \mathbb{R}_+^N the vector of claims
such that D=\sum_{i \in N} d_i\ge E. We denote the class of claims problems with set of
claimants N by C^N. Given z\in\mathbb{R}^N, S\in 2^N and N'\subset N\in\mathcal{N}, let z(S)=\sum_{j\in S}z_j, z_{N'}=(z_i)_{i\in N'} and denote z_{-i}=z_{N\backslash \{i\}}\in \mathbb{R}^{N\backslash \{i\}}. For simplicity, we will write z=(z_{-i},z).
The minimal right of claimant i\in N in (E,d) is defined by m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\}.
Let m(E,d)=(m_1(E,d),\dots,m_n(E,d)) be the vector of minimal rights.
The truncated claim of claimant i\in N in (E,d) is defined by
t_i(E,d)=\min\{d_i,E\}. Let t(E,d)=(t_1(E,d),\dots,t_n(E,d)) be the vector of truncated claims.
Given (E,d)\in C^N and k\in\mathbb{N}, we say that (E',d') is a k-replica of (E,d) if E'=k E, N'\supset N, |N'|=k|N|, and there is a partition (N^i)_{i\in N} of N' such that for each i\in N and each j\in N^i, |N^i|=k and d'_j=d_i.
A vector x=(x_1,\dots,x_n) is an awards vector for the claims problem (E,d)\in C^N if 0\le x \le d
and satisfies the balance requirement, that is, x(N)=E. Let X(E,d) be the set of awards vectors for the problem (E,d).
A rule is a function that assigns to each claims problem (E,d) an awards vector.
A rule \mathcal{R} satisfies:
1) Anonymity, if for each (E,d)\in C^N, each bijection f from N into itself, and each i\in N, we have that \mathcal{R}_i(E,d)=\mathcal{R}_{f(i)}(E,(d_{f(i)})_{i\in N}).
2) Continuity, if for each sequence \{(E^{\ell},d^{\ell})\} of elements of C^N and each (E,d)\in C^N, if (E^{\ell},d^{\ell})\rightarrow (E,d) then \mathcal{R}(E^{\ell},d^{\ell})\rightarrow \mathcal{R}(E,d).
3) Homogeneity, if for each (E,d)\in C^N and each \rho>0, then \mathcal{R}(\rho E,\rho d)=\rho \mathcal{R}(E,d).
4) \mathbf{\tfrac{1}{|N|}}-truncated-claims lower bounds on awards, if for each (E,d)\in C^N, then \mathcal{R}(E,d) \geq \frac{1}{|N|} t(E,d).
5) \mathbf{\tfrac{1}{|N|}}-min-of-claims-and-deficit lower bounds on losses, if for each (E,d)\in C^N, then d-\mathcal{R}(E,d) \geq \frac{1}{|N|} t(D-E,d).
6) Equal treatment of equals, if for each (E,d)\in C^N and each pair \{i,j\}\in N, if d_i=d_j, then \mathcal{R}_i(E,d)=\mathcal{R}_j(E,d).
7) Equal treatment of equal groups, if for each (E,d)\in C^N and each pair \{N',N''\} of subsets of N, if d(N')=d(N''), then \sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N''}\mathcal{R}_i(E,d).
8) Order preservation, if for each (E,d)\in C^N and each pair \{i,j\} \subset N, if d_i \leq d_j, then \mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d) (in awards) and d_i-\mathcal{R}_i(E,d)\leq d_j-\mathcal{R}_j(E,d) (in losses).
9) Group order preservation, if for each (E,d)\in C^N and each pair \{N',N''\} of subsets of N, if d(N') \leq d(N''), then \sum_{i\in N'}\mathcal{R}_i(E,d)\leq \sum_{i\in N''}\mathcal{R}_i(E,d) (in awards) and \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) Conditional null compensation, if for each (E,d)\in C^N and each i\in N, if \sum_{j\in N}\max\{0,d_j-d_i\}\geq E, then \mathcal{R}_i(E,d)=0.
11) Conditional full compensation, if for each (E,d)\in C^N and each i\in N, if \sum_{j\in N}\min\{d_j,d_i\} \leq E, then \mathcal{R}_i(E,d)=d_i.
12) Linked claim-endowment monotonicity, if for each (E,d)\in C^N, each i\in N, and each \delta>0, we have \mathcal{R}_i(E+\delta,(d_{-i},d_i+\delta))-\mathcal{R}_i(E,d)\leq \delta.
13) Claim monotonicity, if for each (E,d)\in C^N, each i \in N, and each d'_i \geq d_i, \mathcal{R}_i(E,(d_{-i}, d'_i))\geq \mathcal{R}_i(E,d).
14) Population monotonicity, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, we have \mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E,d_{N'}).
15) Linked endowment-population monotonicity, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, if d(N')\geq E, we have \mathcal{R}(E,d_{N'})\leq \mathcal{R}_{N'}(E+d(N\backslash N'),d).
16) Progressivity, if for each (E,d)\in C^N and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, we have \frac{\mathcal{R}_i(E,d)}{d_i}\leq \frac{\mathcal{R}_j(E,d)}{d_j}.
17) Regressivity, if for each (E,d)\in C^N and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, we have \frac{\mathcal{R}_i(E,d)}{d_i}\geq \frac{\mathcal{R}_j(E,d)}{d_j}.
18) No transfer paradox, if for each (E,d)\in C^N, each pair \{i,j\}\subset N, each d'_i>d_i, and each 0\leq d'_j<d_j, if d'_i+d'_j=d_i+d_j, then \mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}})) \geq \mathcal{R}_i(E,d) and \mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq \mathcal{R}_j(E,d).
19) Bounded impact of claims transfer, if for each (E,d)\in C^N, each pair \{i,j\}\subset N, each d'_i>d_i, and each 0\leq d'_j<d_j, if d'_i+d'_j=d_i+d_j, then \mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))- \mathcal{R}_i(E,d)\leq d'_i-d_i and \mathcal{R}_j(E,d)-\mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq d_j-d'_j.
20) Concavity, if for each (E,d)\in C^N, each triple \{E,E',E''\}\subset \mathbb{R}^+ such that 0<E<E'<E''\leq D, and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, then \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) Convexity, if for each (E,d)\in C^N, each triple \{E,E',E''\}\subset \mathbb{R}^+ such that 0<E<E'<E''\leq D, and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, then \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) Endowment monotonicity, if for each (E,d)\in C^N and each E' >E, if D \geq E' then \mathcal{R}(E',d) \geq \mathcal{R}(E,d).
23) Order preservation under endowment variations, if for each (E,d)\in C^N, each pair \{i,j\}\subset N and each E'>E, if D\geq E' and d_i\leq d_j, then \mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E',d)-\mathcal{R}_j(E,d).
24) Order preservation under claims variations, if for each (E,d)\in C^N with |N|\geq 3, each i\in N, each d'_i>d_i, and each pair \{j,k\}\subset N\backslash\{i\}, if d_j\leq d_k, then \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) Minimal rights first, if for each (E,d)\in C^N, \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) Claims truncation invariance, if for each (E,d)\in C^N, \mathcal{R}(E,d) =\mathcal{R}( E, t(E,d) ).
27) Composition down, if for each (E,d)\in C^N and each E'<E, we have \mathcal{R}(E',d)=\mathcal{R}(E',\mathcal{R}(E,d)).
28) Composition up, if for each (E,d)\in C^N and each E'>E such that D \geq E', we have \mathcal{R}(E',d)=\mathcal{R}(E,d)+\mathcal{R}(E'-E,d-\mathcal{R}(E,d)).
29) Midpoint property, if for each (E,d)\in C^N such that E=\frac{1}{2}D, then \mathcal{R}(E,d)=\frac{d}{2}.
30) Self-duality, if for each (E,d)\in C^N, \mathcal{R}(E,d) =d- \mathcal{R}\bigl(D-E,d\bigr).
31) Claims separability, if for each pair (E,d),(E',d')\in C^N and each N'\subset N, if d_{N'}=d'_{N'}, E=E', and \sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d'), then \mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d').
32) Claims-and-endowment separability, if for each pair (E,d),(E',d')\in C^N and each N'\subset N, if d_{N'}=d'_{N'} and \sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d'), then \mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d').
33) Endowment convexity, if for each d\in\mathbb{R}^N_+, each pair \{E,E'\}\subset \mathbb{R}_+ such that D\geq \max\{E,E'\}, and each \lambda\in[0,1], \mathcal{R}(\lambda E+(1-\lambda)E',d)=\lambda\mathcal{R}(E,d)+(1-\lambda)\mathcal{R}(E',d).
34) Claims-and-endowment uniformity, if for each pair (E,d),(E',d')\in C^N, and each N'\subset N, if d_{N'}=d'_{N'}, then either \mathcal{R}_{N'}(E,d)\geq \mathcal{R}_{N'}(E',d') or \mathcal{R}_{N'}(E,d)\leq \mathcal{R}_{N'}(E',d').
35) Endowment-and-population uniformity, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each (E',d')\in C^{N'}, if d_{N'}=d', then either \mathcal{R}_{N'}(E,d)\geq \mathcal{R}(E',d') or \mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E',d').
36) No advantageous transfer, if for each (E,d)\in C^N, each N'\subset N, and each (d'_i)_{i\in N'}, if d'(N')=d(N'), then \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) Summation independence, if for each (E,d)\in C^N, each i\in N, and N'=N\backslash \{i\}, for each (d'_j)_{j\in N'}\in\mathbb{R}^{N'}_+, if d'(N')=d(N'), then \mathcal{R}_i(E,(d_i,(d'_j)_{j\in N'}))=\mathcal{R}_i(E,d).
38) Consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N if x=\mathcal{R}(E,d), then x_{N'}=\mathcal{R}(x(N'),d_{N'}).
39) Bilateral consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N and |N'|=2, and each (E,d)\in C^N if x=\mathcal{R}(E,d), then x_{N'}=\mathcal{R}(x(N'),d_{N'}).
40) Converse consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N and |N'|=2, each (E,d)\in C^N, and each x\in X(E,d), we have x_{N'}=\mathcal{R}(x(N'),d_{N'}), then x=\mathcal{R}(E,d).
41) Null claims consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N and each (E,d)\in C^N, if d_{N\backslash N'}=0, then \mathcal{R}_{N'}(E,d)= \mathcal{R}(E,d_{N'}).
42) Null compensation consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, if \mathcal{R}_{N\backslash N'}(E,d)=0, then \mathcal{R}_{N'}(E,d)=\mathcal{R}(E,d_{N'}).
43) Full compensation consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, if \mathcal{R}_{N\backslash N'}(E,d)=d_{N\backslash N'}, then \mathcal{R}_{N'}(E,d)=\mathcal{R}(E-d(N\backslash N'),d_{N'}).
44) No advantageous merging, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each d'\in \mathbb{R}^{N'}_+, if there is i\in N' such that d'_i=d_i+d(N\backslash N') and for each k\in N'\backslash\{i\}, d'_k=d_k, then \mathcal{R}_i(E,d')\leq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d).
45) No advantageous splitting, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each d'\in \mathbb{R}^{N'}_+, if there is i\in N' such that d'_i=d_i+d(N\backslash N') and for each k\in N'\backslash\{i\}, d'_k=d_k, then \mathcal{R}_i(E,d')\geq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d).
46) Order preservation under population variations, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each pair \{i,j\}\subset N', if d(N')\geq E and d_i\leq d_j, then \mathcal{R}_i(E,d_{N'})-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d_{N'})-\mathcal{R}_j(E,d).
47) Division invariance, if for each pair N,N' of subsets of \mathcal{N} such that N'\supset N, each (E,d)\in C^N and each k\in \mathbb{N}. Let (E',d') be a k-replica of (E,d) with associated partition (N^i)_{i\in N}, and \mathcal{R}(E',d') be a k-replica of some awards vector x\in X(E,d) associated with the same partition, we have \mathcal{R}(E,d)=x.
48) Replication invariance, if for each pair N,N' of subsets of \mathcal{N} such that N'\supset N, each (E,d)\in C^N and each k\in \mathbb{N}. Let (E',d') be a k-replica of (E,d) with associated partition (N^i)_{i\in N}, then for each i\in N and each j\in N^i we have \mathcal{R}_j(E',d')=\mathcal{R}_i(E,d).
Value
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.
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.
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_2 emissions. Annals of Operations Research 336, 1435-1459.
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.
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.
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.
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]
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| axioms | R Documentation |
Properties of the rules
Description
This function shows which of the main properties the rules satisfy.
Usage
axioms(Rules = "All", Properties = "All")
Arguments
Rules |
The rules: AA, APRO, CE, CEA, CEL, AV, MO, PIN, PRO, RA, Talmud. By default, |
Properties |
The properties listed in the description section. By default, |
Details
Let \mathcal{N} be the set of all finite nonempty subsets of the natural numbers \mathbb{N}. Let N=\{1,\ldots,n\} be the set of claimants, E\ge 0 the endowment to be divided and d\in \mathbb{R}_+^N the vector of claims
such that D=\sum_{i \in N} d_i\ge E. We denote the class of claims problems with set of
claimants N by C^N. Given z\in\mathbb{R}^N, S\in 2^N and N'\subset N\in\mathcal{N}, let z(S)=\sum_{j\in S}z_j, z_{N'}=(z_i)_{i\in N'} and denote z_{-i}=z_{N\backslash \{i\}}\in \mathbb{R}^{N\backslash \{i\}}. For simplicity, we will write z=(z_{-i},z).
The minimal right of claimant i\in N in (E,d) is defined by m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\}.
Let m(E,d)=(m_1(E,d),\dots,m_n(E,d)) be the vector of minimal rights.
The truncated claim of claimant i\in N in (E,d) is defined by
t_i(E,d)=\min\{d_i,E\}. Let t(E,d)=(t_1(E,d),\dots,t_n(E,d)) be the vector of truncated claims.
Given (E,d)\in C^N and k\in\mathbb{N}, we say that (E',d') is a k-replica of (E,d) if E'=k E, N'\supset N, |N'|=k|N|, and there is a partition (N^i)_{i\in N} of N' such that for each i\in N and each j\in N^i, |N^i|=k and d'_j=d_i.
A vector x=(x_1,\dots,x_n) is an awards vector for the claims problem (E,d)\in C^N if 0\le x \le d
and satisfies the balance requirement, that is, x(N)=E. Let X(E,d) be the set of awards vectors for the problem (E,d).
A rule is a function that assigns to each claims problem (E,d) an awards vector.
A rule \mathcal{R} satisfies:
1) Anonymity, if for each (E,d)\in C^N, each bijection f from N into itself, and each i\in N, we have that \mathcal{R}_i(E,d)=\mathcal{R}_{f(i)}(E,(d_{f(i)})_{i\in N}).
2) Continuity, if for each sequence \{(E^{\ell},d^{\ell})\} of elements of C^N and each (E,d)\in C^N, if (E^{\ell},d^{\ell})\rightarrow (E,d) then \mathcal{R}(E^{\ell},d^{\ell})\rightarrow \mathcal{R}(E,d).
3) Homogeneity, if for each (E,d)\in C^N and each \rho>0, then \mathcal{R}(\rho E,\rho d)=\rho \mathcal{R}(E,d).
4) \mathbf{\tfrac{1}{|N|}}-truncated-claims lower bounds on awards, if for each (E,d)\in C^N, then \mathcal{R}(E,d) \geq \frac{1}{|N|} t(E,d).
5) \mathbf{\tfrac{1}{|N|}}-min-of-claims-and-deficit lower bounds on losses, if for each (E,d)\in C^N, then d-\mathcal{R}(E,d) \geq \frac{1}{|N|} t(D-E,d).
6) Equal treatment of equals, if for each (E,d)\in C^N and each pair \{i,j\}\in N, if d_i=d_j, then \mathcal{R}_i(E,d)=\mathcal{R}_j(E,d).
7) Equal treatment of equal groups, if for each (E,d)\in C^N and each pair \{N',N''\} of subsets of N, if d(N')=d(N''), then \sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N''}\mathcal{R}_i(E,d).
8) Order preservation, if for each (E,d)\in C^N and each pair \{i,j\} \subset N, if d_i \leq d_j, then \mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d) (in awards) and d_i-\mathcal{R}_i(E,d)\leq d_j-\mathcal{R}_j(E,d) (in losses).
9) Group order preservation, if for each (E,d)\in C^N and each pair \{N',N''\} of subsets of N, if d(N') \leq d(N''), then \sum_{i\in N'}\mathcal{R}_i(E,d)\leq \sum_{i\in N''}\mathcal{R}_i(E,d) (in awards) and \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) Conditional null compensation, if for each (E,d)\in C^N and each i\in N, if \sum_{j\in N}\max\{0,d_j-d_i\}\geq E, then \mathcal{R}_i(E,d)=0.
11) Conditional full compensation, if for each (E,d)\in C^N and each i\in N, if \sum_{j\in N}\min\{d_j,d_i\} \leq E, then \mathcal{R}_i(E,d)=d_i.
12) Linked claim-endowment monotonicity, if for each (E,d)\in C^N, each i\in N, and each \delta>0, we have \mathcal{R}_i(E+\delta,(d_{-i},d_i+\delta))-\mathcal{R}_i(E,d)\leq \delta.
13) Claim monotonicity, if for each (E,d)\in C^N, each i \in N, and each d'_i \geq d_i, \mathcal{R}_i(E,(d_{-i}, d'_i))\geq \mathcal{R}_i(E,d).
14) Population monotonicity, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, we have \mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E,d_{N'}).
15) Linked endowment-population monotonicity, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, if d(N')\geq E, we have \mathcal{R}(E,d_{N'})\leq \mathcal{R}_{N'}(E+d(N\backslash N'),d).
16) Progressivity, if for each (E,d)\in C^N and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, we have \frac{\mathcal{R}_i(E,d)}{d_i}\leq \frac{\mathcal{R}_j(E,d)}{d_j}.
17) Regressivity, if for each (E,d)\in C^N and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, we have \frac{\mathcal{R}_i(E,d)}{d_i}\geq \frac{\mathcal{R}_j(E,d)}{d_j}.
18) No transfer paradox, if for each (E,d)\in C^N, each pair \{i,j\}\subset N, each d'_i>d_i, and each 0\leq d'_j<d_j, if d'_i+d'_j=d_i+d_j, then \mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}})) \geq \mathcal{R}_i(E,d) and \mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq \mathcal{R}_j(E,d).
19) Bounded impact of claims transfer, if for each (E,d)\in C^N, each pair \{i,j\}\subset N, each d'_i>d_i, and each 0\leq d'_j<d_j, if d'_i+d'_j=d_i+d_j, then \mathcal{R}_i(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))- \mathcal{R}_i(E,d)\leq d'_i-d_i and \mathcal{R}_j(E,d)-\mathcal{R}_j(E,(d'_i,d'_j,d_{N\backslash \{i,j\}}))\leq d_j-d'_j.
20) Concavity, if for each (E,d)\in C^N, each triple \{E,E',E''\}\subset \mathbb{R}^+ such that 0<E<E'<E''\leq D, and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, then \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) Convexity, if for each (E,d)\in C^N, each triple \{E,E',E''\}\subset \mathbb{R}^+ such that 0<E<E'<E''\leq D, and each pair \{i,j\}\subset N, if 0<d_i\leq d_j, then \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) Endowment monotonicity, if for each (E,d)\in C^N and each E' >E, if D \geq E' then \mathcal{R}(E',d) \geq \mathcal{R}(E,d).
23) Order preservation under endowment variations, if for each (E,d)\in C^N, each pair \{i,j\}\subset N and each E'>E, if D\geq E' and d_i\leq d_j, then \mathcal{R}_i(E',d)-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E',d)-\mathcal{R}_j(E,d).
24) Order preservation under claims variations, if for each (E,d)\in C^N with |N|\geq 3, each i\in N, each d'_i>d_i, and each pair \{j,k\}\subset N\backslash\{i\}, if d_j\leq d_k, then \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) Minimal rights first, if for each (E,d)\in C^N, \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) Claims truncation invariance, if for each (E,d)\in C^N, \mathcal{R}(E,d) =\mathcal{R}( E, t(E,d) ).
27) Composition down, if for each (E,d)\in C^N and each E'<E, we have \mathcal{R}(E',d)=\mathcal{R}(E',\mathcal{R}(E,d)).
28) Composition up, if for each (E,d)\in C^N and each E'>E such that D \geq E', we have \mathcal{R}(E',d)=\mathcal{R}(E,d)+\mathcal{R}(E'-E,d-\mathcal{R}(E,d)).
29) Midpoint property, if for each (E,d)\in C^N such that E=\frac{1}{2}D, then \mathcal{R}(E,d)=\frac{d}{2}.
30) Self-duality, if for each (E,d)\in C^N, \mathcal{R}(E,d) =d- \mathcal{R}\bigl(D-E,d\bigr).
31) Claims separability, if for each pair (E,d),(E',d')\in C^N and each N'\subset N, if d_{N'}=d'_{N'}, E=E', and \sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d'), then \mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d').
32) Claims-and-endowment separability, if for each pair (E,d),(E',d')\in C^N and each N'\subset N, if d_{N'}=d'_{N'} and \sum_{i\in N'}\mathcal{R}_i(E,d)=\sum_{i\in N'}\mathcal{R}_i(E',d'), then \mathcal{R}_{N'}(E,d)=\mathcal{R}_{N'}(E',d').
33) Endowment convexity, if for each d\in\mathbb{R}^N_+, each pair \{E,E'\}\subset \mathbb{R}_+ such that D\geq \max\{E,E'\}, and each \lambda\in[0,1], \mathcal{R}(\lambda E+(1-\lambda)E',d)=\lambda\mathcal{R}(E,d)+(1-\lambda)\mathcal{R}(E',d).
34) Claims-and-endowment uniformity, if for each pair (E,d),(E',d')\in C^N, and each N'\subset N, if d_{N'}=d'_{N'}, then either \mathcal{R}_{N'}(E,d)\geq \mathcal{R}_{N'}(E',d') or \mathcal{R}_{N'}(E,d)\leq \mathcal{R}_{N'}(E',d').
35) Endowment-and-population uniformity, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each (E',d')\in C^{N'}, if d_{N'}=d', then either \mathcal{R}_{N'}(E,d)\geq \mathcal{R}(E',d') or \mathcal{R}_{N'}(E,d)\leq \mathcal{R}(E',d').
36) No advantageous transfer, if for each (E,d)\in C^N, each N'\subset N, and each (d'_i)_{i\in N'}, if d'(N')=d(N'), then \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) Summation independence, if for each (E,d)\in C^N, each i\in N, and N'=N\backslash \{i\}, for each (d'_j)_{j\in N'}\in\mathbb{R}^{N'}_+, if d'(N')=d(N'), then \mathcal{R}_i(E,(d_i,(d'_j)_{j\in N'}))=\mathcal{R}_i(E,d).
38) Consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N if x=\mathcal{R}(E,d), then x_{N'}=\mathcal{R}(x(N'),d_{N'}).
39) Bilateral consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N and |N'|=2, and each (E,d)\in C^N if x=\mathcal{R}(E,d), then x_{N'}=\mathcal{R}(x(N'),d_{N'}).
40) Converse consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N and |N'|=2, each (E,d)\in C^N, and each x\in X(E,d), we have x_{N'}=\mathcal{R}(x(N'),d_{N'}), then x=\mathcal{R}(E,d).
41) Null claims consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N and each (E,d)\in C^N, if d_{N\backslash N'}=0, then \mathcal{R}_{N'}(E,d)= \mathcal{R}(E,d_{N'}).
42) Null compensation consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, if \mathcal{R}_{N\backslash N'}(E,d)=0, then \mathcal{R}_{N'}(E,d)=\mathcal{R}(E,d_{N'}).
43) Full compensation consistency, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, and each (E,d)\in C^N, if \mathcal{R}_{N\backslash N'}(E,d)=d_{N\backslash N'}, then \mathcal{R}_{N'}(E,d)=\mathcal{R}(E-d(N\backslash N'),d_{N'}).
44) No advantageous merging, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each d'\in \mathbb{R}^{N'}_+, if there is i\in N' such that d'_i=d_i+d(N\backslash N') and for each k\in N'\backslash\{i\}, d'_k=d_k, then \mathcal{R}_i(E,d')\leq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d).
45) No advantageous splitting, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each d'\in \mathbb{R}^{N'}_+, if there is i\in N' such that d'_i=d_i+d(N\backslash N') and for each k\in N'\backslash\{i\}, d'_k=d_k, then \mathcal{R}_i(E,d')\geq \mathcal{R}_i(E,d)+\sum_{j\in N\backslash N'} \mathcal{R}_j(E,d).
46) Order preservation under population variations, if for each pair N,N' of subsets of \mathcal{N} such that N'\subset N, each (E,d)\in C^N, and each pair \{i,j\}\subset N', if d(N')\geq E and d_i\leq d_j, then \mathcal{R}_i(E,d_{N'})-\mathcal{R}_i(E,d)\leq \mathcal{R}_j(E,d_{N'})-\mathcal{R}_j(E,d).
47) Division invariance, if for each pair N,N' of subsets of \mathcal{N} such that N'\supset N, each (E,d)\in C^N and each k\in \mathbb{N}. Let (E',d') be a k-replica of (E,d) with associated partition (N^i)_{i\in N}, and \mathcal{R}(E',d') be a k-replica of some awards vector x\in X(E,d) associated with the same partition, we have \mathcal{R}(E,d)=x.
48) Replication invariance, if for each pair N,N' of subsets of \mathcal{N} such that N'\supset N, each (E,d)\in C^N and each k\in \mathbb{N}. Let (E',d') be a k-replica of (E,d) with associated partition (N^i)_{i\in N}, then for each i\in N and each j\in N^i we have \mathcal{R}_j(E',d')=\mathcal{R}_i(E,d).
Value
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.
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.
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_2 emissions. Annals of Operations Research 336, 1435-1459.
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.
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.
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.
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]
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