Package 'ClaimsProblems'

Average of awards rule

Description

This function returns the awards vector assigned by the average of awards rule (AA) to a claims problem.

Usage

AA(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

A vector $x=(x_1,\dots,x_n)$ is an awards vector for the claims problem $(E,d)$ if $0\le x \le d$ and satisfies the balance requirement, that is, $\sum_{i=1}^{n}x_i=E$ the sum of its coordinates is equal to $E$. Let $X(E,d)$ be the set of awards vectors for $(E,d)$.

The average of awards rule assigns to each claims problem $(E,d)$ the expectation of the uniform distribution defined over the set of awards vectors, that is, the centroid of $X(E,d)$.

Let $\mu$ be the (n-1)-dimensional Lebesgue measure and $V(E,d)=\mu (X(E,d))$ the measure (volume) of the set of awards $X(E,d)$. The average of awards rule assigns to each problem $(E,d)$ the awards vector given by:

$AA(E,d)=\frac{1}{V(E,d)}\int_{X(E,d)} x d\mu$

The average of awards rule corresponds to the core-center of the associated coalitional (pessimistic) game.

Value

The awards vector selected by the AA rule. If name = TRUE, the name of the function (AA) as a character string.

References

Gonzalez-Díaz, J. and Sánchez-Rodríguez, E. (2007). A natural selection from the core of a TU game: the core-center. International Journal of Game Theory, 36(1), 27-46.

Mirás Calvo, M.Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2022). The average-of-awards rule for claims problems. Soc Choice Welf. doi:10.1007/s00355-022-01414-6

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2020). An algorithm to compute the core-center rule of a claims problem with an application to the allocation of CO2 emissions. Working paper.

See Also

allrules, CD, setofawards, coalitionalgame

Examples

E=10
d=c(2,4,7,8)
AA(E,d)
#The average of awards rule is self-dual: AA(E,d)=d-AA(D-E,d)
D=sum(d)
d-AA(D-E,d)

---

Summary of the division rules

Description

This function returns the awards vectors selected, for a given claims problem, by the rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, and Talmud.

Usage

allrules(E, d, draw = TRUE, col = NULL)

Arguments

E	The endowment.
d	The vector of claims.
draw	A logical value.
col	The colours (useful only if draw=TRUE). If col=NULL then the sequence of default colours is: c("red", "blue", "green", "yellow", "pink", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that \sum_{i=1}^{n} d_i\ge E,\ the sum of claims exceeds the endowment.

A vector $x=(x_1,\dots,x_n)$ is an awards vector for the claims problem $(E,d)$ if: no claimant is asked to pay ($0\le x$); no claimant receives more than his claim ($x\le d$); and the balance requirement is satisfied, that is, the sum of the awards is equal to the endowment ($\sum_{i=1}^{n} x_i= E$).

A rule is a function that assigns to each claims problem $(E,d)$ an awards vector for $(E,d)$, that is, a division between the claimants of the amount available.

The formal definitions of the main rules are given in the corresponding function help.

Value

A data-frame with the awards vectors selected by the main division rules. If draw = TRUE, it displays a mosaic plot representing the data-frame.

References

Mirás Calvo, M.Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2022). The average-of-awards rule for claims problems. Soc Choice Welf. doi:10.1007/s00355-022-01414-6

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.

See Also

AA, APRO, CD, CE, CEA, CEL , DT, MO, PIN, PRO, RA, Talmud, verticalruleplot

Examples

E=10
d=c(2,4,7,8)
allrules(E,d)

---

Adjusted proportional rule

Description

This function returns the awards vector assigned by the adjusted proportional rule (APRO) to a claims problem.

Usage

APRO(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,$ the sum of claims exceeds the endowment.

For each subset $S$ of the set of claimants $N$, let $d(S)=\sum_{j\in S}d_j$ be the sum of claims of the members of $S$ and let $N\backslash S$ be the complementary coalition of $S$.

The minimal right of claimant $i$ in $(E,d)$ is whatever is left after every other claimant has received his claim, or 0 if that is not possible:

$m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\},\ i=1,\dots,n.$

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 $(E,d)$ is the minimum of the claim and the endowment:

$t_i(E,d)=\min\{d_i,E\},\ i=1,\dots,n.$

Let $t(E,d)=(t_1(E,d),\dots,t_n(E,d))$ be the vector of truncated claims.

The adjusted proportional rule first gives to each claimant the minimal right, and then divides the remainder of the endowment $E'=E-\sum_{i=1}^n m_i(E,d)$ proportionally with respect to the new claims. The vector of the new claims $d'$ is determined by the minimum of the remainder and the lowered claims, $d_i'=\min\{E-\sum_{j=1}^n m_j(E,d),d_i-m_i\},\ i=1,\dots,n$. Therefore:

$APRO(E,d)=m(E,d)+PRO(E',d').$

The adjusted proportional rule corresponds to the $\tau$-value of the associated (pessimistic) coalitional game.

Value

The awards vector selected by the APRO rule. If name = TRUE, the name of the function (APRO) as a character string.

References

Curiel, I. J., Maschler, M., and Tijs, S. H. (1987). Bankruptcy games. Zeitschrift für operations research, 31(5), A143-A159.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.

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.

See Also

allrules, CD, PRO, coalitionalgame

Examples

E=10
d=c(2,4,7,8)
APRO(E,d)
#The adjusted proportional rule is self-dual: APRO(E,d)=d-APRO(D-E,d)
D=sum(d)
d-APRO(D-E,d)

---

Concede-and-divide rule

Description

This function returns the awards vector assigned by the concede-and-divide (CD) rule to a two-claimant problem.

Usage

CD(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of two claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d=(d_1,d_2)\in \mathcal{R}^2$ the vector of claims with $d\ge 0$ and such that the sum of the two claims exceeds the endowment $d_1+d_2 \ge E$.

The concede-and-divide rule first assigns to each of the two claimants the difference between the endowment and the other agent’s claim (or 0 if this difference is negative), and divides the remainder equally.

$CD_1(E,d)=\max\{E-d_2,0\}+\frac{E-\max\{E-d_1,0\}-\max\{E-d_2,0\}}{2}$

$CD_2(E,d)=\max\{E-d_1,0\}+\frac{E-\max\{E-d_1,0\}-\max\{E-d_2,0\}}{2}$

Several rules are extensions of the concede-and-divide rule to general populations: AA, APRO, MO, RA, and Talmud.

Value

The awards vector selected by the CD rule. If name = TRUE, the name of the function (CD) as a character string.

References

Aumann, R. and Maschler, M., (1985). Game theoretic analysis of a bankruptcy problem from the Talmud. J. Econ. Theory 36, 195–213.

Mirás Calvo, M. Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2021). Analyzing rules that extend the concede-and-divide principle. 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.

See Also

allrules, pathawards, AA, APRO, MO, RA, Talmud

Examples

E=10
d=c(7,8)
CD(E,d)
# Talmud, RA, MO, APRO, and AA coincide with CD for two-claimant problems
Talmud(E,d)
RA(E,d)
MO(E,d)
APRO(E,d)
AA(E,d)

---

Constrained egalitarian rule

Description

This function returns the awards vector assigned by the constrained egalitarian rule (CE) rule to a claims problem.

Usage

CE(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

Assume that the claims are ordered from small to large, $0 \le d_1 \le...\le d_n$. The constrained egalitarian rule coincides with the constrained equal awards rule (CEA) applied to the problem $(E, d/2)$ if the endowment is less or equal than the half-sum of the claims $D/2$. Otherwise, any additional unit is assigned to claimant $1$ until she/he receives the minimum of the claim and half of $d_2$. If this minimun is $d_1$, she/he stops there. If it is not, the next increment is divided equally between claimants $1$ and $2$ until claimant $1$ receives $d_1$ (in this case she drops out) or they reach $d_3/2$. If claimant $1$ leaves, claimant $2$ receives any aditional increment until she/he reaches $d_2$ or $d_3/2$. In the case that claimant $1$ and $2$ reach $d_3/2$, any additional unit is divided between claimants $1$, $2$, and $3$ until the first one receives $d_1$ or they reach $d_4/2$, and so on.

Therefore:

If $E \le D/2$ then $CE(E,d) = CEA(E,d/2)=(\min\{\frac{d_i}{2},\lambda\})_{i\in N}$ where $\lambda \ge 0$ is chosen so as to achieve balance.

If $E \ge D/2$ then the CE rule assigns to claimant $i$ the maximum of two quantities: the half-claim and the minimum of the claim and a value $\lambda \ge 0$ chosen so as to achieve balance.

$CE_i(E,d)=\max\{\frac{d_i}{2},\min\{d_i,\lambda\}\},\ i=1,\dots,n, \ where \ \sum_{i=1}^{n} CE_i(E,d)=E.$

Value

The awards vector selected by the CE rule. If name = TRUE, the name of the function (CE) as a character string.

References

Chun, Y., Schummer, J., Thomson, W. (2001). Constrained egalitarianism: a new solution for claims problems. Seoul J. Economics 14, 269–297.

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.

See Also

allrules, CEA, Talmud, PIN

Examples

E=10
d=c(2,4,7,8)
CE(E,d)

---

Constrained equal awards rule

Description

This function returns the awards vector assigned by the constrained equal awards rule (CEA) to a claims problem.

Usage

CEA(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and let $d\in \mathcal{R}^n$ be the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,$ the sum of claims exceeds the endowment.

The constrained equal awards rule (CEA) equalizes awards under the constraint that no individual's award exceeds his/her claim. Then, claimant $i$ receives the minimum of the claim and a value $\lambda \ge 0$ chosen so as to achieve balance.

$CEA_i(E,d)=\min\{d_i,\lambda\},\ i=1,\dots,n, \ such \ that \ \sum_{i=1}^{n} CEA_i(E,d)=E.$

The constrained equal awards rule corresponds to the Dutta-Ray solution to the associated (pessimistic) coalitional game. The CEA and CEL rules are dual.

Value

The awards vector selected by the CEA rule. If name = TRUE, the name of the function (CEA) as a character string.

References

Maimonides, Moses, 1135-1204. Book of Judgements, Moznaim Publishing Corporation, New York, Jerusalem (Translated by Rabbi Elihahu Touger, 2000).

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.

See Also

allrules, CE, CEL, PIN, Talmud

Examples

E=10
d=c(2,4,7,8)
CEA(E,d)
# CEA and CEL are dual: CEA(E,d)=d-CEL(D-E,d)
D=sum(d)
d-CEL(D-E,d)

---

Constrained equal losses rule

Description

This function returns the awards vector assigned by the constrained equal losses rule (CEL) to a claims problem.

Usage

CEL(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and let $d\in \mathcal{R}^n$ be the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,$ the sum of claims exceeds the endowment.

The constrained equal losses rule (CEL) equalizes losses under the constraint that no award is negative. Then, claimant $i$ receives the maximum of zero and the claim minus a number $\lambda \ge 0$ chosen so as to achieve balance.

$CEL_i(E,d)=\max\{0,d_i-\lambda\},\ i=1,\dots,n, \ such \ that \ \sum_{i=1}^n CEL_i(E,d)=E.$

CEA and CEL are dual rules.

Value

The awards vector selected by the CEL rule. If name = TRUE, the name of the function (CEL) as a character string.

References

Maimonides, Moses, 1135-1204. Book of Judgements, Moznaim Publishing Corporation, New York, Jerusalem (Translated by Rabbi Elihahu Touger, 2000).

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.

See Also

allrules, CEA

Examples

E=10
d=c(2,4,7,8)
CEL(E,d)
# CEL and CEA are dual: CEL(E,d)=d-CEA(D-E,d)
D=sum(d)
d-CEA(D-E,d)

---

Coalitional game associated with a claims problem

Description

This function returns the pessimistic and optimistic coalitional games associated with a claims problem.

Usage

coalitionalgame(E, d, opt = FALSE, lex = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
opt	Logical parameter. If opt = TRUE, both the pessimist and optimistic associated coalitional games are given. By default, opt = FALSE, and only the associated pessimistic coalitional game is computed.
lex	Logical parameter. If lex = TRUE, coalitions of claimants are ordered lexicographically. By default, lex = FALSE, and coalitions are ordered using their binary representations.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

For each subset $S$ of the set of claimants $N$, let $d(S)=\sum_{j\in S}d_j$ be the sum of claims of the members of $S$ and let $N\backslash S$ be the complementary coalition of $S$.

Given a claims problem $(E,d)$, its associated pessimistic coalitional game is the game $v_{pes}:2^N\rightarrow \mathcal{R}$ assigning to each coalition $S\in 2^N$ the real number:

$v_{pes}(S)=\max\{0,E-d(N\backslash S)\}.$

Given a claims problem $(E,d)$, its associated optimistic coalitional game is the game $v_{opt}:2^N\rightarrow \mathcal{R}$ assigning to each coalition $S\in 2^N$ the real number:

$v_{opt}(S)=\min\{E,d(S)\}.$

The optimistic and the pessimistic coalitional games are dual games, that is, for all $S\in 2^N$:

$v_{opt}(S)=E-v_{pes}(N\backslash S).$

An efficient way to represent a nonempty coalition $S\in 2^N$ is by identifying it with the binary sequence $a_{n}a_{n-1}\dots a_{1}$ where $a_i=1$ if $i\in S$ and $a_i=0$ otherwise. Therefore, each coalition $S$ is represented by the number associated with its binary representation: $\sum_{i\in T}2^{i-1}$. Then coalitions can be ordered by their associated numbers.

Alternatively, coalitions can be ordered lexicographically.

Given a claims problem $(E,d)$, its associated coalitional game $v$ can be represented by the vector whose coordinates are the values assigned by $v$ to all the nonempty coalitions. For instance. if $n=3$, the associated coalitional game can be represented by the vector of the values of all the 7 nonempty coalitions, ordered using the binary representation:

$v = [v(\{1\}),v(\{2\}),v(\{1,2\}),v(\{3\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]$

Alternatively, the coordinates can be ordered lexicographically:

$v = [v(\{1\}),v(\{2\}),v(\{3\}),v(\{1,2\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]$

When $n=4$, the associated coalitional game can be represented by the vector of the values of all the 15 nonempty coalitions, ordered using the binary representation:

$v = [v(\{1\}),v(\{2\}),v(\{1,2\}),v(\{3\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\}),v(\{4\}),$

$v(\{1,4\}),v(\{2,4\}),v(\{1,2,4\}),v(\{3,4\}),v(\{1,3,4\}),v(\{2,3,4\}),v(\{1,2,3,4\})]$

Alternatively, the coordinates can be ordered lexicographically:

$v=[v(\{1\}),v(\{2\}),v(\{3\}),v(\{4\}),v(\{1,2\}),v(\{1,3\}),v(\{1,4\}),v(\{2,3\}),\dots$

$\dots v(\{2,4\}),v(\{3,4\}),v(\{1,2,3\}),v(\{1,2,4\}),v(\{1,3,4\}),v(\{2,3,4\}),v(\{1,2,3,4\})]$

Value

The pessimistic (and optimistic) associated coalitional game(s).

References

O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2:345–371.

See Also

setofawards

Examples

E=10
d=c(2,4,7,8)
v=coalitionalgame(E,d,opt=TRUE,lex=TRUE)
#The pessimistic and optimistic coalitional games are dual games
v_pes=v$v_pessimistic_lex
v_opt=v$v_optimistic_lex
v_opt[1:14]==10-v_pes[14:1]

---

Cumulative awards curve

Description

The graphical representation of the cumulative curves of a rule (or several rules) with respect to a given rule, for a claims problem.

Usage

cumawardscurve(E, d, Rule = PRO, Rules, col = NULL, legend = TRUE)

Arguments

E	The endowment.
d	The vector of claims.
Rule	Principal Rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud. By default, Rule = PRO.
Rules	The rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
col	The colours. If col = NULL then the sequence of default colours is: c("red", "blue", "green", "yellow", "pink", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").
legend	A logical value. The colour legend is shown if legend = TRUE.

Details

Let $E> 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that the sum of claims $D=\sum_{i=1}^{n} d_i\ge E$ exceeds the endowment.

Rearrange the claims from small to large, $0 \le d_1 \le...\le d_n$. The cumulative curve allows us to compare the division recommended by a specific rule $R$ with the division the division recommended by another specific rule $S$.

The cumulative awards curve of a rule $S$ with respect of a rule $R$ for the claims problem $(E,d)$ is the polygonal path connecting the $n+1$ points

$(0,0), (\frac{R_1}{E},\frac{S_1}{E}),\dots,(\frac{\sum_{i=1}^{n-1}R_i}{E},\frac{\sum_{i=1}^{n-1}S_i}{E}),(1,1).$

The cumulative awards curve fully captures the Lorenz ranking of rules: if a rule $R$ Lorenz-dominates a rule $S$ then, for each claims problem, the cumulative curve of $R$ lies above the cumulative curve of $S$. If $R = PRO$, the cumulative curve coincides with the cumulative claims-awards curve.

$cumulativecurve$ function of version 0.1.0 returned the cumulative claims-awards curve with respect to the proportional rule.

Value

The graphical representation of the cumulative curves of a rule (or several rules) for a claims problem.

References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association, 9(70), 209-219.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi:10.1007/s10058-022-00300-y

See Also

deviationindex, indexgpath, lorenzcurve, giniindex, lorenzdominance, allrules.

Examples

E=10
d=c(2,4,7,8)
Rule=PRO
Rules=c(AA,RA,Talmud,CEA,CEL)
cumawardscurve(E,d,Rule,Rules)

---

Deviation index

Description

This function returns the deviation index and the signed deviation index for a rule with respect to another rule.

Usage

deviationindex(E, d, R, S)

Arguments

E	The endowment.
d	The vector of claims.
R	A rule : AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
S	A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.

Details

Let $E> 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $D=\sum_{i=1}^{n} d_i\ge E$, the sum of claims $D$ exceeds the endowment.

Rearrange the claims from small to large, $0 \le d_1 \le...\le d_n$. The signed deviation index of the rule $S$ with respect to the rule $R$ for the problem $(E,d)$, denoted by $I(R(E,d),S(E,d))$, is the ratio of the area that lies between the identity line and the cumulative curve over the total area under the identity line.

Let $R_0=0$ and $S_0=0$. For each $k=1,\dots,n$ define $X_k=\frac{1}{E} \sum_{j=0}^{k} R_j$ and $Y_k=\frac{1}{E} \sum_{j=0}^{k} S_j$. Then

$I(R(E,d),S(E,d))=1-\sum_{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1}).$

In general $-1 \le I(R(E,d),S(E,d)) \le 1$.

The deviation index of the rule $S$ with respect to the rule $R$ for the problem $(E,d)$, denoted by $I^{+}(R(E,d),S(E,d))$, is the ratio of the area between the line of the cumulative sum of the distribution proposed by the rule $R$ and the cumulative curve over the area under the line $x=y$.

In general $0 \le I^{+}(R(E,d),S(E,d)) \le 1$.

The proportionality deviation index is the deviation index when $R = PRO$. The proportionality deviation index of the proportional rule is zero for all claims problems. The signed proportionality deviation index is the signed deviation index with $R = PRO$.

$proportionalityindex$ function of version 0.1.0 returned the the signed proportionality index.

Value

The deviation index and the signed deviation index of a rule for a claims problem.

References

Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality, 10(3), 421-443.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi:10.1007/s10058-022-00300-y

See Also

indexgpath, cumawardscurve, lorenzcurve, giniindex, lorenzdominance, allrules.

Examples

E=10
d=c(2,4,7,8)
R=CEA
S=AA
deviationindex(E,d,R,S)
#The deviation index of rule R with respect of the rule R is 0.
deviationindex(E,d,PRO,PRO)

---

Dominguez-Thomson rule

Description

This function returns the awards vector assigned by the Dominguez-Thomson rule (DT) to a claims problem.

Usage

DT(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

The truncated claim of claimant $i$ in $(E,d)$ is the minimum of the claim and the endowment.

$t_i(E,d)=\min\{d_i,E\},\ i=1,\dots,n$

Let $t(E,d)=(m_1(E,d),\dots,m_n(E,d))$ be the vector of truncated claims and $b(E,d)=\frac{1}{n}t(E,d)$

The DT rule is defined recursively such that, in each step, each claimant receives the $n$-th part of the truncated claim.

Let $(E^1,d^1)=(E,d)$. For each $k\ge 2$ define:

$(E^k,d^k)=(E^{k-1}-\sum_{i=1}^n b_i(E^{k-1},d^{k-1}),d^{k-1}-b(E^{k-1},d^{k-1})).$

In step 1, the endowment is E and the claims vector is d. For $k \ge 2$, the endowment is the remainder once all the claimants have received the amount of the previous steps and the new vector of claims is readjusted accordingly. Observe that neither the endowment nor each agent's claim ever increases from one step to the next. This recursive process exhausts $E$ in the limit.

For each $(E,d)$ the Dominguez-Thomson rule assigns the awards vector:

$DT(E,d)=\sum_{k=1}^{\infty} b(E^k,d^k)$

Value

The awards vector selected by the DT rule. If name = TRUE, the name of the function (DT) as a character string.

References

Domínguez, D. and Thomson, W. (2006). A new solution to the problem of adjudicating conflicting claims. Economic Theory, 28(2), 283-307.

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.

See Also

allrules

Examples

E=10
d=c(2,4,7,8)
DT(E,d)

---

Dynamic plot

Description

For each claimaint, it plots the awards of the chosen rules for a dynamic model with t periods.

Usage

dynamicplot(
  E,
  d,
  Rules,
  claimant,
  percentage,
  times,
  col = NULL,
  legend = TRUE
)

Arguments

E	The endowment.
d	The vector of claims
Rules	The rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
claimant	A claimant.
percentage	A number in (0,1).
times	Number of iterations.
col	The colours. If col=NULL then the sequence of default colours is: c("red", "blue", "green", "yellow", "pink", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").
legend	A logical value. The colour legend is shown if legend=TRUE.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that \sum_{i=1}^{n} d_i\ge E,\ the sum of claims exceeds the endowment.

A vector $x=(x_1,\dots,x_n)$ is an awards vector for the claims problem $(E,d)$ if: no claimant is asked to pay ($0\le x$); no claimant receives more than his claim ($x\le d$); and the balance requirement is satisfied, that is, the sum of the awards is equal to the endowment ($\sum_{i=1}^{n} x_i= E$).

A rule is a function that assigns to each claims problem $(E,d)$ an awards vector for $(E,d)$, that is, a division between the claimants of the amount available.

The formal definitions of the main rules are given in the corresponding function help.

Given $l$ a natural number, the function solves each claims problem in time $t$, which is $(E_t,d)$, with $E_t=(1-p)^t E$, $p$ $\in$ $(0,1)$ and $t=1,\ldots,l$.

Value

This function represents the awards proposed by different rules for a claimant if the resource decreases in each iteration by a given percentage.

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.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2020). An algorithm to compute the core-center rule of a claims problem with an application to the allocation of CO2 emissions. Working paper.

See Also

allrules, pathawards, pathawards3, schedrule, schedrules

Examples

E=10
d=c(2,4,7,8)
Rules=c(Talmud,RA,AA,PRO)
claimant=1
percentage=0.076
times=10
dynamicplot(E,d,Rules,claimant,percentage,times)

---

Gini index

Description

This function returns the Gini index of any rule for a claims problem.

Usage

giniindex(E, d, Rule)

Arguments

E	The endowment.
d	The vector of claims.
Rule	A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.

Details

Let $E> 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $D=\sum_{i=1}^{n} d_i\ge E$, the sum of claims $D$ exceeds the endowment.

Rearrange the claims from small to large, $0 \le d_1 \le...\le d_n$. The Gini index is a number aimed at measuring the degree of inequality in a distribution. The Gini index of the rule $R$ for the problem $(E,d)$, denoted by $G(R,E,d)$, is the ratio of the area that lies between the identity line and the Lorenz curve of the rule over the total area under the identity line.

Let $R_0(E,d)=0$. For each $k=0,\dots,n$ define $X_k=\frac{k}{n}$ and $Y_k=\frac{1}{E} \sum_{j=0}^{k} R_j(E,d)$. Then

$G(R,E,d)=1-\sum_{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1}).$

In general $0\le G(R,E,d) \le 1$.

Value

The Gini index of a rule for a claims problem and the Gini index of the vector of claims.

References

Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality, 10(3), 421-443.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi:10.1007/s10058-022-00300-y

See Also

lorenzcurve, cumawardscurve, deviationindex, indexgpath, lorenzdominance.

Examples

E=10
d=c(2,4,7,8)
Rule=AA
giniindex(E,d,Rule)
# The Gini index of the proportional awards coincides with the Gini index of the vector of claims
giniindex(E,d,PRO)

---

Index path

Description

The function returns the deviation index path or the signed deviation index path for a rule with respect to another rule for a vector of claims.

Usage

indexgpath(
  d,
  Rule = PRO,
  Rules,
  signed = TRUE,
  col = NULL,
  points = 201,
  legend = TRUE
)

Arguments

d	The vector of claims.
Rule	Principal Rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud. By default, Rule = PRO.
Rules	The rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
signed	A logical value. If signed = FALSE, it draws the deviation index path and, if signed = TRUE it draws the signed deviation index path. By default, signed = TRUE.
col	The colours. If col = NULL then the sequence of default colours is: c("red", "blue", "green", "yellow", "pink", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").
points	The number of endowment values to be drawn.
legend	A logical value. The legend is shown if legend = TRUE.

Details

Let $d\in \mathcal{R}^n$ be a vector of claims rearranged from small to large, $0 \le d_1 \le...\le d_n$.

Given two rules $R$ and $S$, consider the function $J$ that assigns to each $E\in (0,D]$ the value $J(E)=I(R(E,d),S(E,d))$, that is, the signed deviation index of the rules $R$ and $S$ for the problem $(E,d)$. The graph of $J$ is the signed index path of $S$ in function of the rule $R$ for the vector of claims $d$.

Given two rules $R$ and $S$, consider the function $J^{+}$ that assigns to each $E\in (0,D]$ the value $J^{+}(E)=I^{+}(R(E,d),S(E,d))$, that is, the deviation index of the rules $R$ and $S$ for the problem $(E,d)$. The graph of $J^{+}$ is the index path of $S$ in function of the rule $R$ for the vector of claims $d$.

The signed index path and the index path are simple tools to visualize the discrepancy of the divisions recommended by a rule for a vector of claims with respect to the divisions recommended by another rule. If R = PRO, the function draws the proportionality deviation index path or the signed proportionality deviation index path.

$indexpath$ function of version 0.1.0 returned the signed proportionality deviation index path.

Value

This function returns the deviation index path of a rule (or several rules) for a vector of claims.

References

Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality, 10(3), 421-443.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi:10.1007/s10058-022-00300-y

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.

See Also

deviationindex, cumawardscurve, giniindex, lorenzcurve, lorenzdominance, allrules.

Examples

d=c(2,4,7,8)
Rule=PRO
Rules=c(Talmud,RA,AA)
col=c("red","green","blue")
indexgpath(d,Rule,Rules,signed=TRUE,col)

---

The Lorenz curve

Description

This function returns the Lorenz curve of any rule for a claims problem.

Usage

lorenzcurve(E, d, Rules, col = NULL, legend = TRUE)

Arguments

E	The endowment.
d	The vector of claims.
Rules	The rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
col	The colours. If col=NULL then the sequence of default colors is: c("red", "blue", "green", "yellow", "pink", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").
legend	A logical value. The colour legend is shown if legend=TRUE.

Details

Let $E> 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that the sum of claims $D=\sum_{i=1}^{n} d_i\ge E$ exceeds the endowment.

Rearrange the claims from small to large, $0 \le d_1 \le...\le d_n$. The Lorenz curve represents the proportion of the awards given to each subset of claimants by a specific rule $R$ as a function of the cumulative distribution of population.

The Lorenz curve of a rule $R$ for the claims problem $(E,d)$ is the polygonal path connecting the $n+1$ points

$(0,0), (\frac{1}{n},\frac{R_1(E,d)}{E}),\dots,(\frac{n-1}{n},\frac{\sum_{i=1}^{n-1}R_i(E,d)}{E}),(1,1)$

Basically, it represents the cumulative percentage of the endowment assigned by the rule to each cumulative percentage of claimants.

Value

The graphical representation of the Lorenz curve of a rule (or several rules) for a claims problem.

References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association, 9(70), 209-219.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi:10.1007/s10058-022-00300-y

See Also

giniindex, cumawardscurve, deviationindex, indexgpath, lorenzdominance.

Examples

E=10
d=c(2,4,7,8)
Rules=c(AA,RA,Talmud,CEA,CEL)
col=c("red","blue","green","yellow","pink")
lorenzcurve(E,d,Rules,col)

---

Lorenz-dominance relation

Description

This function checks whether or not the awards assigned by two rules to a claims problem are Lorenz-comparable.

Usage

lorenzdominance(E, d, Rules, Info = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
Rules	The two rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
Info	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

A vector $x=(x_1,\dots,x_n)$ is an awards vector for the claims problem $(E,d)$ if $0\le x \le d$ and satisfies the balance requirement, that is, $\sum_{i=1}^{n}x_i=E$ the sum of its coordinates is equal to $E$. Let $X(E,d)$ be the set of awards vectors for $(E,d)$.

Given a claims problem $(E,d)$, in order to compare a pair of awards vectors $x,y\in X(E,d)$ with the Lorenz criterion, first one has to rearrange the coordinates of each allocation in a non-decreasing order. Then we say that $x$ Lorenz-dominates $y$ (or, that $y$ is Lorenz-dominated by $x$) if all the cumulative sums of the rearranged coordinates are greater with $x$ than with $y$. That is, $x$ Lorenz-dominates $y$ if for each $k=1,\dots,n-1$ we have that

$\sum_{j=1}^{k}x_j \geq \sum_{j=1}^{k}y_j$

Let $R$ and $R'$ be two rules. We say that $R$ Lorenz-dominates $R'$ if $R(E,d)$ Lorenz-dominates $R'(E,d)$ for all $(E,d)$.

Value

If Info = FALSE, the Lorenz-dominance relation between the awards vectors selected by both rules. If both awards vectors are equal then cod = 2. If the awards vectors are not Lorenz-comparable then cod = 0. If the awards vector selected by the first rule Lorenz-dominates the awards vector selected by the second rule then cod = 1; otherwise cod = -1. If Info = TRUE, it also gives the corresponding cumulative sums.

References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association, 9(70), 209-219.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi:10.1007/s10058-022-00300-y

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.

See Also

cumawardscurve, deviationindex, indexgpath, lorenzcurve, giniindex.

Examples

E=10
d=c(2,4,7,8)
Rules=c(AA,CEA)
lorenzdominance(E,d,Rules)

---

Minimal overlap rule

Description

This function returns the awards vector assigned by the minimal overlap rule rule (MO) to a claims problem.

Usage

MO(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

The truncated claim of a claimant $i$ is the minimum of the claim and the endowment:

$t_i(E,d)=t_i=\min\{d_i,E\},\ i=1,\dots,n$

Suppose that each agent claims specific parts of E equal to her/his claim. After arranging which parts agents claim so as to “minimize conflict”, equal division prevails among all agents claiming a specific part and each agent receives the sum of the compensations she/he gets from the various parts that he claimed.

Let $d_0=0$. The minimal overlap rule is defined, for each problem $(E,d)$ and each claimant $i$, as:

If $E\le d_n$ then

$MO_i(E,d)=\frac{t_1}{n}+\frac{t_2-t_1}{n-1}+\dots+\frac{t_i-t_{i-1}}{n-i+1}.$

If $E>d_n$ let $s\in (d_k,d_{k+1}]$, with $k\in \{0,1,\dots,n-2\}$, be the unique solution to the equation $\sum_{i \in N} \max\{d_i-s,0\} =E-s$. Then:

$MO_i(E,d)=\frac{d_1}{n}+\frac{d_2-d_1}{n-1}+\dots+\frac{d_i-d_{i-1}}{n-i+1}, \ i\in\{1,\dots,k\}$

$MO_i(E,d)=MO_i(s,d)+d_i-s, \ i\in\{k+1,\dots,n\}.$

Value

The awards vector selected by the MO rule. If name = TRUE, the name of the function (MO) as a character string.

References

Mirás Calvo, M.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.

O'Neill, B. (1982). A problem of rights arbitration from the Talmud. Math. Social Sci. 2, 345-371.

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.

See Also

allrules, CD.

Examples

E=10
d=c(2,4,7,8)
MO(E,d)

---

The path of awards for two claimants

Description

This function returns the graphical representation of the path of awards of any rule for a claims vector and a pair of claimants.

Usage

pathawards(d, claimants, Rule, col = "red", points = 201)

Arguments

d	The vector of claims.
claimants	Two claimants.
Rule	The rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
col	The colour.
points	The number of values of the endowment to draw the path.

Details

Let $d\in \mathcal{R}^n$, with $d\ge 0$, be a vector of claims and denote $D=\sum_{i=1}^{n} d_i$ the sum of claims.

The path of awards of a rule $R$ for two claimants $i$ and $j$ is the parametric curve:

$p(E)=\{(R_i(E,d),R_j(E,d))\in \mathcal{R}^2:\;E\in[0,D]\}.$

Value

The graphical representation of the path of awards of a rule for the given claims and a pair of claimants.

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.

See Also

pathawards3, schedrule, schedrules, verticalruleplot

Examples

d=c(2,4,7,8)
claimants=c(1,2)
Rule=Talmud
pathawards(d,claimants,Rule)
# The path of awards of the concede-and-divide rule
pathawards(c(2,3),c(1,2),CD)
#The path of awards of the DT rule for d=(d1,d2) with d2<2d1
pathawards(c(1,1.5),c(1,2),DT,col="blue",points=1001)
#The path of awards of the DT rule for d=(d1,d2) with d2>2d1
pathawards(c(1,2.5),c(1,2),DT,col="blue",points=1001)

---

The path of awards for three claimants

Description

This function returns the graphical representation of the path of awards of any rule for a claims vector and three claimants.

Usage

pathawards3(d, claimants, Rule, col = "red", points = 300)

Arguments

d	The vector of claims.
claimants	Three claimants.
Rule	The rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
col	The colour of the path, by default, col="red".
points	The number of values of the endowment to draw the path.

Details

Let $d\in \mathcal{R}^n$, with $d\ge 0$, be a vector of claims and denote $D=\sum_{i=1}^{n} d_i$ the sum of claims.

The path of awards of a rule $R$ for three claimants $i$, $j$, and $k$ is the parametric curve:

$p(E)=\{(R_i(E,d),R_j(E,d),R_k(E,d))\in \mathcal{R}^3:\;E\in[0,D]\}.$

Value

The graphical representation of the path of awards of a rule for the given claims and three claimants.

See Also

pathawards, schedrule, schedrules, verticalruleplot

Examples

d=c(2,4,7,8)
claimants=c(1,3,4)
Rule=Talmud
pathawards3(d,claimants,Rule)

---

Piniles' rule

Description

This function returns the awards vector assigned by the Piniles' rule (PIN) to a claims problem.

Usage

PIN(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $D=\sum_{i=1}^{n} d_i\ge E$, the sum of claims $D$ exceeds the endowment.

The Piniles' rule coincides with the constrained equal awards rule (CEA) applied to the problem $(E, d/2)$ if the endowment is less or equal than the half-sum of the claims, $D/2$. Otherwise it assigns to each claimant $i$ half of the claim, $d_i/2$ and, then, it distributes the remainder with the CEA rule. Therefore:

If $E \le \frac{D}{2}$ then,

$PIN(E,d) = CEA(E,d/2).$

If $E \ge \frac{D}{2}$ then,

$PIN(E,d)=d/2+CEA(E-D/2,d/2).$

Value

The awards vector selected by the PIN rule. If name = TRUE, the name of the function (PIN) as a character string.

References

Piniles, H.M. (1861). Darkah shel Torah. Forester, Vienna.

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.

See Also

allrules, CEA, Talmud

Examples

E=10
d=c(2,4,7,8)
PIN(E,d)

---

Plot of an awards vector

Description

This function plots an awards vector in the set of awards vectors for a claims problem.

Usage

plotrule(E, d, Rule = NULL, awards = NULL, set = TRUE, col = "blue")

Arguments

E	The endowment.
d	The vector of claims.
Rule	A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
awards	An awards vector.
set	A logical value.
col	The colour.

Details

Let $E\ge 0$ be the endowment to be divided and $d\in \mathcal{R}^n$ the vector of claims with $d\ge 0$ and such that $\sum_{i=1}^{n} d_i\ge E,\;$ the sum of claims exceeds the endowment.

A vector $x=(x_1,\dots,x_n)$ is an awards vector for the claims problem $(E,d)$ if $0\le x \le d$ and satisfies the balance requirement, that is, $\sum_{i=1}^{n}x_i=E$ the sum of its coordinates is equal to $E$. 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 for $(E,d)$, that is, a division between the claimants of the amount available.

Value

If set = TRUE, the function creates a new figure plotting both the set of awards vectors for the claims problem and the given awards vector. Otherwise, it just adds to the current picture the point representing the given awards vector. The function only plots one awards vector at a time.

The awards vector can be introduced directly as a vector. Alternatively, we can provide a rule and then the awards vector to be plotted is the one selected by the rule for the claims problem. Therefore, if Rule = NULL it plots the given awards vector. Otherwise, it plots the awards vector selected by the given rule for the claims problem. In order to plot two (or more) awards vectors, draw the first one with the option set = TRUE and add the others, one by one, with the option set = FALSE.

See Also

setofawards, allrules

Examples

E=10
d=c(2,4,7,8)
plotrule(E,d,Rule=AA,col="red")
# Plotting the awards vector (1,3,5,1) and the AA rule
# First, plot the awards vector (1,3,5,1) and the set of awards
plotrule(E,d,awards=c(1,3,5,1),col="green")
# Second, add the AA rule with the option set=FALSE
plotrule(E,d,Rule=AA,set=FALSE,col="red")

---