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Genuine android action active for all permission setting complete
The proportional rule is a division rule for solving bankruptcy problems . According to this rule, each claimant should receive an amount proportional to their claim. In the context of taxation, it corresponds to a proportional tax .[1]
[ edit ]
There is a certain amount of money to divide, denoted by
E
{\displaystyle E}
(=Estate or Endowment). There are n claimants . Each claimant i has a claim denoted by
c
i
{\displaystyle c_{i}}
. Usually,
∑
i
=
1
n
c
i
>
E
{\displaystyle \sum _{i=1}^{n}c_{i}>E}
, that is, the estate is insufficient to satisfy all the claims.
The proportional rule says that each claimant i should receive
r
⋅
c
i
{\displaystyle r\cdot c_{i}}
, where r is a constant chosen such that
∑
i
=
1
n
r
⋅
c
i
=
E
{\displaystyle \sum _{i=1}^{n}r\cdot c_{i}=E}
. In other words, each agent gets
c
i
∑
j
=
1
n
c
j
⋅
E
{\displaystyle {\frac {c_{i}}{\sum _{j=1}^{n}c_{j}}}\cdot E}
.
Examples
[ edit ]
Examples with two claimants:
P
R
O
P
(
60
,
90
;
100
)
=
(
40
,
60
)
{\displaystyle PROP(60,90;100)=(40,60)}
. That is: if the estate is worth 100 and the claims are 60 and 90, then
r
=
2
/
3
{\displaystyle r=2/3}
, so the first claimant gets 40 and the second claimant gets 60.
P
R
O
P
(
50
,
100
;
100
)
=
(
33.333
,
66.667
)
{\displaystyle PROP(50,100;100)=(33.333,66.667)}
, and similarly
P
R
O
P
(
40
,
80
;
100
)
=
(
33.333
,
66.667
)
{\displaystyle PROP(40,80;100)=(33.333,66.667)}
.
Examples with three claimants:
P
R
O
P
(
100
,
200
,
300
;
100
)
=
(
16.667
,
33.333
,
50
)
{\displaystyle PROP(100,200,300;100)=(16.667,33.333,50)}
.
P
R
O
P
(
100
,
200
,
300
;
200
)
=
(
33.333
,
66.667
,
100
)
{\displaystyle PROP(100,200,300;200)=(33.333,66.667,100)}
.
P
R
O
P
(
100
,
200
,
300
;
300
)
=
(
50
,
100
,
150
)
{\displaystyle PROP(100,200,300;300)=(50,100,150)}
.
Characterizations
[ edit ]
The proportional rule has several characterizations . It is the only rule satisfying the following sets of axioms:
Self-duality and composition-up;[2]
Self-duality and composition-down;
No advantageous transfer;[3] [4] [5]
Resource linearity;[5]
No advantageous merging and no advantageous splitting.[5] [6] [7]
Truncated-proportional rule
[ edit ]
There is a variant called truncated-claims proportional rule , in which each claim larger than E is truncated to E , and then the proportional rule is activated. That is, it equals
P
R
O
P
(
c
1
′
,
…
,
c
n
′
,
E
)
{\displaystyle PROP(c_{1}',\ldots ,c_{n}',E)}
, where
c
i
′
:=
min
(
c
i
,
E
)
{\displaystyle c'_{i}:=\min(c_{i},E)}
. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:
T
P
R
O
P
(
100
,
200
,
300
;
100
)
=
(
33.333
,
33.333
,
33.333
)
{\displaystyle TPROP(100,200,300;100)=(33.333,33.333,33.333)}
, since all claims are truncated to 100;
T
P
R
O
P
(
100
,
200
,
300
;
200
)
=
(
40
,
80
,
80
)
{\displaystyle TPROP(100,200,300;200)=(40,80,80)}
, since the claims vector is truncated to (100,200,200).
T
P
R
O
P
(
100
,
200
,
300
;
300
)
=
(
50
,
100
,
150
)
{\displaystyle TPROP(100,200,300;300)=(50,100,150)}
, since here the claims are not truncated.
Adjusted-proportional rule
[ edit ]
The adjusted proportional rule [8] first gives, to each agent i , their minimal right , which is the amount not claimed by the other agents. Formally,
m
i
:=
max
(
0
,
E
−
∑
j
≠
i
c
j
)
{\displaystyle m_{i}:=\max(0,E-\sum _{j\neq i}c_{j})}
. Note that
∑
i
=
1
n
c
i
≥
E
{\displaystyle \sum _{i=1}^{n}c_{i}\geq E}
implies
m
i
≤
c
i
{\displaystyle m_{i}\leq c_{i}}
.
Then, it revises the claim of agent i to
c
i
′
:=
c
i
−
m
i
{\displaystyle c'_{i}:=c_{i}-m_{i}}
, and the estate to
E
′
:=
E
−
∑
i
m
i
{\displaystyle E':=E-\sum _{i}m_{i}}
. Note that that
E
′
≥
0
{\displaystyle E'\geq 0}
.
Finally, it activates the truncated-claims proportional rule, that is, it returns
T
P
R
O
P
(
c
1
,
…
,
c
n
,
E
′
)
=
P
R
O
P
(
c
1
″
,
…
,
c
n
″
,
E
′
)
{\displaystyle TPROP(c_{1},\ldots ,c_{n},E')=PROP(c_{1}'',\ldots ,c_{n}'',E')}
, where
c
i
″
:=
min
(
c
i
′
,
E
′
)
{\displaystyle c''_{i}:=\min(c'_{i},E')}
.
With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:
A
P
R
O
P
(
60
,
90
;
100
)
=
(
35
,
65
)
{\displaystyle APROP(60,90;100)=(35,65)}
. The minimal rights are
(
m
1
,
m
2
)
=
(
10
,
40
)
{\displaystyle (m_{1},m_{2})=(10,40)}
. The remaining claims are
(
c
1
′
,
c
2
′
)
=
(
50
,
50
)
{\displaystyle (c_{1}',c_{2}')=(50,50)}
and the remaining estate is
E
′
=
50
{\displaystyle E'=50}
; it is divided equally among the claimants.
A
P
R
O
P
(
50
,
100
;
100
)
=
(
25
,
75
)
{\displaystyle APROP(50,100;100)=(25,75)}
. The minimal rights are
(
m
1
,
m
2
)
=
(
0
,
50
)
{\displaystyle (m_{1},m_{2})=(0,50)}
. The remaining claims are
(
c
1
′
,
c
2
′
)
=
(
50
,
50
)
{\displaystyle (c_{1}',c_{2}')=(50,50)}
and the remaining estate is
E
′
=
50
{\displaystyle E'=50}
.
A
P
R
O
P
(
40
,
80
;
100
)
=
(
30
,
70
)
{\displaystyle APROP(40,80;100)=(30,70)}
. The minimal rights are
(
m
1
,
m
2
)
=
(
20
,
60
)
{\displaystyle (m_{1},m_{2})=(20,60)}
. The remaining claims are
(
c
1
′
,
c
2
′
)
=
(
20
,
20
)
{\displaystyle (c_{1}',c_{2}')=(20,20)}
and the remaining estate is
E
′
=
20
{\displaystyle E'=20}
.
With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are
(
0
,
0
,
0
)
{\displaystyle (0,0,0)}
and thus the outcome is equal to TPROP, for example,
A
P
R
O
P
(
100
,
200
,
300
;
200
)
=
T
P
R
O
P
(
100
,
200
,
300
;
200
)
=
(
20
,
40
,
40
)
{\displaystyle APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)}
.
See also
[ edit ]
References
[ edit ]
^ Moulin, Hervé (1985). "Egalitarianism and Utilitarianism in Quasi-Linear Bargaining" . Econometrica . 53 (1 ): 49–67. doi :10.2307/1911723 . ISSN 0012-9682 . JSTOR 1911723 .
^ Moulin, Hervé (1985-06-01). "The separability axiom and equal-sharing methods" . Journal of Economic Theory . 36 (1 ): 120–148. doi :10.1016/0022-0531(85 )90082-1 . ISSN 0022-0531 .
^ a b c Chun, Youngsub (1988-06-01). "The proportional solution for rights problems" . Mathematical Social Sciences . 15 (3 ): 231–246. doi :10.1016/0165-4896(88 )90009-1 . ISSN 0165-4896 .
^ O'Neill, Barry (1982-06-01). "A problem of rights arbitration from the Talmud" . Mathematical Social Sciences . 2 (4 ): 345–371. doi :10.1016/0165-4896(82 )90029-4 . hdl :10419/220805 . ISSN 0165-4896 .
^ de Frutos, M. Angeles (1999-09-01). "Coalitional manipulations in a bankruptcy problem" . Review of Economic Design . 4 (3 ): 255–272. doi :10.1007/s100580050037 . hdl :10016/4282 . ISSN 1434-4750 . S2CID 195240195 .
^ Curiel, I. J.; Maschler, M.; Tijs, S. H. (1987-09-01). "Bankruptcy games" . Zeitschrift für Operations Research . 31 (5 ): A143–A159. doi :10.1007/BF02109593 . ISSN 1432-5217 . S2CID 206811949 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Proportional_rule_(bankruptcy)&oldid=1195803120 "
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