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The '''proportional rule''' is a division rule for solving [[bankruptcy problem]]
== Formal definition ==
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== Examples ==
Examples with two claimants:
* <math>PROP(60,90; 100) = (40,60)</math>. That is: if the estate is worth 100 and the claims are 60 and 90, then <math>r = 2/3</math>, so the first claimant gets 40 and the second claimant gets 60.
* <math>PROP(50,100; 100) = (33.333,66.667)</math>, and similarly <math>PROP(40,80; 100) = (33.333,66.667)</math>.
Examples with three claimants:
* <math>PROP(100,200,300; 100) = (16.667, 33.333, 50)</math>.
* <math>PROP(100,200,300; 200) = (33.333, 66.667, 100)</math>.
* <math>PROP(100,200,300; 300) = (50, 100, 150)</math>.
== Characterizations ==
== Truncated-claims variant ==▼
The proportional rule has several [[Characterization (mathematics)|characterizations]]. It is the only rule satisfying the following sets of axioms:
There is a variant called '''truncated-claims proportional rule''', in which each claim larger than ''E'' is truncted to ''E'', and then the proportional rule is activated. That is, it equals <math>PROP(c_1',\ldots,c_n',E)</math>, where <math>c'_i := \min(c_i, E)</math>. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:▼
* Self-duality and composition-up;<ref>{{Cite journal|last=Young|first=H. P|date=1988-04-01|title=Distributive justice in taxation|url=https://dx.doi.org/10.1016/0022-0531%2888%2990007-5|journal=Journal of Economic Theory|language=en|volume=44|issue=2|pages=321–335|doi=10.1016/0022-0531(88)90007-5|issn=0022-0531}}</ref>
* Self-duality and composition-down;
* No advantageous transfer;<ref>{{Cite journal|last=Moulin|first=Hervé|date=1985|title=Egalitarianism and Utilitarianism in Quasi-Linear Bargaining|url=https://www.jstor.org/stable/1911723|journal=Econometrica|volume=53|issue=1|pages=49–67|doi=10.2307/1911723|jstor=1911723 |issn=0012-9682}}</ref><ref>{{Cite journal|last=Moulin|first=Hervé|date=1985-06-01|title=The separability axiom and equal-sharing methods|url=https://dx.doi.org/10.1016/0022-0531%2885%2990082-1|journal=Journal of Economic Theory|language=en|volume=36|issue=1|pages=120–148|doi=10.1016/0022-0531(85)90082-1|issn=0022-0531}}</ref><ref name=":0">{{Cite journal|last=Chun|first=Youngsub|date=1988-06-01|title=The proportional solution for rights problems|url=https://dx.doi.org/10.1016/0165-4896%2888%2990009-1|journal=Mathematical Social Sciences|language=en|volume=15|issue=3|pages=231–246|doi=10.1016/0165-4896(88)90009-1|issn=0165-4896}}</ref>
* Resource linearity;<ref name=":0" />
*No advantageous merging and no advantageous splitting.<ref name=":0" /><ref name=":5">{{Cite journal|last=O'Neill|first=Barry|date=1982-06-01|title=A problem of rights arbitration from the Talmud|url=https://dx.doi.org/10.1016/0165-4896%2882%2990029-4|journal=Mathematical Social Sciences|language=en|volume=2|issue=4|pages=345–371|doi=10.1016/0165-4896(82)90029-4|issn=0165-4896|hdl=10419/220805|hdl-access=free}}</ref><ref>{{Cite journal|last=de Frutos|first=M. Angeles|date=1999-09-01|title=Coalitional manipulations in a bankruptcy problem|url=https://doi.org/10.1007/s100580050037|journal=Review of Economic Design|language=en|volume=4|issue=3|pages=255–272|doi=10.1007/s100580050037|issn=1434-4750|hdl=10016/4282|s2cid=195240195 |hdl-access=free}}</ref>
▲There is a variant called '''truncated-claims proportional rule''', in which each claim larger than ''E'' is
* <math>TPROP(100,200,300; 100) = (33.333, 33.333, 33.333)</math>, since all claims are truncated to 100;
* <math>TPROP(100,200,300; 200) = (40, 80, 80)</math>, since the claims vector is truncated to (100,200,200).
* <math>TPROP(100,200,300; 300) = (50, 100, 150)</math>, since here the claims are not truncated.
== Adjusted-proportional rule ==
The '''adjusted proportional rule'''<ref>{{Cite journal|last1=Curiel|first1=I. J.|last2=Maschler|first2=M.|last3=Tijs|first3=S. H.|date=1987-09-01|title=Bankruptcy games|url=https://doi.org/10.1007/BF02109593|journal=Zeitschrift für Operations Research|language=en|volume=31|issue=5|pages=A143–A159|doi=10.1007/BF02109593|s2cid=206811949 |issn=1432-5217}}</ref> first gives, to each agent ''i'', their ''minimal right'', which is the amount not claimed by the other agents. Formally, <math>m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. Note that <math>\sum_{i=1}^n c_i \geq E</math> implies <math>m_i \leq c_i</math>.
Then, it revises the claim of agent ''i'' to <math>c'_i := c_i - m_i</math>, and the estate to <math>E' := E - \sum_i m_i</math>. Note that that <math>E' \geq 0</math>.
Finally, it activates the truncated-claims proportional rule, that is, it returns <math>TPROP(c_1,\ldots,c_n,E') = PROP(c_1'',\ldots,c_n'',E')</math>, where <math>c''_i := \min(c'_i, E')</math>.
With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:
* <math>APROP(60,90; 100) = (35,65)</math>. The minimal rights are <math>(m_1,m_2) = (10,40)</math>. The remaining claims are <math>(c_1',c_2') = (50,50)</math> and the remaining estate is <math>E'=50</math>; it is divided equally among the claimants.
* <math>APROP(50,100; 100) = (25,75)</math>. The minimal rights are <math>(m_1,m_2) = (0,50)</math>. The remaining claims are <math>(c_1',c_2') = (50,50)</math> and the remaining estate is <math>E'=50</math>.
* <math>APROP(40,80; 100) = (30,70)</math>. The minimal rights are <math>(m_1,m_2) = (20,60)</math>. The remaining claims are <math>(c_1',c_2') = (20,20)</math> and the remaining estate is <math>E'=20</math>.
With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are <math>(0,0,0)</math> and thus the outcome is equal to TPROP, for example, <math>APROP(100,200,300; 200) = TPROP(100,200,300; 200) = (20, 40, 40)</math>.
== See also ==
* [[Proportional division]]
* [[Proportional representation]]
== References ==
{{Reflist}}
[[Category:Bankruptcy theory]]
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