Proportional rule (bankruptcy): Difference between revisions
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⚫ | The '''proportional rule''' is a division rule for solving [[bankruptcy problem]]s. 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]].<ref name=":1">{{Cite journal|last=William|first=Thomson|date=2003-07-01|title=Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey|url=https://www.sciencedirect.com/science/article/abs/pii/S0165489602000707|journal=Mathematical Social Sciences|language=en|volume=45|issue=3|pages=249–297|doi=10.1016/S0165-4896(02)00070-7|issn=0165-4896}}</ref> |
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{{Short description|Genuine android action active for all permission setting complete}} |
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⚫ | The '''proportional rule''' is a division rule for solving [[bankruptcy problem]]s. 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]].<ref name=":1">{{Cite journal|last=William|first=Thomson|date=2003-07-01|title=Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey|url=https://www.sciencedirect.com/science/article/abs/pii/S0165489602000707|journal=Mathematical Social Sciences|language=en|volume=45|issue=3|pages=249–297|doi=10.1016/S0165-4896(02)00070-7|issn=0165-4896}}</ref> |
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== Formal definition == |
== Formal definition == |
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* 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> |
* 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> |
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* Resource linearity;<ref name=":0" /> |
* Resource linearity;<ref name=":0" /> |
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*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}}</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> |
*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> |
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== Truncated-proportional rule == |
== Truncated-proportional rule == |
Latest revision as of 18:44, 27 July 2024
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]
Formal definition
[edit]There is a certain amount of money to divide, denoted by (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by . Usually, , that is, the estate is insufficient to satisfy all the claims.
The proportional rule says that each claimant i should receive , where r is a constant chosen such that . In other words, each agent gets .
Examples
[edit]Examples with two claimants:
- . That is: if the estate is worth 100 and the claims are 60 and 90, then , so the first claimant gets 40 and the second claimant gets 60.
- , and similarly .
Examples with three claimants:
- .
- .
- .
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 , where . The results are the same for the two-claimant problems above, but for the three-claimant problems we get:
- , since all claims are truncated to 100;
- , since the claims vector is truncated to (100,200,200).
- , 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, . Note that implies .
Then, it revises the claim of agent i to , and the estate to . Note that that .
Finally, it activates the truncated-claims proportional rule, that is, it returns , where .
With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:
- . The minimal rights are . The remaining claims are and the remaining estate is ; it is divided equally among the claimants.
- . The minimal rights are . The remaining claims are and the remaining estate is .
- . The minimal rights are . The remaining claims are and the remaining estate is .
With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are and thus the outcome is equal to TPROP, for example, .
See also
[edit]References
[edit]- ^ William, Thomson (2003-07-01). "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey". Mathematical Social Sciences. 45 (3): 249–297. doi:10.1016/S0165-4896(02)00070-7. ISSN 0165-4896.
- ^ Young, H. P (1988-04-01). "Distributive justice in taxation". Journal of Economic Theory. 44 (2): 321–335. doi:10.1016/0022-0531(88)90007-5. ISSN 0022-0531.
- ^ 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.