March & April  2015, 2(3&4): 363-382. doi: 10.3934/jdg.2015011

For claims problems, another compromise between the proportional and constrained equal awards rules

1. 

Department of Economics, University of Rochester, Rochester, NY 14627, United States

Received  July 2015 Revised  October 2015 Published  November 2015

For the problem of adjudicating conflicting claims, we propose to compromise in the two-claimant case between the proportional and constrained equal awards rules by taking, for each problem, a weighted average of the awards vectors these two rules recommend. We allow the weights to depend on the claims vector, thereby generating a large family of rules. We identify the members of the family that satisfy particular properties.We then ask whether the rules can be extended topopulations of arbitrary sizes by imposing ``consistency": the recommendation made foreach problem should be ``in agreement" with the recommendation madefor each reduced problem that results when some claimants have received theirawards and left. We show that only the proportional and constrained equal awards rules qualify.We also study a dual compromise between the proportional and constrained equal losses rules.
Citation: William Thomson. For claims problems, another compromise between the proportional and constrained equal awards rules. Journal of Dynamics & Games, 2015, 2 (3&4) : 363-382. doi: 10.3934/jdg.2015011
References:
[1]

R. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud,, J. Econ. Theory, 36 (1985), 195. doi: 10.1016/0022-0531(85)90102-4. Google Scholar

[2]

K. Bosmans and L. Lauwers, Lorenz comparisons of nine rules for the adjudication of conflicting claims,, Int. J. Game Theory, 40 (2011), 791. doi: 10.1007/s00182-010-0269-z. Google Scholar

[3]

A. Cappelen, R. I. Luttens, E. Sorensen and B. Tungodden, Fairness in Bankruptcy Situations: An Experimental Study,, mimeo, (2015). doi: 10.2139/ssrn.2649022. Google Scholar

[4]

C. Chambers and J. Moreno-Ternero, Taxation and poverty,, Soc. Choice Wel., (2015), 1. doi: 10.1007/s00355-015-0905-4. Google Scholar

[5]

C. Chambers and W. Thomson, Group order preservation and the proportional rule for bankruptcy problems,, Math. Soc. Sci., 44 (2002), 235. doi: 10.1016/S0165-4896(02)00038-0. Google Scholar

[6]

S. Chen, Systematic favorability in claims problems with indivisibilities,, Soc. Choice Welf., 44 (2015), 283. doi: 10.1007/s00355-014-0828-5. Google Scholar

[7]

Y. Chun, The proportional solution for rights problem,, Math. Soc. Sci., 15 (1988), 231. doi: 10.1016/0165-4896(88)90009-1. Google Scholar

[8]

I. Curiel, M. Maschler and S. H. Tijs, Bankruptcy games,, Zeitschrift für Op. Research, 31 (1987). doi: 10.1007/BF02109593. Google Scholar

[9]

N. Dagan, R. Serrano and O. Volij, A non-cooperative view of consistent bankruptcy rules,, Games Econ. Behavior, 18 (1997), 55. doi: 10.1006/game.1997.0526. Google Scholar

[10]

N. Dagan and O. Volij, The bankruptcy problem: A cooperative bargaining approach,, Math. Soc. Sci., 26 (1993), 287. doi: 10.1016/0165-4896(93)90024-D. Google Scholar

[11]

S. Ertemel and R. Kumar, Ex-ante versus ex-post proportional rules for state contingent claims,, mimeo, (2014). Google Scholar

[12]

K. Flores-Szwagrzak, Priority classes and weighted constrained equal awards rules for the claims problem,, J. Econ. Theory, 160 (2015), 36. doi: 10.1016/j.jet.2015.08.008. Google Scholar

[13]

J. M. Giménez-Gómez and J. Peris, A proportional approach to claims problems with a guaranteed minimum,, European J. Oper. Res., 232 (2014), 109. doi: 10.1016/j.ejor.2013.06.039. Google Scholar

[14]

P. Harless, Generalized proportional rules for adjudicating conflicting claims,, mimeo, (2015). Google Scholar

[15]

C. Herrero and A. Villar, Sustainability in bankruptcy problems,, TOP, 10 (2002), 261. doi: 10.1007/BF02579019. Google Scholar

[16]

T. Hokari and W. Thomson, On properties of division rules lifted by bilateral consistency,, J. Math. Econom., 44 (2008), 1057. doi: 10.1016/j.jmateco.2008.01.001. Google Scholar

[17]

J. L. Hougaard and L. Thorlund-Peterson, Bankruptcy rules, inequality, and uncertainty,, mimeo, (2001). Google Scholar

[18]

B.-G. Ju, E. Miyagawa and T. Sakai, Non-manipulable division rules in claims problems and generalizations,, J. Econ. Theory, 132 (2007), 1. doi: 10.1016/j.jet.2005.08.003. Google Scholar

[19]

J. Moreno-Ternero and A. Villar, The Talmud rule and the securement of agents' awards,, Math. Soc. Sci., 47 (2004), 245. doi: 10.1016/S0165-4896(03)00087-8. Google Scholar

[20]

J. Moreno-Ternero and A. Villar, The TAL-family of rules for bankruptcy problems,, Soc. Choice Welf., 27 (2006), 231. doi: 10.1007/s00355-006-0121-3. Google Scholar

[21]

J. Moreno-Ternero and A. Villar, On the relative equitability of a family of taxation rules,, J. Pub. Econ. Theory, 8 (2006), 283. doi: 10.1111/j.1467-9779.2006.00264.x. Google Scholar

[22]

H. Moulin, Equal or proportional division of a surplus, and other methods,, Int. J. Game Theory, 16 (1987), 161. doi: 10.1007/BF01756289. Google Scholar

[23]

H. Moulin, Priority rules and other asymmetric rationing methods,, Econometrica, 68 (2000), 643. doi: 10.1111/1468-0262.00126. Google Scholar

[24]

B. O'Neill, A problem of rights arbitration from the Talmud,, Math. Soc Sci., 2 (1982), 345. doi: 10.1016/0165-4896(82)90029-4. Google Scholar

[25]

J. Stovall, Collective rationality and monotone path division rules,, J. Econ. Theory, 154 (2014), 1. doi: 10.1016/j.jet.2014.08.003. Google Scholar

[26]

W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey,, Math. Soc. Sci., 45 (2003), 249. doi: 10.1016/S0165-4896(02)00070-7. Google Scholar

[27]

W. Thomson, How To Divide When There Isn't Enough,, mimeo, (2006). Google Scholar

[28]

W. Thomson, On the existence of consistent rules to adjudicate conflicting claims: A geometric approach,, Rev. Econ. Design, 11 (2007), 225. doi: 10.1007/s10058-007-0027-2. Google Scholar

[29]

W. Thomson, Two families of rules for the adjudication of conflicting claims,, Soc. Choice Welf., 31 (2008), 667. doi: 10.1007/s00355-008-0302-3. Google Scholar

[30]

W. Thomson, Lorenz rankings of rules for the adjudication of conflicting claims,, Econ. Theory, 50 (2012), 547. doi: 10.1007/s00199-010-0575-5. Google Scholar

[31]

W. Thomson, On the axiomatics of resource allocation: Interpreting the consistency principle,, Econ. Phil., 28 (2012), 385. doi: 10.1017/S0266267112000296. Google Scholar

[32]

W. Thomson, Consistent Allocation Rules,, mimeo, (2012). Google Scholar

[33]

W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey,, Math. Social Sci., 45 (2013), 249. doi: 10.1016/S0165-4896(02)00070-7. Google Scholar

[34]

W. Thomson, For claims problems, compromising between the proportional and constrained equal awards rules,, Econ. Theory, 60 (2015), 495. doi: 10.1007/s00199-015-0888-5. Google Scholar

[35]

J. Xue, Claim uncertainty and egalitarian division with wastage,, mimeo, (2015). Google Scholar

[36]

P. Young, On dividing an amount according to individual claims or liabilities,, Math. Op. Research, 12 (1987), 398. doi: 10.1287/moor.12.3.398. Google Scholar

[37]

P. Young, Distributive justice in taxation,, J. Econ .Theory, 44 (1988), 321. doi: 10.1016/0022-0531(88)90007-5. Google Scholar

show all references

References:
[1]

R. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud,, J. Econ. Theory, 36 (1985), 195. doi: 10.1016/0022-0531(85)90102-4. Google Scholar

[2]

K. Bosmans and L. Lauwers, Lorenz comparisons of nine rules for the adjudication of conflicting claims,, Int. J. Game Theory, 40 (2011), 791. doi: 10.1007/s00182-010-0269-z. Google Scholar

[3]

A. Cappelen, R. I. Luttens, E. Sorensen and B. Tungodden, Fairness in Bankruptcy Situations: An Experimental Study,, mimeo, (2015). doi: 10.2139/ssrn.2649022. Google Scholar

[4]

C. Chambers and J. Moreno-Ternero, Taxation and poverty,, Soc. Choice Wel., (2015), 1. doi: 10.1007/s00355-015-0905-4. Google Scholar

[5]

C. Chambers and W. Thomson, Group order preservation and the proportional rule for bankruptcy problems,, Math. Soc. Sci., 44 (2002), 235. doi: 10.1016/S0165-4896(02)00038-0. Google Scholar

[6]

S. Chen, Systematic favorability in claims problems with indivisibilities,, Soc. Choice Welf., 44 (2015), 283. doi: 10.1007/s00355-014-0828-5. Google Scholar

[7]

Y. Chun, The proportional solution for rights problem,, Math. Soc. Sci., 15 (1988), 231. doi: 10.1016/0165-4896(88)90009-1. Google Scholar

[8]

I. Curiel, M. Maschler and S. H. Tijs, Bankruptcy games,, Zeitschrift für Op. Research, 31 (1987). doi: 10.1007/BF02109593. Google Scholar

[9]

N. Dagan, R. Serrano and O. Volij, A non-cooperative view of consistent bankruptcy rules,, Games Econ. Behavior, 18 (1997), 55. doi: 10.1006/game.1997.0526. Google Scholar

[10]

N. Dagan and O. Volij, The bankruptcy problem: A cooperative bargaining approach,, Math. Soc. Sci., 26 (1993), 287. doi: 10.1016/0165-4896(93)90024-D. Google Scholar

[11]

S. Ertemel and R. Kumar, Ex-ante versus ex-post proportional rules for state contingent claims,, mimeo, (2014). Google Scholar

[12]

K. Flores-Szwagrzak, Priority classes and weighted constrained equal awards rules for the claims problem,, J. Econ. Theory, 160 (2015), 36. doi: 10.1016/j.jet.2015.08.008. Google Scholar

[13]

J. M. Giménez-Gómez and J. Peris, A proportional approach to claims problems with a guaranteed minimum,, European J. Oper. Res., 232 (2014), 109. doi: 10.1016/j.ejor.2013.06.039. Google Scholar

[14]

P. Harless, Generalized proportional rules for adjudicating conflicting claims,, mimeo, (2015). Google Scholar

[15]

C. Herrero and A. Villar, Sustainability in bankruptcy problems,, TOP, 10 (2002), 261. doi: 10.1007/BF02579019. Google Scholar

[16]

T. Hokari and W. Thomson, On properties of division rules lifted by bilateral consistency,, J. Math. Econom., 44 (2008), 1057. doi: 10.1016/j.jmateco.2008.01.001. Google Scholar

[17]

J. L. Hougaard and L. Thorlund-Peterson, Bankruptcy rules, inequality, and uncertainty,, mimeo, (2001). Google Scholar

[18]

B.-G. Ju, E. Miyagawa and T. Sakai, Non-manipulable division rules in claims problems and generalizations,, J. Econ. Theory, 132 (2007), 1. doi: 10.1016/j.jet.2005.08.003. Google Scholar

[19]

J. Moreno-Ternero and A. Villar, The Talmud rule and the securement of agents' awards,, Math. Soc. Sci., 47 (2004), 245. doi: 10.1016/S0165-4896(03)00087-8. Google Scholar

[20]

J. Moreno-Ternero and A. Villar, The TAL-family of rules for bankruptcy problems,, Soc. Choice Welf., 27 (2006), 231. doi: 10.1007/s00355-006-0121-3. Google Scholar

[21]

J. Moreno-Ternero and A. Villar, On the relative equitability of a family of taxation rules,, J. Pub. Econ. Theory, 8 (2006), 283. doi: 10.1111/j.1467-9779.2006.00264.x. Google Scholar

[22]

H. Moulin, Equal or proportional division of a surplus, and other methods,, Int. J. Game Theory, 16 (1987), 161. doi: 10.1007/BF01756289. Google Scholar

[23]

H. Moulin, Priority rules and other asymmetric rationing methods,, Econometrica, 68 (2000), 643. doi: 10.1111/1468-0262.00126. Google Scholar

[24]

B. O'Neill, A problem of rights arbitration from the Talmud,, Math. Soc Sci., 2 (1982), 345. doi: 10.1016/0165-4896(82)90029-4. Google Scholar

[25]

J. Stovall, Collective rationality and monotone path division rules,, J. Econ. Theory, 154 (2014), 1. doi: 10.1016/j.jet.2014.08.003. Google Scholar

[26]

W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey,, Math. Soc. Sci., 45 (2003), 249. doi: 10.1016/S0165-4896(02)00070-7. Google Scholar

[27]

W. Thomson, How To Divide When There Isn't Enough,, mimeo, (2006). Google Scholar

[28]

W. Thomson, On the existence of consistent rules to adjudicate conflicting claims: A geometric approach,, Rev. Econ. Design, 11 (2007), 225. doi: 10.1007/s10058-007-0027-2. Google Scholar

[29]

W. Thomson, Two families of rules for the adjudication of conflicting claims,, Soc. Choice Welf., 31 (2008), 667. doi: 10.1007/s00355-008-0302-3. Google Scholar

[30]

W. Thomson, Lorenz rankings of rules for the adjudication of conflicting claims,, Econ. Theory, 50 (2012), 547. doi: 10.1007/s00199-010-0575-5. Google Scholar

[31]

W. Thomson, On the axiomatics of resource allocation: Interpreting the consistency principle,, Econ. Phil., 28 (2012), 385. doi: 10.1017/S0266267112000296. Google Scholar

[32]

W. Thomson, Consistent Allocation Rules,, mimeo, (2012). Google Scholar

[33]

W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey,, Math. Social Sci., 45 (2013), 249. doi: 10.1016/S0165-4896(02)00070-7. Google Scholar

[34]

W. Thomson, For claims problems, compromising between the proportional and constrained equal awards rules,, Econ. Theory, 60 (2015), 495. doi: 10.1007/s00199-015-0888-5. Google Scholar

[35]

J. Xue, Claim uncertainty and egalitarian division with wastage,, mimeo, (2015). Google Scholar

[36]

P. Young, On dividing an amount according to individual claims or liabilities,, Math. Op. Research, 12 (1987), 398. doi: 10.1287/moor.12.3.398. Google Scholar

[37]

P. Young, Distributive justice in taxation,, J. Econ .Theory, 44 (1988), 321. doi: 10.1016/0022-0531(88)90007-5. Google Scholar

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