# American Institute of Mathematical Sciences

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 and 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), 195-213. doi: 10.1016/0022-0531(85)90102-4. [2] K. Bosmans and L. Lauwers, Lorenz comparisons of nine rules for the adjudication of conflicting claims, Int. J. Game Theory, 40 (2011), 791-807. doi: 10.1007/s00182-010-0269-z. [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. [4] C. Chambers and J. Moreno-Ternero, Taxation and poverty, Soc. Choice Wel., forthcoming, 2015, 1-23. doi: 10.1007/s00355-015-0905-4. [5] C. Chambers and W. Thomson, Group order preservation and the proportional rule for bankruptcy problems, Math. Soc. Sci., 44 (2002), 235-252. doi: 10.1016/S0165-4896(02)00038-0. [6] S. Chen, Systematic favorability in claims problems with indivisibilities, Soc. Choice Welf., 44 (2015), 283-300. doi: 10.1007/s00355-014-0828-5. [7] Y. Chun, The proportional solution for rights problem, Math. Soc. Sci., 15 (1988), 231-246. doi: 10.1016/0165-4896(88)90009-1. [8] I. Curiel, M. Maschler and S. H. Tijs, Bankruptcy games, Zeitschrift für Op. Research, 31 (1987), A143-A159. doi: 10.1007/BF02109593. [9] N. Dagan, R. Serrano and O. Volij, A non-cooperative view of consistent bankruptcy rules, Games Econ. Behavior, 18 (1997), 55-72. doi: 10.1006/game.1997.0526. [10] N. Dagan and O. Volij, The bankruptcy problem: A cooperative bargaining approach, Math. Soc. Sci., 26 (1993), 287-297. doi: 10.1016/0165-4896(93)90024-D. [11] S. Ertemel and R. Kumar, Ex-ante versus ex-post proportional rules for state contingent claims, mimeo, 2014. [12] K. Flores-Szwagrzak, Priority classes and weighted constrained equal awards rules for the claims problem, J. Econ. Theory, 160 (2015), 36-55. doi: 10.1016/j.jet.2015.08.008. [13] J. M. Giménez-Gómez and J. Peris, A proportional approach to claims problems with a guaranteed minimum, European J. Oper. Res., 232 (2014), 109-116. doi: 10.1016/j.ejor.2013.06.039. [14] P. Harless, Generalized proportional rules for adjudicating conflicting claims, mimeo, 2015. [15] C. Herrero and A. Villar, Sustainability in bankruptcy problems, TOP, 10 (2002), 261-273. doi: 10.1007/BF02579019. [16] T. Hokari and W. Thomson, On properties of division rules lifted by bilateral consistency, J. Math. Econom., 44 (2008), 1057-1071. doi: 10.1016/j.jmateco.2008.01.001. [17] J. L. Hougaard and L. Thorlund-Peterson, Bankruptcy rules, inequality, and uncertainty, mimeo, 2001. [18] B.-G. Ju, E. Miyagawa and T. Sakai, Non-manipulable division rules in claims problems and generalizations, J. Econ. Theory, 132 (2007), 1-26. doi: 10.1016/j.jet.2005.08.003. [19] J. Moreno-Ternero and A. Villar, The Talmud rule and the securement of agents' awards, Math. Soc. Sci., 47 (2004), 245-257. doi: 10.1016/S0165-4896(03)00087-8. [20] J. Moreno-Ternero and A. Villar, The TAL-family of rules for bankruptcy problems, Soc. Choice Welf., 27 (2006), 231-249. doi: 10.1007/s00355-006-0121-3. [21] J. Moreno-Ternero and A. Villar, On the relative equitability of a family of taxation rules, J. Pub. Econ. Theory, 8 (2006), 283-291. doi: 10.1111/j.1467-9779.2006.00264.x. [22] H. Moulin, Equal or proportional division of a surplus, and other methods, Int. J. Game Theory, 16 (1987), 161-186. doi: 10.1007/BF01756289. [23] H. Moulin, Priority rules and other asymmetric rationing methods, Econometrica, 68 (2000), 643-684. doi: 10.1111/1468-0262.00126. [24] B. O'Neill, A problem of rights arbitration from the Talmud, Math. Soc Sci., 2 (1982), 345-371. doi: 10.1016/0165-4896(82)90029-4. [25] J. Stovall, Collective rationality and monotone path division rules, J. Econ. Theory, 154 (2014), 1-24. doi: 10.1016/j.jet.2014.08.003. [26] W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey, Math. Soc. Sci., 45 (2003), 249-297. doi: 10.1016/S0165-4896(02)00070-7. [27] W. Thomson, How To Divide When There Isn't Enough, mimeo, 2006. [28] W. Thomson, On the existence of consistent rules to adjudicate conflicting claims: A geometric approach, Rev. Econ. Design, 11 (2007), 225-251. doi: 10.1007/s10058-007-0027-2. [29] W. Thomson, Two families of rules for the adjudication of conflicting claims, Soc. Choice Welf., 31 (2008), 667-692. doi: 10.1007/s00355-008-0302-3. [30] W. Thomson, Lorenz rankings of rules for the adjudication of conflicting claims, Econ. Theory, 50 (2012), 547-569. doi: 10.1007/s00199-010-0575-5. [31] W. Thomson, On the axiomatics of resource allocation: Interpreting the consistency principle, Econ. Phil., 28 (2012), 385-421. doi: 10.1017/S0266267112000296. [32] W. Thomson, Consistent Allocation Rules, mimeo, 2012c. [33] W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey, Math. Social Sci., 45 (2013), 249-297. doi: 10.1016/S0165-4896(02)00070-7. [34] W. Thomson, For claims problems, compromising between the proportional and constrained equal awards rules, Econ. Theory, 60 (2015), 495-520. doi: 10.1007/s00199-015-0888-5. [35] J. Xue, Claim uncertainty and egalitarian division with wastage, mimeo, 2015. [36] P. Young, On dividing an amount according to individual claims or liabilities, Math. Op. Research, 12 (1987), 398-414. doi: 10.1287/moor.12.3.398. [37] P. Young, Distributive justice in taxation, J. Econ .Theory, 44 (1988), 321-335. doi: 10.1016/0022-0531(88)90007-5.

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##### References:
 [1] R. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud, J. Econ. Theory, 36 (1985), 195-213. doi: 10.1016/0022-0531(85)90102-4. [2] K. Bosmans and L. Lauwers, Lorenz comparisons of nine rules for the adjudication of conflicting claims, Int. J. Game Theory, 40 (2011), 791-807. doi: 10.1007/s00182-010-0269-z. [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. [4] C. Chambers and J. Moreno-Ternero, Taxation and poverty, Soc. Choice Wel., forthcoming, 2015, 1-23. doi: 10.1007/s00355-015-0905-4. [5] C. Chambers and W. Thomson, Group order preservation and the proportional rule for bankruptcy problems, Math. Soc. Sci., 44 (2002), 235-252. doi: 10.1016/S0165-4896(02)00038-0. [6] S. Chen, Systematic favorability in claims problems with indivisibilities, Soc. Choice Welf., 44 (2015), 283-300. doi: 10.1007/s00355-014-0828-5. [7] Y. Chun, The proportional solution for rights problem, Math. Soc. Sci., 15 (1988), 231-246. doi: 10.1016/0165-4896(88)90009-1. [8] I. Curiel, M. Maschler and S. H. Tijs, Bankruptcy games, Zeitschrift für Op. Research, 31 (1987), A143-A159. doi: 10.1007/BF02109593. [9] N. Dagan, R. Serrano and O. Volij, A non-cooperative view of consistent bankruptcy rules, Games Econ. Behavior, 18 (1997), 55-72. doi: 10.1006/game.1997.0526. [10] N. Dagan and O. Volij, The bankruptcy problem: A cooperative bargaining approach, Math. Soc. Sci., 26 (1993), 287-297. doi: 10.1016/0165-4896(93)90024-D. [11] S. Ertemel and R. Kumar, Ex-ante versus ex-post proportional rules for state contingent claims, mimeo, 2014. [12] K. Flores-Szwagrzak, Priority classes and weighted constrained equal awards rules for the claims problem, J. Econ. Theory, 160 (2015), 36-55. doi: 10.1016/j.jet.2015.08.008. [13] J. M. Giménez-Gómez and J. Peris, A proportional approach to claims problems with a guaranteed minimum, European J. Oper. Res., 232 (2014), 109-116. doi: 10.1016/j.ejor.2013.06.039. [14] P. Harless, Generalized proportional rules for adjudicating conflicting claims, mimeo, 2015. [15] C. Herrero and A. Villar, Sustainability in bankruptcy problems, TOP, 10 (2002), 261-273. doi: 10.1007/BF02579019. [16] T. Hokari and W. Thomson, On properties of division rules lifted by bilateral consistency, J. Math. Econom., 44 (2008), 1057-1071. doi: 10.1016/j.jmateco.2008.01.001. [17] J. L. Hougaard and L. Thorlund-Peterson, Bankruptcy rules, inequality, and uncertainty, mimeo, 2001. [18] B.-G. Ju, E. Miyagawa and T. Sakai, Non-manipulable division rules in claims problems and generalizations, J. Econ. Theory, 132 (2007), 1-26. doi: 10.1016/j.jet.2005.08.003. [19] J. Moreno-Ternero and A. Villar, The Talmud rule and the securement of agents' awards, Math. Soc. Sci., 47 (2004), 245-257. doi: 10.1016/S0165-4896(03)00087-8. [20] J. Moreno-Ternero and A. Villar, The TAL-family of rules for bankruptcy problems, Soc. Choice Welf., 27 (2006), 231-249. doi: 10.1007/s00355-006-0121-3. [21] J. Moreno-Ternero and A. Villar, On the relative equitability of a family of taxation rules, J. Pub. Econ. Theory, 8 (2006), 283-291. doi: 10.1111/j.1467-9779.2006.00264.x. [22] H. Moulin, Equal or proportional division of a surplus, and other methods, Int. J. Game Theory, 16 (1987), 161-186. doi: 10.1007/BF01756289. [23] H. Moulin, Priority rules and other asymmetric rationing methods, Econometrica, 68 (2000), 643-684. doi: 10.1111/1468-0262.00126. [24] B. O'Neill, A problem of rights arbitration from the Talmud, Math. Soc Sci., 2 (1982), 345-371. doi: 10.1016/0165-4896(82)90029-4. [25] J. Stovall, Collective rationality and monotone path division rules, J. Econ. Theory, 154 (2014), 1-24. doi: 10.1016/j.jet.2014.08.003. [26] W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey, Math. Soc. Sci., 45 (2003), 249-297. doi: 10.1016/S0165-4896(02)00070-7. [27] W. Thomson, How To Divide When There Isn't Enough, mimeo, 2006. [28] W. Thomson, On the existence of consistent rules to adjudicate conflicting claims: A geometric approach, Rev. Econ. Design, 11 (2007), 225-251. doi: 10.1007/s10058-007-0027-2. [29] W. Thomson, Two families of rules for the adjudication of conflicting claims, Soc. Choice Welf., 31 (2008), 667-692. doi: 10.1007/s00355-008-0302-3. [30] W. Thomson, Lorenz rankings of rules for the adjudication of conflicting claims, Econ. Theory, 50 (2012), 547-569. doi: 10.1007/s00199-010-0575-5. [31] W. Thomson, On the axiomatics of resource allocation: Interpreting the consistency principle, Econ. Phil., 28 (2012), 385-421. doi: 10.1017/S0266267112000296. [32] W. Thomson, Consistent Allocation Rules, mimeo, 2012c. [33] W. Thomson, Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: A survey, Math. Social Sci., 45 (2013), 249-297. doi: 10.1016/S0165-4896(02)00070-7. [34] W. Thomson, For claims problems, compromising between the proportional and constrained equal awards rules, Econ. Theory, 60 (2015), 495-520. doi: 10.1007/s00199-015-0888-5. [35] J. Xue, Claim uncertainty and egalitarian division with wastage, mimeo, 2015. [36] P. Young, On dividing an amount according to individual claims or liabilities, Math. Op. Research, 12 (1987), 398-414. doi: 10.1287/moor.12.3.398. [37] P. Young, Distributive justice in taxation, J. Econ .Theory, 44 (1988), 321-335. doi: 10.1016/0022-0531(88)90007-5.
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