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Dividing goods and bads under additive utilities
Cornell university, arXiv.org
,
2016.
Bogomolnaia A., Moulin H., Sandomirskiy F., Yanovskaya E. B.
When utilities are additive, we uncovered in our previous paper (Bogomolnaia et al. "Dividing Goods or Bads under Additive Utilities") many similarities but also surprising differences in the behavior of the familiar Competitive rule (with equal incomes), when we divide (private) goods or bads. The rule picks in both cases the critical points of the product of utilities (or disutilities) on the efficiency frontier, but there is only one such point if we share goods, while there can be exponentially many in the case of bads.
We extend this analysis to the fair division of mixed items: each item can be viewed by some participants as a good and by others as a bad, with corresponding positive or negative marginal utilities. We find that the division of mixed items boils down, normatively as well as computationally, to a variant of an all goods problem, or of an all bads problem: in particular the task of dividing the non disposable items must be either good news for everyone, or bad news for everyone.
If at least one feasible utility profile is positive, the Competitive rule picks the unique maximum of the product of (positive) utilities. If no feasible utility profile is positive, this rule picks all critical points of the product of disutilities on the efficient frontier.
Language:
English
Keywords: fair division of goodsfair division of badscompetitive equilibrium with equal incomesNash productenvy-freenessзадачи справедливого распределения благзадачи справедливого распределения антиблагконкуррентное равновесие с равными доходамипроизведение Нэшаотсутствие зависти
Publication based on the results of:
Bogomolnaia A., Moulin H., Sandomirskiy F. et al., Econometrica 2017 Vol. 85 No. 6 P. 1847-1871
A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others.
If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is ...
Added: October 14, 2016
Bogomolnaia A., Moulin H., Sandomirskiy F. et al., / Cornell university, arXiv.org. Series arXiv:1608.01540 "Computer Science". 2016.
The Competitive Equilibrium with Equal Incomes is an especially appealing efficient and envy-free division of private goods when utilities are additive: it maximizes the Nash product of utilities and is single-valued and continuous in the marginal rates of substitution. The CEEI to divide bads captures similarly the critical points of the Nash product in the ...
Added: October 14, 2016
Bogomolnaia A., Moulin H., Sandomirskiy F. et al., / Высшая школа экономики. Series EC "Economics". 2016. No. 153.
When utilities are additive, we uncovered in our previous paper (Dividing Goods or Bads Under Additive Utilities) many similarities but also surprising dierences in the behavior of the familiar Competitive rule (with equal incomes), when we divide (private) goods or bads. The rule picks in both cases the critical points of the product of utilities ...
Added: November 14, 2016
Bogomolnaia A., Moulin H., Sandomirskiy F. et al., / Высшая школа экономики. Series EC "Economics". 2016. No. 147.
The Competitive Equilibrium with Equal Incomes is an especially appealing efficient and envy-free division of private goods when utilities are additive: it maximizes the Nash product of utilities and is single-valued and continuous in the marginal rates of substitution. The CEEI to divide bads captures similarly the critical points of the Nash product in the ...
Added: August 19, 2016
Bogomolnaia A., Sandomirskiy F., Moulin H. et al., / Высшая школа экономики. Series EC "Economics". 2017. No. 158.
A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it ...
Added: March 2, 2017
Bogomolnaia A., Moulin H., Sandomirskiy F. et al., Social Choice and Welfare 2019 Vol. 52 No. 3 P. 395-417
We compare the Egalitarian rule (aka Egalitarian Equivalent) and the Competitive rule (aka Competitive Equilibrium with Equal Incomes) to divide bads (chores). They are both welfarist: the competitive disutility profile(s) are the critical points of their Nash product on the set of efficient feasible profiles. The C rule is Envy Free, Maskin Monotonic, and has ...
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