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Social Choice and Welfare

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01-02-2015

Systematic favorability in claims problems with indivisibilities

Author: Siwei Chen

Published in: Social Choice and Welfare | Issue 2/2015

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Abstract

We study claims problems with indivisible goods. Due to indivisibilities, in certain situations, two agents with equal claims may have to receive unequal amounts. Our main goal is to find rules that deal with these situations in a consistent way. We propose three “systematic favorability” properties. We define a subfamily of up (down) rules studied in Moulin and Stong (Math Oper Res 27:1–30, 2002). We show that the subfamily are the only rules satisfying our first systematic favorability property, composition up, and bilateral consistency. Another family of rules we study are the sequential priority rules. Given a priority order over agents, we satisfy their claims one agent at a time until the resource runs out. These rules are the only ones that satisfy our second systematic favorability property, composition down, and bilateral consistency. Using duality, we also provide another characterization of the sequential priority rules, with our third systematic favorability property. Besides, we provide an alternative characterization of the rules studied in Herrero and Martinez (Soc Choice Welf 30:603–617, 2008).

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Also see Moulin (2002) and Thomson (2003).

In claims problems with infinite divisibility, the random arrival rule is defined by taking average over all possible orders of agents. But it is not well-defined in this model.

I am indebted to an anonymous referee for advices on the model setting.

This property is defined by Herrero and Villar (2002) under the name of sustainability.

This property is defined by Herrero and Villar (2002) under the name of independence of residual claims.

In claims problems with infinite divisibility, the hypothesis is \(c_{i} \ge c_{j}\), and order preservation of awards implies equal treatment of equals. The property is modified for this model because equal treatment of equals is not feasible in the context of claims problems with indivisibilities.

The notation \(c_{-i} \in {\mathbb {Z}}_{+}^{N \setminus \{i\}}\) denotes the profile that results from deletion of the \(i\)-th coordinate \(c_{i}\) from \(c\).

This property is introduced in the context of taxation by Young (1988) under the name of “composition principle”. The name here is taken from Thomson (2003).

This property is introduced by Moulin (1987) under the name of “path independence”. The name here is also taken from Thomson (2003). A property sharing the same idea, “step-by-step negotiations”, is introduced by Kalai (1977) for bargaining. It is also reminiscent of the “path independence” property for choice functions (Plott 1973).

For a survey on consistency, see Thomson (2011).

See Thomson (2008) and Thomson (2011).

I am indebted to an anonymous referee for the suggestion of weakening this property by requiring the claims of all the other agents to be fixed in the two problems.

I would like to thank the associate editor for the suggestion on naming.

For two-agent problems, this property implies that the only case in which \(j\) is fully compensated is \(E' = c'_{i} + c'_{j}\).

It is introduced by Young (1994).

Herrero and Martinez (2008) use the terminology “monotonic standard of comparison”.

For claims problem with infinite divisibility, conditional full compensation and composition down characterize the constrained equal awards rule (Herrero and Villar 2002). This result is strengthened by Yeh (2006), replacing composition down by the much weaker notion of claims monotonicity.

The idea of duality is introduced by Aumann and Maschler (1985).

\(\left( c, \sum _{i \in N}c_{i} - E\right) \) is a well-defined claims problem, because \( \sum _{i \in N} c_{i} \ge \sum _{i \in N}c_{i} - E \ge 0\) and for each \(i \in N, c_{i} \ge c_{i} - \varphi _{i}\left( c, \sum _{i \in N}c_{i} - E\right) \ge 0\).

These results, except for 2(e), are proved in the infinitely divisible case. See Moulin (2000), Herrero and Villar (2002), Moreno-Ternero and Villar (2006), Thomson (2008), and Thomson and Yeh (2008). The same arguments apply to our model with indivisibilities.

Although the lemmas are results for claims problems with infinite divisibility, they are valid in this model too.

Duality implies that null-compensation favorability, composition up and bilateral consistency also characterize this family.

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Title: Systematic favorability in claims problems with indivisibilities
Author: Siwei Chen
Publication date: 01-02-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0828-5

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