Mathematical Social Sciences 40 2000 131–155 www.elsevier.nl locate econbase
‘Hydraulic’ rationing
a,b ,
Marek M. Kaminski
a
Department of Politics , New York University, 715 Broadway, New York, NY 10003, USA
b
Central Eastern European Economic Research Centre CEEERC, Warsaw University, Warsaw, Poland
Received May 1998; received in revised form May 1999; accepted September 1999
Abstract
The problem of distributing a single homogeneous divisible good among a variable set of agents, or the ‘rationing problem,’ is analyzed. Examples of rationing include bankruptcy,
taxation, claims settlement, cost allocation, surplus sharing, and social choice problems. Agents are described by their personal characteristics, or types. A type may be an agent’s utility function,
]] preference ordering, claim to an estate, financial record, etc. A rule of division that can be
represented as a system of connected vessels is called hydraulic. For separable spaces of types and ]]]
continuous rules, this property is equivalent to obeying the fundamental axioms of symmetry and ]]]
consistency. A universal criterion is presented for deciding when a bilateral rule has a consistent ]]]
extension.
2000 Elsevier Science B.V. All rights reserved.
1. Introduction
Assignment of taxes, division of an estate among creditors, or mass privatization of state-owned enterprises are examples of the rationing of a single homogeneous divisible
]]] good among a variable set of agents. Rationing problems in the context of bankruptcy
were first considered by O’Neill 1982, and their mathematical analysis was developed by Aumann and Maschler 1985. Young 1994 introduced a general framework with
the central concept of ‘type,’ or all relevant attributes of a claimant. For apportionment and bankruptcy for transplant for example, types were assumed to be the populations of
the states and the agents’ claims, respectively. Young obtained a number of axiomatic characterizations of allocation rules for such specific contexts and results that were valid
for any indivisible goods.
The current paper introduces Young’s framework in a unified way. The rationing
Tel.: 11-212-998-8504; fax: 11-212-995-4184. E-mail address
: marek.kaminskinyu.edu M.M. Kaminski 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
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.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
problem is described by three variables, types of agents, the amount of the good, and the restriction on the amount a type can get. A generic type may be interpreted as any piece
of information about an individual. The type of a person may include welfarist information, i.e. preferences or the utility function, and or non-welfarist characteristics,
e.g. the size of a claim, a person’s health record, tax form entries, or sex, race and age. In the main results of this paper, a type is assumed to be an element of any relational
structure a set with relations or any separable topological space. This general setting allows for various non-isomorphic interpretations. A result obtained in such a setting
remains valid for every specific context. Various interpretation of types produce the results, or their generalizations, of Aumann and Maschler 1985, Dagan and Volij
1997, and Young 1987, 1994, or lead to new results for multi-claim bankruptcy or social choice.
Types and the maximal portion a type can get define a specific model of rationing, or the set of all rationing problems of a given kind. Solutions to rationing problems are
provided by rules. For every problem, a rule specifies how a good is allocated among ]]
agents. Many rules are generated by standards, a concept invented by Young 1994. A ]]]
standard R is a weak ordering over all pairs consisting of a type t and an amount of a
i
]]] good g . We read ‘
t ,g Rt ,g ’ as: type t has at least as high priority to g as type t
i i
i j
j i
i j
has to g . A standard is numerical if there is a real-valued function r such that
j
]]] t ,g Rt ,g iff rt ,g rt ,g Young, 1994.
i i
j j
i i
j j
Consider the following solution to a rationing problem. Every agent gets one vessel from a system of connected vessels that are linked to a central reservoir through a
system of pipes. The amount of water in a central reservoir is equal to the amount of good to be rationed. The allocation is obtained by opening the main sluice-gate and
letting the water flow down and fill the vessels. When the water stops flowing, agents close their taps and enjoy the portions they received. Since the vessels are connected, the
water levels in all vessels will be equal to one another. Solutions to all rationing problems are generated in a ‘hydraulic’ way, by the assignment of vessels to agents Fig.
1.
The hydraulic metaphor conveniently enables one to visualize a numerical standard. Every such a standard R can be imagined as a specific system of connected vessels. In
Fig. 1, different agents are assigned vessels that reflect their type. The representation of R assigns to a type
t and his possible portion g , a number representing t ’s priority to
i i
i
g , r t ,g . This numerical representation has a simple graphical representation. Let us
i i
i
mount all vessels below some reference level, say, the bottom of main reservoir. Consider the vessel
t filled with g of water. The numerical priority of t to get g can
4 4
4 4
be represented by the distance from the surface of water in the vessel t to the reference
4
level. The farther from the surface to the reference level, the higher the priority assigned to
t ,g . Obviously, many different numerical representations of the same standard
4 4
exist. The present paper investigates the relationship between rules and standards. Following
the metaphor of a system of connected vessels, every rule that allocates the good in the same way as a certain system of connected vessels allocates water is called ‘hydraulic.’
The formal definition is given in Section 2. The adjective ‘hydraulic’ denotes merely the way the rule can be represented. One can also inquire about minimal desirable
M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
133
Fig. 1. Hydraulic metaphor. Note: a problem t , t , t , t ; g is solved by assigning to types t , t , t , t
1 2
3 4
1 2
3 4
appropriate vessels and pouring g water into the main reservoir.
properties a rule should satisfy. Among the most fundamental properties of rationing rules, which in fact are satisfied by almost all real-world rules, are continuity, symmetry
and consistency. It turns out that for the class of continuous rules in a separable topology, a rule is symmetric and consistent if and only if it is hydraulic. If continuity is
substituted by monotonicity, then, for any space of types a rule is symmetric, monotone, and consistent if and only if it is ‘semi-hydraulic,’ i.e. if it is generated by a standard
which is not necessarily hydraulic. This result leads to a corollary that provides simple necessary and sufficient conditions in order for any bilateral rationing rule to have a
unique extension to a consistent rule.
The plan of the paper is as follows. Sections 2 and 3 introduce the formal framework and present the main results. Section 4 describes examples of hydraulic rules in claims
settlement and bankruptcy, and develops applications to social choice. Section 5 discusses the relationship between the present results and the work of Dagan and Volij
1997, and Young 1987. The last section concludes with some general comments. Proofs omitted in the text are given in Appendix A.
2. Model
