Additive rules in discrete allocation problems

Abstract

In this paper, we study allocation problems and other related problems where a discrete estate should be divided among agents who have claims on it. We characterize the set of rules satisfying additivity on the estate along with additivity on the estate and the claims. These results complete the characterizations given by Bergantiños and Vidal-Puga (Mathematical Social Sciences 47 (2004), 87–101) in the continuous case.

Introduction

Many economic and political situations can be modelled as a problem of how to divide a resource among agents who have claims on it. The way in which this division should take place may depend on the particular problem to be studied.

In some problems agents have legitimate rights over a scarce good. In this case no agent should receive more than his claim, but neither less than nothing. These problems are called bankruptcy problems (O’Neill, 1982, Aumann and Maschler, 1985, Young, 1987).

Other problems are those which arise when the agents collectively contribute in a common project which generates a surplus. The claims of the agents should then be interpreted as their contribution to the project. For example, a group of families may collaborate to build a row of terraced houses for living. In these circumstances, the only restriction is that nobody should receive less than nothing. We call these problems surplus problems (Moulin, 1987).

In allocation problems (Chun, 1988, Herrero et al., 1999) there are no restrictions in what an agent may receive. In loss problems (Bergantiños and Vidal-Puga, 2004) nobody should receive more than his claim.

All these papers are devoted to the continuous problem, i.e. the resource is perfectly divisible (for instance, money). Nevertheless, there are some practical situations in which the resources come in indivisible units. We call the class of such problems discrete problems.

We now give some examples of discrete problems. The assignment of seats in a parliament, queuing problems where the demands consist of finite number of “jobs”, or secretaries to departments in a university, can be modelled as a discrete bankruptcy problem.¹ Consider a group of families that collaborate to build a row of terraced houses for living. This situation corresponds to a discrete surplus problem. Assume now the terraced houses in the previous example are built not by families but by building contractors. They just need to distribute the houses but they may have other (real or projected) houses on their own. In this case, the assignment may be such that some contractors cede some of their own houses. This situation corresponds to a discrete allocation problem.

Finally, assume that the joint project by the contractors fails, and there are less houses than claimed. Since the joint project failed, no contractor should get more houses than those he has rights on. Thus, we can model this situation as a discrete loss problem.

Even though most of the literature is devoted to the continuous problem, during the last years many papers study the discrete problem. We can mention, for instance, Moulin, 2000, Moulin, 2002a, Moulin and Stong, 2002, Herrero and Martínez, 2004. Also, in Balinsky and Young (1982) it is possible to find an extensive literature on discrete allocation problems in other settings.

We can study these problems from two different approaches. One of them is the axiomatic characterizations of rules. The idea is to propose desirable properties and find out which of them characterize every rule. Properties often help agents to compare different rules and to decide which rule is preferred for a particular situation. Another approach is to study what the rules satisfying a set of properties are. For instance, Young (1988) characterizes the rules satisfying continuity, symmetry, and consistency; De Frutos (1999) characterizes the rules satisfying non-manipulability; and Moulin (2000) characterizes the rules satisfying consistency, composition up, composition down, and scale invariance. Thomson, 2003, Moulin, 2002b give a survey of this literature. In this paper, we follow the second approach and concentrate on the property of additivity.

Additivity is a widely used property. For instance, the Shapley value, the most important value in cooperative games with transferable utility, is characterized by additivity and other properties. If we compare the Shapley value with other prominent values (for example the nucleolus) we realize that these values satisfy all the properties characterizing the Shapley value (efficiency, null player, and symmetry) except additivity.

Bergantiños and Vidal-Puga (2004) study the property of additivity in the continuous problem. In this paper we study this property in the discrete problem. We must note that even though the results of this paper are related with those of Bergantiños and Vidal-Puga (2004), the proofs are completely different and this paper is not a consequence of the previous one. We use two definitions of additivity: additivity on the estate (A1) (Moulin, 1987, Chun, 1988), and additivity on the estate and the claims (A2) (Bergantiños and Méndez-Naya, 2001).

In this paper we characterize the rules satisfying A1 and A2 in each of the four problems mentioned before. The rules satisfying A1 are as follows: In allocation problems they are characterized by the product of the estate and a claims-dependent function. In surplus problems all the estate is given to an agent, who is selected depending on the claims. In loss and bankruptcy problems there are no rules.

The rules satisfying A2 are as follows: In allocation problems they are characterized by the sum of two parts, one depending on the estate and the other depending on the claims. In surplus problems, the estate is always given to a fixed agent. In loss problems, all the loss is always suffered by a fixed agent. There is no bankruptcy rule satisfying this additivity.

The paper is organized as follows. Section 2 introduces the problems studied in this paper. Section 3 studies the rules which satisfy these additivity properties.