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Article Excerpt
Abstract We explore the implications of no-envy (Foley 1967) in the context of queueing problems. We identify an easy way of checking whether a rule satisfies efficiency and no-envy. The existence of such a rule can easily be established. Next, we ask whether there is a rule satisfying efficiency and no-envy together with an additional solidarity requirement: how agents should be affected as a consequence of changes in the waiting costs. However, there is no rule satisfying efficiency, no-envy, and either one of two cost monotonicity axioms. To remedy the situation, we propose modifications of no-envy, adjusted no-envy and backward/forward no-envy. Finally, we discuss whether three fairness requirements, no-envy, the identical preferences lower bound, and egalitarian equivalence, are compatible in this context.
Keywords Queueing problem * No-envy * Cost monotonicity * Identical preferences lower bound * Egalitarian equivalence
JEL Classification Numbers D63 * D71
1 Introduction
Consider a group of agents who must be served in a facility. The facility can handle only one agent at a time and agents incur waiting costs. We are interested in finding the order in which to serve agents and the (positive or negative) monetary compensations they should receive. We assume that an agent's waiting cost is constant per unit of time, but that agents differ in their waiting costs. Each agent's utility is equal to his monetary compensations minus his total waiting cost. This queueing problem has been studied extensively from the incentive perspective (Dolan 1978; Suijs 1996; Mitra 2001, 2002). However, it has received only a limited attention from the normative perspective. The only exceptions are recent works, Maniquet (2003) and Chun (2004), which discuss the properties of rules obtained by applying the Shapley (1953) value.
In this paper, we investigate the implications of no-envy in queueing problems. No-envy, introduced by Foley (1967), requires that no agent should end up with a higher utility by consuming what any other agent consumes. Although its implications have been studied for a wide class of problems, it has not been the object of any study in queueing problems.
We identify an easy way of checking whether a rule satisfies efficiency and no-envy. It can be described in a simple way: choose any efficient queue, and then check the difference of transfers between any two neighboring agents. If the difference is not greater than the higher waiting cost of the two agents and is not smaller than the lower waiting cost of the two agents, it then passes the no-envy test. Of course, it is an immediate consequence of no-envy that an agent served earlier should receive a smaller transfer than an agent served later. The existence of such a rule can easily be established.
Next, we investigate whether there is a rule satisfying efficiency and no-envy together with an additional solidarity requirement: how agents should be affected as a consequence of changes in the waiting costs. Negative cost monotonicity (Maniquet 2003) requires that an increase in an agent's waiting cost should cause other agents to weakly lose. On the other hand, positive cost monotonicity (Chun 2004) requires that it should cause other agents to weakly gain. We show that if the society consists of more than two agents, then there is no rule satisfying efficiency, no-envy, and either negative cost monotonicity or positive cost monotonicity. (1)
Faced with the impossibility results, we propose modifications of no-envy. To apply no-envy, each agent is supposed to reevaluate what any other agent consumes. In the queueing problem, an allocation consists of agents' positions in the queue and their transfers, and a rule determines the transfers by agents' positions and waiting costs. If two agents interchange their positions in the queue, then their transfers would not be the same as before because their waiting costs are different. Adjusted no-envy requires that an agent should not envy the other agents after making the adjustment in transfers. Our second modification of no-envy requires that an agent should not envy the other agents at least in one direction. More specifically, backward no-envy requires that an agent should not envy the agents with lower waiting costs, whereas forward no-envy requires that an agent should not envy the agents with higher waiting costs.
Other fairness requirements widely discussed in the literature are: the identical preferences lower bound requires that each agent should be at least as well off as he would be, under efficiency and equal treatment of equals, if all other agents had the same preferences, and egalitarian equivalence requires that there should be a reference bundle such that each agent enjoys the same welfare between his bundle and that reference bundle. We investigate whether the three requirements are compatible in the current context. First, it is easy to show that efficiency and no-envy together imply the identical preferences lower bound. Also, we can show an existence of a rule satisfying efficiency, egalitarian equivalence, and the identical preferences lower bound. However, if we have more than three agents, then there is no rule satisfying efficiency, egalitarian equivalence, and no-envy together.
2 Preliminaries
Let I [equivalent to] {1, 2,...} be an (infinite) universe of "potential" agents, and N be the family of non-empty subsets of I. Each agent i [member of] I is characterized by his unit waiting cost, [[theta].sub.i] [greater than or equal to] 0. Given N [member of] N, each agent i [member of] N is assigned a position [[sigma].sub.i] [member of] N in a queue and a positive or negative transfer [t.sub.i] [member of] R. The agent who is served first incurs no waiting cost. Each agent needs the same amount of service time. If agent i [member of] N is served in the [[sigma].sub.i]th position, his waiting cost is ([[sigma].sub.i] - 1)[[theta].sub.i]. Each agent i [member of] N has a quasi-linear utility function, so that his utility from consuming the bundle ([[sigma].sub.i], [t.sub.i]) is given by u([[sigma].sub.i], [t.sub.i]; [[theta].sub.i]) = [t.sub.i] - ([[sigma].sub.i] - 1)[[theta].sub.i].
A queueing problem is defined as a list q = (N, [theta]) where N [member of] N is the set of agents and [theta] is the vector of unit waiting costs. Let...
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