Monotonicity implications for the ranking of rules for airport problems

• Published date: 01 December 2016
• Author: Carmen Quinteiro Sandomingo,Estela Sánchez Rodríguez,Miguel Ángel Mirás Calvo
• Date: 01 December 2016
• DOI: http://doi.org/10.1111/ijet.12101

doi: 10.1111/ijet.12101

Monotonicity implications for the ranking of rules for

airport problems

Miguel ´

Angel Mir´

as Calvo,∗Carmen Quinteiro Sandomingo†and Estela S´

anchez Rodr´

ıguez‡

The airport problem is a classic cost allocation problemthat has been w idely studied. Several rules

have been proposed to divide the total cost among the agents, attending tothe characteristics of

the problem or via game theory. The axiomatic approach provides a wayto choose among rules.

Our main goal is to provide some tools to evaluate how rules differentially treat larger airlines as

compared to smaller airlines. Weuse the Lorenz and no-subsidy orderings to compare rules. We

introduce some monotonicity and boundedness properties that imply a specific ranking with

respect to the nucleolus and the Shapley value.

Key wor ds airport problem, allocation rule, monotonicity axiom, ranking of rules

JEL classification C71

Accepted 19 February 2016

1 Introduction

The airport problem, introduced by Littlechild and Owen (1973), is concernedw ith dividing the total

cost of building an airstrip among several airlines that need runways of different lengths. The larger a

plane, the longer the airstrip it needs. An airstrip that accommodates a given plane can accommodate

all smaller planes at no extra cost. A way of associating with each problem of this type an allocation of

the cost among the companies involvedis called a rule. Basically, rules have been proposed from three

different perspectives: firstly,by a direct definition that has some sort of intuitive appeal; secondly, by

associating to each airport problem a cooperative game, an airport game, and using game theory to

come up with a solution; and thirdly, by formulating a set of properties, or axioms, and selecting the

rules according to the properties that they satisfy or violate. For more details on all these approaches

and the attention that this problem has generated in the literature we refer the reader to the survey

by Thomson (2007).

There is an important family of properties that specify how a rule should respond to changes in

the cost parameters of an airport problem: the monotonicity properties. Generally, these properties

are concerned with the effect that a variation of an agent cost parameter, or the cost parameters of

a particular group, has on the contribution of that agent, or group of agents. In this paper we study

properties that reflect whether or not a variation in a particular agent’s cost is beneficial to the other

∗Department of Mathematics, University of Vigo,Vigo, Spain.

†Department of Mathematics, Faculty of Economic and Business Sciences, University of Vigo, Vigo, Spain. Email:

quinteir@uvigo.es

‡Department of Statistics and Operational Research, University of Vigo,Vigo, Spain.

Authors acknowledge the financial support of the Spanish government, Ministerio de Econom´

ıa y Competitividad,

through grants MTM2014-53395-C3-3-P and ECO2012-38860-C02-02. We are grateful to two anonymousreferees for

their comments and suggestions.

International Journal of Economic Theory 12 (2016) 379–400 © IAET 379

International Journal of Economic Theory

Ranking of rules for airport problems Miguel ´

Angel Mir´

as Calvo et al.

agents. An example of this type is the others-oriented monotonicity property that requires that if an

agent’s cost parameter increases, each of the other agents should pay at most as much as initially.

Although this property is satisfied by several rules such as the sequential equal contributions rule

(the Shapley value), it is violated by the slack maximizer’s rule (the nucleolus). We propose two new

monotonicity properties along these lines. The higher cost decreasing monotonicity property states

that if a single cost ciincreases then all agents with cost higher than cishould not pay more than

initially. On the other hand, the lower cost increasing monotonicity property states that if a single

cost ciincreases then all agents with cost lower than cishould not pay less than initially. In economic

terms, if an airline contributes more due to its cost increase, then all the higher cost airlines enjoy a

positive externality and all the lower cost airlines suffer a negative externality.

Boundedness conditions have a long tradition in models of distributive justice (see, for instance,

Moulin 1991; Maniquet 1996). In airport problems there are two natural bounds that are imposed

on any rule: non-negativity of the cost allocations, as a trivial lower bound; and cost boundedness,

that is, any allocation should be bounded above by the cost vector, as a trivial upper bound. Another

basic requirement, satisfied by all the rules in the literature, is that no group of agents should

contribute more than what it would have to pay on its own, because otherwise, the group would

unfairly “subsidize” the other agents. These restrictions, called the no-subsidy constraints, place an

upper bound on each agent’s contributions. Naturally, one can impose extra bounds. We introduce

two additional bounds for rules. The downstream subtraction lower bound property requires that

each agent should contribute as least as much as an equal proportion of its cost in the downstream

subtraction reduced problem. On the other hand, the downstream subtraction upper bound property

requires that each agent should contribute at most as much as the first coordinate of the nucleolus

of the downstream subtraction reduced problem.

Our main goal in this paper is to provide some tools to evaluate how rules differentially treat

larger airlines as compared to smaller airlines. To address this question a way of ranking allocations

is needed. The Lorenz order is commonly used in situations of this kind. It is widely accepted

as embodying a set of minimal ethical judgments that should be made in carrying out inequality

comparisons. In order tocompare a pair of allocations, xand y, with the Lorenz criterion, first one has

to rearrange the coordinates of each allocation in a non-decreasing order.Then we say that xLorenz

dominates yif all the cumulative sums of the rearranged coordinates are greater with xthan with y.

In airport problems, airlines are naturally ordered in terms of the need they have for the runway. So

we consider making the comparison of two allocations without rearranging the coordinates. We call

this ranking criterion “no-subsidy dominance.” Trivially, for all rules satisfying order preservation for

contributions, both Lorenz dominance and no-subsidy dominance are equivalent. The Lorenz and

no-subsidy orders are partial orderings on the set of all allocations. Then one should not expect to be

able to make either Lorenz or no-subsidy comparisons of any two rules generally. The comparisons

may depend on the number of airlines, the specific costvalues, ...Infact,foreachparticularproblem,

the contribution vectors chosen by two rules maynot be comparable. Thus, finding results that allow

us to rank rules, at least within specific subclasses, is a challenging goal.

We will show that the monotonicity and boundedness properties introducedin our study imply

a certain ranking of rules. Among all the rules satisfying the downstream subtraction lower bound,

the Shapley value is the one that best treats (is no-subsidy dominated by) the airlines with lower

costs. Among all the rules satisfying the downstream subtraction upper bound, the nucleolus is the

one that best treats (no-subsidy dominates) the airlines with higher costs. We establish necessary

and sufficient conditions for a rule to satisfy these boundedness properties in terms of monotonicity

properties and provide examples and counterexamples of rules that fail them. The principal results

are illustrated in Figure 1.

380 International Journal of Economic Theory 12 (2016) 379–400 © IAET