An Introduction to Allocation Rules

163 Pages·2009·1.08 MB·English

by  Jens Leth Hougaard (auth.)

An Introduction to Allocation Rules
Jens Leth Hougaard
An Introduction
to Allocation Rules
ABC
Prof.Dr.JensLethHougaard
DepartmentofFoodandResourceEconomics
UniversityofCopenhagen
Rolighedsvej25
1958FrederiksbergC
Denmark
[email protected]
ISBN978-3-642-01827-5 e-ISBN978-3-642-01828-2
DOI10.1007/978-3-642-01828-2
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2009928107
(cid:176)c Springer-VerlagBerlinHeidelberg2009
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To Pernille and Ida
Preface
This book contains a systematic analysis of allocation rules related to cost
and surplus sharing problems. Broadly speaking, it examines various types
of rules for allocating a common monetary value (cost) between individual
members of a group (or network) when the characteristics of the problem
are somehow objectively given. Without being an advanced text it offers
a comprehensive mathematical analysis of a series of well-known allocation
rules. The aim is to provide an overview and synthesis of current knowl-
edge concerning cost and surplus sharing methods. The text is accompanied
by a description of several practical cases and numerous examples designed
to make the theoretical results easily comprehensible for both students and
practitioners alike.
The book is based on a series of lectures given at the University of
Copenhagen and Copenhagen Business School for graduate students joining
the math/econ program.
I am indebted to numerous colleagues, conference participants and stu-
dents who during the years have shaped my approach and interests through
collaboration,commentsandquestionsthatweregreatlyinspiring.Inparticu-
lar,IwouldliketothankHansKeiding,MauriceKoster,TobiasMarkeprand,
Juan D. Moreno-Ternero, Herv´e Moulin, Bezalel Peleg, Lars Thorlund-
Petersen, Jørgen Tind, Mich Tvede and Lars Peter Østerdal.
vii
Contents
1 Motivation and a Few Cases........................................ 1
1.1 Introduction ........................................................ 1
1.2 Motivation .......................................................... 2
1.2.1 Fairness and Economic Efficiency......................... 2
1.2.2 Management Decisions and Incentives.................... 7
1.3 Some Practical Cases .............................................. 10
1.3.1 The Tennessee Valley Project ............................. 10
1.3.2 Farmer’s Irrigation Costs .................................. 12
1.3.3 Regulating Public Monopolies............................. 13
1.4 Limits of Scope..................................................... 15
References ................................................................ 15
2 Simple Sharing Problems ........................................... 17
2.1 Introduction ........................................................ 17
2.2 Rationing Problems................................................ 18
2.2.1 Four Rationing Rules ...................................... 19
2.2.2 Inequality Comparisons.................................... 22
2.2.3 Axiomatic Characterizations .............................. 26
2.2.4 Manipulation................................................ 29
2.2.5 Comments................................................... 34
2.3 Cost Sharing with Joint Cost Function........................... 35
2.3.1 Rules Based on Equality and Proportionality............ 35
2.3.2 Rules Based on the Serial Principle....................... 39
2.3.3 Rules Based on the Incremental Principle................ 55
2.3.4 Comments................................................... 57
2.4 Summary............................................................ 58
References ................................................................ 59
3 Cost Allocation as Cooperative Games.......................... 61
3.1 Introduction ........................................................ 61
3.2 The Model .......................................................... 62
3.2.1 Some Applications.......................................... 64
3.3 The Stand-alone Cost Principle................................... 66
ix
x Contents
3.4 Four Allocation Rules.............................................. 74
3.4.1 The Nucleolus............................................... 75
3.4.2 The Lorenz Allocation ..................................... 77
3.4.3 The Shapley Value ......................................... 79
3.4.4 The τ-Value................................................. 81
3.5 Monotonicity Vs. the Stand-alone Cost Principle................ 84
3.6 Axiomatic Characterizations ...................................... 87
3.6.1 Consistency and Reduced Problems ...................... 87
3.6.2 Characterizing the Nucleolus.............................. 89
3.6.3 Characterizing the Shapley Value......................... 90
3.6.4 Characterizing the Lorenz Allocation..................... 91
3.6.5 Characterizing the τ-Value ................................ 92
3.7 Comments........................................................... 92
3.8 Summary............................................................ 93
References ................................................................ 94
4 General Sharing Problems.......................................... 97
4.1 Introduction ........................................................ 97
4.2 The Model .......................................................... 99
4.3 Three Allocation Rules ............................................ 100
4.3.1 The Aumann–Shapley Rule................................ 100
4.3.2 The Shapley–Shubik Rule.................................. 104
4.3.3 The Friedman–Moulin Rule ............................... 105
4.4 Axiomatic Characterization ....................................... 109
4.4.1 Non-additive Extensions ................................... 114
4.5 Practical Application by Convex Envelopment of Cost Data... 116
4.5.1 Cost Allocation Using Data Envelopment................ 117
4.6 Comments........................................................... 121
4.7 Summary............................................................ 122
References ................................................................ 123
5 Sharing in Networks.................................................. 125
5.1 Introduction ........................................................ 125
5.2 Modeling Networks as Graphs..................................... 127
5.3 Minimum Cost Spanning Trees.................................... 127
5.3.1 Cost Allocation for Specific Spanning Trees.............. 131
5.3.2 Characteristics of Cost Allocation Rules ................. 134
5.4 Demand-Based Cost Allocation................................... 140
5.5 Efficient Network Structure ....................................... 144
5.5.1 The Model .................................................. 144
5.5.2 Stability and Efficiency .................................... 146
5.5.3 Allocation Rules Respecting Network Structure ......... 148
5.6 Comments........................................................... 149
Contents xi
5.7 Summary............................................................ 150
References ................................................................ 151
Index ......................................................................... 153
Chapter 1
Motivation and a Few Cases
1.1 Introduction
Considertwopersonssharingacabonthewayhomefromaparty.Typically,
the one who gets off first either tries to avoid payment or pay some arbi-
trary amount (corresponding to whatever is left of cash) to the remaining
passenger.
Now, analysing this scenario as a cost allocation problem between two
agents it turn out that, despite the triviality of the situation, the problem is
actually rather complex. In some sense it seems fair that the cost of the first
partofthetrip(thatis,untilthefirstpersonleavesthecab)shouldbeshared
bythetwowhiletheremainingcostshouldbecoveredsolelybytheremaining
passenger. However, maybe the cab has to drive a longer route because the
first passenger who gets off lives somewhat off the natural route going to
the remaining passenger’s destination and why should he pay for this? The
remainingpassengermayevenreasonasfollows:thepersonwhogetsofffirst
should have paid the first part of the trip by himself if they did not share
the cab (his stand-alone cost) so in some sense this person must be willing
to pay more than half of the amount – in fact, his stand-alone cost can be
viewed as the upper limit of his willingness to pay. So clearly the situation is
not simple at all although the persons involved of course are forced to find a
fast and practical solution.
This book is about analysing such sharing problems in a systematic
fashion. Broadly speaking, it examines various types of rules for allocating a
common monetary value (cost) between individual members of a group (or
network) when the characteristics of the problem are somehow objectively
given. The monetary value (cost) must be allocated exactly, that is, with no
profit or deficit (typically referred to as a requirement of budget-balance).
Anallocationruleisageneralallocationprinciplethatisusedwithrespect
to an entire class of similarly structured allocation problems for which there
is no objective way to attribute value (cost) to specific members.
J.L.Hougaard, An Introduction to Allocation Rules, 1
DOI10.1007/978-3-642-01828-2 1, (cid:2)c Springer-VerlagBerlinHeidelberg 2009