Subsidize-free cost allocation method for infrastructure ... - Ozyrys

Subsidize-free cost allocation method
for infrastructure market game
Mariusz Kaleta
Abstract—On complex infrastructure markets the limitations
for free commodities trade occur. Usually, these limitations
result from physical nature of commodities and required
infrastructure, e.g. transmission network in the telecom and
energy sectors, and constitute so called infrastructure costs. In
the paper we present a new method for infrastructure costs
allocation which creates the incentives for market participants
to mitigate the system constraints. Problem is modelled as
a cooperative game, but we show that classical game-theory
based approaches do not produce subsidize-free allocation. To
achieve allocation free form subsidies we propose multicriteria
linear programme. The concept of equitable rational preference
relation is used to nd nondominated allocation fairly treating
infrastructure constraints of the same or similar strength.
I. INTRODUCTION
Competitive market designing for infrastructure sectors
meets difculties due to free commodity trade limitations.
We assume that system resources needed to physical commodities
ow are assured by an exchange-like central mechanism,
e.g. auction, exchange, real-time balancing market.
Most of practical examples come from sectors, which have
been functioned traditionally as state-owned or private monopolies
and which are now being deregulating and transforming
into competitive markets. These include electrical
energy, telecommunication, rail, water, urban transport and
other sectors.
System resources can be represented by a graph, where
turnover limitations can be modelled by ow capacities
(maximal and minimal) on the arcs or minimal/maximal
quantities of commodity delivered or produced at nodes. For
instance on electrical energy markets transmission lines have
limited capacity and in each node of power grid the minimal
and maximal voltage levels are dened.
Market entities submit their buy or sell offers. At xed
time the market is balanced by central institution (market
operator), some of the offers are accepted, and nally commodities
ow in the system is obtained. Then, the global
economic benet ˆQ arisen from market entities trade must
be distributed among the entities. If system resources are
unlimited, the market uniform price is a sufcient tool for
fairly economic benet distribution. However, if the system
constraints become active and economic benet becomes
lower, then more sophisticated costs allocation mechanism
is necessary. In that case each market entity pays or is
The research was partially supported by the Ministry of Science and
Higher Education of Poland under grant 3T11C 005 27 ”Models and
Algorithms for Efcient and Fair Resource Allocation in Complex Systems”
M. Kaleta is with Institute of Control and Computation Engineering,
Warsaw University of Technology, Nowowiejska 15/19 00-665 Warsaw,
Poland m.kaleta@ia.pw.edu.pl
paid according to market price, but also has to pay a
varying component of infrastructure usage charge. Good cost
allocation mechanism should assure good economic signals
and local market power mitigation.
Basic model for single commodity market clearing under
constraints can be formulated as a linear programming
problem (LP model) with objective function that maximizes
the prot (economic wealth) arising from the trade [14]. In
order to assure the proper commodities delivery conditions
the infrastructure resources from set I = {1, 2, ..., I} must
be allocated. Consider the vector of available resources
b = (b 1 , . . . , b I ) ∈ R I and global prot function ˆQ(b),
i.e. optimal economic wealth that results from the market
trade. Model for problem of single commodity infrastructure
market balancing (further referred as SCIM) is the following:
subject to
ˆQ(b) = max ( ∑ e m d m − ∑ s l p l ) (1)
p,d
m∈B
l∈S
∑
l∈S
p l ≥ ∑ m∈B
d m (2)
f i (p, d) ≤ b i i ∈ I (3)
0 ≤ d m ≤ d max
m m ∈ B (4)
0 ≤ p l ≤ p max
l l ∈ S (5)
Offer l from set of sell offers S is described by maximal
offered volume p max
l
and unit price s l . Variable p l is an
accepted level of offer l. Offer m from set of buy offers
B is described by maximal offered volume d max
m and unit
price e m . Variable d m is an accepted level of offer m. The
objective ˆQ(b) is a global economic benets coming form the
commodity turnover. Inequalities (3) represent infrastructure
resource constraints, which form set of I system constraints,
where b i is an availability of resource i, f i is an afne
function of vectors p and d which describes demands for
resource i.
Problem SCIM becomes a typical pure exchange model
when there are sufciently much resources to create no
limits for commodity exchange freedom. Denote the global
prot in case of ”no infrastructure constraints” as ˆQ ±∞ =
lim b→(±∞) ˆQ(b), where at position i of vector (±∞) there
is +∞ or −∞ depending on which direction enlarges
admissible solution set of problem (1)-(5). Then the joint
cost of infrastructure constraints C(b) for available level of
resources b is equal C(b) = ˆQ ±∞ − ˆQ(b). Since the market
prot ˆQ±∞ are constant we can substitute the objective
function (1) with C(b) = min p,d ( ˆQ ±∞ − ˆQ(b)) and solve

problem (1)-(5) to nd the joint infrastructure constraints
cost.
Usually, in many practical cases the simplistic arbitrary
cost-allocation rules are used. For instance, in a majority of
the European electric energy national markets the cost of
system infrastructure utilization is covered by arbitrary at
”postage stamp” fees – identical for all players or group of
players, irrespectively of how much each player contributes
to the costs of utilizing the system. In this approach any
player who contributes to higher costs of resource utilization
in fact achieves extra benets at the cost of other players –
that's it, he is subsidized by other players.
Schweppe et al. [11] introduced and Hogan [5] extended
a concept of locational marginal nodal pricing (LMP) for
electric energy transmitted over the network. The LMP
method allows one to calculate marginal prices of energy
at individual nodes of the infrastructure network. Price
diversication between two adjacent nodes occurs when the
transmission capacity between two nodes is limited and
exhausted. Recently, the marginal nodal pricing became
more and more popular method for designing the electrical
real-time energy balancing markets. However, nodal pricing
suffers from high sensitivity – in each node marginal player
with local market power appears [12]. Finally, this is not a
complete market solution – it is designed for systems where
only transmission constraints are dominant.
Recently, above drawbacks made researchers moved towards
approaches based on game theory, which were successfully
applied for cost allocation for other problems [2],
[4], [6], [12], [13]. In this case it is assumed, that the
contribution of each entity in every resource constraint (3)
is known. For instance one can compute power ow in the
transmission system using ow tracking methods and obtain
share of each entity in each constrained transmission line [1].
Thus we only need to distribute infrastructure cost among the
infrastructure resources constraints.
Cooperative game theory approach is a natural framework
for problems of cost allocation among collaborating
agents [16]. However, most of so far researches assumed
that game characteristic function C(b) is subadditive. In
a typical problem this assumption is a consequence of
problem nature – voluntariness of players participation in a
common business. However, the problem of cost allocation
for infrastructure market concerns the system constraints
which usually are results of technical or physical conditions
and must absolutely be satised. Thus, we are searching for
fair solution for agents who must participate in a common
business. Under this assumption characteristic function C(b)
may not be subadditive, as it happens e.g. on the electrical
energy balancing markets, but also in other industries.
Classical game theory cost allocation methods, including
Shapley value, Aumann-Shapley pricing, SCRB (Separable
Cost Remaining Benets) are based on subadditivity assumption
of C(b) [16]. More over, even for some subadditive functions
C(b) these methods may produce solution which are
not free from so called subsidizing phenomena. Subsidizing
means that player entering into common business causes cost
increase higher then the value he will be charged – thus, other
entities must cover some portion of this cost. It is obvious
that subsidizing phenomena open the doors for speculative,
parasitical strategies which limit effective market functioning
and is clearly undesired.
Classical game theory approaches distribute exactly cost
C(˜b), where ˜b represents actual resources availability. This is
known as an efciency or break-even properties (or axiom).
In the paper we show that subsidize-free allocation cannot
meet efciency (break-even) condition. The main achievement
of our investigation is a new method for allocation costs
of market infrastructure which assures that allocation is free
from subsidies. Obviously, it may produce an allocation that
breaks the efciency axiom.
II. GAME-THEORY MODEL FOR ALLOCATION
PROBLEM
We assume that during market balancing the resource
constraints with right hand side equal ˜b i , i ∈ I must be
satised. Total costs related to the resource constraints should
be assigned to each constraint on the basis of constraint
impact on market balance. Then assigned costs can be
distributed among market entities according to given resource
utilization degree. Charges should discourage market entities
to take advantage of system constraints and their local market
power as a tool for unfounded extra benets.
The problem can be described as a cooperative game
of I players related to i ∈ I system constraints. It is
articial game in the sense that players decisions are already
known and game theory is only to logically justify admitted
allocation. The aim is to nd game rules (allocation rule)
under which players decisions, which are known apriori,
form equilibrium point.
Formally, let I = {1, 2, ..., I} be a set of resource
constraints indices, I = |I|, cost function is a function
C : R I → R, with C((±∞)) = 0. An infrastructure
market game, in short IMG, is a pair (C, ˜b), where C is a
joint cost function described by a linear programme C(b) =
min p,d ( ˆQ ±∞ − ˆQ(b)) subject to (2)-(5) and ˜b ∈ R I is a
vector of actual available levels of resources. Classically, cost
allocation method is a function ϕ(C, ˜b) = (x 1 , ..., x I ) ∈ R I
and ∑ i∈I x i = C(˜b) (efciency condition). In this paper we
will omit the efciency condition and we will refer to a cost
allocation method just as a function ϕ(C, ˜b).
Most of known cost allocation methods, including Shapley
value, Aumann-Shapley pricing, SCRB method, core of the
game, assume that an allocation must satisfy break-even
property, which means the following condition is satised:
∑
x i = C(˜b). (6)
i∈I
Game players' decisions are known, however, cost allocation
is based on hypothetical players' behaviors which potentially
could mitigate or eliminate system constraints related to
them. i − th resource can be considered in a continues range
of 〈−∞; ˜b i 〉 or 〈˜b i ; +∞〉, however two marginal cases are of
special importance: resource constraint is completely relaxed

and resource constraint is limited by ˜b i . Game players form
group S ∈ I of players called coalition if only constraints
related to them are xed with their values from vector ˜b
and all other constraints are completely relaxed. We denote
infrastructure cost for coalition S as a C(S). Further we use
notation C(S) for infrastructure cost of subset S of players
and C(b) for infrastructure cost for resources availability at
level b, where C(S) = C(b ′ ), b ′ i = ˜b i for i ∈ S, b ′ i = ∞ or
−∞ respectively for i ∉ S.
Denition 1. Allocation is free from subsidizing if it
satises incremental cost test
a(S) ≥ C(N) − C(N \ S), (7)
where a(S) = ∑ i∈S x i.
Denition 2. Allocation rule ϕ(C, ˜b) is free from subsidizing
for class of cost functions C if it produces allocation
free from subsidizing for every admissible ˜b and C ∈ C.
Coalition S is encouraged to dissolve if cost allocated on
coalition S is lower then marginal benet related to coalition
S dissolving, thus equations (7) are satised. Only in this
case cost allocated on system constraints are sufciently
incentive to motivate players to remove their constraints
(at least not to make them more restrictive). Let's assume,
that condition (7) is not satised for subset S. Eliminating
constraints (7) reveals costs C(N)−C(N \S). Nevertheless,
if allocation a(S) is lower then joint costs C(N)−C(N \S)
then coalition S must be subsidized by other players.
Notice, that efciency property (6) together with (7)
constitute denition of game core. Core of the game is
an nonempty set for many problems from important wellknown
class of games pNAD. Set of games pNAD is a set
of non-atomic, additive games of bounded variation with
polynomial, continuously differentiable cost function [10].
Games class pNAD is important due to unique allocation
scheme (Aumann-Shapley pricing) which meets widely acknowledged
set of axioms for allocation rule. Unfortunately,
IMG does not belong to pNAD, because C is not continuously
differentiable and game is not additive.
Lemma 1. Break-even and subsidize-free allocation rule
exists only if characteristic function C satises
m∑
C(I) ≤ [
i=1
C(I \ S i )]/(m − 1) (8)
Proof. Let's
⋃split set I into m disjoint subsets
S 1 , S 2 , . . . , S m , m
i=1 S i = I, S j ∩ S k = ∅, ∀j, k ∈
{1, . . . , m}, j ≠ k. According to (7) an allocation is free
from subsidizing if the following conditions are satised
C(I) − C(I \ S i ) ≤ a(S i ) i = 1, . . . , m (9)
By summing up the inequalities (9) we obtain
m∑
m∑
mC(I) − C(I \ S i ) ≤
∑
i=1
i=1
a(S i ) i = 1, . . . , m (10)
An allocation satises break-even property if the equalities
i∈I x i = ∑ m
i=1 a(S i) = C(I) are met. By substituting this
equation in the inequality (10) we obtain
m∑
C(I) ≤ [
i=1
C(I \ S i )]/(m − 1) (11)
Thus, the function C must satisfy conditions (8) to make
the free from subsidizing property preserving possible. □
From the lemma 1 it comes that there is no universe
free from subsidizing and break-even allocation rule for all
possible functions C. For m = 2 this condition transforms
into subadditivity denition. For m > 2 this condition is
even more stronger.
Theorem 1. Any break-even allocation rule for IMG is
not subsidize-free.
Proof comes straightly from previous lemma when we
observe that characteristic function of IMG is not subadditive
and does not meet condition (8).
From theorem 1 it comes that only allocation rules that
break the efciency axiom can be subsidize-free. Nevertheless,
when core of game is empty, many researches relax
the conditions (7) and use formulation of ɛ-core (e.g. for bin
packing game see [8]) keeping the efciency axiom. ɛ-core
is dened by the following conditions
a(S) ≥ (1 − ɛ)(C(N) − C(N \ S)), ɛ ∈ 〈0; 1〉 (12)
Sentence 1. Any allocation in ɛ-core, ɛ > 0, but not in
0-core is not subsidize-free.
Proof of the sentence is evident, because inequalities (7)
are in fact 0-core denition (12). Since for ɛ > 0 this
obviously violates the conditions for subsidize-free allocation
in denition 1, we are interested in holding 0-core (original,
not relaxed core denition) property.
Thus, the only way to achieve subsidize-free allocation is
to relax efciency condition. Because majority of methods
known in the literature assume that allocation must be
break-even, they cannot allow to nd free from subsidizing
allocation.
Theoretically, subsidize free allocation can be easily obtained
– it is easy to show that it is sufcient to allocate high
enough cost on each player. In practice we are interested in
the possible small charges which would assure freedom from
subsidizing.
III. MASIT – A NEW COST ALLOCATION
METHOD
Under the assumption, that incremental cost test (7) must
be satised for each coalition, we obtain the following system
assuring subsidize-free allocation:
∑
x i ≥ C(I) − C(I \ S) ∀S ⊆ I (13)
i∈S
Each market entity responsible for system constraints demands
the lowest possible charges. Thus, the following
multi-criteria problem MASIT (Minimal Allocation Satisfying
Incremental cost Test) can be formulated.
Problem MASIT:
min (x 1 , x 2 , . . . , x n )
x 1 ,x 2 ,...,x n
(14)

subject to
∑
i∈S
x i ≥ C(I) − C(I \ S) ∀S ⊆ I (15)
x i ≥ 0 ∀i ∈ I (16)
For formulas simplication, further we will use
transformed objective max(−x 1 , −x 2 , . . . , −x n ) =
min(x 1 , x 2 , . . . , x n ) and variables ¯x = −x =
(−x 1 , −x 2 , . . . , −x n ) = (¯x 1 , ¯x 2 , . . . , ¯x n ). Solution of
minimization function (14) over simplex formed by set
of incremental cost test inequalities usually is not unique.
Simple scalarizing function, e.g. minimization of mean or
maximum allocation, can result with an allocation easy to
question from the point of view of general acknowledged
allocation fairness. Particulary, two players with the same
impact on the global costs may be charged different
values which minimize scalarizing function. If optimal
solution equally treating both of the players exists, it
should be strictly preferred. This lead to equitable rational
preference relation concept based on equitable Pigou-Dalton
shifts axiom. Equitable shift consists in worsening better
(lower) allocation ¯x i and simultaneously decreasing higher
allocation ¯x j by relatively small value ε > 0. Allocation
vector ¯x − εe i + εe j arising from equitable shift is strictly
preferred then original vector ¯x. Allocation vector ¯x ′
equitable dominates vector ¯x ′′ if it is strictly preferred
according to the rational equitable preference relation
¯x ′ ≻ w ¯x ′′ .
Let Θ : R n → R n denotes an operator of non-decreasing
ordering the coordinates of vector ¯x, it means that Θ(¯x) =
(Θ 1 (¯x), Θ 1 (¯x), . . . , Θ n (¯x)), where Θ 1 (¯x) ≤ Θ 2 (¯x) ≤ · · · ≤
Θ n (¯x). Let ¯Θ = ( ¯Θ 1 , ¯Θ 2 , . . . , ¯Θ n ) be an operator of cumulative
ordering, where ¯Θ i = ∑ i
l=1 Θ l(¯x) for i = 1, 2, . . . , n.
Successive coordinates of vector ¯Θ(¯x) signify the highest
allocated value, sum of two highest allocated values, sum of
three highest allocated values and so on. Then admissible
solution x of task (14)-(16) is equitable effective if and only
if it is solution of the following multi-criteria problem [7]:
max{ ¯Θ(¯x)} (17)
subject to constraints (15), (16) and ¯x = −x. This problem
can be solved by transforming it into single criteria problem
by weighing the criteria. This approach is equivalent to OWA
aggregation (Ordered Weighted Average) applied to task of
Θ(x) maximization subject to (15) and (16) [15]. OWA aggregation
can be depicted in computational convenient form
of maximization linear combination of cumulative ordered
criteria which can be expressed by linear formulas. Finally,
the following linear programme can be formulated:
MASIT OWA problem:
max
n∑
w k (kv k −
k=1
n∑
i=1
d ki ) (18)
∑
i∈S
x i ≥ C(I) − C(I \ S) ∀S ⊆ I (19)
v k + x i ≤ d ki ∀i, k ∈ I (20)
d ki , x i ≥ 0 ∀i, k ∈ I (21)
where w k are nonnegative coefcients, v k are unlimited
variables and d ki are nonnegative variables which represent
bottom deviation form v k .
Notice, that in the consequence of equitable rational preference
relation properties any solution of (18)-(21) satises
allocation symmetry (anonymous) condition. Thus, the allocation
is not sensitive for players (constraints) renumbering.
Constraints (19) result in incremental cost test satisfaction.
Also property of positive cost allocation on inuential player
and no cost allocation on insignicant player (dummy player)
are satised. Therefore, according to the condition (7), a
solution of problem (18)-(21) is an allocation free from
subsidizing.
IV. ILLUSTRATIVE EXAMPLE
We consider simple example of single commodity turnover
among four sellers and one buyer. Offer data are presented
in table I. Decisions related to the market game result
from solving model OPT (1)-(5), where constraints (3) are
following:
−p 1 ≥ −50 (22)
p 3 ≥ 100 (23)
p 4 ≥ 50 (24)
p 1 + p 3 ≥ 140 (25)
If constraints (22)-(25) are relaxed, the economical benets
are equal 13 thousands. When resource constraints are
considered, benets fall to 9 thousands. Thus, the joint
infrastructure cost of resource constraints is 4 thousands.
An allocation according to the Shapley value is (1; 1.066;
1.8; 0.133)∗10 3 respectively to the successive constraints
(22)-(25). Let us notice, that relaxing third constraint causes
economical benets increase by 2 thousands, while charge
for this constraint is only 1.8 thousand. Incentives for this
constraint intensication appear, if positive part of difference
0.2 thousand can be directly or indirectly (as result of collusion)
intercepted by player responsible for this constraint.
Notice also, that there are many allocation vectors which
are equitable effective solutions of multi-criteria task MASIT,
e.g. allocations (1; 1.2; 2; 0.8), (1.2; 1; 2; 0.8), (1.1; 1.1; 2;
0.8). Because rst and second constraints of (22)-(25) are
redundant, ona may expect that their costs will not differ
signicantly between them. Therefore, an allocation (1.1;
1.1; 2; 0.8) is preferred to the rest two allocations. This
allocation is also a solution of problem MASIT OWA (18)-
(21) and is symmetric equitable solution for any values of
w k .
V. CONCLUSIONS
In various practical problems, including infrastructure cost
allocation in the competitive market conditions, there is

l
TABLE I
OFFER DATA
p max
l
s l
1 100 MWh 80 $/MWh
2 50 MWh 100 $/MWh
3 100 MWh 120 $/MWh
4 100 MWh 140 $/MWh
m
d max
m
e m
1 200 MWh 160 $/MWh
a need to allocate joint costs onto many agents, where
characteristic function C(S) is not subadditive. Then known
allocation methods, which assume distributing value C(I)
accurately (break-even property), do not prevent the subsidize
phenomena. We relax condition for break-even allocation
property of total cost C(I), but prevent to break
the incremental cost tests which are acknowledged as a
subsidize-free allocation conditions. Subsidize-free allocation
can be obtained by multi-criteria model MASIT. From
the set of effective solutions of the problem MASIT the
solutions which equally treat players of the similar impact
on infrastructure costs are preferred. These solutions can
be obtained due to equitable rational preference relation.
In computational practice the problem can be formulated as
a multi-criteria linear programme MASIT OWA, for which
effective solutions are also equitable solutions of problem
MASIT.
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183–190.
[16] H. P. Young, Cost Allocation: Methods, Principles, Applications,
Elseviers Science Publishers B.V., 1985.
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