Existence of Fractional Solutions in NTU DEA Game
Jing Fu
1
and Shigeo Muto
2
1
Academy for Co-creative Education of Environment and Energy Science, Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
2
Department of Social Engineering, Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
Keywords:
NTU Coalitional Game, DEA, Fractional Solutions, α-core, β-core.
Abstract:
This paper deals with the problem of fairly allocating a certain amount of benefit among individuals or or-
ganizations with multiple criteria for their performance evaluation. It is an extension work of our paper on
game theoretic approaches to weight assignments in data envelopment analysis (DEA) problems (Sekine et al.,
2014). One of the main conclusions in our previous work is that the core of the TU (transferable utility) DEA
game is non-empty if and only if the game is inessential, that is, the evaluation indices are identical for all the
criteria for each player. This condition is equivalent to a trivial single-criterion setting, which motivates us to
turn to the NTU (non-transferable utility) situation and check the existence of the fractional solutions. In this
study, we contribute on showing the existence of α-core, and giving two sufficient conditions such that β-core
exists and is identical to α-core in NTU DEA game. Our discussion is also interesting in light of a direction
to improve the robustness of the β-core existence condition by relaxing the inessential condition in TU DEA
game.
1 INTRODUCTION
The problem dealt in this paper is the consensus-
making among individuals or organizations in shar-
ing a fixed amount of benefit with multiple criteria
for evaluating their performance. DEA problem origi-
nates from the bankruptcy problem by (O’Neil, 1982),
which has been subsequently analyzed in a variety
of contexts. Among them the multi-issue allocation
game by (Calleja et al., 2005) has the most similar
context to our DEA game. The example given in their
paper is
“The central government has to decide how to allocate the
taxpayers’ money to various public services. The system of
government is such that it does not allocate this money di-
rectly to these services, but indirectly through various gov-
ernment departments. Each department (player) has a num-
ber of claims on the amount of money available, arising
from those public services (issues) for which it has respon-
sibility. Some of these services are provided by just a sin-
gle department (i.e., tax collection vs. the Department of
Finance), while more departments may be responsible for
other services (i.e., foreign trade vs. the Department of Eco-
nomic Affairs, Foreign Affairs and Defense). If we were to
add up all the claims of a department into one single claim,
an ordinary bankruptcy problem would arise.”
As argued by (Calleja et al., 2005), the underly-
ing issues should play a role in determining an out-
come. If the departmental claims are combined, how-
ever, this crucial information is lost. DEA game dif-
fers from the multi-issue allocation game in two as-
pects: first, what a player directly claims is a strategy
based on his performance on different criteria, instead
of an amount of the benefit available. Here a criterion
is the standard to evaluate the performance of a player,
in other words, it is an attribute of a player’s perfor-
mance. Whilst in the multi-issue allocation game, an
issue is an act a player involved in and gives rise to
a claim. Second, with the concept of DEA, the al-
location of the benefit is determined endogenously
based on all the players’ strategy combination. Let
us explain it this way. Each player is empowered to
vote for the importance of different criteria (strategy),
and our mechanism determines the allocation by re-
ferring to this voting result (strategy combination).
It is more “democratic” and can potentially increase
the perceived fairness of the players on the allocation
result. On the contrary, the basic assumption in the
multi-issue allocation game is that once the central or-
ganization is paying out money according to one par-
ticular issue, this issue must first be fully dealt with
before moving on to the next issue. Hence they have
107
Fu J. and Muto S..
Existence of Fractional Solutions in NTU DEA Game.
DOI: 10.5220/0005206701070115
In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 107-115
ISBN: 978-989-758-075-8
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Table 1: Pollution Emission Level Comparison between IGCC
and Conventional PCT.
SO
X
NO
X
Particulate
IGCC
1
1.0 ppm 3.4 ppm < 0.1 mg/m
3
N
PCT
2
< 17.7 ppm < 49.4 ppm < 20 mg/m
3
N
1
Joban Joint Power: http://www.joban-power.co.jp/igccdata/en/
2
Shenhua Guohua Beijing Cogeneration: http://www.shghrd.
com/HistoryData.aspx
to define the allocation rule either by a proportional
game or a queue game exogenously.
To illustrate our model, consider the following
case. According to a statement released after the State
Council’s executive meeting presided by Premier Li
Keqiang, Chinese central government has laid out a
number of detailed measures that identify large cities
and regions with the most frequent smog and haze as
key areas in the battle against air pollution. A spe-
cial fund of 10 billion yuan ($1.65 billion) is set up
in 2014 to reward efforts to curb air pollution in the
key areas. Suppose the central government is provid-
ing a fixed amount of clean air fund (benefit) and calls
for proposals from both domestic and overseas orga-
nizations. Successful applicants (players) have ad-
vantages in different criteria, namely, the sustainabil-
ity indicators. For example, applicant A (i.e., Joban
Joint Power, Japan) owns IGCC (integrated gasifica-
tion combined cycle) technology, which is a type of
gasification technology that turns coal into gas, and
can potentially improve the efficiency of coal-fired
power compared to conventional pulverized coal tech-
nology (PCT) as well as the environmental perfor-
mance (Table 1). Applicant B (i.e., Shenhua Guohua
Beijing Cogeneration, China) has advantage in good
source of clean coal with low sulfur content and low
installation cost of new coal-fired power plant. The
central government has to decide a reasonable alloca-
tion of the fund to these successful applicants, which
should reach the consensus among them. In DEA
game, as assumed by (Nakabayashi and Tone, 2006),
the successful applicants are egoistic and each of them
sticks to his superiority regarding the sustainability
indicators, i.e., efficiency, cleaner resource, pollution
emission, installation cost, maintenance cost and etc.
The process above is similar to a two-stage tender-
ing with technical and financial proposals submitted
separately. In the first stage, organizations A, B, C ...
submit their technical proposals without fund claim,
in accordance with the specifications by the central
government and their respective performance on dif-
ferent criteria. The technical proposals are evaluated
and successful applicants are announced. The second
stage is to invite the successful applicants to vote for
the importance of all the criteria. The fund allocation
is endogenously determined by DEA game with ref-
erence to this voting result. Our paper focuses on the
second stage.
In the literature, DEA is generally introduced as a
mathematical programming approach for measuring
relative efficiencies of decision-making units (DMUs,
in DEA game we use “player” instead), where multi-
ple inputs and multiple outputs are present. The in-
terest of this paper can be best justified by the latest
technical note by (Cook et al., 2014), they emphasized
that
“Ultimately DEA is a method for performance evaluation
and benchmarking against best-practice. The inputs are
usually the ‘less-the-better’ type of performance measures
and the outputs are usually the ‘more-the-better’ type of per-
formance measures. This case is particularly relevant to the
situations where DEA is employed as a MCDM (multiple
criteria decision making) tool. DEA then can be viewed
as a multiple-criteria evaluation methodology where DMUs
are alternatives, and DEA inputs and outputs are two sets
of performance criteria where one set (inputs) is to be min-
imized and the other (outputs) is to be maximized.”
In the case above, the central government has
taken the trouble to apply multiple criteria to deter-
mine the allocation of the fund, as a single criterion is
hardly to become an acceptable basis of a fair judg-
ment. Different interest group, either a single appli-
cant, or a coalition, has different opinion on the im-
portance of each criterion for estimating a reasonable
allocation. Hence DEA is employed as a MCDM tool,
where the DMUs are the successful applicants, and
output-to-input ratio is the relative contribution of one
DMU compared to the total contribution of all DMUs.
Here “contribution” is equivalent to “performance”,
which is quantitatively measured by the score given
to each DMU by different criteria. This representa-
tion, presented in the next section, is not a typical effi-
ciency measure within the context of maximizing the
ratio of weighted outputs to weighted inputs subject
to constraints reflecting the performance of the other
DMUs; whilst it is the ratio of the fund each DMU
may receive by integrating the performance of other
DMUs in the objective function. (Nakabayashi et al.,
2009) justified this representation by addressing the
fundamental concept of DEA–variable weights. The
weights are derived directly from the existing data
and they are chosen in a manner that assigns a best
set of weights to each DMU. The term “best” means
that the resulting output-to-input ratio for each DMU
is maximized relative to all other DMUs when these
weights are assigned to these outputs and inputs for
every DMU.
(Nakabayashi and Tone, 2006) first proposed
DEA game to solve this kind of multi-criteria benefit
allocation problem. Their DEA max game was, how-
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
108

ever, subadditive. Namely, players gain less in total,
or lose their power when they cooperate. To make the
game superadditive, they took the dual of the game,
called DEA min game, in which each player or each
coalition picks up the weight that minimizes their
evaluation. No reasonable justification was given in
their paper for picking up the minimizing weight un-
der the assumption that players are egoistic and want
to maximize their own evaluation. (Sekine et al.,
2014) have improved the DEA game above by reas-
signing the total weights for the coalition members,
and studied different solution concepts of TU DEA
game and the equilibria of the strategic form DEA
game. In particular, we have proved that the core of
the TU DEA game is non-empty if and only if the
game is inessential, which is equivalent to a trivial
single-criterion situation as the inessentiality requires
the evaluation indices for all the criteria to be identical
for each player. Hence the core of the TU DEA game
is generally empty, and this extension paper turns the
focus to the NTU coalitional game associated with
the strategic form DEA game (NTU DEA game for
short).
An NTU coalitional game is a specification of
payoff vectors attainable by members of each coali-
tion through some joint course of action. A classical
example is an exchange economy, where coalitions
can reach certain payoff distributions that constitute
the feasible set for that coalition by pooling and redis-
tributing their initial endowments. The players con-
front the problem of choosing a payoff or solution
that is feasible for the group as a whole. This paper
explores two fractional solutions for the NTU DEA
game, namely, the α-core and β-core.
The rest of the paper is organized as follows. In
section 2, we review the basic DEA game scheme in-
cluding the DEA model in (Nakabayashi and Tone,
2006), and the strategic form DEA game in (Sekine
et al., 2014). Section 3 defines the NTU DEA game
in both α and β fashion. Section 4 gives our main
analysis results regarding the existence of α-core and
β-core, and section 5 discusses a direction to improve
the robustness of β-core existence condition. Finally,
section 6 closes this paper with some concluding re-
marks.
2 A REVIEW ON THE DEA
GAME SCHEME
This section reviews the DEA game scheme before
we proceed to the formulation of NTU DEA game.
2.1 The DEA Model
Let E(> 0) denote the fixed amount of benefit to be
allocated to players 1, ... , n. Players’ contributions
are evaluated by multiple criteria and summarized as
the score matrix C = (c
i j
)
i=1,...,m, j=1,...,n
, where c
i j
is
player j’s contribution measured by criterion i, called
the evaluation index. The problem is to find a weight
vector on the criteria (strategy) determined endoge-
nously by players themselves, and reasonable alloca-
tions of E based on the weight vector combinations
(strategy combinations). Following the DEA analy-
sis, each player k chooses an nonnegative weight vec-
tor w
k
= (w
k
1
,. ..,w
k
m
) such that
∑
m
i=1
w
k
i
= 1, w
k
i
≥
0 ∀i = 1,... ,m, where w
k
i
is the weight given to cri-
terion i by player k. Then the contribution of player
k relative to the total contribution of all players mea-
sured by the weight vector w
k
is given by
∑
m
i=1
w
k
i
c
ik
∑
m
i=1
w
k
i
(
∑
n
j=1
c
i j
)
Player k chooses the weight vector that maximizes
this ratio. The weight vector is found by solving the
following fractional program
max
w
k
∑
m
i=1
w
k
i
c
ik
∑
m
i=1
w
k
i
(
∑
n
j=1
c
i j
)
s.t.
m
∑
i=1
w
k
i
= 1, w
k
i
≥ 0 ∀i = 1,.. . ,m
Each of the other players similarly maximizes the ra-
tio produced by his own weight vector. This repre-
sentation deviates from the output-to-input ratio in
the traditional DEA approach, as the weighted sum
of player k’s contribution is hardly to say as “output”,
and the weighted sum of all players’ total contribu-
tions is not a typical “input” as well. However, it is
classical in nature as each egoistic player is trying to
present to the authority that he contributes more or
performs better than other players under his own val-
uation system, in order to win the consensus of the
authority and others that his ability and effort worth
more benefit. Hence player k’s contribution is the
“more-the-better”, whereas the total contribution of
all players is the “less-the-better”, consistent with the
argument by (Cook et al., 2014).
This maximization problem can be reformulated
as the following much simpler form. For each row
(c
i1
,. ..,c
in
), i = 1,... , m, divide each element by the
row-sum
∑
n
j=1
c
i j
. Let
c
0
i j
= c
i j
/
n
∑
j=1
c
i j
i = 1,.. .,m
ExistenceofFractionalSolutionsinNTUDEAGame
109

The matrix C
0
= (c
0
i j
)
i=1,...,m, j=1,...,n
is called the nor-
malized score matrix and
∑
n
j=1
c
0
i j
= 1 is satisfied.
By Charnes-Cooper transformation (Charnes et al.,
1978), the maximization problem is not affected by
this operation. Then the fractional maximization pro-
gram above can be expressed as the following linear
maximization program.
max
w
k
m
∑
i=1
w
k
i
c
0
ik
s.t.
m
∑
i=1
w
k
i
= 1, w
k
i
≥ 0 ∀i = 1,.. . ,m
Let c(k) be the maximal value of the program. Appar-
ently the maximal value is attained by letting w
k
i(k)
= 1
for the criterion i(k) such that c
0
i(k)k
= max
i=1,...,m
c
0
ik
and letting w
k
i
= 0 for all other criteria i 6= i(k). Thus
c(k) is the highest relative contribution of player k,
namely
c(k) = max
i=1,...,m
c
0
ik
For simplicity, we assume that the score ma-
trix is given in the normalized form. That is,
C = (c
i j
)
i=1,...,m, j=1,...,n
, where
∑
n
j=1
c
i j
= 1 ∀i =
1,. ..,m; c
i j
≥ 0 ∀i = 1,.. . ,m, ∀ j = 1, ... , n.
(Nakabayashi and Tone, 2006) showed that if each
player k claims the portion c(k) of E, the sum of the
claims generally exceeds the total benefit E. Then
the problem arises: how to allocate E reasonably to
players? To find a fair allocation of E, they proposed
to apply cooperative game theory. As is mentioned
above, we have improved their DEA game in (Sekine
et al., 2014). Let us first review the definition of the
strategic form DEA game, and then proceed to intro-
duce the NTU DEA game.
2.2 Strategic Form DEA Game
Let N = {1,.. .,n} be the set of players and M =
{1,. ..,m} be the set of criteria. The basic DEA model
is as follows. Each player j ∈ N chooses a weight vec-
tor w
j
= (w
j
1
,. ..,w
j
m
) with w
j
1
+ ... + w
j
m
= 1,w
j
i
≥
0 ∀i ∈ M on the criteria so as to maximize his relative
contribution,
∑
m
i=1
w
j
i
c
i j
. The fixed amount of bene-
fit E is shared by the players in accordance with their
relative contribution.
Therefore the strategic form game reflecting the
DEA model can be defined as
G
DEA
:= (N,{W
j
}
j∈N
,{ f
j
}
j∈N
)
where N = {1,..., n} is the set of players, W
j
=
{w
j
= (w
j
1
,. ..,w
j
m
)|w
j
1
+...+w
j
m
= 1,w
j
i
≥ 0 ∀i ∈ M}
is the strategy set of player j ∈ N, and f
j
: W =
W
1
× .. . × W
n
→ ℜ is the payoff function of player
j ∈ N, which is given by
f
j
(w
1
,. ..,w
n
) = (
m
∑
i=1
(
1
n
n
∑
j=1
w
j
i
)c
i j
)E
Namely, the reward E is shared by players in propor-
tion to the weighted sum of their evaluation indices
where the weights are the average weights over all
players. Suppose that there are 4 out of 5 players re-
ceiving the highest evaluation on criterion i(k). Ob-
viously egoistic them will all put their whole weight
on i(k), and i(k) will receive an average weight of at
least 0.8 out of 1. This payoff function well reflects
the overall ranking on the importance of different cri-
teria by all the players, and can potentially increase
the perceived fairness on the allocation result. Here-
after we call this game the strategic form DEA game.
3 NTU DEA GAME
NTU coalitional game is somewhat more complicated
than the TU one by the fact that there is no standard
definition, and many results are sensitive to details in
the definition of the game. First let us present the def-
inition we stick to in this research.
Definition 3.1 (NTU Coalitional Game). The pair
(N,V ) is called NTU coalitional game if and only if V
is a correspondence from any coalition S ⊆ N into a
set of real vectors V (S) ⊆ ℜ
N
satisfying the following
conditions:
(1) If S 6=
/
0,V (S) is an non-empty closed subset of
ℜ
N
; and V (
/
0) =
/
0.
(2) ∀x,y ∈ ℜ
N
, if x ∈ V (S), and x
j
≥ y
j
∀ j ∈ S, then
y ∈ V (S).
(3) ∀ j ∈ N,∃V
j
∈ ℜ such that ∀x ∈ ℜ
N
: x ∈ V ({ j})
if and only if x
j
≤ V
j
.
(4) {x ∈ V (N) : x
j
≥ V
j
} is a compact set.
The interpretation of the NTU coalitional game
(N,V ) is that V (S) is the set of feasible payoff vec-
tors for the coalition S if that coalition forms. Only
the coordinates for players j ∈ S in elements of V (S)
matter. A consequence of property (2) is that, if x
is feasible for S, then any y ≤ x is feasible for S as
well; this property is often called comprehensiveness,
and it can be interpreted as free disposability of utility.
Property (3) says that for any player j ∈ N, its feasible
payoff is always upper bounded. Property (4) ensures
that the individually rational part of V (N) is bounded.
Next, based on the strategic form DEA game, the
NTU DEA game can be defined in both α and β fash-
ion depending on a coalition’s reactions against the
deviations by its counter-coalition.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
110

Definition 3.2 (α-coalitional NTU DEA Game).
Suppose a strategic form DEA game G
DEA
:=
(N,{W
j
}
j∈N
,{ f
j
}
j∈N
) is given. The α-coalitional
NTU game (N,V
α
) associated with G
DEA
is a corre-
spondence from any coalition S ⊆ N into a set of real
vectors V
α
(S) ⊆ ℜ
N
, which satisfies
(1) For all non-empty S ( N, x ∈ V
α
(S) if and only
if there exists w
S
∈ W
S
such that, for all u
N\S
∈
W
N\S
and for all j ∈ S, x
j
≤ f
j
(w
S
,u
N\S
).
(2) x ∈ V
α
(N) if and only if there exists w
N
∈W
N
such
that, for all j ∈ N,x
j
≤ f
j
(w
N
).
where W
S
=
∏
j∈S
W
j
. Coalition S ⊆ N is said to α-
improve upon x ∈ V
α
(S) if there exists w
S
∈ W
S
such
that for all u
N\S
∈ W
N\S
, x
j
< f
j
(w
S
,u
N\S
) ∀ j ∈ S
is satisfied. The α-core is the set of payoff vectors
x ∈ V
α
(N) upon which no coalition α-improves.
Definition 3.3 (β-coalitional NTU DEA Game).
Suppose a strategic form DEA game G
DEA
:=
(N,{W
j
}
j∈N
,{ f
j
}
j∈N
) is given. The β-coalitional
NTU game (N,V
β
) associated with G
DEA
is a corre-
spondence from any coalition S ⊆ N into a set of real
vectors V
β
(S) ⊆ ℜ
N
, which satisfies
(1) For all non-empty S ( N, x ∈ V
β
(S) if and only if
for all u
N\S
∈ W
N\S
, there exists w
S
∈ W
S
such
that for all j ∈ S, x
j
≤ f
j
(w
S
,u
N\S
).
(2) x ∈ V
β
(N) if and only if there exists w
N
∈ W
N
such
that, for all j ∈ N,x
j
≤ f
j
(w
N
).
where W
S
=
∏
j∈S
W
j
. Coalition S ⊆ N is said to β-
improve upon x ∈ V
β
(S) if for all u
N\S
∈ W
N\S
, there
exists w
S
∈ W
S
such that x
j
< f
j
(w
S
,u
N\S
) ∀ j ∈ S
is satisfied. The β-core is the set of payoff vectors
x ∈ V
β
(N) upon which no coalition β-improves.
Corollary 3.1. From the definition of the α- and β-
coalitional NTU DEA game, it directly follows that
β − core ⊆ α − core
4 MAIN RESULTS
For the convenience of the reader, we first introduce
the beautiful theorem by (Scarf, 1971).
Theorem 4.1 (Scarf). Assume that for each j ∈ N,
W
j
is a compact convex set, and f
j
is quasi-concave
in w ∈ W , then the α-core is non-empty.
Applying Scarf’s theorem, we can easily show the
non-emptiness of the α-core in the NTU DEA game.
Theorem 4.2. Suppose a strategic form DEA game
G
DEA
:= (N, {W
j
}
j∈N
,{ f
j
}
j∈N
) is given. The α-core
of the NTU coalitional game (N,V
α
) associated with
G
DEA
is non-empty.
Proof. For all j ∈ N, for all w
j
∈ W
j
, we have w
j
i
∈
[0,1] ∀i ∈ M and
∑
m
i=1
w
j
i
= 1, thus W
j
is a compact
convex set.
From the definition of the strategic form DEA
game, we know that
f
j
(w
1
,. ..,w
n
) = (
m
∑
i=1
(
1
n
n
∑
j=1
w
j
i
)c
i j
)E
Let w,v ∈ W, and λ ∈ (0,1), then
f
j
(λw
1
+ (1 − λ)v
1
,. ..,λw
n
+ (1 − λ)v
n
)
= (
m
∑
i=1
(
1
n
n
∑
j=1
(λw
j
i
+ (1 − λ)v
j
i
))c
i j
)E
= λ f
j
(w
1
,. ..,w
n
) + (1 − λ) f
j
(v
1
,. ..,v
n
)
≥ min{ f
j
(w
1
,. ..,w
n
), f
j
(v
1
,. ..,v
n
)}
Thus f
j
is quasi-concave in w ∈ W . The α-core of the
NTU DEA game is non-empty.
Definition 3.3 says that β-core 3 x of the NTU
DEA game is non-empty if and only if for any coali-
tion S ⊆ N, there exists u
N\S
∈ W
N\S
such that for all
w
S
∈ W
S
, there exists j ∈ S such that
x
j
≥ f
j
(w
S
,u
N\S
)
is satisfied. This two-folded “exists” requirement
makes β-core somehow more intricate than α-core.
(Masuzawa, 2003) presented a specific class of
NTU games with non-empty cores different from that
of (Scarf, 1971), without quasi-concavity of payoff
functions. Although Theorem 4.2 has shown that the
payoff function of the strategic form DEA game is
quasi-concave, his concept of dominant punishment
strategy has high applicability, and we will apply it
to find an existence condition for β-core. First let us
define the dominant punishment strategy.
Definition 4.1 (Dominant Punishment Strategy). Let
/
0 6= S ⊆ N, and w
S
,v
S
∈ W
S
. We say that w
S
is weakly
punishment-dominant over v
S
against N \ S if for all
j ∈ N \ S and for all u
N\S
∈ W
N\S
f
j
(u
N\S
,v
S
) ≥ f
j
(u
N\S
,w
S
)
is satisfied. w
S
is a dominant punishment strategy
against N \ S if w
S
is weakly punishment-dominant
over all v
S
∈ W
S
against N \ S.
If this concept is applied to the fixed benefit al-
location problem such as the DEA game, a domi-
nant punishment strategy w
S
actually secures coali-
tion S the largest portion of the benefit it can receive
whatever strategies the complementary coalition may
choose. Let us give a lemma regarding the existence
of dominant punishment strategy in the strategic form
DEA game.
ExistenceofFractionalSolutionsinNTUDEAGame
111

Lemma 4.1. Suppose a strategic form DEA game
G
DEA
:= (N,{W
j
}
j∈N
,{ f
j
}
j∈N
) is given. Let C
j
=
{i( j) ∈ M | c
i( j) j
≤ c
i j
∀i ∈ M} ∀ j ∈ N. Each
player k ∈ N has a dominant punishment strategy
against others if and only if for all k ∈ N, C
N\{k}
=
C
1
∩ . . . ∩C
k−1
∩C
k+1
∩ . . . ∩C
n
6=
/
0.
Proof. Sufficiency: from the definition, we know that
C
j
is the set of criteria giving player j the lowest eval-
uation. Let k ∈ N, C
N\{k}
6=
/
0 means that there is at
least one common criterion i( j) ∈ M giving all the
player j ∈ N \ {k} the lowest evaluation. Assume for
all k ∈ N, w
k
corresponds to a strategy such that player
k puts his whole weight on criterion i( j) ∈ C
N\{k}
.
Then for all j ∈ N \ {k}, for all v
k
∈ W
k
m
∑
i=1
(v
k
i
− w
k
i
)c
i j
≥ 0
is satisfied. Hence for all j ∈ N \ {k}, for all u
N\{k}
∈
W
N\{k}
and for all v
k
∈ W
k
f
j
(u
N\{k}
,v
k
) − f
j
(u
N\{k}
,w
k
)
= (
m
∑
i=1
(
1
n
(v
k
i
− w
k
i
))c
i j
)E
≥ 0
w
k
weakly punishment dominates over all v
k
∈ W
k
and is the dominant punishment strategy of player k.
Necessity: suppose that there exists player k ∈ N
such that C
N\{k}
=
/
0. Let w
k
∈ W
k
be a strategy such
that for certain player j ∈ N \ {k}, for all u
N\{k}
∈
W
N\{k}
and for all v
k
∈ W
k
f
j
(u
N\{k}
,v
k
) − f
j
(u
N\{k}
,w
k
)
= (
m
∑
i=1
(
1
n
(v
k
i
− w
k
i
))c
i j
)E
≥ 0
is satisfied. Then w
k
must correspond to a strategy
such that player k puts his whole weight on the crite-
rion (criteria) i( j) ∈ C
j
. However, as C
N\{k}
=
/
0, there
must exist player j
0
∈ N \{k} and j
0
6= j such that cri-
terion (criteria) i( j) /∈ C
j
0
. Let w
k
( j
0
) be a strategy
such that player k puts his whole weight on the cri-
terion (criteria) i( j
0
) ∈ C
j
0
, then for player j
0
, for all
u
N\{k}
∈ W
N\{k}
f
j
0
(u
N\{k}
,w
k
( j
0
)) − f
j
0
(u
N\{k}
,w
k
)
= (
m
∑
i=1
(
1
n
(w
k
i
( j
0
) − w
k
i
))c
i j
0
)E
< 0
in which w
k
( j
0
) punishes player j
0
more severely than
w
k
. Hence player k has no dominant punishment strat-
egy.
Each player k ∈ N has a dominant punishment
strategy against others if and only if for all k ∈ N,
C
N\{k}
6=
/
0.
Now we will give a proposition showing the exis-
tence of β-core with the concept of dominant punish-
ment strategy.
Proposition 4.1. Suppose a strategic form DEA game
G
DEA
:= (N, {W
j
}
j∈N
,{ f
j
}
j∈N
) is given. If each
player j ∈ N has a dominant punishment strategy
against others, the β-core of the NTU coalitional
game (N,V
β
) associated with G
DEA
exists and is iden-
tical to the α-core.
Proof. With Theorem 4.2 and Corollary 3.1, it suf-
fices to show that if each player j ∈ N has a domi-
nant punishment strategy against others, α − core ⊆
β − core.
To see this, assume a payoff vector x ∈ V
β
(S) can
be β-improved. Then, for coalition S ⊆ N, for all
u
N\S
∈ W
N\S
, there exists v
S
∈ W
S
such that x
j
<
f
j
(v
S
,u
N\S
) ∀ j ∈ S is satisfied.
Because each player j ∈ N has a dominant pun-
ishment strategy against others, with Lemma 4.1, it
can be easily verified that each non-empty coalition
S ⊆ N has a dominant punishment strategy against
N \ S. Then for all j ∈ S, for all v
S
∈ W
S
, for all
u
N\S
∈ W
N\S
, there exists d
N\S
∈ W
N\S
such that
f
j
(v
S
,u
N\S
) ≥ f
j
(v
S
,d
N\S
)
It follows that for d
N\S
, there exists w
S
∈ W
S
such that
x
j
< f
j
(w
S
,d
N\S
). Then, there exists w
S
∈ W
S
such
that for all u
N\S
∈ W
N\S
x
j
< f
j
(w
S
,d
N\S
) ≤ f
j
(w
S
,u
N\S
) ∀ j ∈ S
is satisfied. Hence x can be α-improved as well, and
β − core = α − core.
Even though it is a bit strong, next we will present
a sufficient and necessary condition under which each
non-empty coalition S ⊆ N has a dominant strategy.
Similar to the dominant punishment strategy, let us
first give the definition of the dominant strategy.
Definition 4.2 (Dominant Strategy). Let
/
0 6= S ⊆ N,
and w
∗S
∈ W
S
. We say that w
∗S
is a dominant strategy
of S if for all u
N\S
∈ W
N\S
, for all w
S
∈ W
S
, and for
all j ∈ S
f
j
(w
∗S
,u
N\S
) ≥ f
j
(w
S
,u
N\S
)
is satisfied.
Lemma 4.2. Suppose a strategic form DEA game
G
DEA
:= (N,{W
j
}
j∈N
,{ f
j
}
j∈N
) is given. Let
¯
C
j
=
{i( j) ∈ M | c
i( j) j
≥ c
i j
∀i ∈ M} ∀ j ∈ N. Each non-
empty coalition S ⊆ N has a dominant strategy if and
only if
¯
C
N
=
¯
C
1
∩
¯
C
2
∩ . . . ∩
¯
C
n
6=
/
0.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
112

Proof. Sufficiency: from the definition, we know that
¯
C
j
is the set of criteria giving player j the highest
evaluation.
¯
C
N
6=
/
0 means that there is at least one
common criterion i( j) ∈ M giving all the player j ∈ N
the highest evaluation. Assume for each non-empty
coalition S ⊆ N, w
∗S
=
∏
j∈S
w
∗ j
corresponds to a
strategy combination such that each player j ∈ S puts
his whole weight on the criterion (criteria) i( j) ∈
¯
C
N
.
Then for all u
N\S
∈ W
N\S
, for all w
S
∈ W
S
and for all
k ∈ S
f
k
(w
∗S
,u
N\S
) − f
k
(w
S
,u
N\S
)
= (
m
∑
i=1
(
1
n
(
∑
j∈S
w
∗ j
i
−
∑
j∈S
w
j
i
))c
ik
)E
≥ 0
Hence w
∗S
is the dominant strategy of coalition S.
Necessity: suppose
¯
C
N
=
/
0. Take the grand coali-
tion N, and let w
∗N
=
∏
j∈N
w
∗ j
be a strategy com-
bination such that for certain player k ∈ N, for all
w
N
∈ W
N
f
k
(w
∗N
) − f
k
(w
N
)
= (
m
∑
i=1
(
1
n
(
∑
j∈N
w
∗ j
i
−
∑
j∈N
w
j
i
))c
ik
)E
≥ 0
is satisfied. Then w
∗N
must be a strategy combi-
nation such that each player j ∈ N puts his whole
weight on the criterion (criteria) i(k) ∈
¯
C
k
. How-
ever, as
¯
C
N
=
/
0, there must exist player k
0
∈ N and
k
0
6= k such that i(k) /∈
¯
C
k
0
. With the strategy com-
bination w
N
(k
0
) =
∏
j∈N
w
j
(k
0
) such that each player
j ∈ N puts his whole weight on the criterion (criteria)
i(k
0
) ∈
¯
C
k
0
, we have
f
k
0
(w
∗N
) − f
k
0
(w
N
(k
0
))
= (
m
∑
i=1
(
1
n
(
∑
j∈N
w
∗ j
i
−
∑
j∈N
w
j
i
(k
0
)))c
ik
0
)E
< 0
The grand coalition N has no dominant strategy.
Each non-empty coalition S ⊆ N has a dominant
strategy if and only if
¯
C
N
6=
/
0.
The following proposition gives another condition
for an non-empty β-core with the concept of dominant
strategy.
Proposition 4.2. Suppose a strategic form DEA game
G
DEA
:= (N, {W
j
}
j∈N
,{ f
j
}
j∈N
) is given. If each
non-empty coalition S ⊆ N has a dominant strategy,
the β-core of the NTU coalitional game (N,V
β
) asso-
ciated with G
DEA
exists and is identical to the α-core.
Proof. Again with Theorem 4.2 and Corollary 3.1,
it suffices to show that if each non-empty coalition
S ⊆ N has a dominant strategy, α − core ⊆ β −core.
Assume a payoff vector x ∈ V
β
(S) can be β-
improved. Then, for coalition S ⊆ N, for all
u
N\S
∈ W
N\S
, there exists w
S
∈ W
S
such that x
j
<
f
j
(w
S
,u
N\S
) ∀ j ∈ S is satisfied.
As S has a dominant strategy, then there exists
w
∗S
∈ W
S
such that for all u
N\S
∈ W
N\S
, for all w
S
∈
W
S
, for all j ∈ S
f
j
(w
∗S
,u
N\S
) ≥ f
j
(w
S
,u
N\S
)
is satisfied. It follows that there exists w
∗S
∈ W
S
such
that for all u
N\S
∈ W
N\S
, for all j ∈ S
x
j
< f
j
(w
S
,u
N\S
) ≤ f
j
(w
∗S
,u
N\S
)
Hence x can be α-improved as well, and β − core =
α − core.
Propositions 4.1 and 4.2 establish two important
conditions under which the β-core of NTU DEA game
is non-empty. Formally, we give the following theo-
rem.
Theorem 4.3. Suppose a strategic form DEA game
G
DEA
:= (N, {W
j
}
j∈N
,{ f
j
}
j∈N
) is given. For all
j ∈ N, let C
j
= {i( j) ∈ M | c
i( j) j
≤ c
i j
∀i ∈ M}, and
¯
C
j
= {i( j) ∈ M | c
i( j) j
≥ c
i j
∀i ∈ M}. If either condi-
tion A or B below is satisfied, the β-core of the NTU
coalitional game (N,V
β
) associated with G
DEA
exists
and is identical to the α-core.
A. C
N\{ j}
= C
1
∩ ... ∩ C
j−1
∩ C
j+1
∩ ... ∩ C
n
6=
/
0 ∀ j ∈ N
B.
¯
C
N
=
¯
C
1
∩
¯
C
2
∩ . . . ∩
¯
C
n
6=
/
0
This theorem states that in NTU DEA game,
if each player has a dominant punishment strategy
against others, or each non-empty coalition has a
dominant strategy, β-core is non-empty and identical
to α-core. However, both A and B are sufficient con-
ditions and criticized to be a bit strong. In the next
section, we will discuss a direction to improve the ro-
bustness of the β-core existence condition.
5 DISCUSSION
Although we haven’t yet proved a robust existence
condition for the β-core, we believe that the algorithm
in the proposition below opens up new lines of con-
sideration in finding the β-core with relaxation of the
inessential condition in the TU DEA game. For the
convenience of the reader, we re-state the inessential
condition.
ExistenceofFractionalSolutionsinNTUDEAGame
113

Theorem 5.1 (Sekine et al.). A TU DEA game is
inessential if and only if, for all j ∈ N, c
i j
= c
i
0
j
holds
for all i,i
0
= 1,. . .,m.
(Sekine et al., 2014) has shown that the core of
TU DEA game is non-empty if and only if the evalu-
ation indices for all the criteria are identical for each
player, which lost the crucial information provided by
different criteria. Our proposition below concerns the
existence of β-core of a special case.
Proposition 5.1. Suppose a strategic form DEA game
G
DEA
:= (N,{W
j
}
j∈N
,{ f
j
}
j∈N
) is given. Let
¯
C
j
=
{i( j) ∈ M | c
i( j) j
≥ c
i j
∀i ∈ M} ∀ j ∈ N; and for proof
simplicity, assume m = n, the number of criteria is
equal to the number of players. If
¯
C
N
=
¯
C
1
∩
¯
C
2
∩.. .∩
¯
C
n
=
/
0, and
¯
C
N\{ j}
=
¯
C
1
∩ ... ∩
¯
C
j−1
∩
¯
C
j+1
∩ ... ∩
¯
C
n
6=
/
0 ∀ j ∈ N, then the β-core of the NTU coalitional
game (N,V
β
) associated with G
DEA
is non-empty and
x = (1/n)
n
E ∈ β − core.
Proof. It follows directly from the assumption that for
each player k ∈ N, |
¯
C
k
| = m−1 is satisfied; and for all
i(k), i
0
(k) ∈
¯
C
k
, c
i(k)k
= c
i
0
(k)k
is satisfied. Here |
¯
C
k
| =
m−1 is a relaxation of the inessential condition which
requires |
¯
C
k
| = m.
Let us use i
(k) to denote the criterion i(k) ∈ M \
¯
C
k
, and try to find the β-core following the definition.
Suppose x ∈ V
β
(N) is a feasible allocation satisfying
group rationality and its corresponding strategy pro-
file is w
N
. Then for all k ∈ N
x
k
= f
k
(w
N
)
= (
m
∑
i=1
(
1
n
∑
j∈N
w
j
i
)c
ik
)E
= (c
i(k)k
−
1
n
(c
i(k)k
− c
i(k)k
)
∑
j∈N
w
j
i(k)
)E
Our basic algorithm to find the β-core is to shrink the
number of players in S from n to 1 with n steps; at
each step, let the corresponding strategy profile w
N
meet the criterion by Definition 3.3; and finally find a
feasible allocation satisfying all the criteria.
Criterion 1: we begin from the grand coalition
N. As is known that
∑
k∈N
x
k
= E, there does not exist
v
N
∈ W
N
such that x
k
< f
k
(v
N
) ∀k ∈ N. Because the
increase of one player’s payoff will definitely reduce
some other players’ payoff in this fixed benefit allo-
cation problem. Hence for the moment, any payoff
vector x ∈ V
β
(N) is in the β-core.
Criterion 2: next let us examine N \{p} ∀p ∈ N.
No β-improvement on x ∈ V
β
(N) by N \ {p} requires
that there exists u
p
∈ W
p
such that for all v
N\{p}
∈
W
N\{p}
, there should exist at least one k ∈ N \ {p}
with f
k
(w
N
) ≥ f
k
(v
N\{p}
,u
p
) satisfied, which means
f
k
(w
N
) − f
k
(v
N\{p}
,u
p
)
=
1
n
(c
i(k)k
− c
i(k)k
)(
∑
j∈N\{p}
v
j
i(k)
+ u
p
i(k)
−
∑
j∈N
w
j
i(k)
)E
≥ 0
The inequality constraint above is equivalent to
∑
j∈N\{p}
v
j
i(k)
+ u
p
i(k)
−
∑
j∈N
w
j
i(k)
≥ 0
Suppose u
p
i(k)
= 1 and consider the worst situation that
∑
j∈N\{p}
v
j
i(k)
= 0, at this point x ∈ V
β
(N) should be
corresponding to a strategy profile w
N
such that there
exists k ∈ N \ {p} with
∑
j∈N
w
j
i(k)
≤ 1 satisfied.
.. .
Criterion n: following similar procedures, we
continue shrinking the number of players in S one
by one until the single player coalition {p} ∀p ∈ N.
No β-improvement on x ∈ V
β
(N) by {p} requires
that there exists u
N\{p}
∈ W
N\{p}
such that for all
v
p
∈ W
p
, f
p
(w
N
) ≥ f
p
(v
p
,u
N\{p}
) is satisfied. w
N
should be corresponding to a strategy profile such that
∑
j∈N
w
j
i(p)
≤ n −1.
With all the criteria above, it can be easily proved
that the β-core of the NTU DEA game is non-empty
and x = (1/n)
n
E ∈ β − core.
Example 5.1. For a clearer understanding of Propo-
sition 5.1, here is an numerical example with 3 × 3
score matrix, and let us find its β-core following the
algorithm above.


0.45 0.40 0.30
0.45 0.20 0.35
0.10 0.40 0.35


Solution. Criterion 1: the weight matrix should be


w
1
1
w
2
1
w
3
1
w
1
2
w
2
2
w
3
2
w
1
3
w
2
3
w
3
3


in which
∑
3
i=1
w
j
i
= 1 ∀ j ∈ {1, 2,3}.
Criterion 2: consider the two-player coalition
{1,2}. Following criterion 2, either w
1
3
+w
2
3
+w
3
3
≤ 1
or w
1
2
+ w
2
2
+ w
3
2
≤ 1 should be satisfied. Similar for
coalitions {1, 3} and {2,3}, thus at least two out of
the following three conditions have to be satisfied.
(1) w
1
1
+ w
2
1
+ w
3
1
≤ 1
(2) w
1
2
+ w
2
2
+ w
3
2
≤ 1
(3) w
1
3
+ w
2
3
+ w
3
3
≤ 1
Criterion 3: consider the single player coalition,
the following three conditions should all be satisfied.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
114

(1) w
1
1
+ w
2
1
+ w
3
1
≤ 2
(2) w
1
2
+ w
2
2
+ w
3
2
≤ 2
(3) w
1
3
+ w
2
3
+ w
3
3
≤ 2
The multivariate linear inequalities above can be
solved by MATLAB. One of the feasible strategy
combinations
w = w
1
× w
2
× w
3
=


0 0 1
0 1 0
1 0 0


satisfies all three criteria above, and hence x =
(1/3,1/3, 1/3)E is in the β-core. 
We can see that even if the dominant punish-
ment strategy and the dominant strategy for each non-
empty coalition do not exist, β-core may still exist
under certain assumptions. With relaxation of the
inessential condition in TU DEA game, proposition
5.1 shows a new direction in light of improving the
robustness of the β-core existence condition.
6 SUMMARY
In this extension paper, we contribute on studying two
of the fractional solutions in NTU DEA game. We
have shown the existence of the α-core with Scarf’s
theorem, and given two sufficient conditions under
which the β-core is non-empty and identical to the
α-core (Theorem 4.3). This paper also indicates a di-
rection to find a more robust existence condition for β-
core by relaxing the inessential condition in TU DEA
game, and we will follow this line to improve our cur-
rent work.
Even though our study in DEA game is predom-
inantly theoretical and derivation-based, we are now
doing benefit allocation comparison study under dif-
ferent DEA game schemes with the data from Joban
Joint Power and Shenhua Guohua Beijing Cogener-
ation, which will be included in the future work as
well.
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