The new value AL for balanced games is discussed, which is based on averaging of lexicographic maxima of the core.Exactifications of games play a special role to find interesting relations of AL with other solution concepts for various classes of games as convex games, big boss games, simplex games etc. Also exactifications are helpful to associate fully defined games to partially defined games and to develop solution concepts there.cooperative games;average lexicographic value;exact games;partially defined games

Tijs, S.H.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2005–123 
 
THE FIRST STEPS WITH ALEXIA, THE AVERAGE 
LEXICOGRAPHIC VALUE 
 
By Stef Tijs 
 
November 2005 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 The ﬁrst steps with Alexia, the average
lexicographic value
Stef Tijs
CentER and Department of Econometrics and Operations Research,
Tilburg University;
Department of Mathematics, University of Genoa
Abstract
The new value AL for balanced games is discussed, which is based on averaging
of lexicographic maxima of the core. Exactiﬁcations of games play a special role to
ﬁnd interesting relations of AL with other solution concepts for various classes of
games as convex games, big boss games, simplex games etc. Also exactiﬁcations
are helpful to associate fully deﬁned games to partially deﬁned games and to
develop solution concepts there.
Keywords: cooperative game, average lexicographic value, exact game,
partially deﬁned game.
JEL code: C712
1 Introduction
Two solution concepts are dominant in game theory: the Nash equilibrium set (NE-
set) and the core. The importance of the ﬁrst concept was recognized immediately, the
history of the core is more complex. It turns out that there are remarkable similarities
i fo n el o o k sa tt h er o l eo ft h eN E - s e ti nn o n - c o o p e r a t i v eg a m et h e o r ya n dt h er o l eo ft h e
core in cooperative game theory. Non-cooperative games without Nash equilibria as well
as cooperative games with an empty core are not very attractive for a game theorist and
also not in practice. In such cases one can still hope for the existence of approximate
Nash equilibria or approximate core elements. In case of a determined non-cooperative
game an agreement of the players on some NE gives a certain stability because in a
play none of the players can proﬁt in unilaterally deviating from the agreed equilibrium.
Similarly, if in a game a core element is proposed no subgroup of players can perform
better in splitting oﬀ. In case the NE-set is large or the core is large there is room for a
selection theory or a reﬁnement theory. To ﬁnish with the similarities, for both solution
concepts, there are axiomatizations using consistency and converse consistency.
In this paper we concentrate on balanced cooperative games, which are games with a
non-empty core [4], and introduce for such games a new core selection, the AL-value.
Just as in the deﬁnition of the well-known Shapley value [8] an averaging of n! vectors
takes place which correspond to the n! possible orders of the players in an n-person
game. For the Shapley value the vectors are the marginal vectors of the game, for the
AL-value the vectors are the lexicographical optimal points in the core.
The outline of the paper is as follows. In section 2 I introduce the AL-value, discuss
some interesting properties and treat some examples. It turns out that on the cone of
convex games [9] the AL-value and the Shapley value coincide. Section 3 deals with
exact games and the exactiﬁcation operator on games. The AL-value is an additive
function on the cone of exact games. Characteristic for the AL-value is the INVEX-
property (the invariance w.r.t. exactiﬁcation). Further, for simplex games [1, 11] and
also for dual simplex games or 1-convex games [1, 3] it is shown that the AL-value
of such a game coincides with the Shapley value of the exactiﬁcation of the game. In
section 4 the AL-value is studied for the cone of big boss games [5] and it coincides there
with the τ-value [10] and the nucleolus [6]. In section 5 the exactiﬁcation operator is
adapted to treat a family of partially deﬁned games, which gives a possibility to deﬁne
for such games also the AL-value and other values. Section 6 indicates topics for further
research.3
2 The average lexiographic value AL
Given a balanced n-person <N ,v>and given an ordering σ =( σ(1),σ(2),...,σ(n))
of the players in N, the lexicographic maximum of the core C(v) with respect to σ
is denoted by Sσ(v). It is the unique point of the core C(v) with the properties:
(Sσ(v))σ(1) =m a x {xσ(1)|x ∈ C(v)}, (Sσ(v))σ(2) =m a x {xσ(2)|x ∈ C(v) with xσ(1) =
(Sσ(v))σ(1)},...,(Sσ(v))σ(n) =m a x {xσ(n)|x ∈ C(v) with (xσ(1),x σ(2),...,x σ(n−1))=
((Sσ(v))σ(1),(Sσ(v))σ(2),...,(Sσ(v)σ(n−1)))}.
Note that Sσ(v) is an extreme point of the core for each σ.
The average lexicographic value of <N ,v>is the average over all Sσ(v) i.e. AL(v)=
1
n!

σ∈Π(N)
Sσ(v), where Π(N) denotes the set of n! orderings of N.
Example 2.1. Let <N , v>be a 2-person balanced game with N = {1,2}. Then
v(1,2) ≥ v(1) + v(2) and C(v)= conv({f1,f2}) with f1 =( v(N) − v(2),v(2)),f 2 =
(v(1),v(N) − v(1)). Further Π(N)={(1,2),(2,1)},S (1,2)(v)=f1,S (2,1)(v)=f2, So,
AL(v)=1
2(f1 + f2)=( v(1) + 1
2(v(1,2) − v(1) − v(2)),v (2) + 1
2(v(1,2) − v(1) − v(2)),
the standard solution for the 2-person game <N ,v>.
Example 2.2. Let <N ,v>be the 3-person convex game with N = {1,2,3},v (i)=0
for each i ∈ N, v(S)=1 0if |S| =2and v(N)=3 0 . Then S(1,2,3)(v)=( 2 0 ,10,0) =
m(3,2,1)(v),S (1,3,2)(v)=( 2 0 ,0,10) = m(2,3,1)(v),...,S(3,2,1)(v)=( 0 ,10,20) = m(1,2,3)(v).
Here mσ(v) is the marginal vector w.r.t. σ with
m
σ
σ(k)(v)=v(σ(1),...,σ(k)) − v(σ(1),σ(2),...,σ(k − 1)) for each k ∈ N.
So, AL(v)=( 1 0 ,10,10) = 1
3!

σ∈Π(N)
Sσ(v)= 1
3!

σ∈Π(N)
m¯ σ(v)=φ(v) where
¯ σ =( σ(3),σ(2),σ(1)), the reverse order of σ, and φ(v) is the Shapley value of <N ,v>.
Theorem 2.3. For each convex game <N ,v> : AL(v)=φ(v).
Proof. Note that for each σ ∈ Π(N):Sσ(v)=m¯ σ(v), where ¯ σ =( σ(n),σ(n −
1),...σ(2),σ(1)). 
Theorem 2.4. Let <N ,v>be a balanced simplex game [1,11] i.e. a game where C(v)
is equal to the non-empty imputation set I(v)={x ∈ Rn|
n 
i=1
xi = v(N),x i ≥ v({i}) for
each i ∈ N}. Then AL(v)=CIS(v), the center of the imputation set.
Proof. Note that I(v)=conv{f1(v),f2(v),...,fn(v)} and CIS(v)= 1
n
n 
k=1
fk(v) where
(fk(v))i = v(i) for i ∈ N\{k} and (fk(v))k = v(N) −

i∈N\{k}
v(i). Because Sσ(v)=4
fσ(1)(v) for each σ ∈ Π(N) we obtain AL(v)= 1
n!

σ∈Π(N)
fσ(1)(v)=1
n
n 
k=1
fk(v)=CIS(v).

For dual simplex games (also called 1-convex games) (see [3],[11]) we give without proof
the following results.
Theorem 2.5. Let <N , v>be a balanced n-person game with C(v)=I∗(v)=
conv({g1(v),g 2(v),...,g n(v)}), where
(g
k(v))i = v
∗(k)=v(N) − v(N\{k}) for i  = k and
(g
k(v))k = v(N) −

i∈N\{k}
v
∗(i).
Then AL(v)=ENSR(v), where ENSR is the rule which splits equally the non-
separable rewards. AL(v) is also equal to the nucleolus [6] and the τ-value [10] of
(N,v).
It will be clear that AL satisﬁes the following properties: IR (Individual rationality),
EFF (eﬃciency), S-equivalence, CS (core selection) and SYM (symmetry). Also DUM
(the dummy property) holds for AL because for each balanced game <N , v>the
AL-value AL(v) is an element of the core and for each x ∈ C(v):
v(i) ≤ xi =
n 
k=1
xk −

k∈N\{i}
xk ≤ v(N) − v(N\{i})
So, if i is a dummy player, then xi = v(i) for each core element and, especially ALi(v)=
v(i) for a dummy player. In the next section we consider two other properties of AL :
INVEX, ADDE.
3E x a c t g a m e s
Exact games are introduced by Schmeidler [7] and they play an interesting role in this
section. Recall that a game is an exact game if for each coalition S ∈ 2N\{φ} there is
an element xS ∈ C(v) such that

i∈S
xS
i = v(S). Let us denote by EXN the set of exact
games with player set N. In fact EXN is a cone of games and one easily sees that AL :
EXN → Rn is additive. We call this interesting property ADDE :A L (v+w)=AL(v)+
AL(w) for each v,w ∈ EXN.5
Note that for each balanced game <N , v>t h e r ei sau n i q u ee x a c tg a m e<N , v E >
with the same core as the original game. This exactiﬁcation <N ,v E > of <N ,v>is
deﬁned by vE(φ)=0and
v
E(S)=m i n {

i∈S
xi|x ∈ C(v)} for each S ∈ 2
N\{φ}
So, C(vE)=C(v) for each balanced game <N , v>and vE = v iﬀ <N , v>is
exact. Note that an interesting property for AL is: if for <N ,v> ,<N ,w>we have
C(v)=C(w)  = φ, then AL(v)=AL(w). This property is equivalent with the property
INVEX : AL(v)=AL(vE) for each balanced game <N ,v> ,where INVEX stands for
’invariant w.r.t. exactiﬁcation’.
In view of theorem 2.3 this INVEX-property of AL gives the possibility to prove that for
some games <N ,v>t h eA L - v a l u eo f<N ,v>coincides with the Shapley value φ(vE)
of the exactiﬁcation <N ,v E > of <N ,v>.T h i si st h ec a s ef o rt h o s eg a m e<N ,v>
for which the exactiﬁcation is convex. This holds e.g. for simplex games, dual simplex
games and also for 2- and 3-person balanced game. So we obtain
Theorem 3.1.
(i) If <N ,v>is a balanced 2-person game or a 3-person game, then AL(v)=φ(vE).
(ii) For each simplex game <N ,v>we have AL(v)=φ(vE).
(iii) For each dual simplex game <N ,v>we have AL(v)=φ(vE).
Proof of (ii) only. Let <N ,v>be a simplex game. Then C(v)=
I(v)=conv{f1(v),f2(v),...,fn(v)}. So vE(N)=v(N) and for each S ∈ 2N\{φ,N} :
vE(S)=m i n {

i∈S
xi|x ∈ C(v)} =m i n {

i∈S
fk
i |k ∈{ 1,2,...,n}} =m i n {

i∈S
v(i),v (N) −

i∈N\S
v(i)} =

i∈S
v(i). This implies that vE is a sum of convex games namely vE =
n 
i=1
v(i)u{i}+(v(N)−
n 
k=1
v(k))uN (where uS denotes the unanimity game with uS(T)=1
if S ⊂ T and uS(T)=0otherwise). So, <N , v E > is a convex game and AL(v)=
AL(vE)=φ(vE). 
Now we give a 4-person exact game <N ,v> , where φ(v)=φ(vE)  = AL(v). This game
is a slight variant of an example in [2] on p. 91.6
Example 3.2. Let ε ∈ (0,1] and let <N ,v>be the game with N = {1,2,3,4},
v(S)=7 ,12,22 if |S| =2 ,3,4 respectively and
v(1) = ε, v(2) = v(3) = v(4) = 0.
Note that <N ,v>is not convex because
v(1,2,3) − v(1,2) = 5 <v (1,3) − v(1) = 7 − ε.
Note further that Ext(C(v)) has the maximum number of 24 extreme points:
(i) 12 extreme points which are permutations of (10,5,5,2),
(ii) 9 extreme points which are permutations of (7,7,8,0) but with ﬁrst coordinate
unequal to 0,
(iii) (ε,7 − ε,7 − ε,8+ε),(ε,7 − ε,8 − ε,7 − ε) and (ε,8 − ε,7 − ε,7 − ε).
From this follows that <N ,v>is an exact game, and that each lexicographic maximum
Sσ(v) is equal to a permutation of the vector (10,5,5,2), where each such permutation
corresponds to two orders.
So, AL(v)=( 5 1
2,51
2,51
2,51
2) and is unequal to
φ(v
E)=φ(v)=( 5 1
2 + 1
4ε,51
2 − 1
12ε,51
2 − 1
12ε,51
2 − 1
12ε).
4 Big boss games and the average lexicographic value
Big boss games are introduced in [5] and further discussed in [1] and [11]. Recall that
an n-person game <N ,v>is a big boss game with n as big boss if the following three
conditions hold:
1. Big boss property: v(S)=0for all S with n/ ∈ S.
2. Monotonicity property: v(S) ≤ v(T) for all S,T ∈ 2N with S ⊂ T.
3. Union property: v(N) − v(S) ≥

i∈N\S
Mi(v) for each S ∈ 2N with n ∈ S.7
It is well-known that the core of a big boss game with n as big boss is given by
C(v)={x ∈ R
n|0 ≤ xi ≤ Mi(v) for each i ∈ N\{n},
n 
i=1
xi = v(N)}
and the τ-value by τ(v)=( 1
2M1(v), 1
2M2(v),...,1
2Mn−1(v),v (N) − 1
2
n−1 
i=1
Mi(v)).
The extreme points of a big boss game <N , v>with n as big boss are of the form
P T where T ⊂ N\{n} and P T
i = Mi(v) if i ∈ T, PT
i =0if i ∈ N\T ∪{ n} and
P T
n = v(N) −

i∈T
Mi(v). For each σ ∈ Π(N) the lexiocographic maximum Sσ(v) equals
P T(σ), where T(σ)={i ∈ N\{n}|σ(i) <σ (n)}.
Theorem 4.1. Let <N ,v>be a big boss game with n as big boss. Then AL(v)=τ(v).
Proof. For each i ∈ N\{n} : ALi(v)= 1
n!

σ∈Π(N)
(Sσ(v))i = 1
n!

σ∈Π(N)
(PT(σ))i =
1
n!Mi(v)|{σ ∈ Π(N)|σ(i) <σ (n)}| = 1
2Mi(v)=τi(v)
By EFF of τ a n dA Lt h e na l s oA L n(v)=τn(v). 
L e tu sl o o ka tt h ee x a c t i ﬁ c a t i o n<N ,v E > of the big boss game <N ,v>with n as a
big boss.
(i) For S ⊂ N\{n} we have
v
E(S)= m i n
T⊂N\{n}

i∈S
P
T
i =

i∈S
P
φ
i =0
(ii) For S with n ∈ S we have
v
E(S)= m i n
T⊂N\{n}

i∈S
P
T
i =m i n
T⊂N\{n}
(v(N) −

i∈T\S
Mi(v))
=

i∈S
P
N\{n}
i =( v(N) −
n−1 
i=1
Mi(v)) +

i∈S
Mi(v).
This implies that vE is a non-negative combination of convex unanimity games:
v
E =( v(N) −
n−1 
i=1
Mi(v))u{n} +

i∈N\{n}
Mi(v)u{i,n}
So, vE is a convex game (and also a big boss game) and the extreme points of C(v) and
of C(vE) coincide. So we obtain τ(v)=AL(v)=AL(vE)=φ(vE).
Theorem 4.2. The AL-value of a big boss game equals the Shapley value of the
exactiﬁcation of the big boss game.8
5 An approach to handle partially deﬁned games
Cases where a player set N is confronted with the problem of dividing v(N), where not
for each subcoalition of N the worth is given, are discussed extensively in the literature.
I will consider special balanced partially deﬁned games. These are games <N ,v ,F >,
where N is the player set, F is a subset of 2N, containing N and φ and v : F→R has
the properties v(φ)=0and
CF(v): ={x ∈ R
n|
n 
i=1
xi = v(N),

i∈S
xi ≥ v(S) for all F∈2
N}
is a non-empty and bounded set.
For such a balanced F-game v one can study the exact ’extension’ ¯ v :2 N → R where
¯ v(S)=m i n {

i∈S
xi|x ∈ CF(v)}
where we have a real extension if <N ,v ,F > has the exactness property: v(S)=¯ v(S)
for S ∈F . Given a solution Ψ for games <N , v>one can deﬁne a solution ¯ Ψ for
balanced partially deﬁned game by
¯ Ψ(N,v,F)=Ψ ( N,¯ v)
It is interesting to study AL in such situations.
6 Concluding remarks
Further research on the average lexicographic value will include
(i) monotonicity properties of AL,
(ii) continuity properties of AL,
(iii) consistency properties of AL,
(iv) axiomatizations,
(v) numerical aspects,
(vi) cones with a perfect kernel system and AL,
(vii) relations with other core selections,
(viii) extensions of the AL-value for non-balanced games,
(ix) more relations with other solution concepts.9
References
[1] Branzei, R., Tijs, S. (2001). Additivity regions for solutions in cooperative game
theory, Libertas Mathematica 21: 155-167.
[2] Derks, J., Kuipers,J. (2002). On the number of extreme points of the core of a
transferable utility game, in: ’Chapters in Game Theory’ (Eds. P. Borm and H. Peters),
Kluwer Academic Publishers, pp. 83-97.
[3] Driessen, T., Tijs, S. (1983). The τ-value, the nucleolus and the core for a subclass
of games, Methods of Operations Research 46: 395-406.
[4] Gillies, D.B. (1953). Some theorems on n-person games. Dissertation, Department
of Mathematics, Princeton University.
[5] Muto, S., Nakayama, M., Potters, J., Tijs, S. (1988). On big boss games, The
Economic Studies Quarterly 39: 303-321.
[6] Schmeidler, D. (1969). The nucleolus of a characteristic function game, SIAM Journal
of Applied Mathematics 17: 1163-1170.
[7] Schmeidler, D. (1972). Cores of exact games, Journal of Mathematical Analysis and
Applications 40, 214-225.
[8] Shapley, L. (1953). A value for n-person games, in: ’Contributions to the Theory
of Games II’ (Eds. H.W. Kuhn and A.W. Tucker), Annals of Mathematics Studies No.
28, Princeton University Press, pp. 307-317.
[9] Shapley, L. (1971). Cores of convex games, International Journal of Game Theory
1: 11-26.
[10] Tijs, S. (1981). Bounds of the core of a game and the τ-value, in: ’Game Theory
and Mathematical Economics (Eds. O. Moeschlin and D. Pallaschke), North-Holland
Publishing Company, pp. 123-132.
[11] Tijs, S., Branzei, R. (2002). Additive stable solutions on perfect cones of cooperative
games, International Journal of Game Theory 31: 469-474.

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