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3On Quasi-cores, the Shapley Value and the Semivalues
Irinel Dragan
University of Texas at Arlington, Department of Mathematics,
Arlington, Tx. 76019-0408, USA
Abstract. The aim of the present paper is that of introducing a new concept of coalitional rationality for values of cooperative TU games, called w—coalitional rationality, such that this becomes the usual coalitional rationality in the case of an efficient value. As a motivation for our new concept, the class of Semivalues, which are in general non efficient values, was considered, and we proved necessary and sufficient conditions for the w—coalitional rationality of Semivalues. It is well known that the only efficient Semivalue is the Shapley value. The basic idea to be followed here is the fact that the Semivalues are not connected to the core, because the efficiency is missing, but may be connected to some quasi-core. (in particular, to the Shapley-Shubik weak e—core, from which the w is borrowed).
Shapley and Shubik (1966) have introduced the quasi-cores in connection with the market games and have discussed the non-emptiness of two types of quasi-cores (see the paper by Kannai in Handbook of Game Theory, vol.I, 1992).
Keywords: Coalitional rationality, Shapley value, Semivalues, Average per capita
formulas, Quasi-cores.
Introduction
In the first section we introduced a more general concept of quasi-core and the symmetric quasi-core, which will be our basic tool; as a byproduct, we provided also necessary and sufficient conditions for the non-emptiness of the symmetric quasi-core.
The Semivalues of a game are Shapley values of an easily obtained auxiliary game, as shown in a previous paper [Dragan, 2005]; this was proved by using the fact that any Least Square value is a Shapley value [Dragan, 2006], and any Semivalue is a Least Square value [Ruiz et al.,1998]. Therefore, in the second section we proved some necessary and sufficient conditions for the appurtenance of the Shapley value to the
symmetric quasi-core, by using a proof suggested by the case of the appurtenance of the Shapley value to the core.
After introducing a new concept of coalitional rationality for any value, the connection between Semivalues and the Shapley value was allowing us to reduce the problem of finding necessary and sufficient conditions of coalitional rationality for a Semivalue, to the problem of appurtenance of the Shapley value to a symmetric quasi-core. To be able to do this, in the third section we proved an Average per capita formula for the Efficient normalization of a Semivalue. This was possible due to previous results obtained in a joint paper by Dragan and Martinez-Legaz (2001), where an Average per capita formula for the Semivalues was obtained. Note that in that paper an alternative definition of coalitional rationality was given, and the case of Semivalues was also considered, following that definition.
The necessary and sufficient conditions for w-coalitional rationality of a Semivalue, are given in the last section. Some examples discuss the application of the results to the Semivalues of a three-person cooperative game, and compare the findings to the previously found similar conditions for the Shapley value. Throughout the paper we sketched the proofs of the previous results needed.
1. Quasi-cores
A cooperative transferable utility game (TU-game) is a pair (N,v), where N is a finite set of players, n = \N|, and v : 2N ^ R is a real function defined on the set of subsets of N, denoted by 2N, with v(0) = 0. Any non-empty subset S of N is called a coalition and, for all S C N, the number v(S) is thought to represent the outcome available to S, independent of the actions of players not in S, in case that S is formed. This way of thinking implies that any vector x = (x*) of outcomes, where x* is the win of player i, is not acceptable to some coalition S, if we have x(S) < v(S). (it is used to denote by x(S) the sum of all components for i G S). Indeed, if this is the case, then the players in S would break the coalition(s) in which they belong and will form the coalition S in which they can win more. The opposite situation, when we have
x(S) > v(S), VS C N, (1)
for the outcome vector x G Rn, is described by saying that x is coalitionally rational.
For the grand coalition there are no players outside N, so that the outcome should
satisfy
x(N )= v(N). (2)
This is called the Pareto optimality condition, or efficiency condition, while (1) for singletons are called individual rationality conditions. The set of outcomes
CO(N, v) = {x G Rn : x(N) = v(N), x(S) > v(S), VS C N}, (3)
is the Core of the game (N,v), due to D.B. Gillies (1953, 1959). The Core is considered a solution of the game, in the sense that every element x in the Core is
acceptable all players, because no one alone or in a group can improve upon the outcome x. However, it is clear that sometimes the Core may be empty. For example, if B = {Sj C N : j =1, 2,...,k} is a partition of N, and the core is non-empty, then for each block Sj we should have x(Sj) > v(Sj), so that by summing up these inequalities and using x(N) = v(N) we should have
j=k
v(N) >£ v(Sj). (4)
j=i
Therefore, if one of such inequalities does not hold for some partition, then the Core should be empty. Moreover, (4) are only necessary conditions for non-emptiness.
The characterization of games with non-empty cores was given by O. Bondareva (1963) and L.S. Shapley (1967), and it is based upon the concept of balanced family of coalitions. A collection of coalitions {Si, S2,...,Sk}, not necessarily a partition, which has a set of non-negative numbers {Ai, A2 ,...,Xk} called the weights, such that
]T Aj = 1, Vi G N, (5)
Sj -AE Sj
is called a balanced family of coalitions. The Bondareva-Shapley theorem (see [Bondareva, 1963], [Shapley, 1967]) says that a game has a non-empty Core if, and only if, for any collection of balanced sets {S1, S2,...,Sk} with the set of weights {Ai, A2,..., Ak} we have
j=k
J^Ajv(Sj) < v(N). (6)
j=i
The excellent survey by Y. Kannai (1992) is discussing the existence of the Core for TU-games and more general classes of games, as well as the connected results. A TU-game with a non-empty core is called a balanced game. The concept of balanced game and the Bondareva-Shapley theorem stated above is needed in the following.
Now, consider a cooperative TU-game (N,v), and a set of 2N — 2 nonnegative weights
{Ss G R : S C N,Ss G [as,fa ]} (7)
with aS > 0, where each interval [aS,pS] contains the number 1. By extension of the way of thinking discussed above, where all Ss were one, a coalition S C N may consider that a vector x = (x*) of outcomes is not acceptable for S if the total win x(S) does not exceed a fraction Ss of v(S), that is
x(S) < Ssv(S). (8)
Of course, S would prefer those vectors x G Rn of outcomes for which
x(S) > Ssv(S), VS C N.
(9)
Such a payoff vector x, satisfying (9), will be called S-coalitionally rational. For the grand coalition, there are no players outside N, so that the outcome should still satisfy the condition (2).
Definition 1. The set of outcomes
COs (N,v) = {x G Rn : x(N) = v(N),x(S) > Ssv(S), VS C N}, (10)
where x(S) is the sum of components for players in S, will be called the S— quasi-core of the game (N, v)
Note that for a set of weights and a game (N,v) we can define a new associate TU-game (N,wg) by
ws(S)= Ssv(S), VS C N, (11)
where Sn = 1 and the other components are satisfying (7); obviously, the S-quasi-core of (N, v) is the usual core of (N, wg). After a careful reading of Shapley-Shubik paper [Shapley, 1966], as well as of the Kannai survey [Kannai, 1992], the above definition of a quasi-core is not surprising. Indeed, if for a game with all v(S) > 0 we take in (10)
Ss = 1--t^,VScN, (12)
v(S)
where e is a positive number, we get
COs(N, v) = {x G Rn : x(N) = v(N), x(S) > v(S) — e, VS C N}, (13)
that is the Shapley-Shubik strong e—core (see [20], p.812). Similarly, if for a positive game we take in (10)
Ss=1~^ VSCN, (14)
v(S)
where e > 0 and \S \ is the cardinality of S, we get
COs (N,v) = {x G Rn : x(N) = v(N),x(S) > v(S) — e \S\, VS C N}, (15)
that is the Shapley-Shubik weak e—core (see [Shapley, 1966], p. 812). Moreover, the Shapley-Shubik paper gives reasonable meanings for these types of quasi-cores and is discussing the existence of the strong e-core, ([Shapley, 1966], section 12), and the existence of the weak e-core, ([Shapley, 1966], section 8), for market games. Accordingly, taking into account the above remark, inspired by the introduction of the game (N,wg) in (11), the non-emptiness of the S-quasi-core of the game (N,v), defined in (10), can be reduced to the non- emptiness of the usual core of the game (N, wg). Therefore, a byproduct of this idea and of the Bondareva-Shapley theorem, is the following result:
Theorem 1. A game (N,v) has a non-empty S-quasi-core, associated with a set of weights (7), if and only if for any balanced collection {Si, S2, ..., Sk} with the set of ba,la,ncing weights {Ai, A2,...,Ak}, we have
Y^AjSsjv(Sj) < v(N). (16)
Example 1. A 3-person cooperative TU-game has 5 minimal balanced sets with the following balancing weights
Bi = {(1), (2), (3)} ^ Ai = A2 = A3 = 1,
B2 = {(1), (2, 3)} ^ Ai = A23 = 1,
B3 = {(2), (1, 3)} ^ A2 = Ai3 = 1,
B4 = {(3), (1, 2)} ^ A3 = Ai2 = 1,
B5 = {(1, 2), (1, 3), (2, 3)} —> A12 = A13 = A23 = -•
(17)
2
(see [G.Owen, 1992], p. 230, for a complete discussion, showing that we can confine ourselves in (16) to minimal balanced collections).
The S-quasi-core associated with the set of weights {Si, S2, S3, Si2, Si3, S23} is nonempty for a game
v(1) = v(2) = v(3) = 0, v(1, 2)= a, v(1, 3) = b,v(2, 3) = c, v(1, 2, 3) = 1, (18)
if, and only if, we have
max(Si2a, Si3b, S23C) < 1, aSi2 + bSi3 + CS23 < 2. (19)
Note that if we have Si = S2 = S3 = S(1), and Si2 = Si3 = S23 = S(2), then for the same game the non-emptiness conditions become
12 max(a,6, c) < ——, a + b + c<——. (20)
S(2) S(2)
In the next section we consider only the S-quasi-cores associated with weights depending on the size of the coalition, that is
Ss = S(\S\), VS c N, (21)
because such quasi-cores will be interesting in the present paper. Of course, in this case a weight vector S G Rn, called a symmetric weight vector, would give all weights by (21), and we take Sn = 1. For this type of S-quasi-core we shall consider the problem: find out necessary and sufficient conditions for the appurtenance of the Shapley value of a TU-game (N, v) to the S-quasi-core associated with a system of weights satisfying (21); this will be done in the next section. There this type of quasicores will be called symmetric S-quasi-cores. Notice that the strong e-core and the weak e-core are not symmetric S-quasi-cores, because in (12) and (14), respectively, the weight vector S has components depending on the coalitions, not only on the sizes of coalitions; therefore, a comparison of our results to those given in [Kannai,
1992], or in [Shapley, 1966], can not be made. The conditions to be found in the next section will be further used in the last two sections, in order to derive necessary and sufficient conditions of “coalitional rationality” for a Semivalue, a concept to be defined also in the following.
2. Symmetric quasi-cores and the Shapley value
A game (N,v), and a set of symmetric weights S G Rn, with Sn = 1, are given. Recall that the S-quasi-core of (N,v) was defined as
COg (N,v) = {x G Rn : x(N) = v(N),x(T) > S(t)v(T), VT C N}, (22)
where x(T) = x%, VT C N. Let us introduce the notations
ieT
(S)
-y-, Vi G N, s = 1, 2,...,n — 1, (23)
i.e. vs is the average worth of all coalitions of size s, and v\ is the average worth of all coalitions of size s which do not contain player i. A formula for the well known concept of solution, the Shapley value [Shapley, 1953], which was called an Average per capita formula, is expressing the value in terms of the above averages as follows:
v(S)
|S| = s
E <
|S|=s,i/S n1
v
s
n
s
s
Theorem 2 [Dragan]. For all players i G N we have
SHi(N,v) = '^ + J2'^^, (24)
ns
s=i
where SH is the well known linear operator called the Shapley value, and the numbers vs and v\, Vi G N are defined by (10) for s = 1, 2,...,n — 1.
In the joint work [Dragan, 2001], the problem of the appurtenance of a Semivalue to what was called the Power Core was considered. As the Shapley value is a Semivalue, and the Shapley value is efficient, from the necessary and sufficient conditions obtained, we derived necessary and sufficient conditions for the appurtenance of the Shapley value to the Core (see [Dragan, 2001], p.135), that is the S-quasi-core with
unitary weights. Now, we consider the problem of the appurtenance of the Shapley
value to the symmetric S-quasi-core, which is an interesting problem by itself, especially in the cases when the Core of the game is empty. As the Shapley value is efficient, we should have to satisfy in (9) only the inequalities
x(T) > S(t)v(T), VT C N. (25)
However, the strategy used for the proof in [Dragan, 2001] still works, and we shall give it here below in the proof of the following:
Theorem 3. The Shapley value of the TU game (N, v) belongs to the symmetric S-quasi-core of (N,v) associated with the weight vector S G Rn with S(n) = 1, if and only if we have
^+x:H:))'z|5n.J'T~'<t'(s)-^‘.(D.vrcw. (20
s=1 V V 7 7 |S|=s
Proof.
Consider any coalition T C N and let us write (12) for the Shapley value by using the Average per capita formula (11). For any coalition T C N we get
n_i tvs — vi
YSHi(N,v)=t^ + Y------------— (27)
^ D Q.
n s
ieT s =1
Now, to return to the coalitional form, use (10) and
Evs = (n — M E (t — s n ti)v(s)
icT V / I CI _ o
(28)
in (14), and it is easy to see that we obtain (13). Obviously, the theorem holds in more general cases, if S(t) is replaced by ST.
Note that if we check the appurtenance of the Shapley value to a symmetric S-quasi-core by using Theorem 3, then in (13) for each fixed T C N we should take for each s = 1, 2,...,n — 1, all coalitions of size s, even though for some we may have IS n Ti = 0, because S n T =0, so that only the second term of the numerator occurs.
Note that Theorem 3 will be used in the last section in order to derive necessary and sufficient conditions of “’coalitional rationality” for a Semivalue. However, for this purpose a different equivalent form of (13) is more appropriate. The formula (13) of Theorem 3 was written such that it is pretty close to the corresponding formula of our work [Dragan, 2001]. If we use the simple equality
st ( n ^ \ 1 = 1 (s-1)!(n-s~1)! = i_ n-i (29)
\ s J J n — s nt (n — 1)! nt s ’
where 7n_1 are the Shapley coefficients for the marginal contributions of coalitions of size s in an n — 1-person cooperative TU-game, then (13) becomes
—+ 7E^1lE (\SnT\--t)v(S)}>d-^v(T),\/TcN. (30)
n t n t
s = 1 |S|=s
These will be the conditions to be used in our last section to prove the main result (Theorem 8) and to compare the necessary and sufficient conditions of appurtenance
of a Semivalue to the Core with the necessary and sufficient conditions of appurtenance of the Shapley value to the Core, obtained from (17) for S(t) = 1, yT C N.
On the other hand, it is clear that (13) of Theorem 2.2, or (17), is a sufficient condition for the non-emptiness of the symmetric S-quasi-core. Of course, an interesting exercise would be to prove that (17) implies (16) of Theorem 1.
Example 2. Return to the game considered in Example 1 and let us write the coalitional rationality conditions for the symmetric S-quasi-core, that is (12), when the outcome is the Shapley value. By Theorem 3, we can use the equivalent conditions (13). For coalitions T, which are singletons, we obtain
—a — b + 2c < 2, —a + 2b — c < 2, 2a — b — c < 2, (31)
and for coalitions T with IT| = 2, we obtain
—a — b +2c > 6S(2)c — 4, —a + 2b — c > 6S(2)b — 4, 2a — b — c > 6S(2)a — 4. (32)
Note that S(1) does not occur in (18) because the worth of each singleton equals zero.
Obviously, if the Shapley value is in the symmetric S-quasi-core, then this is nonempty. For the game (18) of Example 1, we show easily that (18) and (19) imply (20). Indeed, by pairing the inequalities (18) with the corresponding inequalities (19), we get S(2) max(a, b, c) < 1, and by adding up the inequalities (17) we get S(2)(a + b + c) < 2. Of course, we may also write (19) to get the same results, where the Shapley coefficients will appear and to explain the form of conditions (18).
In the next section, after introducing a new concept of coalitional rationality for a value of a TU-game, in order to apply this concept to a Semivalue, we have to derive an Average per capita formula for the value called Efficient normalization of a Semivalue due to Ruiz et al. (1998). They introduced the Efficient normalization of a Semivalue, in order to establish the relationship between the Semivalues and the new family of values called the Least Square Values introduced in their paper, (see also [Dragan, 2006]. Here, we have a different purpose, namely, we intend to derive a relationship between the Efficient normalization of a Semivalue, and the Shapley value, with no need of Least Square Values. Such an Average per capita formula for the Efficient normalization of a Semivalue, appeared in a different form in [Dragan, 2005], where we discussed the possibility of computation of a Semivalue by computing a Shapley value. To make the paper self-contained we show shortly the corresponding proof, after taking the Average per capita formula for Semivalues from the joint work [Dragan, 2001].
3. A concept of coalitional rationality. The Efficient normalization of a Semivalue.
Let Gn be the space of cooperative TU-games with the set of players N. If n = IN I, then it is well known that this space is identified with the Euclidian space R2 _1. Let G be the union of all spaces Gn for different sets of players. A value is a
functional ^ which for each game (N,v) G Gn gives an outcome vector ^(N, v) G Rn. The value is efficient, if we have for all functionals v defined on 2N the equality
E *j (N,v)= v(N). (33)
jeN
In general, the values are not efficient and there are several methods to make the value efficient.
Definition 2. The Efficient normalization of a value ^ is the value E^ given by E^s(N,v) = ^s(N,v)+a(N,v), Vi G N, (34)
where
a(N,v) = ±[v(N)-'52*j(N,v)]. (35)
n jeN
Of course, the normalization term a(N,v) depends on ^, but it does not depend on i; however, we do not think that any confusion may occur if we do not mention it.
In words, after giving to each player j G N his payoff provided by ^j(N,v), the reminder is shared equally by the players, if a is positive, or returned in equal shares if a is negative. Obviously, if ^ is efficient, then a = 0. Note that there are other methods to derive an efficient value from a non-efficient one. Now, we introduce a new concept of coalitional rationality for a value.
Definition 3. A value which is not necessarily efficient, is w-coalitionally rational, if we have
Y/E^j(N,v) > v(S), VS C N. (36)
jes
In words, the new concept of coalitional rationality is requiring that the Efficient normalization of ^ belongs to the Core of the game (N,v). The efficiency was not mentioned in Definition 3 because it is automatically satisfied for the Efficient normalization. Note that if the value ^ is efficient, then the w-coalitional rationality reduces to the usual coalitional rationality. Note also that the conditions (36), which define the w-coalitional rationality, based upon (35), maybe rewritte nunder the form
Y'^j(N,v) > v(S) — ISI.a(N,v), VS C N.
jes
These are the coalitional rationality conditions imposed to a value which belongs to the weak a-core (if a is positive), as it was shown by (15) in the first section. This was the reason of calling the new concept a w-coalitional rationality. The reader may compare this concept of coalitional rationality to the concept introduced in the previous joint paper [Dragan, 2001]. We consider that both have the same
merit, because they are reduced to the usual concept of coalitional rationality for efficient values. As we have seen in that paper, the concept was successfully used to derive necessary and sufficient conditions of coalitional rationality for Semivalues, and the same thing will happen here, as it will be seen below. However, to write the conditions (36) for a Semivalue, we need an Average per capita formula for the Efficient normalization of a Semivalue; this is our aim here below.
The Semivalues, introduced axiomatically in a more general framework by Dubey et al. (1981), may be defined on Gn by the formula
SEi(N,v) = E Pn[v(S) — v(S — {i})], Vi G N, (37)
S:ieSCN
where pn = (pn,fin,... ,Prn)T is a non-negative weight vector satisfying the normalization condition _
E( n — 11) pn = 1. (38)
S=1 ' '
Note that (37) is also the definition of a Semivalue for arbitrary sets N, hence for sets of players of different sizes we should have different weight vectors. In other words, to define a Semivalue on the union of all spaces Gn , a sequence of weight vectors p1 ,p2,... ,pn,... should be given, all of them satisfying normalization conditions like
(38). Moreover, following the authors mentioned above, we assume that recursive relationships connect these vectors
pn_1 = pn + pn+1, s = 1, 2,...,n — 1. (39)
We call (39) the inverse Pascal triangle relations. Obviously, from (38) and (39) it follows that the normalization conditions are satisfied by pl, for all t < n — 1 as
soon as they are satisfied for t = n; moreover, if pn satisfying (38) is given, then all
pl,t < n, are uniquely determined by (39). Now, the Semivalues are defined on G by the same formula (37), where (N,v) is arbitrary in G. Obviously, the Shapley value [Shapley, 1953] belongs to the family of the Semivalues, for pn = (n!)_1 (s — 1)!(n — s)!, s = 1, 2,... ,n.
In the previous work [Dragan and Martinez, 2001], we proved an Average per capita formula for Semivalues, which will be given here without a proof. Consider again the notations (10) for the average worth of coalitions of size s and the average worth of coalitions of size s which do not contain the player i, for s = 1, 2,...,n— 1, and all i G N. We use also vn = v(N). Then, we have:
Theorem 4 [Th. 1]. The Semivalue associated with the non-negative weight vector pn satisfying (38) is given by
n_1 n n 1 i SEi(N,v)=q^ + Y<lsVs~qs Vs,VieN, (40)
n n s
where the non-negative vector q is defined by
„n_ Pns (s — 1)!(n - s)!
with H------------------------■’ s = l,2,...,n. (41)
Yn n!
n — 1
Note that in (40) we had </"_1 = Psn~i, where and 7"_1 were given by
(37); also, y7 are the weights which give in (39) the Shapley value, and satisfy (39). Note that from (40) for these weights, that is pn = Yn, s = 1, 2,...,n, we get (11). However, (40) applies also to other Semivalues, for example, to the Banzhaf value (J.F. Banzhaf, III, 1965), obtained for other weight vectors pn satisfying (38) and
(39), precisely pn = 21-n, s = 1, 2,...,n. It is clear that the new weight vectors qn satisfy other normalization conditions and other inverse Pascal triangle relations.
In the same paper was proved a formula for the computation of what was called the Power game, which may be used to compute the worth of the grand coalition in that game. Calvo and Santos (1997) and Sanchez (1997) call this Power game an auxiliary game. We defined the Power game, (N, ), of the given game (N, v), by
nf (T) = £ (T,v), VT C N. (42)
jeT
The value ^ was considered coalitional rational in [5] if ^(N, v) belongs to the Core
of the Power game of ^. Here, we need the worth of the grand coalition of the Power
game, but we do not use that formula which will be obtained independently, from
(40) shown above.
To derive from the new concept, called w-coalitional rationality, introduced above by Definition 3, necessary and sufficient conditions of w-coalitional rationality for a Semivalue, we need an Average per capita formula for the Efficient normalization of a Semivalue.
Consider the Average per capita formula (40) of Theorem 4. After adding up all components in (40), we get
n 1 n n 1
r,n-1 ____
Vj)
„ s ' s
jeN s = 1 s = 1 jeN
Y se3 (n,v) = qyn+«EV-EV(E^ (43)
Now, by using the equality ^ vS = nvs, valid for all s =1, 2,...,n — 1, we proved
jeN
Theorem 5. Let SE be the Semivalue associated with the non-negative weight vector pn, and qn be the new weight vector introduced by (39). Then we have
<E(N) = ]T SE^N, v) = qyn +nJ2 1)V&- (44)
jeN s=1 s
Note that a similar formula would give the worth of the other coalitions in the Power game, but the averages should be taken in the subgames of (N, v); we do not need
them here. From (44) we intend to compute the Average per capita formula for the Efficient normalization of a Semivalue. Taking into account Theorem 5, we have to compute only the efficiency term a(N, v), as seen in the proof of the following
Theorem 6. Let SE be a Semivalue associated with a non-negative weight vector pn,n > 2, and qn be the non-negative weight vector defined by (41). Then, the Efficient normalization of the Semivalue, denoted by ESE, is given by
n-1 i
ESEi(N,v) = — + Yjq^lVs~Vs, VieN. (45)
n
1
Proof.
From (35) and (44) we compute the efficiency term
(1 - CK ^ (<£ - <£-V
a = --------- '
Now, by using (34) for ^ = SE, from (35) and (46) we get (45).
Formula (45) is the Average per capita formula for the Efficient normalization of a Semivalue, and it will be used, together with Theorem 3, and another result about the relationship of a Semivalue with the Shapley value, in the last section, in order to derive necessary and sufficient conditions of w—coalitional rationality for a Semivalue.
4. Necessary and sufficient conditions for w-coalitional rationality
A relationship between the Efficient normalization of a Semivalue and the Shapley value is easy to obtain by looking at the Average per capita formulas of these two values, namely (11) and (45). It is given by
Theorem 7. Consider a TU game (N, v), and a nonnegative weight vector pn, n > 2. Let qn-1 be the non-negative weight vector defined by
pn I pn
n-1 = Ps----/Vfi = 1 2 1 (47)
s 7?+7s”+i ^
where YSn are the coefficients of terms in the formula of the Shapley value.Then, the Efficient normalization of a Semivalue for the game (N,v) is the Shapley value of the game (N,w) obtained from (N,v) by means of the rescaling:
w(S) = qn-1v(S), VS C N, w(N) = v(N). (48)
n
Proof.
From (48) and (10) we have
vn = wn, qrn-1vs = ws, qrn-1 v\ = w\, Vi G N, s = 1, 2,...,n — 1. (49)
These equalities used in (45) and the Average per capita formula for the Shapley value (10) show that we have
the result stated in Theorem 7.
Returning to the w-coalitional rationality introduced by Definition 2, we see that the following result follows:
Corollary. The Semivalue SE of the TU game (N,v) associated with the nonnegative
game (N, w) defined by (49) belongs to the symmetric 5-quasi-core of the game (N, w), where 5 G Rn is given by 5(s) = (qn-1)-1, s = 1, 2,...,n — 1, and 5(n) = 1, and the weight vector qn-1 is given in terms of pn by (47).
From Definition 2 with ^ = SE,taking into account Theorem 7, expressed by (50), and the equalities (48), we get
that is the fact that the Shapley value of the game (N, w) belongs to the symmetric 5-quasi-core with 5 shown in (51).
Now, to obtain necessary and sufficient conditions of w—coalitional rationality it is enough to write the necessary and sufficient conditions of Theorem 3 for the appurtenance of the Shapley value of the game (N, w) to the symmetric 5—quasicore of this game, with 5(s) = ~^=r, s = 1, 2,..., n — 1, and S(n) = 1. Hence, it is enough to write (17) for the game (N,w), with this weight vector 5 and replace w in terms of v, as shown by (48). By Corollary, we obtain necessary and sufficient conditions for the w—coalitional rationality of a Semivalue, i.e. we proved the main result of this paper:
Theorem 8. The Semivalue SE associated with the nonnegative weight vector pn is w-coalitionally rational, if, and only if, for all coalitions T C N we have
ESE(N,v) = SH(N,w),
(50)
weight vector pn is w-coalitionally rational if, and only if, the Shapley value of the
Proof.
where the weight vector pn 1 is given in terms of pn by the inverse Pascal triangle relationships.
Proof.
By Corollary, the necessary and sufficient conditions (13) for the appurtenance of the Shapley value of the game (N, w) to the symmetric ^-quasi-core, written under the form (17), are
— + 7Z>r1[£ (\SnT\--t)w(S)} > 1 w(T),VTciV, (53)
n T s = 1 |S|=s n tqs
and by using (48) in (53), where (41) have also been used, we obtain (52).
Note that (52) is looking very similar to the necessary and sufficient condition for the appurtenance of the Shapley value to the Core, obtained from (17) by taking all S(t) = 1, for t = 1, 2,...,n — 1. Indeed, that condition is the particular case of (52) for pin-1 = y7-1, s = 1, 2,...,n — 1 (see also [5], formula (29), where Yl-1 have been replaced by their formula shown in (41)).
Example 3. Return to the 3-person game considered in the Example 1. Suppose that we would like to find the necessary and sufficient conditions for the w-coalitional rationality of a Semivalue defined by the weight vector p3 = (pl,p3,p3)T, where pi + 2p2 + p3 = 1, and the weight vector p2 = (p1,p2)T is derived from p3 by the inverse Pascal triangle relations (39). Then, by using (52) we get the conditions
a b — 2c ^ —7), a — 26 -f c ^ —7), —2(2 b c ^ —7),
p22 p22 p22
for coalitions T with \T\ = 1, and
3a — 2 ^ 3b — 2 , „ 3c — 2
cz — b — c ^ ---7y—, —a -\- 26 — c ^ ------7y—, —a — b -\- 2c ^ ^,
p22 p22 p22
for coalitions T with \T\ = 2. Now, if the Semivalue is the Shapley value, then the w-coalitional rationality conditions are the usual coalitional conditions, that is the conditions of the appurtenance of the Shapley value to the Core, because the Shapely value is efficient. In this case, we have p\ = so that from the above inequalities we get the conditions
—a — b + 2c < 2, —a + 2b — c < 2, 2a — b — c < 2,
a + b + 4c < 4, a + 4b + c < 4, 4a + b + c < 4,
which can also be obtained from the conditions proved in the second section for
<*(2) = 1.
Notice that (52) may be used for other Semivalues, for example the Banzhaf value. In this case, p\ = p\ = p\ = so that again we have p>\ = It follows that the conditions are the same as for the Shapley value, but the Banzhaf value is not efficient, we can compute a(N,v) = -^(1 — a — b — c), which in general is not zero; further, we can compute the Efficient normalization of the Banzhaf value and check that this value happens to be equal to the Shapley value for such games. This
explains why the conditions are the same, even though they represent the conditions for the appurtenance of the Efficient normalization to the Core, in the last case. Note that we may also take a Semivalue defined by the weight vector p3 = (|, |)T,
for which we have p2 = (j, |)T, so that the conditions for w-coalitional rationality will be different. Obviously, this will be a linear system of inequalities in three unknowns which may be consistent, or not. The fact that the system may not be consistent follows from the well known fact that sometimes the Shapley value, which is a Semivalue, does not belong to the Core.
Aknowledgement
This work has been done during the visits of I.Dragan at University Autonoma of Barcelona, Department of Economics, in 2003 and 2005, where helpful discussions have been carried out with our previous coauthor Professor Juan Enrique Martinez-Legaz.
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