CC BY
4Elena Yanovskaya
St.Petersburg Institute for Economics and Mathematics,
Russian Academy of Sciences, 1, ul.Tchaikovskogo 191187, St.Petersburg, Russia E-mail address: eyanov@emi.nw.ru
Abstract. The class of TU games whose maximal per capita characteristic function values are attained on the grand coalition. Three axiomatic characterizations of the Shapley value - Shapley’s original axiomatization, Sobolev’s axiomatization with the aid of consistency, and Young’s axiomatization by means of marginality - are used for the corresponding axiomatization of the Shapley value on the class of totally cooperative games. It is shown that only two last axiomatizations characterize the Shapley value uniquely, and Shapley’s axiomatization leads to linear combinations of the Shapley value and the egalitarian value.
Keywords: maximal per capita values, Shapley value, axiomatic characterization. Introduction
Axioms describing properties of cooperative game solutions can be divided into two classes: “within games” and “between games” axioms. The axioms from the first class formulate properties of a solution/value for single games from some class of games (such are the symmetry, the efficiency and the dummy axioms). They can be considered for every class of games, because the axioms are formulated for each game from the class separately. The “between games” axioms connect between themselves the solutions of different games. For example, all independence axioms belong to this class. Hence, the classes of games axiomatized with the help of between games axioms cannot be arbitrary. For example, if we define invariance or covariance of a solution w.r.t. some transformations of games, then all transformed games, together with the initial game, should belong to the class.
3 This work was supported by the NWO grant 047.017.017 and by Russian Foundation for Basic Research, grant 05-01-89005.
In the original axiomatization of the Shapley value with the aid of efficiency, dummy, symmetry, and additivity the first three axioms are “within games” ones, and the last one - a “between games” axiom.
A player faced with the prospect of playing a cooperative game can easily answer whether he agrees or not with some within games axioms, but only for this game. He may not have an opinion about these axioms with regard to other games. He may not even imagine to be a player of this or that game (e.g. a rich man does not represent himself as a poor one and vice versa), and in this case he cannot accept or reject a within games axiom on the whole class of games under consideration.
It is still more reasonable to consider a player not being sure to accept or not the “between games” axioms.
On the other hand, if a value satisfies a certain set of “within games” axioms on some class of games, then it trivially satisfies those axioms on every subclass of games. As for ’’between games” axioms, then this fact is only true when all games being compared belong to the class. For example, consistency axioms require that the reduced games belong to the same class as the original game.
Therefore, the smaller the subclass of games under axiomatization, the more convincing the corresponding values turn out to be for games from this class, but the less likely these axioms will determine a unique value on it. The ultimate goal of the values’ axiomatizations should be the finding of minimal classes of games admitting the axiomatic characterization of a value with the help of certain system of axioms.
In this paper we consider the famous Shapley value and its axiomatizations on a subclass of cooperative games with transferable utilities (TU).
The first axiomatization of the Shapley value was obtained by Shapley (1953) for the class of superadditive games, but his proof can also be used to characterize the Shapley value on the space of all games with finite player sets. Weber (1988) obtained the characterization of the Shapley value with the same axioms for the class of monotonic games. Dubey (1975) characterized the Shapley value on the class of monotonic simple games by means of Shapley’s axioms where additivity was replaced by modularity (transfer): the sum of solutions of two simple games equals to the sum of solutions of two simple games being the minimum and the maximim of the initial games.
Neyman (1989) found the minimal class of games characterized by Shapley’s axioms. It is the additive class spanned by a single game, i.e. this class consists of the initial game and all linear combinations of the restricted on coalitions games. Neyman obtained the corresponding characterizations of the Shapley value on these classes of games. This result is a unique result characterizing the Shapley value by means of Shapley’s axioms on the minimal class of games.
The consistency axioms establish connections between the solution vector of a cooperative game and those of its reduced game. The last one is obtained from the initial game by removing one or more players and by giving them the payoffs according to a specific principle (e.g. a proposed payoff vector). Consistency of a value means that the restriction of the value payoff vector of the initial game to any
coalition is the value payoff vector of the corresponding reduced game. Therefore, to characterize a value by means of consistency, it is necessary to define a universal set of players N and then to consider collections of all classes of TU games with the player sets N С N.
Since reduced games are defined not uniquely, there are several definitions of value’s consistency. In particular, the Shapley value has two axiomatizations with the help of consistency: in the sense of definitions by Sobolev [Sobolev, 1975] and by Hart-Mas-Colell [Hart and Mas-Colell, 1989]. The first definition requires the unboundedness of the universal set of players. The second one considers only games with at most n players. In both definitions consistency and standardness of a value for two-person games uniquely characterize the Shapley value on the corresponding classes of TU games.
One more axiomatization of the Shapley value for games with a fixed player set was obtained by Young (1985). His axiomatization of the Shapley value on the space of all TU games with a fixed player set consists of three axioms: efficiency, symmetry, and marginality. The last axiom states the dependence of players’ payoffs only on their marginal contributions to the characteristic function of the game. Khmelnitskaya (2003) applied these axioms to the characterization of the Shapley value on the class of TU constant sum games.
In the paper a subclass of TU games with high per capita values of the grand coalition is considered. For games from this class the maximal per capita characteristic function value is attained on the grand coalition, so in some sense the players are interested in the complete cooperation. In fact, on this class Dutta’s egalitarian solution [Dutta, 1990] is unique as well as for the convex games, and coincides with the egalitarian value assigning equal shares of the total payoff to all players. It turns out that the mentioned systems of axioms (except for the Hart-Mas-Colell ax-iomatization) characterize the Shapley value on the class under consideration either uniquely, or, together with the Shapley value, they describe the set of values being linear combinations of the Shapley and the egalitarian values.
The paper is organized as follows. In Section 1 we give a list of properties of cooperative games solutions (values) used in axiomatizations of the Shapley value. Three main axiomatizations of the Shapley value on the classes of all TU games with a fixed and variable player sets are given.
The main results are contained in Section 2. Here the class of games with high per capita grand coalition values is considered and three axiomatizations of the Shapley value cited in Section 1 are applied to the characterization of the same value on the class of games with high per capita grand coalition values. It turns out that Shapley’s original axiomatization leads to the set of values being linear combinations of the Shapley and the egalitarian values. Consistency a la Hart-Mas-Colell cannot be applied to the class under consideration because subgames of games from this class (which take part in the definition of the reduced games) may not belong to it. Other two axiomatizations characterize the Shapley value uniquely on this class.
1. Properties of TU values and the known axiomatizations of the Shapley value
Let N be an arbitrary finite set. We denote by Gn a class of TU games with the player set N. Let r = (,N,v) gGN denote an arbitrary game, x G X(r) denote an arbitrary payoff vector, where
X(r) = {y G IRN | £ yi < v(N)},
i£N
the set of payoff vectors of the game r, and
Y(r) = {y G IR" | £ yi = v(N)}
ieN
be the set of efficient (Pareto optimal) payoff vectors.
For any x G IRN, S C N denote by xS the projection of x on the space IRS, and by x(S) - the sum ^ieS xi, with a convention x(0) = 0.
A solution to a class GN is a mapping a, assigning to each game r gGN a subset a(r) C X(r) of its payoff vectors.
If la(N, v) = 11 for a solution a and for each game {N, v) G GN, then the solution a is called a value.
Give now some properties of values, which will be applied in the following characterizations of the Shapley value on some classes of TU games (Section 4).
Recall that the player i G N in the game (N,v) is a dummy, if v(S U {i}) = = v(S) + v({i}) for all S cN,iG S.
A value $ to a class GN
- is efficient, if ieN $i(N,v) = v(N) for every TU-game {N,v);
- is anonymous, if $n(i)(N,nv) = $i(N,v) for all games {N,v), all i G N and
every permutation n of N. Here the game {N, nv) is defined by (nv)(nS) := v(S) for all S C N;
- is symmetric, if for each {N,v) G GN, such that v(S U {i}) = v(S U {'}) for
some i,j G N, and all S C N,i,j G S
$i(N,v)=$j (N,v).
It is known that anonymity implies symmetry for values.
- is (weakly) translation covariant, if $(N,v + a) = $(N, v) + a for all games
{N, v), and a = (ai)ieN G IRN (ai = aj, i,j G N). Here the game {N, v + a) is
defined by (v + a)(S) := v(S) + Xjes aj for all S C N;
- is scale covariant, if $(N, av) = a$(N, v) for all a > 0;
- is covariant, if $ is both translation and scale covariant;
- is additive, if $(N, v) + $(N, w) = $(N,v + w), where for every S C N (v + w)(S) = v(S) + w(S).
- is linear if and for all a, 3 G IR and for all games {N, v), {N, w) gGN the game {N, av + 3w) G GN, and
$(N, av + 3w) = a ■ $(N, v)+ 3 ■ $(N, w).
Here the game {av + 3w) is defined by (av + 3w)(S) := a ■ v(S) + 3 ■ w(S) for all S C N.
- satisfies the dummy axioms, if for every {N, v) gGn where i G N is a dummy player, it holds $i(N, v) = v({i}).
- is marginalist, or satisfies the marginality axiom, if from {N,v), {N,w) G GN, v(S U {i}) — v(S) = w(S U {i}) — w(S) for some i G N and all coalitions S $ i, it follows
$i(N,v) = $i(N,w).
The following axioms connect between themselves the values of games with different player sets. Hence, to formulate them, we should consider the collections of classes of TU games G = UNcAf GN for some infinite universal set N such that
N cN =^ N' cN for all N' c N. (1)
- A value $ on a class G of TU games is consistent or satisfies the reduced game property if from {N,v) G G,x = $(N,v) it follows that for every N' C N the reduced game {N', vx) or {N', v®) belong to the class G as well, and
$N(N, v) = $(N', vx) (= $(N', v*)).
In this definition different notations for reduced games are given. This follows from non-uniqueness of the definitions of the reduced games. For the axiomatization of the Shapley value the following definitions of the reduced games are applied:
- A linear reduced game of the game {N, v) G G on the player set N \ {i} and w.r.t. a payoff vector x is the game {N \ {i}, vN\{i}), whose characteristic function v'N\{i} is defined by
'v(N) — x, if S = N \{i},
x (S)=<>0, if S = 0, (2)
N\{i} I wn,sv(S) + (1 — wn,s)(v(S U {i}) — x)
otherwise.
Here the numbers wn,s G [0,1] are weights, depending on the numbers of players in the initial game and in the corresponding coalition.
The reduced games on arbitrary player sets N' C N are defined by a consecutive elimination of players from the game. To obtain the same result for different permutations of players leaving the game, the weights wns should satisfy the equalities
wn-1,s(1 — wn,s) = wn,s + l(1 — wn-1,s) for all N, s < n — 2
[Yanovskaya and Driessen, 2002].
The following definition due to Hart-Mas-Colell [Hart and Mas-Colell, 1990] define the reduced games w.r.t. some value F to the class |JN,cN GN> for arbitrary finite N.
- A reduced game in the sense of Hart-Mas-Colell of the game {N,v) on the player set N' C N is the game {N' ,vF), where the characteristic function vF is defined as follows:
I v(N) — £iEN\n' Fi(N,v), for S = N',
vF(S) = | v(S U (N \ N')) — £iEN\N' Fi(S U (N \ N'), v), (3)
for other S,
where {S U N \ N', v) is the subgame of {N, v).
Shapley’s amazing result consisted in the fact that four axioms4 from the list above characterize a value uniquely.
Theorem [Shapley, 1953]. For each finite set N there exists a unique value on the class Gn satisfying the efficiency, symmetry, dummy, and additivity axioms: it is the Shapley value given as follows:
$i(N, v) = (n~s)!(s~1)! (y(S U {*}) - v(S)j for all i G N. (4)
S:S3i n'
The most appealing property following from (4) is that the Shapley value is a marginalist value. Young (1985) obtained another characterization of the Shapley value with the help of marginality axiom.
Theorem [Young, 1985]. For each finite N there exists a unique value on the class Gn satisfying efficiency, symmetry, and marginality. It is the Shapley value.
Consider now axiomatizations of the Shapley value on some classes with variable player sets by some consistency axioms. The first such a characterization was given by A. Sobolev (1975) who used the linear consistency axiom with the weights
n s 1
n — 1
(5)
wn.s —
4 In the original paper (Shapley, 1953) Shapley used a unique carrier axiom instead of both the efficiency and the dummy axioms.
Theorem [Sobolev, 1975]. The Shapley value is the unique value on the class G, satisfying efficiency, symmetry, and linear consistency with the weight defined in (5).
The analogous result, but for the classes of games with at most n players was obtained by Hart and Mas-Colell for their definition of consistency.
Theorem [Hart, Mas-Colell, 1989]. For each finite set N the Shapley value is the unique value on the class |Jn/cn Gn' satisfying efficiency, symmetry, and consistency in the sense of the definition (3) of the reduced games.
In the two last Theorems axioms efficiency and symmetry can be replaced by property of standardness for two-person games. Recall that a value $ is standard on a class of two-person games, if for every game {{i,j},v) from the class and every player k G {i,j}
**№,:/},«) = w(W) + - «({*}) = 1,2.
The standard value is efficient, linear and symmetric. The class of all efficient, linear, and symmetric values for two-person games consists of A-standard values, A G IR. A value is A-standard for a class of two-person games if for every game {{i,j},v) and every player k G {i,j}
= A '«(W) + - A '«({*}) - A •«({.?'})•
It is clear that A-standard values are linear combinations of the egalitarian value and the standard value.
Up to the present there are many other axiomatizations of the Shapley value (e.a. Myerson (1977), Chun (1989), Brink (2001)), but in this paper we will apply only the systems of axioms used in the cited theorems to characterize the Shapley value on a subclass of games.
3. Games with high per capita grand coalition values
3.1. Definition and properties
Consider the class GN of TU games with the player set N, such that the maximum in coalitions of the per capita characteristic function values ^p- is attained on the grand coalition N:
{N, v) G Q% <*=> ^ for all ScN, (6)
n s
where n = \N\, s = \S| - are the numbers of players in the coalitions.
We call Gh the class of games with high per capita grand coalition values. In this section we show the uniqueness or nonuniqueness of the Shapley value on the classes Gh and Gh = UNCN GN in the axiomatizations given in Section 2.
The class GN has some attractive properties:
Proposition 1. The class GN is balanced.
Proof.
Let {N, v) G GN be an arbitrary game. Then the egalitarian payoff vector • • •, ^p^j , evidently, belongs to the core of (N, v).
Proposition 2. If i G N is a dummy in a game {N, v) G GN, then
v(N)
№» = —• (7)
v
Proof.
Equality (7) follows from the following system of inequalities and equality:
v({i}) + v(N \ {i}) = v(N), v(N)
*({*}) <
v(N\{j}) < v(N)' n — 1 _ n
Moreover, the egalitarian value on the class GN is the unique value of constrained egalitarianism [Dutta, 1990]. The last value was defined on the class of convex games, because for each game from this class there is a unique coalitions of maximal size on which the maximum of per capita values is attained. This property provides the single-valuedness of the constrained egalitarian solution.
Games from the class GN may not be convex. However, if we apply Dutta’s algorithm to any game from we obtain the unique payoff vector (^p, • • •, ^p Let N be an infinite universal set of players satisfying the condition (1).
Denote G1 = UncN GN. For the classes GN with finite N, and Gh we give the analogs of Theorems given in Section 2 characterizing the Shapley value for the classes of all TU games with fixed player sets GN for each finite N, and for the class G.
3.2. Shapley’s axiomatization
We begin with Shapley’s axiomatization [Shapley, 1953]. Note that, besides the Shapley value itself, the egalitarian solution also satisfies the Shapley axioms on the class GN because of (7).
First, let us find the general form of efficient, linear, and symmetric values on the class GN.
Lemma 1. If a value Psi on the class GN is efficient, linear, and symmetric, then
*i(N,v) = ^ + V %(S) - V ~^v(S) for all (N,v) G (8)
n s n — s
S:S3i S'.S^i
SCN
Proof.
Note that (8) gives the general form of efficient, linear, and symmetric values on
the class of all TU games GN (see, e.g., [Ruiz et al., 1998]).
Therefore, the value ^ in (8) is efficient, linear, and symmetric on the class GN as well.
Let now ^ be an arbitrary efficient, linear, and symmetric value on the class GN. For any number A denote by va the following characteristic function:
va(S) = {A if S = N, (9)
AV 7 [0, for other S. V 7
Then {N,va) G GN for A > 0, and for each game {N,v) G GN there is A > 0 such that {N,v + va) G GN. Denote wa = v + va (or wa(v), if we would like to fix the game v). Define the value $ on the class GN as follows:
$(N,v) = ^(wa ) — ^(va). (10)
Equality (10) is well-defined, since $ does not depend on A by additivity of ^. In fact, if v = wa — va = wb — vb , then
wa + vb = wb + va =^ ^(N,wa) + ^(N,vb) = ^(N,wb) + ^(N,va).
Thus, the value $ is determined on the whole class GN. It is clear that it is efficient and symmetric.
Let us show additivity:
Consider two games {N,v1), {N,v2) G GN. Let A1,A2 be arbitrary numbers such that {N,wa1 (vi)), {N,wa2 (v2)) G GN. Denote A = Ai+A2. Then {N, (vi+v2)+va) G GN. By definition of $
$(N,vi + v2) = ^(N, wa (vi + v2)) — ^(N,va),
wa (vi + v2) = vi + v2 + va = vi + v2 + va1 + va2 ,
from where by additivity of ^ we have $( N, vi + v2) =
= ^(N, vi + vai ) + ^(N, v2 + va2) — ^(N, vai ) — ^(N,va2) = $(N, vi) + $(N, v2).
Positive homogeneity of $ also follows from that of ^. Therefore, the value $ has the form (8). By symmetry ^(N, va) = (A/n, ...,A/n), and, hence, ^(N, wa) can be also represented by (8). An arbitrary characteristic function w such that {N,w) G GN, can be represented as w = v + va,A > 0,v G GN, i.e. w = wa(v), therefore, the value ^ on the class GN has the form (8).
Theorem 1. Values on the class GN satisfying efficiency, linearity, symmetry, and dummy form an one-parametric family of values, being linear combinations of the Shapley value and the egalitarian value.
Proof.
Let us show the formula for coefficients in (8):
(s _____ 1)!(n __ s)!
as = ^--->- for all a G IR. (11)
n!
It is clear that the value ^ over the class Gn, determined in (8) with coefficients (11) verifies the conditions of the Theorem.
Now let ^ be an arbitrary value on the class Gn , verifying the conditions of the Theorem. Then by Lemma 1 it is defined by (8) for some as. Let (N, v) € GN be an
arbitrary game with the dummy player i. Then = «({*}) = ^p, and (8)
implies
E ~(V(S \ «)+«)) - E —<s) = °- (!2)
* j Q < J n — C
(«(^ \ {*})+«№»)- E —
s n — s
S:S3i S'.S^i
SÇN
for all v(S), S $ i. The numbers v(S), S C N satisfy the following equalities: v({i}) = = v(p_ = v_i^xm „(S')+„({*}) = ,(su{i}). Put in (12) v(N\{i}) = (n-l>({*}). Then in this equality the values v(S), S $ i,S = N \ {i} and v(i) may be arbitrary, only satisfying inequalities v(S) < sv({i}), and (12) becomes an identity w.r.t. such v(S). Therefore, the coefficients in v(S),S $ i,S = N \ {i}, and in bv({i}) may be nullified, as = s!("7s)! up to an arbitrary multiplier a.. Putting these values as in (8) we obtain
/
*i(N,v) = ^l + a
S:S3i S:S^i
\SÇN
, ,v(N) v-^ (s — 1)!(n — s)! , , , r o ^ /1 ^ v(N)
= (1 - a) h a E j (V(S) ~V(S\ {*})) = (1 “ a) H
n n! n
S:S3i
+ aShi(N, v).
3.3. Axiomatization with the help of consistency
Definition (3) of reduced games w.r.t. some value $ can not be applied for axiomatizations of values on the class Gn , because subgames of a game from Gn w.r.t. the Shapley value may not belong to this class. For example, consider the three-person game (N, v) € G%, N = {1, 2, 3} such that
v(N) = 30, v({1}) = 0, v({2}) = 5, v({3}) = 0,
v(1, 2) = 10, v(2, 3) = 9, v(1, 3) = 20.
The Shapley value Sh(N, v) = (12, 9, 9).
Consider the reduced game ({1, 2},vSh) in the sense of Hart-Mas-Colell. We obtain
vSh(1, 2) = 21, vSh({1}) = v(1, 3) - Sh3(1, 3) = 20 - 5 = 15,
Since y' < 15, the game {{\,2},vsh) £ Q
Thus, the Hart-Mas-Colell theorem [Hart and Mas-Colell, 1989] cannot be extended to classes of games whose subgames do not belong to these classes.
Let us consider linear consistency determined by the reduced games given by (2) where the weights wn,s are defined in (5). To deal with linear consistency we should
admit an infinite universal set N of players and to consider the class Gh = UncM Gn, where the union is taken up on all finite subsets of N.
Denote by $a the value for the class Gn , being the linear combination of the Shapley value and the egalitarian value:
$a(N, v) = a$(N, v) + (1 - a)EG(N, v) for all games (N, v) € Gn,
and let xa = $a(N,v) for some arbitrary game (N, v) € GN. Consider the reduced game of (N,v) on the player set N \ {i} for some i € N w.r.t. the payoff xa. The reduced game will be denoted by (N \ {i}, va). Then by (2)
f v(N) - $a(N,v), for S = N \{i},
va(S)= I n_s_ 1 s
------7-v(S) +------~(v(S U {*}) - (AT,v)), for SCN\ {i}.
n-1 n-1
It turns out that all values permit the axiomatization with the help of linear consistency with the coefficients s 1.
h
Theorem 2. The values are the unique values on the class GN satisfying linear
s i
the class of two-person games Gn, |N| = 2.
consistency with the coefficients wn¡s = nr,s_11 and being the a-standard values on
Proof.
First, let us show that the linear reduced games (N \ {i}, vx), i € N of the games from the class Gn on the players set N \ {i} and w.r.t. arbitrary payoff vector x, belong to the class GN\{*}.
By the definition of the linear reduced games for any S C N \ {i} he have
f v(N) - xi, if S = N \{i},
,,-(s) = « /(Su{t} iy ltscWU!}. <13>
n-1 n-1
Equalities (13) imply that the difference
vx(N \{i}) vx(S)
n — 1
s
does not depend on x, and from (N, v) € Gn it follows that the reduced game
Therefore, the Shapley value is linear consistent (with the coefficients i n,s = 1; s = 1; 2, < n — 2, n = 3,4,...) on the class Qh. It is clear that the egalitarian value is also linear consistent on this class, and their linear combinations are also linear consistent on the same class.
Now let ^ be an arbitrary value that is linear consistent on the class Gh and such that ^ = $a on the class of two-person games from Gh. The a-standardness of the value $a and linear consistency of ^ imply that there is at most one value satisfying the conditions of the Theorem [Yanovskaya and Driessen, 2002]5 As it has been already proved, it is the value $a.
If in the conditions of the Theorem we put a =1, then we obtain the next characterization of the Shapley value:
Corollary 1. The Shapley value is a unique value on the class Gh that satisfies linear consistency with the coefficients w„jS = and is standard on the class
G^f, |N| = € of two-person games.
3.4. Axiomatization with the help of marginality
3.4.1 Some properties of marginalist values
Given a TU game (N,v) we denote the differences aiS = v(S) - v(S \ {i}), i € € S C N. Every such a difference is called the marginal contributions of the player i to the coalition S. We will denote the vectors of marginal contributions of the player i by ai(v), or simply by a1 when it is clear what a game (N, v) is meant.
In this section we consider marginalist values $ for TU games.
It is convenient to consider such values in the dividend form. Let (N, v) be a TU game in the characteristic function form. Then the Mobius transform
permits to define the game (N,v) in the dividend form (N, A), where the Harsanyi dividends AS ,S C N are interpreted as values that the coalition S C N should pay for the entry in a larger coalition. Note that A{i} = v({i}) for all i € N.
As it has been already determined in the previous section a marginalist value for a class GN of TU game is a mapping $ associating with each game (N,v) € GN a vector $ € IRN whose coordinates $i, i € N depend only on the differences aiS =
N \ {*}, vx) e GN\{i} for any x.
as = £(—i)s-tv(T )
T CS
(14)
T CS
v(S) — v(S e S.
5 In the cited paper the class of all TU game was considered, however, the proof of uniqueness of the value was fulfilled separately for the arbitrary game, including, certainly, games from the
class Gh.
From (14) it follows that if a game (N,v) is given in the dividend form (N, A), then a marginalist value is defined as a mapping $ : IR2 ^ IRn such that for
each game (N, A) the coordinates of the vector $(N, A) depend only on the values AS, S 3 i, we denote them by AS.
First, give some properties of the marginalist values .
Proposition 3. Let $ be an efficient and marginalist value on some class Gn. Then for every game (N,v) € Gn and player i € N
$i(N, v) = v(i) + Fi({AS}S3i,S=i) for some function Fi. Here AS are Harsanyi’s dividends for the function v.
Proof.
By marginality of $ $i(N,v) = $i({AS}s3i), and by efficiency £ieN $i(N,v) = = v(N) , i.e.
£$i({AS}S3i) = £ as. ieN ScN
In the left-hand side of this equality only the component $i depends on Ai. Put other components in the right-hand side. Then we obtain
$i(N,v) = Ai + £ As - £ $j(N,v).
ScN,S=i jeN,j=i
Denote by Fi two sums in the right-hand side:
Fi = £ As - £ $j(N,v).
ScN,S=i j=i
By marginality of $ Fi depends only on AS, S 3 i, but it does not depend on Ai,
i.e. Fi = Fi ({As } S3i,s=i).
Corollary 2. If Gn is a class of games closed under identical translations v(S) ^ v(S) + f ■ s for every number f, then $i(N, v + f) = $i(N, v)+ /3 for all i € N.
Proof.
Let a game (N, v) and, hence, the games (N, v + f) € GN. The dividends of (N, v) and (N, v + f) differ only on one-element coalitions. Therefore, by Proposition 1
$i(N, v) = v(i)+Fi({As }s3i,s=i) =
= $i(N, v + f) = v(i) + f + Fi({AS } S3i,S=i).
These equalities prove the Corollary.
The result of Corollary 2 can be strengthen. The proofs of two following Corollaries coincide with that of Corollary 2.
Corollary 3. Let for some vector b = (bi)ieN (N,v), (N,v + b) € GN, where (v+ +b)(S) = v(S) + ieS bi (though the class Gn may be not closed under covariant transformations). Then if $ is efficient and marginalist value for the class Gn, then $i(N, v + b) = $i(N, v) + fi for all i € N.
Corollary 4. Let (N,v) € Gn, and there is another game (N,v') € Gn such that Ai = Ai + f, AS = As for other S 3 i. Then for every efficient and marginalist value $
$i(N,v)+f =$i(N,v').
3.4.2 The class Gn1
We begin to consider marginalist values with a subclass Gn1 C Gn. To define this class it is more convenient to consider games in the dividend’s form. Let (N, v) be an arbitrary game, (N, A) be the same game in the dividend’s form. Let for coalitions Q, S, Q C S the following inequalities hold:
£ At > £ At. (15)
T QCTCN QCTCS
The class Gn1 is defined as follows: a game (N, v) € Gh1 if and only if it belongs to the class Gn and inequalities (15) hold for all Q,S, Q C S (including Q = 0).
It is clear that GN ^ Gn ^ G%n, where Gn is the class of totally positive TU games with the player sets N. In fact, if AS > 0 for all S C N, then inequalities (15) trivially hold. Moreover, this class is monotonic in AN in the sense that
(N, A) € GN1, =^ (N, A1) € GN1,
where
AS = AS for all S C N,
AN ^ an .
Evidently, the class GVn, as well as Gn, is invariant w.r.t. identical translation mappings. Therefore, for the class GNN Corollary 2 holds, and in the sequel we may assume without loss of generality that, given a game (N, v) € GN, its marginal contributions ai(v) > 0 for all i € N.
First, we prove the uniqueness of the Shapley value for the class Gn1 .
Recall that equalities
ais = £ At, S 3 i (16)
T :ieT CSUi
put one-to-one correspondence between marginal contributions of a player i and dividends of coalitions containing this player. So, a marginalist value for any player is completely determined by the dividends of coalitions containing him.
Lemma 2. Let a1 = {aS},S C N, i </ S be a vector of marginal contributions of the player i, for whom the dividends in (16) satisfy (15) for Q 3 i. Then there exists a game (N,vi) G Gh such that a1 = ai(vi) and k(vi) = hi(vi), where k(vi) is the number of non-zero dividends of coalitions S, |S\ > 1 in the game (N,vi), and ki(vi) is the number of non-zero dividends of coalitions S,i G S, S = {i}.
Proof.
Let us find the dividends AS, S C N,i G S, which, together with the dividends AS,i G S, defined by (16), will determine the required game (N,vi). Define these dividends as follows:
>, if s 3 i,S = {j},
Ais = { (17)
a‘
} — 2 T :iET CN AT, of S — {j },j = i.
N\{i}
Then the game (N, vi) has been completely determined. Evidently, its vector of marginal contributions of the player i coincides with {aS}SCN\{i}, and v(N) = na'N\{i},
v(N \{i}) = (n - 1)aN\{i}.
Let us show that (N,vi) € Gn1 . First, check that it belongs to the class GVN. By the definition of the game (N,vl) we have ^p- = and
v(S) as + (s - 1)aN\{i} , v(N) .
< ——-, for S 3 i,
s i s n (18) v(S) _ saN\{i} _ i _ v(N)
Inequalities (18) hold because of (15) for Q = {i} and of equalities aiS\{i} =
= 2t:ieTcS AS. Now let us prove inequalities (15) for AS, V S C N. Consider the following cases:
1) If Q 3 i, then the inequalities hold by the conditions of the Theorem.
2) Q 3 i, IQI > 1,S 3 i. Then
£ AT = £ AT, (19)
T :QCT CS T :QU{i}CT CS
and the corresponding inequality is equivalent to (15) for Q U i C S AT, T 3 i.
3) Q = {j}, S 3 i.
£ AT = Aj + £ AT > Aj + £ AT = £ .
T :jeT CN T:i,jeTCN T :i,jeTCS T :jeT CS
4) i € Q,i € S, IQI > 1. Then by the definition of the game (N, vl) (17)
£ AT = 0.
T :QCT CS
Therefore, we should show that
AT = AT > 0 for every Q 3 i. (20)
T :QCT CN T :QU{i}CTCN
The condition of the Lemma implies
£ AT > £ AT (21)
T :QU{i}CTCN T :QU{i}CTCS
for every S 3 i. Put a coalition S = N \ {j},j € Q, consider inequality (21) for it, and delete identical components in both sides of the inequality. Then we obtain
AT > 0. (22)
T :QU{i,j}CT CN
These inequalities hold for all Q 3 i including Q = 0, and for all j € Q, hence, they imply the required inequalities (20).
5) Q = {j}, S 3 i. In this case
£ AT = Aj + Xt:i,jeTcs AT, (23)
T :jeT CS
53 AT = Aj + Xt:i,jeTCN AT > Aj + Xt:i,jeTCS AT, (24)
T :jeT CN
where inequality (24) follows from the conditions of the Lemma.
Thus, the game (N,vi) € GNn .
The number of non-zero dividends of non one-elements coalitions in the game (N,vi) equals to ki(vi) = k(vi).
This Lemma implies that the individual values $i,i € N for the class Gn1 have
been determined for each vector of marginal contributions aiS, satisfying inequalities
(15). The player i does not know whether the corresponding game belongs to the class Gn1 or not. If the game does not belong to this class, the value $ has been defined as well, but it may turn out inefficient.
Theorem 3. On the class Gn1 the Shapley value is the unique efficient, symmetric, and marginalist value.
Proof.
Evidently, the Shapley value verifies all the axioms in the class Gn1 . Let us check the uniqueness. Let $ be an arbitrary efficient, symmetric, and marginalist value on the class GN!. Then for each game (N, v) € GN1 and player i € N the value $i(N, v) depends only on the vector of its marginal contributions ai, or, that is the same, on the dividends AS,S 3 i for which inequalities (15) hold.
The proof is fulfilled by Young’s induction [Young, 1875], but in the number of non-zero dividends of non one-element coalitions. This number k(v) = |{AS =
0, ISI > 1}I is invariant w.r.t. identical positive translations of the characteristic functions v ^ v + b,b > 0 and the value $ is covariant w.r.t. such translations. Thus, without loss of generality, we can consider only games from Gn1 with nonnegative Ai = v(i) > 0 for all i N.
Let us prove the Theorem for k(v) = 1.
First, consider the case AN = 0, AS = 0 for all S C N, |S| > 1. Then AN > 0. In this case the values $i(N, v) for all i € N depend only on Ai, AN, and, by symmetry of $, $i(N, v) = f (Ai, AN) for some function f, which is the same for all players. Since the value $ on the class GNN is covariant w.r.t. identical positive translations, for an arbitrary number b > 0 the following equality holds:
$i(N,v + b) = $i(N,v)+b =^ f (Ai + b, An) = f (Ai, An)+ b, (25)
where (v + b)(S) = v(S) + bs.
From (25) it follows that for any fixed y > 0 f (x, y) = f (0, y) + x, i.e.
$i(N,v) = Ai + $i(N,v0), (26)
where v0(N) = An,v°(S) = 0 for other S. Since the game (N,v0) is symmetric, &i(N, v°) = for all i G N. Therefore, equalities (26) imply $(N,v) = Sh(N,v).
Let now AS = 0,S = N, and AT = 0 for all T = S, \T| > 1. Then all players j € N \ S are dummies, and $j (N, v) = y>(Aj) for some function ^. Similarly to the case above, covariance of $ w.r.t. identical positive translations implies y>(x + b) = = y>(x) + b, i.e. y>(x) = y>(0) + x for all x > 0.
If AS = 0 for all S, then the game (N, v) is zero, and $i(N, v) = 0 for all i € N.
Therefore, y>(0) = 0, and
$j (N, v) = Aj for all j €N \ S. (27)
For other players i € S we have the first case: $i(N, v) = ^(Ai, AS) for all i € S
and some function $. Similar to that case we obtain
AS
®i(N, v) = Ai + — for all i e S. (28)
s
Equalities (27) and (28) imply $(N, v) = Sh(N,v).
Assume that the Theorem is true for all games (N, v) € GN1 with k(v) < m. Let (N,v) € GN1 be an arbitrary game with k(v) = m. Denote R = p|S |S|>1 As = 0. Then R = N (otherwise we would come to the first case AN = 0, AS = 0 for all S C N, IS| > 1). For i € R consider the game (N,vi), determined in Lemma 4. The Lemma implies that (N,vi) € GNt and
(29)
The number of non-zero dividends of non one-element coalitions in the game (N,v'1) is less than m. Hence, by the inductive hypothesis, $(N, vi) = Sh(N,vi), and this equality together with (29) gives $^N, v) = Shi(N, v) for all i G N \ R. Let R = $, then
$j (N,v)= f (Aj, {As} ^=0, )
SdR,\S\>1
for all j G R and some function f, which is the same for all players from R. By the definition of the set R for different players the arguments of F differ one from another only by the first coordinate, whose domain is set of non-negative numbers IR + (though in total the values Aj,j G R cannot be arbitrary, because we should stay in the class Qh1.) Covariance of $ w.r.t. identical positive translations leads to the equality
f (x + b, {AS}S:As/Q) f (x, {AS}S:As = 0) + b
for all x > 0 b > 0, ..
f (^ {AS}S:As = 0) = f (0, {AS}S:As = 0) + x. (30)
By efficiency of $
]T$i(N,v)=^ Shi(N,v). (31)
iER iER
Equalities (30) and (31) imply
f(0,{As}s:As^o) = l ,
T \i£R J
i.e.
f (0, {As}s:As=o) = Shi(N,v0), (32)
where v0(S) = v(S) — £ieS Ai. At last, (32) and (27) imply
$i(N, v) = f (Ai, {As }s:As=o) = Shi(N, v0 ) + Ai = Shi(N, v) for all i G R.
Note that Lemma 2 implies that the solution $ satisfying the conditions of Theorem 3 is determined for the games from \ as well, only for such games it may
be not efficient. Return to the whole class Qh.
3.4.3 Class ON
We begin with the individual domains of marginal, efficient, and symmetric values for the class Qh. Consider vectors of marginal contributions a = {aS}ScN\{i| satisfying the inequalities
0 < aN\^j > — for all S C N \ {*}.
(33)
Note that an arbitrary vector a1 = {aS}ScN\{i} of marginal contributions can be transformed by an identical translation mapping to the vector ai+b = {aS+b}Scn\{*} satisfying inequalities (33). Therefore, by Lemma 1 (33) are not restrictive when we consider marginal and efficient values for the class GN ■
Lemma 3. Given a vector of marginal contributions a = {as}, S C N, S 3 i satisfying inequalities (33), there exists a game (N, V) e GN such that ai(vi) = a1 ■
Proof.
Define the game (N, V) as follows:
vi(S)
'aS, if S 3 i,S = N,
0, if S 3 i,S = N \{i},
(n - l)aN\{i}, if S = N \{i},
if S = N.
,naN\{¿},
Then in the game {N, v%) alS(v) = aS. Let us show that (N, vi) e GN. We have
vi(N) vi(N \{i})
n — 1
aN\{i}.
For other S either v(S) = 0, or v(S) = as, and by the condition of the Theorem
< aN\{i}- Give the last uniqueness Theorem:
Theorem 4. On the class GN the Shapley value is the unique value that is efficient, anonymous, and marginalist.
Proof.
Since (33) is not essential, the domain of marginalist and efficient values for each
2^ — 1
player i e N is the whole space IR ■ Let $ be an arbitrary marginalist, efficient, and symmetric value for the class Gjy ■ Then by Theorem 3 $(N, v) = Sh(N,v) for
any game (N, v) e GhN■ Assume that $ = Sh■ it means that there exists a game
(N,v>) e G% such that $(N,v') = Sh(N,v')■
For arbitrary A > 0 define the values for the class GN as follows:
A
$f(N,v) = <i>i(N,vA)---------for all i e N,
n
where
(0(5) = {v<S)- f S = N
K J [v(N)+ A, if S = N
It is clear that for any A > 0 (N, vA) e GN, and there exists A0 such that for all A > Ao (N,vA) e GN■
The values $A are marginalist, efficient, and symmetric. Let A be a number such that (N,v' + A) e Gh^ Then for this game $A(N,v' + A) = Sh(N,v' + A) that contradicts Theorem 3.
n
Acknowledgments
The author is grateful to Anna Khmelnitskaya for valuable remarks and discussions.
References
Brink R. van den. 2001. An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory, 30: 309-319.
Chun Y. 1989. A new axiomatization of the Shapley value. Games and Economic behavior. 1: 119-130.
Dubey P. 1975. On the uniqueness of the Shapley value. International Journal of Game Theory, 4: 131-139.
Dutta B. 1990. The egalitarian solution and the reduced game properties in convex games. International Journal of Game Theory, 19: 153-159.
Hart S., Mas-Colell A. 1989. Potential, value, and consistency. Econometrica, 57: 589-614.
Myerson R. 1977. Graphs and cooperation in games. Mathematics of Operations Research, 2: 225-229.
Neyman A. 1989. Uniqueness of the Shapley value. Games and Economic Behavior, 1: 116-118.
Ruiz L.M., Valenciano F., Zarzuelo J. 1998. The family of least square values for transferable utilities games. Games and Economic Behaviour, 4: 109-130.
Shapley L.S. 1953. A value for n-person games. Annals of Mathematics Study, 28: 307-317 (Princeton University Press). Also in: Roth, A.E. (editor) The Shapley value: Essays in honor of Lloyd S’. Shapley. Cambridge University Press. Cambridge, U.S.A., 31-40.
Sobolev A.I. 1975. The characterization of optimality principles in cooperative games by functional equations. In: Mathematical Methods in the Social S’ciences, 6. (Ed. N.N. Vorob’ev). Vilnius: 94-151.
Weber R.J. 1988. Probabilistic values for games. In: Roth A.E. (editor) The Shapley value: Essays in honor of Lloyd S. Shapley. Cambridge University Press. Cambridge, U.S.A., 101-119.
Yanovskaya E., Driessen T. 2001. On linear consistency of anonymous values for TU games. International Journal of Game Theory, 30: 601-609.
Young H.P. 1985. Monotonic solutions of cooperative games. International Journal of Game Theory, 14: 65-72.
