Is the solidarity value close to the equal split value?

Mathematical Social Sciences 65 (2013) 195–202

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Is the solidarity value close to the equal split value?✩ Tadeusz Radzik Institute of Mathematics and Computer Science, Wrocław University of Technology, 50-370 Wrocław, Wybrzeże Wyspiańskiego 27, Poland

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Article history: Received 30 July 2012 Received in revised form 30 November 2012 Accepted 6 December 2012 Available online 20 December 2012

abstract In the paper we study the asymptotic behavior of the solidarity value and the equal split value in different classes of cooperative games. In particular, we discuss three natural definitions of asymptotic equivalence of values for TU-games, and identify for each of them the classes of games for which the solidarity value and the equal split value are equivalent. Also, a computer illustration of the obtained results is given. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Classical cooperative game in the characteristic function form (also called a TU-game) is a function v : 2N → R with a finite set N as the grand coalition of the players (agents, individuals). For each coalition S (a subset of N), v(S ) represents the worth of S (the gain possible to be achieved jointly by all the players from S when they collaborate). One of the essential problems addressed in the papers on this topic is the following. Assuming that all the players in N will collaborate and create (finally) the grand coalition N, how to divide ‘‘fairly’’ the whole worth v(N ) between them? In the literature, many various procedures have been proposed for solving this problem, and the classical Shapley value (Shapley, 1953) is the most prominent and the best known one among them. However this value has the so-called null player property that does not allow for solidarity between the players, assigning zero payoff to every ‘‘unproductive player’’; that is to every player i with all his marginal contributions v(S ∪ i) − v(S ) = 0 for S ⊂ N (called then a null player). On the other hand, there is a very classical value allowing for the total solidarity between the players, which radically rejects the null player property, the socalled equal split value. It awards every player in the grand coalition N, independently of his contributions v(S ∪ i) − v(S ) to coalitions v(N ) S, with the same payoff equal to n where n is the cardinality of N. In the literature, several other values for TU-games ‘‘standing’’ between the Shapley value and the equal split value were also studied. In Nowak and Radzik (1994) the authors defined and axiomatized a new value for TU-games, the so-called solidarity value. The system of axioms used there is analogous to the classical axiomatization of the Shapley value (Shapley, 1953), with the difference

✩ This research was supported by Grant S10095 of the Wrocław University of Technology. E-mail address: [email protected].

0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.12.001

that the null player axiom is replaced by another one, more ‘‘profitable’’ for the null players. In the same year, Joosten et al. (1994) generalized this idea to define and discuss a wide class of values for cooperative games, the so-called socially acceptable values. Next, in Nowak and Radzik (1996) the authors generalized their previous paper, founding an axiomatization of the class of convex combinations of the Shapley value and the solidarity value. Since then up to 2008, the solidarity value (though mentioned in a few dozen articles) has not been an object of serious discussion in the literature, and nobody showed that it can be essentially applicable to any model in game theory. The breakthrough came with the paper of Calvo (2008). He proposed two variations of the non-cooperative model for games in coalitional form, introduced by Hart and Mas-Colell (1996), and found two new, very interesting NTU-values: the random marginal and the random removal values. It turned out that the random marginal value coincides with the Shapley value for TU-games and that, which was completely surprising, the random removal value coincides with the solidarity value for TU-games. Calvo used some suggestions of Hart and MasColell (1996) given there (in case (d) analyzed in Section 6), on how to modify the bargaining procedure. In particular, in the context of the TU-game case, they wrote: ‘‘The resulting solution is different from the previous ones (thus, it is neither the Shapley value nor the equal split solution).1 However, for large n, it is easy to see that it is close to the equal split solution (a minor boundedness condition is needed here)’’. It turned out that in fact, the authors overlooked that the resulting solution found by them was simply the solidarity value, and thereby their real statement was that the solidarity value is somehow ‘‘asymptotically close’’ to the equal split value. Over the last year, several other papers related with some modifications or generalizations of the solidarity value have

1 That is, the equal split value which, by definition, shares the payoff v(N ) of the grand coalition equally between the players.

196

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202

appeared. In the first one of them (Calvo and Gutierrez, 2012) the authors study a new family of values, called the weighted solidarity values, and show that these values arise as associated equilibrium payoffs of some bargaining protocol. It is an interesting fact noticed by them (and overlooked in Nowak and Radzik, 1994) that Sprumont (1990) considered a value (as an example of a population monotonic allocation scheme), the formula of which turned out to be equivalent to the solidarity value. In the next paper Casajus and Huettner (in press) replaced the null player property of the Shapley value by a new axiom that guarantees null players non-negative payoffs whenever the grand coalition’s worth is non-negative (which is also a property of the solidarity value). They obtained a new characterization of the class of egalitarian Shapley values, being convex combinations of the Shapley value and the equal split value. Next Casajus (2012) indicated requirements on fair taxation in TU-games by which solidarity among players is expressed, and obtained a formula allowing for an economic interpretation of the Shapley payoffs in unanimity games in terms of taxing. The last two papers of Chameni Nembua (2012) and Malawski (in press) come up with another interpretation of efficient, linear, and symmetric values, where, in essence, the players’ marginal contributions within a coalition are taxed at a rate depending on its size, while the tax revenue is distributed evenly among the other players in the coalition under consideration. The purpose of this paper is to discuss in more detail the statement of Hart and Mas-Colell mentioned above. We show that the general statement of type ‘‘for large n, one value is close to another one’’ has a very complex structure and can be understood in three different ways. This leads us to three natural types of possible convergence between the solidarity value and the equal split value. For each of them we try to find possibly wide classes of TU-games where such a convergence holds. This is done with the help of new notions, related to the so-called asymptotic equivalence of values for TU-games. We also widely discuss asymptotic behavior of the solidarity value and the equal split value in the context of each of these properties. Also, a computer illustration of the obtained results is given. The organization of the paper is as follows. In Section 2 we provide all the preliminary notation and definitions. In Section 3 three concepts of asymptotic equivalence between values of cooperative games are proposed. Section 4 contains the main theorems of the paper and their wide discussion, while Section 5 gives some computer illustration of the obtained results. Section 6 is devoted to the proofs of the theorems. 2. Preliminaries Let Nn = {1, 2, . . . , n} with n ≥ 1 be a finite set of n players, called the grand coalition, and let Γn denote the linear space of all n-person transferable utility games v on Nn , that is, Γn = {v| v : 2Nn → R} with v(∅) = 0, where R denotes (throughout the paper) the set of real numbers. For any subset (coalition) S in Nn , v(S ) describes the payoff (worth) of coalition S when all the players in S collaborate. A game v on Nn is called normalized if v(Nn ) = 1. If for all coalitions S ⊂ Nn , v(S ) = 0 or v(S ) = 1, then v is called a simple game. N For a coalition T ⊂ Nn , the unanimity game uT n is defined by

uT n (S ) = 1 for S ⊃ T and uT n (S ) = 0 otherwise, for S ⊂ Nn . A value Φ (v) = (Φ1 (v), . . . , Φn (v)) on Γn is thought of as a vector-valued mapping Φ : Γn → Rn , which uniquely determines, for each game v ∈ Γn , a distribution of the total wealth available to all the players 1, 2, . . . , n, through their participation in the game v . The cardinality of a set X will be denoted by |X |. For brevity, throughout the paper, the cardinality of sets (coalitions) M , R, S , T and U will be also denoted by appropriate small letters m, r , s, t N

N

and u, respectively. Also for notational convenience, we will write the singleton {i} as i.  A value Φ is called efficient if i∈Nn Φi (v) = v(Nn ) for all games v , and is called additive if Φ (v + w) = Φ (v) + Φ (w) for any two games v and w . A player i is called a null player in game v if v(S ∪ i) = v(S ) for every coalition S ⊂ Nn \ i. If Φi (v) = 0 in case of any null player i in any game v , we say that value Φ satisfies the null player axiom. If a value Φ satisfies the equality Φπ i (π v) = Φi (v) for all i ∈ Nn and every permutation π of the player set Nn , then we say that Φ satisfies the symmetry axiom (here π v is defined by π v(π (S )) = v(S ) for S ⊂ Nn ). Now we list two values basic to our considerations. The equal split value Φ Eq . By definition, it is a value on Γn defined Eq Eq by Φ Eq (v) = (Φ1 (v), . . . , Φn (v)), where for i ∈ Nn , Eq

Φi (v) =

v(Nn ) n

.

(1)

So, the equal split value divides the worth v(Nn ) of the grand coalition Nn equally between all the players, independently of the worth of other coalitions. The solidarity value Φ So . This is a value on Γn discussed in the paper Nowak and Radzik (1994). It is uniquely determined by the three classical axioms: efficiency, additivity and symmetry, and by a modification of the null player axiom, called A-null player axiom. We quickly recall this axiom. To express it we need to define, for any non-empty coalition T ⊂ Nn and game v on Nn , the quantity Av (S ) =

1 s k∈S

[v(S ) − v(S \ k)],

(2)

where s means the cardinality of S. Clearly, Av (S ) can be seen as the average marginal contribution of a member of a coalition S. The axiom is as follows: Axiom (A-null player): If i ∈ Nn is an A-null player in a game v on Nn , that is, Av (S ) = 0 for every coalition S containing player i, then Φi (v) = 0. It is shown in Nowak and Radzik (1994) that for v ∈ Γn , the solidarity value is of the form Φ So (v) = (Φ1So (v), . . . , ΦnSo (v)), where for i ∈ Nn ,

ΦiSo (v) =

 (n − s)!(s − 1)! n!

S ∋i

Av (S ).

(3)

It is left to the reader to verify with the help of (2) and (3) that the solidarity value can be written in the following equivalent form. For i ∈ Nn

ΦiSo (v) =

v(Nn ) n

−

v(Nn \ i) n2

  s!(n − s − 1)!  v(S ∪ i) v(S ) + − . n! s+2 s+1 S (Nn \i

(4)

It is worth mentioning that the classical Shapley value Φ Sh has a similar form to Φ So , in the sense that if we replace the average marginal contribution Av (S ) of a member of a coalition S in (3) by the marginal contribution v(S ) − v(S \ i) of a member i in coalition S, then we get ΦiSh . Besides, both values Φ So and Φ Sh are uniquely determined by almost the same collection of axioms, that is, the only difference is that the A-null player axiom for Φ So should be replaced by the null player axiom for Φ Sh . Now we examine the equal split value Φ Eq and the solidarity N value Φ Eq in the context of unanimity games uT n and the games

wTNn introduced in Nowak and Radzik (1994), where the family {wT : T ⊂ Nn , T ̸= ∅} was used as a basis for the linear space Γn in obtaining formula (3).

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202

Let ∅ ̸= T ⊂ Nn . Then the game wT n is defined on Nn by the formula: for S ⊂ Nn N

wTNn (S ) =

 s  −1

if S ⊃ T ,

t

and

(5)

w (S ) = 0 if S ̸⊃ T ,  n −1 N N . Besides, uT n (Nn ) = 1. Therefore, in whence wT n (Nn ) = t Nn T

view of (1), we have Eq

Φi (uTNn ) =

1

for i ∈ Nn ,

n

(6)

ΦiSo (uNT n ) > ΦjSo (uNT n ) > 0.

(13)

On the other hand, by the efficiency axiom,



j∈Nn \T

ΦjSo (uNT n ) = 1. Hence, by the symmetry

(

) + )

N So uT n i∈T Φi N axiom, t ΦiSo uT n



(

+ (n − t )ΦjSo (uNT n ) = 1 for any i ∈ T and j ∈ Nn \ T . But the

last equality together with (13), immediately implies inequalities (11) and (12). N We end this section with the formula for Φ So (wT n ) where N

Eq

Φi (wTNn ) =

 n −1 t

/n for i ∈ Nn .

(7)

For the solidarity value, the quantity ΦiSo (uT n ) has a very complex structure. Namely, it can be easily derived from (2) that N for i ∈ T and v = uT n we have: Av (S ) = ts if S ⊃ T and Av (S ) = 0 otherwise. Hence, by (3), for i ∈ T , we conclude that N

(

N uT n

)=

 (n − s)!(s − 1)! t n!

S ⊃T

=

s

  n  (n − s)!(s − 1)! t n − t , n! s s−t s=t

which can be written as

ΦiSo

This can be easily justified as follows. Let i ∈ T and j ∈ Nn \ T . Then, comparing the right-hand sides of (8) and (9), we immediately get the inequality

games wT n are defined by (5). Namely, it is proved in Nowak and Radzik (1994, see formula (2.3) there) that

and

ΦiSo

197

(

N uT n

)=



(n − t ) (n − t )(n − t − 1) + + ··· 2 n n (n − 1) (n − 1)(n − 2)2  (n − t )! + for i ∈ T . (n − 1)(n − 2) · · · (t + 1)t 2 t

1

+

(8)

However if i ̸∈ T , (2) implies that for v = uT n we have: Av (S ) = ts if S ⊃ T ∪ i and Av (S ) = 0 otherwise. Consequently, using (3), for i ̸∈ T we have N

ΦiSo (uNT n ) =

=

 (n − s)!(s − 1)! t n! s S ⊃T ∪i   n  (n − s)!(s − 1)! t n − t − 1 , n! s s−t −1 s=t +1

ΦjSo (uNT n ) =

1

t

/t for i ∈ T ,

and

(14)

3. Definitions of the asymptotic equivalence of values Asymptotic equivalence is often studied in the context of indivisible goods allocation problems (see e.g. Che and Kojima, 2010), but hitherto this notion has not been formalized for cooperative TU-games. In this section we study the problem of possible definitions for asymptotic equivalence between two values defined on families of such games. We have mentioned in Section 1 that Hart and Mas-Colell expressed the statement (HM-statement) that (under a boundedness condition) for games with a large number n of players, the solidarity value is close to the equal split value. However, they did not explain what they understood under the word ‘‘close’’ and what sort of boundedness condition is needed. So, to discuss the HMstatement, we need to formalize the sentence ‘‘for large n, one value is close to another value’’, which turns out not to be quite clear-cut. Then we will discuss if those two values are close in a given sense under different boundedness conditions. Below, in Definitions 1–3, we propose three natural concepts of asymptotic equivalence of two values Φ and Ψ in a class of games F . To do it, we must introduce some necessary objects and assumptions. Let us define ∞ 

Γn .

n =1

In what follows, we extend the definition of a value Φ in the way that Φ is defined as any function on the set Γ with Φ : Γn → Rn for each n ≥ 1, and we will assume that each considered value Φ is linear and efficient on every set of games Γn . In what follows, we will also assume that the classes of games considered F satisfy the following natural condition:

(n − t − 1) (n − t − 1)(n − t − 2) + n n (n − 1)2 (n − 1)(n − 2)2  (n − t − 1)! + ··· + (n − 1)(n − 2) · · · (t + 2)(t + 1)2 t

 n  −1

ΦiSo (wTNn ) = 0 for i ∈ Nn \ T .

Γ :=

which can be written as



ΦiSo (wTNn ) =

+

F ∩ Γn ̸= ∅ for n = 1, 2, . . . .

(10)

The first property of asymptotic equivalence of values Φ and Ψ in a class F is based on the natural assumption that in the case of a large grand coalition Nn , the approximation Φi (vn ) − Ψi (vn ) ≈ 0 holds for every game vn ∈ F ∩ Γn and i ∈ Nn . This leads to the first definition.

but in other cases the quantities ΦiSo (uT n ) for the solidarity value satisfy: for any T , ∅ ̸= T ( Nn ,

Definition 1. Two values Φ and Ψ are said to be asymptotically weakly-equivalent in a class of games F ⊂ Γ if for every sequence {vn } in F with vn ∈ Γn ,

for j ∈ Nn \ T .

(9)

Obviously, for every i ∈ Nn Eq

ΦiSo (uNNnn ) = Φi (uNNnn ) =

1 n

, N

1 n

< ΦiSo (uNT n ) <

1 t

for i ∈ T ,

(11)

and 0<

ΦjSo

(

N uT n

)<

1 n

for j ∈ Nn \ T .

(12)

lim Φi (vn ) − Ψi (vn ) = 0,



n→∞



i = 1, 2, . . . .

(15)

The second property of asymptotic equivalence of two values Φ and Ψ is based on a stronger assumption. Namely, we require that in the case of a large grand coalition Nn , the sum of the players’

198

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202

wealth over any coalition Tn ⊂ Nn should be approximately the same with respect to Φ and Ψ , for any game vn on Nn . Definition 2. Two values Φ and Ψ are said to be asymptotically strongly-equivalent in a class of games F ⊂ Γ if for every sequence of games {vn } in F with vn ∈ Γn , and for every sequence of coalitions {Tn } with Tn ⊂ Nn , lim

n→∞

 

 Φi (vn ) −

i∈Tn



Ψi (vn ) = 0.

(16)

i∈Tn

Theorem 2. The solidarity value and the equal split value are asymptotically weakly-equivalent in any uniformly bounded class of games F ⊂ Γ . Remark 3. The result of Theorem 2 provokes the question if it remains true after replacing the property of asymptotic weakequivalence by the two stronger ones described in Definitions 2 and 3. The next theorem shows that it is not true. Moreover, it says that the solidarity value and the equal split value are not related in any of these two senses even for a much narrower class of games than the uniformly bounded ones.

The third property of asymptotic equivalence of two values Φ and Ψ considers the relationship between the wealth of each single player with respect to Φ and Ψ for a game, and this approach leads to the following definition.

Theorem 3. Let Fsg ⊂ Γ be the class of all normalized simple games in Γ . The solidarity value and the equal split value are neither asymptotically strongly-equivalent nor asymptotically strictly-equivalent in Fsg .

Definition 3. Two values Φ and Ψ are said to be asymptotically strictly-equivalent in a class of games F ⊂ Γ if for every sequence of games {vn } in F with vn ∈ Γn ,

Remark 4. In view of Theorem 2, one may ask the question what subclass of the class Fsg of normalized simple games guarantees the asymptotic strong-equivalence and/or the asymptotic strictequivalence of the solidarity value and the equal split value. The standard subclass of the class Fsg is the set Fug of all unanimity games in Γ . Just those two equivalence criteria are discussed in the next two theorems in the context of the class Fug .

lim

n→∞

Φi (vn ) = 1, Ψi (vn )

i = 1, 2, . . . .

(17)

Remark 1. One can immediately see that a value Φ is asymptotically weakly-equivalent to a value Ψ in a class of games F if Φ is asymptotically strongly-equivalent to Ψ in F . Namely, for k = 1, 2, . . . , it suffices to put (in (16)) the sequence {Tn } of singleton coalitions of the form, Tn = {1} for n < k, and Tn = {k} for n ≥ k. On the other hand, in general, asymptotic strict-equivalence does not imply asymptotic weak-equivalence. However, the implication holds when the values Φi (vn ) and Ψi (vn ) are uniformly bounded over all i and vn ∈ F (that is, when there is a real constant c such that |Φi (vn )| ≤ c and |Ψi (vn )| ≤ c for all i and vn ∈ F ). According to the HM-statement, one could think that (under a boundedness condition) for a large number of players the solidarity value is close to the equal split value when at least one of the properties defined in Definitions 1–3 holds for these two values. Just this problem is studied in the next section. 4. Asymptotic equivalence between the solidarity value and the equal split value

Theorem 4. Let {Sn } and {Tn } be two sequences of non-empty sets with Sn , Tn ⊂ Nn for n = 1, 2, . . .. Then lim

n→∞

 

 (

ΦiSo uNSnn

)−

i∈Tn



Eq Φi

(

N uSnn

) = 0.

(18)

i∈Tn

The above result immediately leads (by Definition 2) to the following corollary. Corollary 1. The solidarity value and the equal split value are asymptotically strongly-equivalent in the class Fug of all unanimity games in Γ . Theorem 5. Let T = {1, 2, . . . , t } be a finite set with t ≥ 1. Then lim

n→∞



ΦiSo (uNT n ) Eq

Φi (uNT n )

 t = t −1 1

Here, by definition

1 0

if i ∈ T

(19)

if i > t .

 = +∞ .

In this section we formulate five main theorems of the paper, the proofs of which will be given in Section 6. Their results allow us to draw a final conclusion about the nature of the HM-statement and its possible validity in different classes of games.

Theorem 5 implies the next result, giving an interesting property of the solidarity value in the context of unanimity games. To see that, it suffices to combine (19) for i ∈ T with the relation Eq N N limn→∞ Φj (uT n )/ΦjSo (uT n ) = 1 for j ̸∈ T following from the sec-

Theorem 1. In the class of all normalized games in Γ , the solidarity value and the equal split value are not asymptotically weakly-equivalent, neither asymptotically strongly-equivalent nor asymptotically strictly-equivalent.

ond case of (19) and the obvious equality Φi (uT n ) = Φj (uT n ). The result of these considerations is the following:

Remark 2. The result of Theorem 1 is that, in general, the solidarity value is not asymptotically convergent to the equal split value, which was distinctively suggested by the HM-statement. However, it turns out (Theorem 2 below) that when we restrict our consideration to a uniformly bounded class (defined below) of games in Γ , the HM-statement is true in the weak sense of Definition 1. Before formulating the next theorem, we introduce a new notion. Namely, we call a class of games F ⊂ Γ uniformly bounded if there is a real constant c such that for all n ≥ 1 and S ⊂ Nn , the inequality |vn (S )| ≤ c holds for every game vn in F ∩ Γn .

Eq

Eq

N

N

Corollary 2. Let T = {1, 2, . . . , t } be a finite set with t ≥ 1. Let i ∈ T and j ∈ Nn \ T (j > t ). Then lim

n→∞

ΦiSo (uNT n ) ΦjSo

(

N uT n

)

=

t t −1

.

(20)

Eq

Remark 5. Theorem 5 implies that limn→∞ Φ1So (u{1n} )/Φ1 (u{1n} ) = +∞ and Φ1So (uNT n ) ≈ t −t 1 Φ1Eq (uNT n ) for large n and t ≥ 2. Obviously, this strongly contradicts the statement that the solidarity value and the equal split value are asymptotically strictly-equivalent, even in the class of unanimity games. But considering that t −t 1 ≈ 1 for large t, we immediately see that equality (19) implies that N

N

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202 Table 1

199

Table 2

Values of differences

|T | |N |

1

2 3 4 10 100 1000

0.250 0.278 0.271 0.193 0.042 0.006



i∈T

2

ΦiSo (uNT ) −

3

0.000 0.111 0.139 0.114 0.018 0.002

0.000 0.063 0.086 0.014 0.001



5

0.054 0.012 0.001

Eq

i∈T

Φi (uNT ).

8

10

0.020 0.010 0.001

0.000 0.010 0.001

ΦiSo (uN ) T

Values of ratio 25

0.008 0.001

50

0.005 0.001

Eq

Φi (uN ) T

600

0.000

for i ∈ T .

|T | |N |

1

2

3

5

8

10

25

50

600

2 3 4 10 100 1000

1.500 1.883 2.083 2.293 5.187 7.485

1.000 1.167 1.278 1.571 1.915 1.987

1.000 1.083 1.286 1.472 1.497

1.108 1.234 1.248

1.025 1.130 1.142

1.099 1.110

1.031 1.041

1.010 1.019

1.001

Eq

ΦiSo (uNT n ) ≈ Φi (uNT n ) for large n and t, and for arbitrarily fixed i ≥ 1. Therefore, also under the criterion of asymptotic strictequivalence, we can say that for large n the solidarity value is close to the equal split value if we limit ourselves to the class of N unanimity games uT n with large T . Remark 6. The equal split value Φ Eq has the following obvious   Eq Eq Nn Nn property: i∈Tn Φi (uTn ) → α and i∈Nn \Tn Φi (uTn ) → 1 − α

→ α , which is an immediate consequence of (1) and the N fact that uTnn (Nn ) = 1 and |Nn | = n. Therefore, it follows from

as

|Tn | n

Theorem 4 that the solidarity value has exactly the same property. It can be seen as completely surprising that in the case of a large grand coalition N, the solidarity value for a game uNT awards the coalition T of ‘‘productive’’ players, and coalition N \ T of ‘‘nonproductive’’ players (approximately) with payoffs proportional to the number of players in coalitions T and N \ T , respectively. With this correlation, that when the number of players  one could think  is large, then for games uNT the solidarity value should award any

single player with a payoff approximately equal to 1n , coinciding with the payoffs offered to each player by the equal split value Φ Eq . But this suggestion turns out not to be true, as we meet the second surprising property of the solidarity value expressed in Corollary 2. It is a real difference in comparison to the behavior of the equal split value. Conclusion. The arguments given in this section show that the HMstatement can be seen as true only in the cases when we restrict our considerations to some subsets of the class of all games Γ . Namely, when we adopt the very weak criterion of Definition 1, one can say (by Theorem 2) that the HM-statement is true in every uniformly bounded class of games. However, when we adopt the stronger criterion of Definition 2, then the HM-statement becomes false even in a narrow class like the set of normalized simple games in Γ (Theorem 3), and it becomes true only for the rather very small class Fug of unanimity games (Corollary 1). As far as the asymptotic strict-equivalence is concerned, it follows from Theorem 5 that the solidarity value and the equal split value are not related in the sense defined by this property even in such a small class of games as Fug . 5. Computer illustration of the obtained results In this section we illustrate the results of Theorems 4 and 5 by some computer calculations, comparing the solidarity value and the equal split value over some concrete unanimity games. This will give us more insights about similarities and differences between these two values.  Theorem 4 implies that for large n and any fixed T , i∈T

ΦiSo (uNT )−

Eq

Φi (uNT ) ≈ 0. Another approximation follows from Eq Theorem 5. Namely, for a fixed t and large n, ΦiSo (uNT )/Φi (uNT ) ≈ Eq t So N N if i ∈ T , and Φj (uT )/Φj (uT ) ≈ 1 if j ∈ N \ T . Below, we t −1



i∈T

illustrate the above three approximations in Tables 1–3.

Table 3 ΦjSo (uN ) T

Values of ratio

Eq

Φj (uN ) T

for j ∈ N \ T .

|T | |N |

1

2

3

5

8

10

25

50

600

2 3 4 10 100 1000

0.500 0.583 0.639 0.786 0.958 0.994

0.667 0.722 0.857 0.981 0.998

0.750 0.878 0.985 0.999

0.892 0.988 0.999

0.899 0.989 0.999

0.989 0.999

0.990 0.999

0.990 0.999

0.999

can easily conclude from Table 1 that the approximation  One So  Eq N N Φ i ( uT ) ≈ i∈T i∈T Φi (uT ) is good enough for relatively small |N | ≥ 10 and |T | ≥ 8. It is easy to notice that the ratio considered in Table 2 increases along each column. So for any fixed T it increases as |N | increases. This reflects a real difference in the awarding of a player from T by the solidarity value and the equal split value in unanimity games uNT . An analysis of Table 3 shows that for coalitions T with |T | ≥ 10, the solidarity value awards each player i ̸∈ T in a unanimity game uNT in almost the same way as the equal split value. 6. Proofs of the theorems In this section we give the proofs of Theorems 1–5. Proof of Theorem . We begin with some notation. For n ≥ 1 and ∅ ̸= Sn ⊂ Nn , we define the game vn on Nn with N the help of game w{1n} of the form (5) by the following:

vn := n · w{N1n} ,

n = 1, 2, . . . .

(21)

In view of (5) and (21), we easily see that

vn (Nn ) = 1,

n = 1, 2, . . . .

(22)

For every n ≥ 1 the game vn has the grand coalition Nn and |Nn | = n. Hence, in view of (1) and (22), Eq

Φi (vn ) =

1 n

for 1 ≤ i ≤ n.

(23)

On the other hand, one can easily see with the help of (14) that N for any n ≥ 1, Φ1So (w{1n} ) = 1n , whence, by (21),

Φ1So (vn ) = 1.

(24)

Therefore Eq

lim [Φ1So (vn ) − Φ1 (vn )] = 1,

n→∞

because of (23) and (24). But this, in view of Definition 1, completes the proof of the part that the solidarity value and the equal split value are not asymptotically weakly-equivalent in the class of all normalized games in Γ , and thereby (see also Remark 1) they are not asymptotically strongly-equivalent in this class.

200

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202

On the other hand, (23) and (24) yield the equality limn→∞ Φ1So (vn ) Eq

Φ1 (vn )

On the other hand, we have

= ∞, which implies that the solidarity value and the equal

split value are also not asymptotically strictly-equivalent in the class of all normalized games in Γ .  Proof of Theorem 2. Let F satisfy the assumption. By the assumption, there is a real constant c such that for all n ≥ 1 and S ⊂ Nn , the inequality |vn (S )| ≤ c holds for every game vn in F ∩ Γn . Therefore, it should be shown that for every sequence {vn } of games in F with vn ∈ F ∩ Γn ,

  Eq lim ΦiSo (vn ) − Φi (vn ) = 0,

i = 1, 2, . . . .

n→∞

|ΦiSo (vn )| ≤

+

+

n!

S ⊂Nn \i

= < = Hence, since

(n + 1)c n2

(n + 1)c

s+1

n2

(n + 1)c n2

(n + 1)c n2

n −1 2c 

+

s+1 s =0

2c



n

+

n

log n limn→∞ n

+

n

1+ s=1

1

|vn (S )| s+1

c

ΦjSo (uNSnn ) −

Bn :=



Eq

Φj (uNSnn ),

(27)

j∈Tn





ΦjSo (uNSnn ) −

j∈N\ Tn

Eq

Φj (uNSnn ),

(28)

j∈N\ Tn

)

bnn−sn −1 n−sn −1

 −1

(29)

  := (1 − an ) bn0 1 + (1 − bn1 ) + (1 − bn2 )2 + · · ·  

x

(30)

and bnk = ns−n k . Since for every n ≥ 1 both values Φ So and Φ Eq are efficient on Γn , therefore one can see that where tn = |Tn |, sn = |Sn |, an =

(1 + log n).

An + Bn = 0

= 0, we get

n

 j∈Tn

gnB

On the other hand, with the help of (1), we have

≤

,

and

(26)

n

An :=

+ (1 −



lim ΦiSo (vn ) = 0.

|vn (Nn )|

4

Proof of Theorem 4. Let us introduce the following notation:



dx

n→∞

|ΦiEq (vn )| =

5

+ (1 − bnn−sn −1 )n−sn −1 − 1 ,

n

2c

4n(2n − 1)

=

which implies (by Definition 3) that the solidarity value and the equal split value are not asymptotically strictly-equivalent in the class Fsg either. 

1



(¯v2n )

2n(5n − 2)

n→∞

gnA := an bn0 1 + (1 − bn1 ) + (1 − bn2 )2 + · · ·

+2

+

= lim



 s!(n − s − 1)! c n! s+1 S ⊂Nn \i   n −1  n − 1 s!(n − s − 1)! c (n + 1)c + 2 = n2 s n! s+1 s=0

≤

Eq Φ1

 

|vn (Nn \ i)|

n2  s!(n − s − 1)!  |vn (S ∪ i)|

n

n→∞

Φ1So (¯v2n )

(25)

Let us fix n ≥ 1, 1 ≤ i ≤ n and a game vn ∈ F ∩ Γn . Using (4), we can conclude as follows:

|v(Nn )|

lim

,

tn n

for n = 1, 2, . . . .

(31)

Let us fix in ∈ Sn and jn ∈ Nn \ Sn for n = 1, 2, . . .. Hence, comparing (8) and (9), we get that ΦiSo (uNSnn ) ≥ ΦjSo (uNSnn ). Besides, n n Eq

it is obvious that the equality (1) implies that j∈Tn Φj (uTnn ) = an . Therefore, for any fixed n ≥ 1 with Sn ̸= Nn , we can conclude with the help of (9) as follows:



An =



ΦjSo (uNSnn ) +

j∈Tn ∩Sn



N

ΦjSo (uNSnn ) − an

j∈Tn \Sn

Eq

whence limn→∞ Φi (vn ) = 0. But this together with (26) immediately implies (25), completing the proof.  Proof of Theorem 3. By definition Nn = {1, 2, . . . , n} and let N2n = {1, 2, . . . , 2n}. For n = 1, 2, . . . , we define the simple games v¯ 2n on N2n by the following: v¯ 2n (N2n ) = 1, v¯ 2n (i) = 1 for 1 ≤ i ≤ n, and v¯ 2n (S ) = 0 otherwise. One can easily verify −2 with the help of (4) that ΦiSo (¯v2n ) = 4n5n for 1 ≤ i ≤ n. But (2n−1) Eq

1 Φi (¯v2n ) = 2n for 1 ≤ i ≤ 2n, because of (1). Hence,     Eq So lim Φi (¯v2n ) − Φi (¯v2n )

n→∞

i∈Nn

 = lim

n→∞

i∈Nn

5n − 2 4(2n − 1)

−

1 2

 =

1 8

,

which implies (by Definition 2) that the solidarity value and the equal split value are not asymptotically strongly-equivalent in the class Fsg of all normalized simple games in Γ .

= |Tn ∩ Sn | ·

ΦiSo n

(

N uSnn

) + |Tn \ Sn | · ΦjSo (uNSnn ) − an n

≥ |Tn ∩ Sn | · ΦjSo (uNSnn ) + |Tn \ Sn | · ΦjSo (uNSnn ) − an n n = tn ΦjSo (uNSnn ) − an n  (n − sn − 1) (n − sn − 1)(n − sn − 2) tn sn 1 + + = n n (n − 1)2 (n − 1)(n − 2)2  (n − sn − 1)! + ··· + − an (n − 1)(n − 2) · · · (sn + 2)(sn + 1)2   n(1 − bn1 ) n(1 − bn1 )(1 − bn2 ) = an bn0 1 + + + ··· n−1 n−2   n(1 − bn1 )(1 − bn2 ) · · · (1 − bnn−sn −1 ) + −1 sn + 1

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202

  ≥ an bn0 1 + n(1 −

+ ≥

gnA

n(1 − bn1 ) n−1

n(1 − bn2 )2

+

)

bnn−sn −1 n−sn −1

n−2



Case 2: lim supk→∞ (n′k − sn′ ) < +∞. k

+ ···

Then there is a natural M such that n′k − sn′ ≤ M for k ≥ 1. Hence, for each i ≥ 0, it follows that



k

n′k −M n ′ −i

n′

≤ bi k for n′k > i,

k

n′

−1

sn + 1

201

and thereby lim infk→∞ bi k ≥ 1. However, on the other hand, n′

lim supk→∞ bi k = lim supk→∞

,

n′k

sn′

k

≤ 1 for each i ≥ 0. Therefore

n′k −i

limk→∞ bi = 1 for each i ≥ 0, which by (29) and the assumption of the considered case implies that limk→∞ gnA′ = 0. Hence, in view

whence An ≥ gnA .

(32)

One can easily check that (32) also holds when Sn = Nn . Namely, then bn0 = 1 and thereby (32) is equivalent to An ≥ 0. But this inequality is true because (10) and (27) with Sn = Nn imply An = 0. In exactly the same way we get Bn ≥ gnB .

(33)

Let {nk }k=1,2,... be an arbitrarily fixed sequence of naturals with n1 < n2 < · · ·. Now we will show that

k

of (35), inequality (34) holds also in Case 2. Now, (34) and (32) immediately lead to the inequality lim inf Ank ≥ 0.

(37)

k→∞

One can easily see that we can repeat the reasoning leading to (37), replacing An , gnA and g A by Bn , gnB and g B , respectively, to finally get lim inf Bnk ≥ 0.

(38)

k→∞

gA :=

lim inf gnAk k→∞

≥ 0.

(34)

′

Let {nk }k=1,2,... be a subsequence of {nk } such that

Now using (31), (37) and (38), we can conclude as follows: 0 = lim inf[Ank + Bnk ] ≥ lim inf Ank + lim inf Bnk ≥ 0, k→∞

lim gnA′ = gA .

k→∞

(35)

k

We need to consider two cases. Case 1: lim supk→∞ (n′k − sn′ ) = +∞.

k→∞

k→∞

which immediately implies lim infk→∞ Ank = 0. But this, in view of the arbitrarity of the sequence {nk } gives limn→∞ An = 0, which is equivalent to (18). Thus the proof of Theorem 4 is completed. 

k

Obviously, there is a subsequence {n′′k }k=1,2,... of the sequence {n′k } such that lim (n′′k − sn′′ ) = +∞ and

lim gnA′′ = gA .

k

k→∞

k→∞

(36)

k

Further, since limk→∞ n′′k = +∞, therefore one can easily check that n′′

n′′

lim (bi k − b0k ) = 0

Proof of Theorem 5. Let t = |T | be fixed and let i ∈ T . We consider two cases. Case 1: t ≥ 2. Then ΦiSo (uNT ) is of the form (8) and can be equivalently rewritten as

ΦiSo (uNT ) =



for i = 1, 2, . . . .

k→∞

(n − k)

k=0

Now, using (36), (29) and the above equality, for any fixed u = n′′1 , n′′2 , . . . , we can conclude as follows g A = lim gnA′′ k→∞

t t −1

×

  n −t  (n − s − 1)(n − s − 2) · · · (n − s − t + 1) n−s

s=0

(39)

Now consider the function

k



n′′



n′′

g ( x) =

n′′

≥ lim inf an′′k · b0k 1 + (1 − b1k ) + (1 − b2k )2

(x − 1)(x − 2) · · · (x − t + 1) x

k→∞

+ · · · + (1 − 

n′′ bu k u

n′′ k

−1 n′′

n′′

1 + (1 − b0k ) + (1 − b0k )2

n′′ b0k u

)



.

This function is positive and strictly increasing on the interval (t − 1, ∞) since it is the product of the functions 1 − 1x , x − 2, x − 3, . . . , x − t + 1, with the same properties. Now, considering the clear integral sums, the following inequalities can be shown:





k→∞

+ · · · + (1 −



)

= lim inf an′′k · b0



n

g (x)dx < t −1

 −1 .

n −t  (n − s − 1)(n − s − 2) · · · (n − s − t + 1)

n−s

s=0

<



n+1

g (x)dx.

(40)

t −1

Consequently,



n′′



n′′

n′′





g A ≥ lim inf an′′ · b0k 1 + (1 − b0k ) + (1 − b0k )2 + · · · − 1 k→∞

.

Hence, by (39), for each n there is a 0 < θn < 1 such that

k

ΦiSo (uNT ) =

= lim inf an′′k (1 − 1) = 0, k→∞

n′′

because of the inequalities 0 ≤ an′′ ≤ 1 and 0 ≤ b0k ≤ 1 for all k k ≥ 1. Thus we have shown that inequality (34) holds.

t G(n + θn ) − G(t − 1) n

t −1



,

(41)

(n − k)

k=1

where G(x) is the indefinite integral G(x) =



g (x)dx.

202

T. Radzik / Mathematical Social Sciences 65 (2013) 195–202

Further, one can easily check that G(x) is of the form G(x) =

x

t −1

t −1

+ (−1)

t −1

But this with (19) (proved above) for i ∈ T lead to the following

(t − 1)! ln x + Wt −2 (x),

lim

n→∞

where Wt −2 (x) is a polynomial of degree t − 2. Hence, a simple analysis leads to the following two equalities: lim

G(n + θn )

n→∞ t −1



=

(n − k)

1

and

t −1

G(t − 1)

lim

n→∞ t −1



k=1

= 0.

(42)

(n − k) Eq

(

uNT

(

uNT

) )

= lim

tG(n + θn ) − tG(t − 1) t −1

n→∞



=

(n − k)

t t −1

.

1 , n

(43)

Thus equality (19) has been shown in case t ≥ 2 and i ∈ T . Case 2: t = 1. Then the formula (8) is equivalent to Φ1So (uNT ) = n 1 1 k=1 k , i ∈ T . But considering clear integral sums leads us to n the obvious inequalities: n 1

1 x

dx <

n  1 k=1

k

<1+

n

 1

1 x

dx,

and thereby, for some 0 < θn < 1,

ΦiSo (uNT ) =

θn + log n n

if t = 1.

(44)

Hence, one can easily see that (19) is also true in the case t = 1 and i ∈ T . Now, let k ̸∈ T , and let i ∈ T . Using the efficiency and symmetry axioms, we can conclude

ΦkSo (uNT ) Eq Φk

(uNT )

=

1 n−t

[1 − t ΦiSo (uNT )] Eq Φi

(uNT )

=

n n−t

−

t

= lim

n→∞

n n−t

−

t n−t

completing the proof of Theorem 5.

·

t t −1



= 1,



Acknowledgments

we

k =1



Eq

Φk (uNT )



The author thanks the Associate Editor and two anonymous referees for essential comments and suggestions.

k=1

Now, using (41), (42) and the obvious equality Φi (uNT ) = can conclude as follows:

Φ So lim iEq n→∞ Φ i

ΦkSo (uNT )

ΦiSo (uNT )

n − t Φ Eq (uN ) i T

.

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