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Weighted Shapley hierarchy levels values
Operations Research Letters 47 (2019) 122–126
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Weighted Shapley hierarchy levels values Manfred Besner Department of Geomatics, Computer Science and Mathematics, HFT Stuttgart, University of Applied Sciences, Schellingstr. 24, D-70174 Stuttgart, Germany
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Article history: Received 23 November 2018 Received in revised form 29 December 2018 Accepted 23 January 2019 Available online 4 February 2019 Keywords: Game theory Cooperative game Consistency Level structure (Weighted) Shapley (levels) value Weighted balanced contributions
a b s t r a c t A new class of values for cooperative games with level structure is introduced. We apply a multi-step proceeding to the weighted Shapley values. For characterization, two well-known axiomatizations of the weighted Shapley values are extended, the first one by efficiency and weighted balanced contributions and the second one by weighted standardness for two-player games and consistency. We get a new axiomatization of the Shapley levels value too. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Presenting his famous value for coalition structures, [14] suggested an extension of his value that ‘‘... deals with union structure hierarchies, i. e., the possibility that, inside each union, there may be some groups who are closer together than the remaining members of the union. For example, some of the union members may belong to a certain clan, and will therefore make a common front against the other union members’’. [20] was the first one who put this idea into action by constructing a model for hierarchical structures, called level structure. Then he defined his value, we call it Shapley levels value, by a set of axioms. His axioms are extensions of axioms of a characterization of the Owen value [14] and so finally of the Shapley value [17]. Many applications are characterized by hierarchical structures. [1] build a bridge between hierarchical structures and level structures by studying different mappings that convert hierarchical structures into level structures. After the mapping, they applied the levels Shapley value to the game with the level structure in a second step. [18] pointed out an interesting connection between cooperative and non-cooperative game theory by extending the bidding and proposing mechanism in [15] for the Shapley value to the Shapley levels value. The non-efficient Banzhaf value [4] was generalized to level structures by [2] who gave logically comparable axiomatic characterizations of the Shapley levels value and the Banzhaf levels value. E-mail address: [email protected]. https://doi.org/10.1016/j.orl.2019.01.007 0167-6377/© 2019 Elsevier B.V. All rights reserved.
In [5] the axiomatization of the Shapley value via efficiency and balanced contributions by [12] and [10] is extended to the Shapley levels value. Keeping this in mind, [19] defined a weighted value for coalition structures and [8] extended it to level structures. They axiomatized their value via efficiency and balanced per capita contributions, an adaption of a particular case of weighted balanced contributions [12]. In the end, the axiomatization in [8] extends a special case of an axiomatization of the weighted Shapley values [16], also presented in [12] and [10]. By this value, all coalitions have been assigned weights given by the cardinality of the coalitions. In each level structure we can define ‘‘games between components’’ in which each component of the same level acts as a player. If two components are symmetric players in such games, the total payoff in the original game to the players of both components by the value in [8] is only equal if the components have the same amount of players, in contrast to the Shapley levels value where the amount of players is arbitrary. But sometimes this approach is not adequate, e. g., if some coalitions have a lack of ability to perform or not the same political power and so on. To be no longer restricted to that limitation, we introduce exogenously given weights for the coalitions. These weights become prominent if the coalitions act as players. So we can provide the weighted Shapley hierarchy levels values, extensions from the weighted Shapley values to level structures in general. It turned out, by defining weights of coalitions appropriately, the axiomatization of the value from [8] can be easily transferred to an axiomatization given by efficiency and weighted balanced group contributions, an extension of weighted balanced contributions [12]. [10] presented another axiomatization of the weighted Shapley values. They used as a first axiom the weighted standardness
M. Besner / Operations Research Letters 47 (2019) 122–126
property: in a two-player game the cooperation benefit has to be divided proportionally to the weights of the players. Their second axiom, called consistency, requires that a player’s payoff in a reduced game is the same as in the original game if the reduced game is defined suitably. As the main part of the paper, we extend this axiomatization to level structures to characterize the weighted Shapley hierarchy levels values. In addition, we obtain a new axiomatization of the Shapley levels value as a corollary. The plan of the paper is as follows. Some preliminaries are given in Section 2, the weighted Shapley hierarchy levels values and an axiomatization in the sense of [5] are introduced in Section 3, Section 4 presents a new consistency property with a related axiomatization and Section 5 gives a short conclusion. 2. Preliminaries Let R be the real numbers, R++ the set of all positive real numbers, U a countably infinite set, the universe of all possible players, and N the set of all finite subsets N ⊆ U, N ̸ = ∅. A TUgame is a pair (N , v ) with a player set N ∈ N and a coalition function v : 2N → R, v (∅) = 0, 2N := {S : S ⊆ N }. We call v (S) the worth of coalition S ⊆ N, we denote by Ω S the set of all non-empty subsets of S and the set of all TU-games with player set N is denoted by VN . (S , v ) is the restriction of (N , v ) to S ∈ Ω N and a player i ∈ N is called a null player in v if v (S ∪ {i}) = v (S), S ⊆ N \{i}. For all S ⊆ N the Harsanyi dividends ∆v (S) [9] are defined ∑ inductively by ∆v (S) := 0, if S = ∅, and ∆v (S) := v (S) − R⊊S ∆v (R), otherwise. A set B := {B1 , . . . , Bm } of coalitions of players is said to be a coalition structure on N if Bk⋃∩ Bℓ = ∅, 1 ≤ k < ℓ ≤ m, Bk ̸ = ∅ m for all k, 1 ≤ k ≤ m, and k=1 Bk = N. We call each B ∈ B a component and B(i) denotes the component that contains the player i ∈ N. A finite sequence B := {B0 , . . . , Bh+1 } of coalition structures Br , 0 ≤ r ≤ h + 1, on N is called a level structure [20] on N if Br is r +1 r r +1 a refinement of B { for each} r , 0 ≤ r ≤ h, i. e., B (i) ⊆ B (i) for all i ∈ N, B0 = {i} : i ∈ N , and Bh+1 = {N }. Br denotes the rth level of B and Br (Bk ) denotes the component of the rth level that contains the component Bk ∈ Bk , 0 ≤ k ≤ r ≤ h + 1. The set of all level structures with player set N is denoted by LN . A triple (N , v, B) consisting of a TU-game (N , v ) ∈ VN and a level structure B ∈ LN is called an LS-game. The collection of all LS-games on N is denoted by VLN . We note that each LSgame (N , v, B0 ) with a trivial level structure B0 := {B0 , B1 } corresponds to a TU-game (N , v ) and each LS-game (N , v, B1 ), B1 := {B0 , B1 , B2 }, corresponds to a game with coalition structure [3,14]. Let (N , v, B) ∈ VLN and B = {B0 , . . . , Bh+1 }:
• From a level structure on N follows a restricted level structure on T ∈ Ω N by eliminating the players in N \T . With coalition structures Br |T := {B ∩ T : B ∈ Br , B ∩ T ̸ = ∅}, 0 ≤ r ≤ h + 1, the new level structure on T is given by B|T := {B0 |T , . . . , Bh+1 |T } ∈ LT . (T , v, B|T ) ∈ VLT is called the restriction of (N , v, B) to player set T . To illustrate the following definition we insert a short example. Let N = {1, 2, 3} and B = {B0 , B1 , B2 } be given by B0 = {{1}, {2}, {3}}, B1 = {{1, 2}, {3}}, and B2 = {N }. If we cancel the zeroth level and regard the components of the first level as players of a new game (the induced first level game), the induced 0 1 0 first level structure B1 = {B1 , B1 } from B is given by B1 = {{ } { }} {{ }} 11 {1, 2} , {3} , and B = {1, 2}, {3} . r0
• We define B := B , . . . , B
r h+1−r
Br
∈ L , 0 ≤ r ≤ h, as the induced rth level structure from B = {B0 , . . . , Bh+1 } by k considering the components B ∈ Br as players where Br := r
{
}
123
{B ∈ Br : B ⊆ B′ } : for all B′ ∈ B , 0 ≤ k ≤ h + 1 − r, is a coalition structure such that each component is a set of all components of the rth level which are subsets of the same (component ) of the (rr + k)th level. The induced rth level game Br , v r , Br ∈ VLB is given by
{
} r +k
⋃
v r (Q) := v (
B) for all Q ⊆ Br .
B∈Q
For the special case r = 1, [14] called such a game a quotient game.
{ } • We define Br := B0 , . . . , Br , {N } ∈ LN , 0 ≤ r ≤ h, as the rth cut level structure from B if we cut out all levels between the rth and the (h + 1)th level. (N , v, Br ) is called the rth cut of (N , v, B). Note that we have for each B = {B0 , . . . , Bh+1 } also B = Bh . A TU-value φ is an operator that assigns a payoff vector φ (N , v ) ∈ RN to any (N , v ) ∈ VN , an LS-value ϕ is an operator that assigns a payoff vector ϕ (N , v, B) ∈ RN to any (N , v, B) ∈ VLN . We define W := {f : U → R++ } with wi := w(i) for all w ∈ W and i ∈ U as the set of all positive weight systems on the universe N of all players. W 2 := {f : 2N \∅ → R++ } with wS := w (S) for all N w ∈ W 2 , S ∈ Ω N , denotes the set of all positive weight systems on all coalitions S ∈ Ω N . The (simply) weighted Shapley values Shw [16] are defined by ∑ w ∑ i ∆v (S) for all i ∈ N and w ∈ W. (1) Shw i (N , v ) := j∈S wj S ⊆N , S ∋i
The Shapley value Sh [17] distributes the dividends ∆v (S) equally among all players of a coalition S and is a special case of a weighted Shapley value, given by Shi (N , v ) :=
∑ ∆v (S) for all i ∈ N . |S |
S ⊆N , S ∋i
Let (N , v, B) ∈ VLN , B = Bh , T ∈ Ω N , T ∋ i, and KT (i) :=
h ∏
KTr (i),
r =0
where KTr (i) :=
1
|{B ∈ Br : B ⊆ Br +1 (i), B ∩ T ̸= ∅}|
.
The Shapley Levels value ShL [20] is defined by ShLi (N , v, B) :=
∑
KT (i)∆v (T ) for all i ∈ N (see [5, eq. (1)]).
T ⊆N , T ∋i
Obviously, ShL coincides with Sh if h = 0 and it is well-known that ShL coincides with the Owen value [14] if h = 1. We refer to the following axioms for TU-values φ on VN and LS-values ϕ on VLN . In the case of using a subdomain, we require an axiom to hold when all games belong to this subdomain. If there are used weights for some coalitions, these weights are still valid in the subdomain. Standardness, ST0 [10]. For all (N , v ) ∈ VN , N = {i, j}, i ̸ = j, we have ] 1[ φi (N , v ) = v ({i}) + v ({i, j}) − v ({i}) − v ({j}) . 2 By standardness, in two-player games cooperating leads to sharing the surplus equally. The next axiom stands for dividing the surplus proportionally to players’ weights in two-player games. w-Weighted standardness, WSw [10]. For all (N , v ) ∈ VN , N = {i, j}, i ̸= j, and w ∈ W, we have ] wi [ φi (N , v ) = v ({i}) + v ({i, j}) − v ({i}) − v ({j}) . wi + wj
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M. Besner / Operations Research Letters 47 (2019) 122–126
ϕi (N , v, B) =
are subsets of the component Bk+1 (i) where the component Bk (i) is replaced by coalition T .
Null player, N. For all (N , v, B) ∈ VLN and i ∈ N a null player in v , we have ϕi (N , v, B) = 0.
Definition 3.2. Let (N , v, B) ∈ VLN , B = Bh , w ∈ W 2 , i ∈ N,
Efficiency, E. For all (N , v, B) ∈ VLN , we have v (N).
∑
i∈N
Additivity, A. For all (N , v, B), (N , v ′ , B) ∈ VLN , we have
ϕ (N , v, B) + ϕ (N , v ′ , B) = ϕ (N , v + v ′ , B). The following axiom states that for two components Bk , Bℓ of the same level which belong to the same component one level higher the contribution of Bℓ to the total payoff to all players from Bk equals the contribution of Bk to the total payoff to all players from Bℓ . Balanced group contributions, BGC [5]. For all (N , v, B) ∈ VLN , B = Bh , Bk , Bℓ ∈ Br , 0 ≤ r ≤ h, such that Bℓ ⊆ Br +1 (Bk ), we have
∑
ϕi (N , v, B) −
i∈Bk
∑
ϕi (N \Bℓ , v, B|N \Bℓ )
i∈Bk
=
∑
ϕi (N , v, B) −
∑
i∈Bℓ
ϕi (N \Bk , v, B|N \Bk ).
i∈Bℓ
The next axiom requires that the sum of all players’ payoff of a component coincides with the component’s payoff in an induced level game where the component is a player. Level game property, LG [20]. For all (N , v, B) ∈ VLN , B = Bh , B ∈ Br , 0 ≤ r ≤ h + 1, we have
∑
ϕi (N , v, B) = ϕB (B , v , B ). r
r
r
i∈B
3. Weighted Shapley hierarchy levels values [14] suggested for his value a two-step approach. The first step consists of using the Shapley value on the quotient game. In the second stage, the Shapley value is applied to the result of the first step again. [19] adapted the same procedure for the value ζ , an extension of a weighted Shapley value to coalition structures where the weights are given by the size of the coalitions. [8] presented an extension to level structures and replaced the two-step approach by a multi-step mechanism as suggested also by [14]. It is still not known and there is no hint in [8] that their value can be universalized easily to an extension of the weighted Shapley values in general. The only problem is to know how to deal with exogenously given weights for coalitions. The substantial idea behind our definition, similar suggested by [14] and implemented as an algorithm by [8], is as follows: In a first step, we distribute the worth of the grand coalition v (N) among the components Bh of the hth level as players, using a weighted Shapley value. In the second step, each payoff to a component Bh from the first step is divided by a weighted Shapley value among all components Bh−1 (the new players) which are subsets of the component Bh and so on for all levels. In the last step, we distribute the payoff to components B1 from the first level among the original players i ∈ N. Therefore we need a weight system for the coalitions. For notational parsimony, we give the following notation. Notation 3.1. Bk (i)
Let (N , v, B) ∈ VLN , B = Bh , i ∈ N, and T ∈
, 0 ≤ k ≤ h. We denote by Ti k := {B ∈ Bk : B ⊆ Bk+1 (i), B ̸= B (i)}∪{T } the set containing all components of the kth level which Ω
k
N
Bk (i)
and for all k, 0 ≤ k ≤ h, T ∈ Ω , Ti k the set from Notation 3.1, let wk,T ∈ W be a weight system with coalitions S as players such k,T that wS := wS , wS ∈ w , for all S ∈ Ti k , v¯ ih+1 := v , and v¯ ik given by k,T
(Ti k , v˜ ik ) for all T ∈ Ω B v¯ ik (T ) := Shw T
k (i)
(2)
where v˜ is specified recursively via k i
v˜ ik (Q) := v¯ ik+1 (
⋃
S) for all Q ⊆ Ti k .
S ∈Q
Then the weighted Shapley hierarchy levels value ShwHL is given by HL Shw (N , v, B) := v¯ i0 ({i}) for all i ∈ N . i
Remark 3.3. ShwHL coincides with Shw if B = B0 and w{i} = wi for all i ∈ N. Since Shw is efficient and additive, it is easy to see that ShwHL meets E, A, and LG. Similar to [5], who characterized the Shapley levels value by efficiency and balanced group contributions, [8] characterized their value by efficiency and balanced per capita contributions. The last axiom is a special case of the following one by limiting the weights to the size of the relevant components.
w-Weighted balanced group contributions, WBGCw . For all N (N , v, B) ∈ VLN , B = Bh , w ∈ W 2 , Bk , Bℓ ∈ Br , 0 ≤ r ≤ h, such that Bℓ ⊆ Br +1 (Bk ), we have ∑ ∑ ϕi (N , v, B) − ϕi (N \Bℓ , v, B|N \Bℓ ) i∈Bk
i∈Bk
wBk
∑ =
i∈Bℓ
ϕi (N , v, B) −
∑
ϕi (N \Bk , v, B|N \Bk )
i∈Bℓ
wBℓ
.
This axiom extends w-weighted balanced contributions [12]. In contrast to BGC, here the contributions of one component to the players of the other component and vice versa are proportional to the weights of the components. Theorem 3.4. Let w ∈ W 2 . ShwHL is the unique LS-value that satisfies E and WBGC w . N
Proof. If we use as weights the cardinality of the coalitions, it is clear that Definition 3.2 is equivalent to the algorithm given by [8, Section 3]. Thus, the rest of the proof of Theorem 3.4 is omitted because it is easy to adapt the proofs of Proposition 7 and Theorem 8 in [8], using w-weighted balanced contributions [10,12] for the weighted Shapley values in Proposition 7 and replacing the given weights (size of the components) by arbitrary weights in Theorem 8. □ Remark 3.5. If all weights are equal, we use in fact in line (2) the Shapley value and WBGC w equals BGC. So, by [5, Theorem 2], Definition 3.2 determines in this case the Shapley levels value ShL . Remark 3.6. As shown by example 1 in [8], ShwHL does not satisfy N in general.
M. Besner / Operations Research Letters 47 (2019) 122–126
4. Level-consistency [10] introduced a reduced game of a TU-game. Definition 4.1. Let (N , v ) ∈ VN , R ∈ Ω N , and φ a TU-value. The φ reduced game (R, vR ) ∈ VR is given for all S ∈ Ω R by
) ∑ ( ) ( φ φj S ∪ Rc , v vR (S) := v S ∪ Rc −
(3)
where R := N \R.
∑ ( ) φj S ∪ Rc , v . j∈S
The interpretation for this reduced game is as follows: A coalition of players Rc leaves the game. In the reduced game, each coalition S which is a subset of the coalition R of the remaining players gets the worth of coalition S ∪ Rc in the original game less the removed players’ payoff in the restricted game on S ∪ Rc . Then a TU-value is said to be consistent if each player in R gets in the reduced and in the original game the same payoff. Consistency, C [10]. For all (N , v ) ∈ VN , R ∈ Ω N , we have φ φi (N , v ) = φi (R, vR ) for all i ∈ R. It turns out that the weighted Shapley values have got an intense connection to this axiom. Theorem 4.3 ([10]). Let w ∈ W. Shw is the unique TU-value that satisfies C and WSw . If we exchange WSw by ST0 , this theorem characterizes the Shapley value [10]. By adding LG, [21] extended this axiomatization for the Shapley value to the Owen value (see proof of Theorem 3 in [21]). Similarly, [11] transferred the related axiomatization of the proportional value [6,13] to his proportional value for coalition structures. Our new reduced game for level structures extends the reduced game for TU-games by [10] too but it is no extension of the reduced game used by [21] and [11] respectively. N N Definition ⋃4.4. Let (N , v, B) ∈ VL , B = Bh , R ∈ Ω such that R = B∈Bh ,B⊆R B, and ϕ an LS-value. The reduced level game ϕ (R, vR , B|R ) ∈ VLR is given for all S ∈ Ω R by
ϕ
∑
vR (S) := v (S ∪ Rc ) −
( ) ϕB Bh |S ∪Rc , v h , Bh |S ∪Rc
(4)
B∈Bh , B⊈R
where Rc := N \R and Bh |S ∪Rc , v h , Bh |S ∪Rc is the induced hth level game restricted to S ∪ Rc . Remark 4.5. If B = B0 , Definition 4.4 coincides with Definition 4.1.
ϕ
ϕS ∩B
Inessential (last) level property, IL. For all (N , v, B) ∈ VLN , B = Bh , h ≥ 1, and Bh = Bh+1 , we have ϕ (N , v, B) = ϕ (N , v, Bh−1 ). The last three properties fit best with the weighted Shapley hierarchy levels values. Theorem 4.8. Let w ∈ W 2 . ShwHL satisfies IL, WSw , and LC. N
N
Proof. Let (N , v, B) ∈ VLN , B = Bh , and w ∈ W 2 :
• IL: If Bh = Bh+1 , we have in Notation 3.1 for all i ∈ N h and T ∈ Ω B (i) , Ti h = {T }. It follows in line (2), v¯ ih (T ) = h , T h Shw ({T }, v˜ ih ) = v (T ) for all T ∈ Ω B (i) . Thus, if h ≥ 1, there T has been no change in line (2) for all k, 0 ≤ k ≤ h − 1, if we drop the hth level and IL is shown.
• WSw : By Definition 3.2 and ⋃ line (1), the claimhis obvious. • LC: Let i ∈ N , R = B∈Bh ,B⊆R B, R ⊇ B (i). By IL, it is sufficient to show LC on level structures Bh with Bh ̸ = Bh+1 . If R = N or |N | = 1, Eq. (5) is trivially satisfied. Let now |N | ≥ 2 and R ̸ = N. By [10], Shw satisfies C for all w ∈ h W. For each T ∈ Ω B (i) we have a set Ti h from Notation 3.1 h and a related Ti |R on the restricted level structure B|R . Thus follows by Eq. (2) for k = h h, T ( h h, T ϕ ) v¯ ih (T ) = Shw (Ti h , v˜ ih ) = Shw Ti |R , (v˜R )hi T T C, Remark 4.5
) Bh |S ∪Rc , v h , Bh |S ∪Rc .
(
ϕ
⋃
N
The following axiom extends C. Level-consistency, LC. For all (N , v, B) ∈ VLN , B = Bh , R = ⋃ B, we have h B∈B ,B⊆R ϕ
ϕ
Theorem 4.9. Let w ∈ W 2 . ShwHL is the unique TU-value that satisfies IL, WSw , and LC.
B∈Bh , B⊆R
ϕi (N , v, B) = ϕi (R, vR , B|R ) for all i ∈ R.
ϕ
where (v˜R )hi is given by (v˜R )hi (Q) := vR ( S ∈Q S) for all Q ⊆ Ti h |R . Thus, there is no difference in Eq. (2) between both games for all k ̸ = h too and LC is shown. □ It transpires that the weighted Shapley hierarchy levels values are characterized by the following theorem.
Remark 4.6. If ϕ is efficient, Eq. (4) can be simplified to
vR (S) :=
To characterize the weighted Shapley hierarchy levels values we need an additional very weak property. It states that if the hth and the h + 1th level are equal, we can remove the hth level from the level structure without consequences.
)
(
∑
Remark 4.7. If B = B0 , LC coincides with C.
w-Weighted standardness, WSw . For all (N , v, B) ∈ VLN with N B = B0 , N = {i, j}, i ̸ = j, and w ∈ W 2 , we have [ ] w{i} ϕi (N , v, B) = v ({i}) + v ({i, j}) − v ({i}) − v ({j}) . w{i} + w{j}
Remark 4.2. If φ is efficient, Eq. (3) can be simplified to φ
worth of coalition S ∪ Rc in the original game less the payoff which the removed components obtain in the induced hth level game restricted to S ∪ Rc . Then an LS-value is level-consistent if each player in R gets in the reduced level game the same payoff as in the original game.
We adapt w-weighted standardness.
j∈Rc c
vR (S) :=
125
(5)
R is a coalition of all players of some components of the hth level. All players of the other components of the hth level leave the game. In the reduced level game each coalition S ∈ Ω R gets the
Proof. By Theorem 4.8, it is sufficient to show uniqueness. Let N (N , v, B) ∈ VLN , B = Bh , w ∈ W 2 , i ∈ N arbitrary, and ϕ an w LS-values that satisfies IL, WS , and LC. By LC, we have ϕ
ϕj (N , v, B) = ϕj (Bh (i), vBh (i) , B|Bh (i) ) for all j ∈ Bh (i) ϕ
where vBh (i) is unique by WSw , LC, Remarks 4.7, 4.5 and Theorem 4.3.
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M. Besner / Operations Research Letters 47 (2019) 122–126
By IL, we have for each restricted kth cut level structure Bk |Bk (i) , k > 0, from B on Bk (i), ϕ
ϕ
ϕ (Bk (i), vˆ Bk (i) , Bk |Bk (i) ) = ϕ (Bk (i), vˆ Bk (i) , Bk−1 |Bk (i) ) ϕ
where vˆ Bk (i) is defined recursively via ϕ
vˆ Bk (i)
⎧ ⎨v ϕ h , if k = h, B (i) := ( ϕ )ϕ ⎩ vˆ , otherwise. Bk+1 (i) Bk (i)
Hence, by induction on the levels k, by IL and LC, it follows that ϕ
ϕj (N , v, B) = ϕj (B1 (i), vˆ B1 (i) , B0 |B1 (i) ) for all j ∈ B1 (i) ϕ
where vˆ B1 (i) is unique by WSw , LC, Remarks 4.7, 4.5, and Theorem 4.3. Therefore, by WSw , LC, Remarks 4.7, 4.5, and Theorem 4.3, ϕ is uniquely defined too. □ To introduce a new axiomatization of the Shapley levels value we adapt ST0 . Standardness, ST. For all (N , v, B) ∈ VLN with B = B0 and N = {i, j}, i ̸ = j, we have
ϕi (N , v, B) = v ({i}) +
1[ 2
] v ({i, j}) − v ({i}) − v ({j}) .
Since ST is a special case of WSw , we obtain the following corollary by Theorem 4.9. Corollary 4.10. ShL is the unique TU-value that satisfies IL, ST, and LC. Remark 4.11. The axioms in Theorem 4.9 are logically independent. N
Proof. Let (N , v, B) ∈ VLN , B = Bh , and w ∈ W 2 :
• IL: The LS-value ϕ defined by ⎧ ⎨ 1 ShwHL , if h ≥ 1, ϕ := 2 ⎩ wHL Sh , otherwise, satisfies WSw and LC but not IL. • WSw : Let v ∈ VLN , v (S) > 0 for all S ∈ Ω N , and B = B1 . If we regard the proportional value for cooperative games with a coalition structure by [11] in the notation for a level structure with B = B1 , this LS-value satisfies IL and LC but not WSw . • LC: The LS-value ϕ defined by ϕ (N , v, B) := Shw (N , v ) with wi := w{i} for all i ∈ N satisfies IL and WSw but not LC. □ 5. Conclusion It would seem reasonable in our view to use weights if operating players are in some sense not symmetric, e. g., if some players have not the same political, financial or military power, contribute more to the result, and so on. Often players act not as lone fighters. They join forces and small groups form together larger groups and so on. Thus the weighted Shapley hierarchy levels values meet with the multi-stage mechanism a widespread idea how hierarchical values have to operate. That these values do not satisfy the null player property does not need to be a disadvantage. A null player receives a compen-
sation since she supports the other members of an acting coalition with her contribution to the weight of this coalition. So a null player acts only in the coalition function and not necessarily in the weight system for coalitions without effect. For further information on this area see also [7] and [8]. An emphasis of this paper is on consistency. Consistency plays an important role in many allocation concepts in cooperative game theory. The differences in the numerous varieties of the reduced games are decisive for the diversity of the different consistency characteristics. So the reduced level game is no extension of the reduced game by [21]. There the original players of the game determine the worths in the coalition function of the reduced game. In the reduced level game the acting coalitions are decisive for the worths of the coalitions. If we have a trivial level structure, the reduced level game coincides with the reduced game, suggested by [10], too.
Acknowledgments We are grateful to André Casajus, Winfried Hochstättler and an anonymous referee for their helpful comments and suggestions.
References [1] M. Álvarez-Mozos, R. van den Brink, G. van der Laan, O. Tejada, From hierarchies to levels: new solutions for games with hierarchical structure, Internat. J. Game Theory 46 (4) (2017) 1089–1113. [2] M. Álvarez-Mozos, O. Tejada, Parallel characterizations of a generalized Shapley value and a generalized Banzhaf value for cooperative games with level structure of cooperation, Decis. Support Syst. 52 (1) (2011) 21–27. [3] R.J. Aumann, J. Drèze, Cooperative games with coalition structures, Internat. J. Game Theory 3 (1974) 217–237. [4] J.F. Banzhaf, Weighted voting does not work: a mathematical analysis, Rutgers Law Rev. 19 (1965) 317–343. [5] E. Calvo, J.J. Lasaga, E. Winter, The principle of balanced contributions and hierarchies of cooperation, Math. Social Sci. 31 (3) (1996) 171–182. [6] B. Feldman, The proportional value of a cooperative game, Manuscript. Chicago: Scudder Kemper Investments, 1999. [7] M. Gómez-Rúa, J. Vidal-Puga, The axiomatic approach to three values in games with coalition structure, European J. Oper. Res. 207 (2) (2010) 795–806. [8] M. Gómez-Rúa, J. Vidal-Puga, Balanced per capita contributions and level structure of cooperation, Top 19 (1) (2011) 167–176. [9] J.C. Harsanyi, A bargaining model for cooperative n-person games, in: A.W. Tucker, R.D. Luce (Eds.), Contributions to the theory of games IV, Princeton University Press, Princeton, 1959, pp. 325–355. [10] S. Hart, A. Mas-Colell, Potential, value, and consistency, Econometrica 57 (3) (1989) 589–614. [11] F. Huettner, A proportional value for cooperative games with a coalition structure, Theory and Decision 78 (2) (2015) 273–287. [12] R.B. Myerson, Conference structures and fair allocation rules, Internat. J. Game Theory 9 (1980) 169–182. [13] K.M. Ortmann, The proportional value for positive cooperative games, Math. Methods Oper. Res. 51 (2) (2000) 235–248. [14] G. Owen, Values of games with a priori unions, in: Essays in Mathematical Economics and Game Theory, Springer, 1977, pp. 76–88. [15] D. Pérez-Castrillo, D. Wettstein, Bidding for the surplus: A non-cooperative approach to the Shapley value, J. Econom. Theory 100 (2) (2001) 274–294. [16] L.S. Shapley, Additive and non-additive set functions, Princeton University, 1953. [17] L.S. Shapley, A value for n-person games, in: H. Kuhn, A. Tucker (Eds.), Contributions to the Theory of Games, II, Princeton University Press, Princeton, 1953, pp. 307–317. [18] J.J. Vidal-Puga, Implementation of the levels structure value, Ann. Oper. Res. 137 (1) (2005) 191–209. [19] J.J. Vidal-Puga, The Harsanyi paradox and the right to talk in bargaining among coalitions, Math. Social Sci. 64 (3) (2012) 214–224. [20] E. Winter, A value for cooperative games with levels structure of cooperation, Internat. J. Game Theory 18 (2) (1989) 227–240. [21] E. Winter, The consistency and potential for values of games with coalition structure, Games Econom. Behav. 4 (1) (1992) 132–144.
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Weighted Shapley hierarchy levels values Manfred Besner Department of Geomatics, Computer Science and Mathematics, HFT Stuttgart, University of Applied Sciences, Schellingstr. 24, D-70174 Stuttgart, Germany
article
info
Article history: Received 23 November 2018 Received in revised form 29 December 2018 Accepted 23 January 2019 Available online 4 February 2019 Keywords: Game theory Cooperative game Consistency Level structure (Weighted) Shapley (levels) value Weighted balanced contributions
a b s t r a c t A new class of values for cooperative games with level structure is introduced. We apply a multi-step proceeding to the weighted Shapley values. For characterization, two well-known axiomatizations of the weighted Shapley values are extended, the first one by efficiency and weighted balanced contributions and the second one by weighted standardness for two-player games and consistency. We get a new axiomatization of the Shapley levels value too. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Presenting his famous value for coalition structures, [14] suggested an extension of his value that ‘‘... deals with union structure hierarchies, i. e., the possibility that, inside each union, there may be some groups who are closer together than the remaining members of the union. For example, some of the union members may belong to a certain clan, and will therefore make a common front against the other union members’’. [20] was the first one who put this idea into action by constructing a model for hierarchical structures, called level structure. Then he defined his value, we call it Shapley levels value, by a set of axioms. His axioms are extensions of axioms of a characterization of the Owen value [14] and so finally of the Shapley value [17]. Many applications are characterized by hierarchical structures. [1] build a bridge between hierarchical structures and level structures by studying different mappings that convert hierarchical structures into level structures. After the mapping, they applied the levels Shapley value to the game with the level structure in a second step. [18] pointed out an interesting connection between cooperative and non-cooperative game theory by extending the bidding and proposing mechanism in [15] for the Shapley value to the Shapley levels value. The non-efficient Banzhaf value [4] was generalized to level structures by [2] who gave logically comparable axiomatic characterizations of the Shapley levels value and the Banzhaf levels value. E-mail address: [email protected]. https://doi.org/10.1016/j.orl.2019.01.007 0167-6377/© 2019 Elsevier B.V. All rights reserved.
In [5] the axiomatization of the Shapley value via efficiency and balanced contributions by [12] and [10] is extended to the Shapley levels value. Keeping this in mind, [19] defined a weighted value for coalition structures and [8] extended it to level structures. They axiomatized their value via efficiency and balanced per capita contributions, an adaption of a particular case of weighted balanced contributions [12]. In the end, the axiomatization in [8] extends a special case of an axiomatization of the weighted Shapley values [16], also presented in [12] and [10]. By this value, all coalitions have been assigned weights given by the cardinality of the coalitions. In each level structure we can define ‘‘games between components’’ in which each component of the same level acts as a player. If two components are symmetric players in such games, the total payoff in the original game to the players of both components by the value in [8] is only equal if the components have the same amount of players, in contrast to the Shapley levels value where the amount of players is arbitrary. But sometimes this approach is not adequate, e. g., if some coalitions have a lack of ability to perform or not the same political power and so on. To be no longer restricted to that limitation, we introduce exogenously given weights for the coalitions. These weights become prominent if the coalitions act as players. So we can provide the weighted Shapley hierarchy levels values, extensions from the weighted Shapley values to level structures in general. It turned out, by defining weights of coalitions appropriately, the axiomatization of the value from [8] can be easily transferred to an axiomatization given by efficiency and weighted balanced group contributions, an extension of weighted balanced contributions [12]. [10] presented another axiomatization of the weighted Shapley values. They used as a first axiom the weighted standardness
M. Besner / Operations Research Letters 47 (2019) 122–126
property: in a two-player game the cooperation benefit has to be divided proportionally to the weights of the players. Their second axiom, called consistency, requires that a player’s payoff in a reduced game is the same as in the original game if the reduced game is defined suitably. As the main part of the paper, we extend this axiomatization to level structures to characterize the weighted Shapley hierarchy levels values. In addition, we obtain a new axiomatization of the Shapley levels value as a corollary. The plan of the paper is as follows. Some preliminaries are given in Section 2, the weighted Shapley hierarchy levels values and an axiomatization in the sense of [5] are introduced in Section 3, Section 4 presents a new consistency property with a related axiomatization and Section 5 gives a short conclusion. 2. Preliminaries Let R be the real numbers, R++ the set of all positive real numbers, U a countably infinite set, the universe of all possible players, and N the set of all finite subsets N ⊆ U, N ̸ = ∅. A TUgame is a pair (N , v ) with a player set N ∈ N and a coalition function v : 2N → R, v (∅) = 0, 2N := {S : S ⊆ N }. We call v (S) the worth of coalition S ⊆ N, we denote by Ω S the set of all non-empty subsets of S and the set of all TU-games with player set N is denoted by VN . (S , v ) is the restriction of (N , v ) to S ∈ Ω N and a player i ∈ N is called a null player in v if v (S ∪ {i}) = v (S), S ⊆ N \{i}. For all S ⊆ N the Harsanyi dividends ∆v (S) [9] are defined ∑ inductively by ∆v (S) := 0, if S = ∅, and ∆v (S) := v (S) − R⊊S ∆v (R), otherwise. A set B := {B1 , . . . , Bm } of coalitions of players is said to be a coalition structure on N if Bk⋃∩ Bℓ = ∅, 1 ≤ k < ℓ ≤ m, Bk ̸ = ∅ m for all k, 1 ≤ k ≤ m, and k=1 Bk = N. We call each B ∈ B a component and B(i) denotes the component that contains the player i ∈ N. A finite sequence B := {B0 , . . . , Bh+1 } of coalition structures Br , 0 ≤ r ≤ h + 1, on N is called a level structure [20] on N if Br is r +1 r r +1 a refinement of B { for each} r , 0 ≤ r ≤ h, i. e., B (i) ⊆ B (i) for all i ∈ N, B0 = {i} : i ∈ N , and Bh+1 = {N }. Br denotes the rth level of B and Br (Bk ) denotes the component of the rth level that contains the component Bk ∈ Bk , 0 ≤ k ≤ r ≤ h + 1. The set of all level structures with player set N is denoted by LN . A triple (N , v, B) consisting of a TU-game (N , v ) ∈ VN and a level structure B ∈ LN is called an LS-game. The collection of all LS-games on N is denoted by VLN . We note that each LSgame (N , v, B0 ) with a trivial level structure B0 := {B0 , B1 } corresponds to a TU-game (N , v ) and each LS-game (N , v, B1 ), B1 := {B0 , B1 , B2 }, corresponds to a game with coalition structure [3,14]. Let (N , v, B) ∈ VLN and B = {B0 , . . . , Bh+1 }:
• From a level structure on N follows a restricted level structure on T ∈ Ω N by eliminating the players in N \T . With coalition structures Br |T := {B ∩ T : B ∈ Br , B ∩ T ̸ = ∅}, 0 ≤ r ≤ h + 1, the new level structure on T is given by B|T := {B0 |T , . . . , Bh+1 |T } ∈ LT . (T , v, B|T ) ∈ VLT is called the restriction of (N , v, B) to player set T . To illustrate the following definition we insert a short example. Let N = {1, 2, 3} and B = {B0 , B1 , B2 } be given by B0 = {{1}, {2}, {3}}, B1 = {{1, 2}, {3}}, and B2 = {N }. If we cancel the zeroth level and regard the components of the first level as players of a new game (the induced first level game), the induced 0 1 0 first level structure B1 = {B1 , B1 } from B is given by B1 = {{ } { }} {{ }} 11 {1, 2} , {3} , and B = {1, 2}, {3} . r0
• We define B := B , . . . , B
r h+1−r
Br
∈ L , 0 ≤ r ≤ h, as the induced rth level structure from B = {B0 , . . . , Bh+1 } by k considering the components B ∈ Br as players where Br := r
{
}
123
{B ∈ Br : B ⊆ B′ } : for all B′ ∈ B , 0 ≤ k ≤ h + 1 − r, is a coalition structure such that each component is a set of all components of the rth level which are subsets of the same (component ) of the (rr + k)th level. The induced rth level game Br , v r , Br ∈ VLB is given by
{
} r +k
⋃
v r (Q) := v (
B) for all Q ⊆ Br .
B∈Q
For the special case r = 1, [14] called such a game a quotient game.
{ } • We define Br := B0 , . . . , Br , {N } ∈ LN , 0 ≤ r ≤ h, as the rth cut level structure from B if we cut out all levels between the rth and the (h + 1)th level. (N , v, Br ) is called the rth cut of (N , v, B). Note that we have for each B = {B0 , . . . , Bh+1 } also B = Bh . A TU-value φ is an operator that assigns a payoff vector φ (N , v ) ∈ RN to any (N , v ) ∈ VN , an LS-value ϕ is an operator that assigns a payoff vector ϕ (N , v, B) ∈ RN to any (N , v, B) ∈ VLN . We define W := {f : U → R++ } with wi := w(i) for all w ∈ W and i ∈ U as the set of all positive weight systems on the universe N of all players. W 2 := {f : 2N \∅ → R++ } with wS := w (S) for all N w ∈ W 2 , S ∈ Ω N , denotes the set of all positive weight systems on all coalitions S ∈ Ω N . The (simply) weighted Shapley values Shw [16] are defined by ∑ w ∑ i ∆v (S) for all i ∈ N and w ∈ W. (1) Shw i (N , v ) := j∈S wj S ⊆N , S ∋i
The Shapley value Sh [17] distributes the dividends ∆v (S) equally among all players of a coalition S and is a special case of a weighted Shapley value, given by Shi (N , v ) :=
∑ ∆v (S) for all i ∈ N . |S |
S ⊆N , S ∋i
Let (N , v, B) ∈ VLN , B = Bh , T ∈ Ω N , T ∋ i, and KT (i) :=
h ∏
KTr (i),
r =0
where KTr (i) :=
1
|{B ∈ Br : B ⊆ Br +1 (i), B ∩ T ̸= ∅}|
.
The Shapley Levels value ShL [20] is defined by ShLi (N , v, B) :=
∑
KT (i)∆v (T ) for all i ∈ N (see [5, eq. (1)]).
T ⊆N , T ∋i
Obviously, ShL coincides with Sh if h = 0 and it is well-known that ShL coincides with the Owen value [14] if h = 1. We refer to the following axioms for TU-values φ on VN and LS-values ϕ on VLN . In the case of using a subdomain, we require an axiom to hold when all games belong to this subdomain. If there are used weights for some coalitions, these weights are still valid in the subdomain. Standardness, ST0 [10]. For all (N , v ) ∈ VN , N = {i, j}, i ̸ = j, we have ] 1[ φi (N , v ) = v ({i}) + v ({i, j}) − v ({i}) − v ({j}) . 2 By standardness, in two-player games cooperating leads to sharing the surplus equally. The next axiom stands for dividing the surplus proportionally to players’ weights in two-player games. w-Weighted standardness, WSw [10]. For all (N , v ) ∈ VN , N = {i, j}, i ̸= j, and w ∈ W, we have ] wi [ φi (N , v ) = v ({i}) + v ({i, j}) − v ({i}) − v ({j}) . wi + wj
124
M. Besner / Operations Research Letters 47 (2019) 122–126
ϕi (N , v, B) =
are subsets of the component Bk+1 (i) where the component Bk (i) is replaced by coalition T .
Null player, N. For all (N , v, B) ∈ VLN and i ∈ N a null player in v , we have ϕi (N , v, B) = 0.
Definition 3.2. Let (N , v, B) ∈ VLN , B = Bh , w ∈ W 2 , i ∈ N,
Efficiency, E. For all (N , v, B) ∈ VLN , we have v (N).
∑
i∈N
Additivity, A. For all (N , v, B), (N , v ′ , B) ∈ VLN , we have
ϕ (N , v, B) + ϕ (N , v ′ , B) = ϕ (N , v + v ′ , B). The following axiom states that for two components Bk , Bℓ of the same level which belong to the same component one level higher the contribution of Bℓ to the total payoff to all players from Bk equals the contribution of Bk to the total payoff to all players from Bℓ . Balanced group contributions, BGC [5]. For all (N , v, B) ∈ VLN , B = Bh , Bk , Bℓ ∈ Br , 0 ≤ r ≤ h, such that Bℓ ⊆ Br +1 (Bk ), we have
∑
ϕi (N , v, B) −
i∈Bk
∑
ϕi (N \Bℓ , v, B|N \Bℓ )
i∈Bk
=
∑
ϕi (N , v, B) −
∑
i∈Bℓ
ϕi (N \Bk , v, B|N \Bk ).
i∈Bℓ
The next axiom requires that the sum of all players’ payoff of a component coincides with the component’s payoff in an induced level game where the component is a player. Level game property, LG [20]. For all (N , v, B) ∈ VLN , B = Bh , B ∈ Br , 0 ≤ r ≤ h + 1, we have
∑
ϕi (N , v, B) = ϕB (B , v , B ). r
r
r
i∈B
3. Weighted Shapley hierarchy levels values [14] suggested for his value a two-step approach. The first step consists of using the Shapley value on the quotient game. In the second stage, the Shapley value is applied to the result of the first step again. [19] adapted the same procedure for the value ζ , an extension of a weighted Shapley value to coalition structures where the weights are given by the size of the coalitions. [8] presented an extension to level structures and replaced the two-step approach by a multi-step mechanism as suggested also by [14]. It is still not known and there is no hint in [8] that their value can be universalized easily to an extension of the weighted Shapley values in general. The only problem is to know how to deal with exogenously given weights for coalitions. The substantial idea behind our definition, similar suggested by [14] and implemented as an algorithm by [8], is as follows: In a first step, we distribute the worth of the grand coalition v (N) among the components Bh of the hth level as players, using a weighted Shapley value. In the second step, each payoff to a component Bh from the first step is divided by a weighted Shapley value among all components Bh−1 (the new players) which are subsets of the component Bh and so on for all levels. In the last step, we distribute the payoff to components B1 from the first level among the original players i ∈ N. Therefore we need a weight system for the coalitions. For notational parsimony, we give the following notation. Notation 3.1. Bk (i)
Let (N , v, B) ∈ VLN , B = Bh , i ∈ N, and T ∈
, 0 ≤ k ≤ h. We denote by Ti k := {B ∈ Bk : B ⊆ Bk+1 (i), B ̸= B (i)}∪{T } the set containing all components of the kth level which Ω
k
N
Bk (i)
and for all k, 0 ≤ k ≤ h, T ∈ Ω , Ti k the set from Notation 3.1, let wk,T ∈ W be a weight system with coalitions S as players such k,T that wS := wS , wS ∈ w , for all S ∈ Ti k , v¯ ih+1 := v , and v¯ ik given by k,T
(Ti k , v˜ ik ) for all T ∈ Ω B v¯ ik (T ) := Shw T
k (i)
(2)
where v˜ is specified recursively via k i
v˜ ik (Q) := v¯ ik+1 (
⋃
S) for all Q ⊆ Ti k .
S ∈Q
Then the weighted Shapley hierarchy levels value ShwHL is given by HL Shw (N , v, B) := v¯ i0 ({i}) for all i ∈ N . i
Remark 3.3. ShwHL coincides with Shw if B = B0 and w{i} = wi for all i ∈ N. Since Shw is efficient and additive, it is easy to see that ShwHL meets E, A, and LG. Similar to [5], who characterized the Shapley levels value by efficiency and balanced group contributions, [8] characterized their value by efficiency and balanced per capita contributions. The last axiom is a special case of the following one by limiting the weights to the size of the relevant components.
w-Weighted balanced group contributions, WBGCw . For all N (N , v, B) ∈ VLN , B = Bh , w ∈ W 2 , Bk , Bℓ ∈ Br , 0 ≤ r ≤ h, such that Bℓ ⊆ Br +1 (Bk ), we have ∑ ∑ ϕi (N , v, B) − ϕi (N \Bℓ , v, B|N \Bℓ ) i∈Bk
i∈Bk
wBk
∑ =
i∈Bℓ
ϕi (N , v, B) −
∑
ϕi (N \Bk , v, B|N \Bk )
i∈Bℓ
wBℓ
.
This axiom extends w-weighted balanced contributions [12]. In contrast to BGC, here the contributions of one component to the players of the other component and vice versa are proportional to the weights of the components. Theorem 3.4. Let w ∈ W 2 . ShwHL is the unique LS-value that satisfies E and WBGC w . N
Proof. If we use as weights the cardinality of the coalitions, it is clear that Definition 3.2 is equivalent to the algorithm given by [8, Section 3]. Thus, the rest of the proof of Theorem 3.4 is omitted because it is easy to adapt the proofs of Proposition 7 and Theorem 8 in [8], using w-weighted balanced contributions [10,12] for the weighted Shapley values in Proposition 7 and replacing the given weights (size of the components) by arbitrary weights in Theorem 8. □ Remark 3.5. If all weights are equal, we use in fact in line (2) the Shapley value and WBGC w equals BGC. So, by [5, Theorem 2], Definition 3.2 determines in this case the Shapley levels value ShL . Remark 3.6. As shown by example 1 in [8], ShwHL does not satisfy N in general.
M. Besner / Operations Research Letters 47 (2019) 122–126
4. Level-consistency [10] introduced a reduced game of a TU-game. Definition 4.1. Let (N , v ) ∈ VN , R ∈ Ω N , and φ a TU-value. The φ reduced game (R, vR ) ∈ VR is given for all S ∈ Ω R by
) ∑ ( ) ( φ φj S ∪ Rc , v vR (S) := v S ∪ Rc −
(3)
where R := N \R.
∑ ( ) φj S ∪ Rc , v . j∈S
The interpretation for this reduced game is as follows: A coalition of players Rc leaves the game. In the reduced game, each coalition S which is a subset of the coalition R of the remaining players gets the worth of coalition S ∪ Rc in the original game less the removed players’ payoff in the restricted game on S ∪ Rc . Then a TU-value is said to be consistent if each player in R gets in the reduced and in the original game the same payoff. Consistency, C [10]. For all (N , v ) ∈ VN , R ∈ Ω N , we have φ φi (N , v ) = φi (R, vR ) for all i ∈ R. It turns out that the weighted Shapley values have got an intense connection to this axiom. Theorem 4.3 ([10]). Let w ∈ W. Shw is the unique TU-value that satisfies C and WSw . If we exchange WSw by ST0 , this theorem characterizes the Shapley value [10]. By adding LG, [21] extended this axiomatization for the Shapley value to the Owen value (see proof of Theorem 3 in [21]). Similarly, [11] transferred the related axiomatization of the proportional value [6,13] to his proportional value for coalition structures. Our new reduced game for level structures extends the reduced game for TU-games by [10] too but it is no extension of the reduced game used by [21] and [11] respectively. N N Definition ⋃4.4. Let (N , v, B) ∈ VL , B = Bh , R ∈ Ω such that R = B∈Bh ,B⊆R B, and ϕ an LS-value. The reduced level game ϕ (R, vR , B|R ) ∈ VLR is given for all S ∈ Ω R by
ϕ
∑
vR (S) := v (S ∪ Rc ) −
( ) ϕB Bh |S ∪Rc , v h , Bh |S ∪Rc
(4)
B∈Bh , B⊈R
where Rc := N \R and Bh |S ∪Rc , v h , Bh |S ∪Rc is the induced hth level game restricted to S ∪ Rc . Remark 4.5. If B = B0 , Definition 4.4 coincides with Definition 4.1.
ϕ
ϕS ∩B
Inessential (last) level property, IL. For all (N , v, B) ∈ VLN , B = Bh , h ≥ 1, and Bh = Bh+1 , we have ϕ (N , v, B) = ϕ (N , v, Bh−1 ). The last three properties fit best with the weighted Shapley hierarchy levels values. Theorem 4.8. Let w ∈ W 2 . ShwHL satisfies IL, WSw , and LC. N
N
Proof. Let (N , v, B) ∈ VLN , B = Bh , and w ∈ W 2 :
• IL: If Bh = Bh+1 , we have in Notation 3.1 for all i ∈ N h and T ∈ Ω B (i) , Ti h = {T }. It follows in line (2), v¯ ih (T ) = h , T h Shw ({T }, v˜ ih ) = v (T ) for all T ∈ Ω B (i) . Thus, if h ≥ 1, there T has been no change in line (2) for all k, 0 ≤ k ≤ h − 1, if we drop the hth level and IL is shown.
• WSw : By Definition 3.2 and ⋃ line (1), the claimhis obvious. • LC: Let i ∈ N , R = B∈Bh ,B⊆R B, R ⊇ B (i). By IL, it is sufficient to show LC on level structures Bh with Bh ̸ = Bh+1 . If R = N or |N | = 1, Eq. (5) is trivially satisfied. Let now |N | ≥ 2 and R ̸ = N. By [10], Shw satisfies C for all w ∈ h W. For each T ∈ Ω B (i) we have a set Ti h from Notation 3.1 h and a related Ti |R on the restricted level structure B|R . Thus follows by Eq. (2) for k = h h, T ( h h, T ϕ ) v¯ ih (T ) = Shw (Ti h , v˜ ih ) = Shw Ti |R , (v˜R )hi T T C, Remark 4.5
) Bh |S ∪Rc , v h , Bh |S ∪Rc .
(
ϕ
⋃
N
The following axiom extends C. Level-consistency, LC. For all (N , v, B) ∈ VLN , B = Bh , R = ⋃ B, we have h B∈B ,B⊆R ϕ
ϕ
Theorem 4.9. Let w ∈ W 2 . ShwHL is the unique TU-value that satisfies IL, WSw , and LC.
B∈Bh , B⊆R
ϕi (N , v, B) = ϕi (R, vR , B|R ) for all i ∈ R.
ϕ
where (v˜R )hi is given by (v˜R )hi (Q) := vR ( S ∈Q S) for all Q ⊆ Ti h |R . Thus, there is no difference in Eq. (2) between both games for all k ̸ = h too and LC is shown. □ It transpires that the weighted Shapley hierarchy levels values are characterized by the following theorem.
Remark 4.6. If ϕ is efficient, Eq. (4) can be simplified to
vR (S) :=
To characterize the weighted Shapley hierarchy levels values we need an additional very weak property. It states that if the hth and the h + 1th level are equal, we can remove the hth level from the level structure without consequences.
)
(
∑
Remark 4.7. If B = B0 , LC coincides with C.
w-Weighted standardness, WSw . For all (N , v, B) ∈ VLN with N B = B0 , N = {i, j}, i ̸ = j, and w ∈ W 2 , we have [ ] w{i} ϕi (N , v, B) = v ({i}) + v ({i, j}) − v ({i}) − v ({j}) . w{i} + w{j}
Remark 4.2. If φ is efficient, Eq. (3) can be simplified to φ
worth of coalition S ∪ Rc in the original game less the payoff which the removed components obtain in the induced hth level game restricted to S ∪ Rc . Then an LS-value is level-consistent if each player in R gets in the reduced level game the same payoff as in the original game.
We adapt w-weighted standardness.
j∈Rc c
vR (S) :=
125
(5)
R is a coalition of all players of some components of the hth level. All players of the other components of the hth level leave the game. In the reduced level game each coalition S ∈ Ω R gets the
Proof. By Theorem 4.8, it is sufficient to show uniqueness. Let N (N , v, B) ∈ VLN , B = Bh , w ∈ W 2 , i ∈ N arbitrary, and ϕ an w LS-values that satisfies IL, WS , and LC. By LC, we have ϕ
ϕj (N , v, B) = ϕj (Bh (i), vBh (i) , B|Bh (i) ) for all j ∈ Bh (i) ϕ
where vBh (i) is unique by WSw , LC, Remarks 4.7, 4.5 and Theorem 4.3.
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M. Besner / Operations Research Letters 47 (2019) 122–126
By IL, we have for each restricted kth cut level structure Bk |Bk (i) , k > 0, from B on Bk (i), ϕ
ϕ
ϕ (Bk (i), vˆ Bk (i) , Bk |Bk (i) ) = ϕ (Bk (i), vˆ Bk (i) , Bk−1 |Bk (i) ) ϕ
where vˆ Bk (i) is defined recursively via ϕ
vˆ Bk (i)
⎧ ⎨v ϕ h , if k = h, B (i) := ( ϕ )ϕ ⎩ vˆ , otherwise. Bk+1 (i) Bk (i)
Hence, by induction on the levels k, by IL and LC, it follows that ϕ
ϕj (N , v, B) = ϕj (B1 (i), vˆ B1 (i) , B0 |B1 (i) ) for all j ∈ B1 (i) ϕ
where vˆ B1 (i) is unique by WSw , LC, Remarks 4.7, 4.5, and Theorem 4.3. Therefore, by WSw , LC, Remarks 4.7, 4.5, and Theorem 4.3, ϕ is uniquely defined too. □ To introduce a new axiomatization of the Shapley levels value we adapt ST0 . Standardness, ST. For all (N , v, B) ∈ VLN with B = B0 and N = {i, j}, i ̸ = j, we have
ϕi (N , v, B) = v ({i}) +
1[ 2
] v ({i, j}) − v ({i}) − v ({j}) .
Since ST is a special case of WSw , we obtain the following corollary by Theorem 4.9. Corollary 4.10. ShL is the unique TU-value that satisfies IL, ST, and LC. Remark 4.11. The axioms in Theorem 4.9 are logically independent. N
Proof. Let (N , v, B) ∈ VLN , B = Bh , and w ∈ W 2 :
• IL: The LS-value ϕ defined by ⎧ ⎨ 1 ShwHL , if h ≥ 1, ϕ := 2 ⎩ wHL Sh , otherwise, satisfies WSw and LC but not IL. • WSw : Let v ∈ VLN , v (S) > 0 for all S ∈ Ω N , and B = B1 . If we regard the proportional value for cooperative games with a coalition structure by [11] in the notation for a level structure with B = B1 , this LS-value satisfies IL and LC but not WSw . • LC: The LS-value ϕ defined by ϕ (N , v, B) := Shw (N , v ) with wi := w{i} for all i ∈ N satisfies IL and WSw but not LC. □ 5. Conclusion It would seem reasonable in our view to use weights if operating players are in some sense not symmetric, e. g., if some players have not the same political, financial or military power, contribute more to the result, and so on. Often players act not as lone fighters. They join forces and small groups form together larger groups and so on. Thus the weighted Shapley hierarchy levels values meet with the multi-stage mechanism a widespread idea how hierarchical values have to operate. That these values do not satisfy the null player property does not need to be a disadvantage. A null player receives a compen-
sation since she supports the other members of an acting coalition with her contribution to the weight of this coalition. So a null player acts only in the coalition function and not necessarily in the weight system for coalitions without effect. For further information on this area see also [7] and [8]. An emphasis of this paper is on consistency. Consistency plays an important role in many allocation concepts in cooperative game theory. The differences in the numerous varieties of the reduced games are decisive for the diversity of the different consistency characteristics. So the reduced level game is no extension of the reduced game by [21]. There the original players of the game determine the worths in the coalition function of the reduced game. In the reduced level game the acting coalitions are decisive for the worths of the coalitions. If we have a trivial level structure, the reduced level game coincides with the reduced game, suggested by [10], too.
Acknowledgments We are grateful to André Casajus, Winfried Hochstättler and an anonymous referee for their helpful comments and suggestions.
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