A difference in the Shapley values between marginal and cumulative approaches on restricted domains Katsushige FUJIMOTO College of Symbiotic Systems Science, Fukushima University, 1 Kanayagwa, Fukushima, 960-1296, Japan [email protected] http://www.sss.fukushima-u.ac.jp/~ fujimoto

Abstract. The Shapley value has been proposed as an allocation rule for TU games. The Shapley value is interpreted as (1) an expected value of marginal contributions and (2) a cumulative value of egalitarian allocations of dividends. In ordinary cases (domains), these two interpretations lead to the same result (value), i.e., the Shapley value, in some restricted domains, although they lead to different results. In this study, we compare the two values which are led by these two approaches. Moreover, we suggest how and in what situations these values should be applied. Keywords: the Shapley value, the M¨ obius transform, the Harsanyi dividends, marginal contribution

1

Introduction and Preliminaries

Throughout the paper, N denotes the universal set of n elements. For convenience, we often number the elements such that N = {1, 2, . . . , n}. A real-valued function v : 2N → R on N with v(∅) = 0 is called a game. A monotone game (i.e., v(A) ≤ v(B) whenever A ⊆ B ⊆ N ) is called a capacity or a fuzzy measure. We often call the pair (N, v), rather than v, a game or a capacity. The set of all games on N is denoted by G N . A real vector-valued function φ : G N → R|N | is called a value. In cooperative game theory, N is considered to be the set of all players. For every subset S of N , often called a coalition, v(S) represents the (transferable) utility/profits that players in S can obtain if they decide to cooperate. For every game (N, v), the value φ(N, v) represents an allocation rule, which provides an assessment of the benefits for each player from participating in a game v. In multicriteria decision making, N is considered to be the set of all criteria describing the preferences of a decision maker. For every subset S of N (i.e., group/combination of criteria), v(S) represents the importance (w.r.t. decision making) of S. An i-th component φi (N, v) of the value φ(N, v) (i.e., the 

This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology-Japan, Grant-in-Aid for Scientific Research (C), 22510134, 2010.

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Katsushige FUJIMOTO

value of a criterion) expresses the overall importance of the criterion in making a decision. For the sake of simplicity, we mainly discuss games as a representative of various set functions (e.g., games, capacities, fuzzy measures, and so forth) on N . To avoid cumbersome notations, we often omit braces for singletons, e.g., by writing v(i), U \i instead of v({i}), U \{i}. Similarly, for pairs, we write ij instead of {i, j}. Furthermore, cardinalities of subsets S, T, . . . , are often denoted by the corresponding lower case letters s, t, . . . , otherwise by the standard notation |S|, |T |,... Definition 11 (the M¨ obius transform [17]) The M¨obius transform of v is a function Δv : 2N → R induced by the following recursive procedures: 1)

Δv (∅) := 0,

2)

Δv (S) := v(S) −



Δv (T )

∀S ∈ 2N \ ∅.

T S

That is, the M¨obius transform is vanishing at the empty set, its worth v(i) for every singleton i ∈ N , while recursively, the M¨ obius transform of every coalition of at least two players is equal to its worth minus the sum of the M¨obius transforms of all its proper subcoalitions. In this sense, the M¨obius transform of a coalition S can be interpreted as an extra contribution of cooperation/synergy among the players in S that they did not already achieve by smaller coalitions. In fact, in the context of interaction indices (e.g.,[5, 7]), the M¨obius transform Δv (S) is called the internal interaction index of S, which represents the magnitude of a type of interaction among the elements in S. The M¨obius transform is also occasionally called the Harsanyi dividends [12]. The following relationship between a game and its M¨ obius transform holds:  (−1)s−t v(T ) for each S ∈ 2N . Δv (S) := T ⊆S

Inversely,

v(S) =



Δv (T ) ∀S ∈ 2N .

T ⊆S

Then, v is called the zeta transform of Δv . Definition 12 (the Shapley value [18]) The Shapley value for a game (N, v) is denoted by φ(N, v) and defined by φi (N, μ) :=

 (t − 1)!(n − t)! [μ(T ) − μ(T \ i)] n!

(1)

T ⊆N T i

and also represented by, via the M¨ obius transform (Harsanyi dividends),  1 Δv (T ) for each i ∈ N. φi (N, v) := t T ⊆N T i

(2)

A difference in the Shapley values on combinatorial structures...

3

The Shapley value in form (1) can be interpreted as an expected value of the marginal contribution v(T ) − v(T \ i) of each player i ∈ N averaged over all coalitions T containing the player i. While, in form (2) it can be interpreted as a cumulative value of egalitarian allocations of dividends (the M¨ obius transforms) Δv (T ) for each i ∈ N over all coalitions T containing i.

2

Games as Functions on Posets

The set 2N can be regarded as the lattice (2N , ⊆), i.e., poset, equipped with the order ⊆ induced by set-inclusion. In this section, we provide some interpretations of the Shapley value of v as a functions on a poset. In order to simplify our discussion, let us consider a special case N := {1, 2, 3}. For example, when N = {1, 2, 3}, 2N can be illustrated as Fig. 1.

Fig. 1. Poset (2{1,2,3} , ⊆)

2.1

An interpretation of the Shapley value represented by Eq. (1)

Let i = 1. Then, φ1 (N, v) is obtained as an expected value of marginal contributions v(T ) − v(T \ 1) of the player 1 averaged over T = {1}, {1, 2}, {1, 3}, and {1, 2, 3}. For example, for T = {1, 2}, we provide an interpretation of the term (|{1, 2}| − 1)! · (|{1, 2, 3}| − |{1, 2}|)! · [μ({1, 2}) − μ({2})] |{1, 2, 3}|!

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in equation (1) through Fig. 1. First, the factor [μ({1, 2}) − μ({2})] corresponds to the path from {2} to {1, 2} in Fig. 1. The denominator |{1, 2, 3}|! coincides with the number of all shortest paths from the empty set ∅ to the grand coalition {1, 2, 3}. The numerators (|{1, 2}|−1)! and (|{1, 2, 3}|−|{1, 2}|)! are the numbers of all shortest paths from ∅ to {2} and from {1, 2} to {1, 2, 3}, respectively. That represents a probability of the case where a path is, (|{1,2}|−1)!·(|{1,2,3}|−|{1,2}|)! |{1,2,3}|! from the empty set ∅ to the grand coalition {1, 2, 3} passes through the path {2} to {1, 2}. Therefore, equation (1) represents an expected value of marginal contributions of each player. 2.2

An interpretation of the Shapley value represented by Eq. (2)

Let i = 1. Then, φ1 (N, v) is obtained as a cumulative value of egalitarian allocations of dividends (the M¨ obius transforms) 1 v Δ (T ) t over all coalitions T containing player 1, i.e., T = {1}, {1, 2}, {1, 3}, and {1, 2, 3}. For example, for T = {1, 2} (resp., T = {1, 2, 3}), we provide an interpretation of the term 1 1 Δv ({1, 2}) (resp., Δv ({1, 2, 3})) |{1, 2}| |{1, 2, 3}| in equation (2) through Fig. 1. First, the factor Δv ({1, 2}) (resp., Δv ({1, 2, 3})) corresponds to the vertex {1, 2} (resp., {1, 2, 3}) in Fig. 1. There are |{1, 2}|! (resp., |{1, 2, 3}|!) tributaries flowing from {1, 2} (resp., {1, 2, 3}) into the empty set ∅ (shortest paths from {1, 2} (resp., {1, 2, 3}) into the empty set ∅). Among them, |{1, 2} \ {1}|! (resp., |{1, 2, 3} \ {1}|!) tributaries pass through the vertex {1}. Then, 1 |{1, 2} \ {1}|! = |{1, 2}|! |{1, 2}|

(resp.,

|{1, 2, 3} \ {1}|! 1 = ) |{1, 2, 3}|! |{1, 2, 3}|

of dividend Δv ({1, 2}) (resp., Δv ({1, 2, 3})) pass through {1}. Therefore, equation (2) represents a cumulative value of egalitarian allocations of dividends for each player. 2.3

Restriction of domains/coalitions

In ordinary cooperative game theory and decision problems described through the use of capacities and/or fuzzy measures, it is implicitly assumed that all subsets S of N can be formed; however, this is generally not the case. Let us elaborate on this, and distinguish several cases [8].

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– Some subsets of N may be not meaningful. When N is the set of political parties, it means that some coalitions of parties are unlikely to occur, or even impossible (a coalition mixing left and right parties). When N is the set of players, in order for players to coordinate their actions, they must be able to communicate [19]. – Subsets of N may be not “black and white”, which means that the membership of an element to N may be not simply resume to a matter of member or nonmember. This is the case with multicriteria decision making when underlying scales are bipolar, which is a demarcation between values considered as “good” or “bad,” the central value being neutral [9]. In a voting situation, it is convenient to consider that players may also abstain, hence each voter has three possibilities [4]. When N is the set of players, one may consider that each player can play at a different level of participation [15]. Therefore, in practice, we should consider and define games/capacities on some restricted domains N ⊆ 2N in the former case. We call a function v : N → R a game/capacity on N when v is a restriction of some game/capacity on 2N to N. In this section, we consider restricted domains on which the Shapley value can be defined on the basis of the above discussions/interpretations. Restriction based on the marginal contribution approach In order to define the Shapley value in the sense of the interpretation of the equation (1) as an expected marginal contribution on restricted domains N ⊆ 2N , following conditions for N are required: 1. Shortest paths from ∅ to N can be defined. 2. Marginal contribution of every player can be defined in any shortest paths from ∅ to N . As a domain satisfying the above conditions, Honda and Grabish [13] have defined regular set systems as follows: Definition 21 (regular set system) A regular set system is a family N ⊆ 2N satisfying the following conditions: 1. ∅ ∈ N, N ∈ N. 2. ∀S(= ∅) ∈ N, ∃σ ∈ Π(N ) s.t. k m   σ(i) ∈ N ∀k ∈ N and S = σ(i) for some m ∈ N , i=1

i=1

where Π(N ) is the set of all permutations on N . That is, a regular set system N ⊆ 2N can be identified with a set of permutations on N . Then, we denote the set of permutations corresponding to N as π(N). For example, π(2N ) = Π(N ).

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Example 21 (examples of regular set systems) We show some examples of regular set systems in Fig. 2. Then, poset (a) illustrates 2{1,2,3} and corresponds to Π({1, 2, 3}). Posets (b), (c), and (d) correspond to sets of permutations {132, 231, 312, 321}, {312, 321}, and {123, 213}, respectively.

{2,3}

{1,2,3} {1,3}

{3}

{1,2,3}

{2,3}

{3}

{2,3} {1,3}

{1,2,3}

{1,2,3} {3}

{1,3}

{1,2}

{1,2} {2}

{2}

{2}

{1} (a)

{1} (b)

{1} (c)

(d)

Fig. 2. Examples of regular set systems

It is easy to see that all these regular set systems (restricted domains) satisfy the requirements to define the Shapley value in the sense of the interpretation of equation (1). Indeed, Fig. 2 (d) expresses N = {∅, {1}, {2}, {1, 2}, {1, 2, 3}}. Then, two shortest paths from the empty set ∅ to the grand coalition {1, 2, 3}: (P1) {∅, {1}, {1, 2}, {1, 2, 3}} and (P2) {∅, {2}, {1, 2}, {1, 2, 3}} are defined in N. In (P1), marginal contributions v({1}) − v(∅) of 1, v({1, 2}) − v({1}) of 2, and v({1, 2, 3}) − v({1, 2}) of 3 are defined, while, in (P2), v({2}) − v(∅) of 2, v({1, 2}) − v({2}) of 1, and v({1, 2, 3}) − v({1, 2}) of 3 are defined. Honda and Fujimoto[14] have axiomatically proposed a type of the Shapley value on regular set systems as follows: Definition 22 (the Shapley value on regular set systems) Let N be a regular set system on N . Then, the Shapley value Φ(N, v) ∈ Rn of a game v on N is defined as follows: ⎛ −1 ⎞ ⎛ −1 ⎞ σ (i) σ (i)−1   v⎝ σ(k)⎠ − v ⎝ σ(k)⎠  k=1 k=1 , (3) Φi (N, v) = |π(N)| σ∈π(N)

where Φi (N, v) represents i-th component of Φ(N, v). Note that φ(N, v) = Φ(N, v) if N = 2N . Restriction based on the cumulative allocations approach In order to define the Shapley value in the sense of the interpretation of the equation (2) as a cumulative value of egalitarian allocations of dividends on restricted domains N ⊆ 2N , following conditions for N are required: 1. Shortest paths from the empty set ∅ to any S ∈ N can be defined.

A difference in the Shapley values on combinatorial structures...

7

2. For any S ∈ N and any i ∈ S, there exists at least one shortest path from ∅ to S passing through {i}. As a domain satisfying the above conditions, we adopt a family induced by an undirected graph on N . Definition 23 (undirected graph on N ) An undirected graph G = (N, L) consists of two sets: a non empty finite set N of elements called nodes and L ⊆ L(N ) := {ij | i ∈ N, j ∈ N, i = j} whose element l ∈ L is called a link. A set of nodes S ⊆ N is called connected in (N, L) whenever any j, k ∈ S satisfies either of the following two conditions: (a) j = k, (b) ∃{i1 , · · · , im } ⊆ S {1, · · · , m − 1}

s.t. j = i1 , k = im , {it , it+1 } ∈ L ∀t ∈

Definition 24 (family induced by (N, L)) Let (N, L) be an undirected graph on N . Then, N(L) := {S ∈ 2N | S : connected in (N, L)} ∪ ∅ is called the family induced by (N, L). That is, for example, N(L(N )) = 2N . Note that N(L) is regarded as a poset equipped with the order ⊆. Example 22 (examples of families induced by undirected graphs) We show some examples of undirected graphs on N := {1, 2, 3} and families induced by them in Fig. 3. The poset as shown in (a) is 2N and is induced by the complete graph ({1, 2, 3}, L({1, 2, 3})). Posets (b), (c), and (d) are families induced by Lb := {13, 23}，Lc := {12}，and Ld := ∅, respectively.

{2,3}

{1,2,3}

{2,3} {1,3}

{3}

{1,2,3} {1,3}

{3}

{3}

{1,2} {2}

{2}

{2}

{1}

1

{3} {1,2} {2}

{1}

{1}

{1}

(a)

(b)

(c)

(d)

3

3

3

3

2

1

2

1

2

1

2

Fig. 3. Examples of families induced by undirected graphs

It is easy to see that all these posets (restricted domains) satisfy the requirements to define the Shapley value in the sense of the interpretation of equation (2).

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Katsushige FUJIMOTO

Definition 25 (the M¨ obius transform on N(L)) Let (N, L) be an undirected graph on N . Then, the M¨obius transform of a game v : N(L) → R is a function Δv on N(L) induced by the following recursive procedures: 1)

Δv (∅) := 0,

2)

Δv (S) := v(S) −



Δv (T )

∀S ∈ N \ ∅.

T S T ∈N

Definition 26 (saturated chain on N) A chain (or a totally ordered set or linear ordered set) is a poset in which any two elements are comparative. That is, a subset C of N(L) is called a chain if S ⊆ T or T ⊆ S for any S, T ∈ C. The chain C of N(L) is saturated (or maximal) if there does not exist W ∈ N(L) \ C such that S  W  T for any S, T ∈ C and that C ∪ W is a chain. So, a saturated chain {C1 , . . . , Cm } ⊆ N(L) is called saturated chain from S ∈ N(L) to T ∈ N(L) and denoted by {S ↑ T } if the following conditions hold: a) b)

C1 = S and Cm = T , Ci  Ci+1 ∀i ∈ {1, . . . , m − 1}.

Fujimoto and Honda [6] have proposed a type of the Shapley value on N(L) as follows: Definition 27 (The Shapley value on N(L)) Let (N, L) be an undirected graph on N . Then, the Shapley value Ψ (N(L), v) ∈ Rn based on cumulative egalitarian allocations of dividends (the Shapley value on N(L), for short in the paper) of a game v on N(L) is defined as follows: Ψi (N(L), v) :=

 Si S∈N(L)

|{i ↑ S}| v Δ (S). |{∅ ↑ S}|

(4)

Note that φ(N, v) = Ψ (N, v) if N(L) = 2N , i.e., L = L(N ).

3

Comparison of the Two Approaches

In this section, we compare the Shapley value Φ based on the marginal contribution approach (for short, marginal approach), defined by equation (3), with Ψ based on the cumulative egalitarian allocations approach (for short, cumulative approach), defined by equation (4). The regular set system in Fig. 2 (a), we denote it by Na , and the family induced by the graph in Fig. 3 (a), which we denote by N(La ), are the same posets 2N , (i.e., the ordinary domain). Therefore, the same games can be defined on 2N , Na , and N(La ), then we have φ(N, v) = Φ(Na , v) = Ψ (N(La ), v).

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9

Similarly, the regular set system Nb in Fig. 2 (b) and the family N(Lb ) induced by the graph Lb in Fig. 3 (b) are the same posets. In general, when (N, L) is a connected undirected graph, N(L) becomes a regular set system. Indeed, (N, La ) and (N, Lb ) are connected, therefore, N(La ) and N(Lb ) are regular set systems; while (N, Lc ) and (N, Ld ) are not connected, therefore, N(Lc ) and N(Ld ) are not regular set systems. Therefore, if a domain N over N is represented as a family induced by some connected graph on N , then the domain N is both a regular set system and a family induced by a graph. On such a domain N, we can compare Φ(N, v) with Ψ (N, v) of a game v on N. In the following examples, we compare Φ(N, v) with Ψ (N, v) of some games on the domain N := {∅, {1}, {2}, {3}, {1, 3},{2, 3},{1, 2, 3}} . The domain is a regular set system induced by the undirected graph L = {13, 23} (Fig. 3 (b)), which represents a situation where player 3 acts as a mediator between players 1 and 2 to form the coalition {1, 2, 3}. Example 31

⎧ ⎪ ⎨0 v1 (S) = 8 ⎪ ⎩ 8

if |S| ≤ 1 if |S| = 2 , if S = N

i.e.,

⎧ ⎪ ⎨ 0 if |S| ≤ 1 v1 Δ (S) = 8 if |S| = 2 ⎪ ⎩ −8 if S = N

Then, we have that Φ(N, v1 ) = (2, 2, 4) and Ψ (N, v1 ) = (2, 2, 4). Example 32

⎧ ⎪ ⎨0 if |S| ≤ 1 v2 (S) = 8 if |S| = 2 , ⎪ ⎩ 16 if S = N

i.e.,

⎧ ⎪ ⎨0 if |S| ≤ 1 v2 Δ (S) = 8 if |S| = 2 ⎪ ⎩ 0 if S = N

Then, we have that Φ(N, v2 ) = (6, 6, 4) and Ψ (N, v2 ) = (4, 4, 8). Example 33

⎧ ⎪ ⎨0 if |S| ≤ 1 v3 (S) = 8 if |S| = 2 , ⎪ ⎩ 24 if S = N

i.e.,

⎧ ⎪ ⎨0 if |S| ≤ 1 v3 Δ (S) = 8 if |S| = 2 ⎪ ⎩ 8 if S = N

Then, we have that Φ(N, v3 ) = (10, 10, 4) and Ψ (N, v3 ) = (6, 6, 12). From the above examples, we can see a large difference in allocation for the mediator 3 between the marginal approach and the cumulative approach. Proposition 31 Let N be a family induced by a connected undirected graph, i.e., a regular set system. Then, for any game v : N → R and any real number α ∈ R, it holds that Φ(N, α · v) = α · Φ(N, v), Ψ (N, α · v) = α · Ψ (N, v),

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Katsushige FUJIMOTO

and, for any two games v1 and v2 , it holds that Φ(N, v1 + v2 ) = Φ(N, v1 ) + Φ(N, v2 ), Ψ (N, v1 + v2 ) = Ψ (N, v1 ) + Φ(N, v2 ). Here, we define a unanimity game uT : N → R as follows: 1 if S ⊇ T uT (S) := 0 if otherwise. Then, it is well known that any game v : N → R can be expressed by  v(S) = Δv (T ) · uT (S) ∀S ∈ N.

(5)

T ∈N

Therefore, it follows from Proposition 31 that  Δv (T ) · Φ(N, uT ), Φ(N, v) = T ∈N

Ψ (N, v) =



Δv (T ) · Ψ (N, uT ).

T ∈N

Moreover, when N is a family induced by some connected undirected graph, so is N(S) := {T ∈ N | T ⊆ S} whenever S ∈ N. Therefore, it suffices to compare Φ(N, uN ) with Ψ (N, uN ) of the unanimity game uN on regular set systems N, induced by various connected undirected graphs on N , to clarify the difference between the marginal and the cumulative approaches. Example 34 Fig.4 shows all connected undirected graphs (up to isomorphism) with 2 ≤ n ≤ 4 nodes. Then, Φ(N(L∗ ), uN ) and Ψ (N(L∗ ), uN ) of the unanimity game uN on domains N(L∗ ) induced by each graphs L∗ on N are listed in Table 1. We can see marked differences in the allocations for central players (e.g., players 2 in Lb , players 2 and 3 in Lc , and player 4 in both Le and Lf ) and terminal players, i.e., marginal players, (e.g., players 1 and 3 in Lb , players 1 and 4 in Lc , and players 1, 2 and 3 in Le ) between the two approaches. In the cumulative approach, the central players are more important than the terminal players. Indeed, especially in star-like graphs (e.g., Lb and Le ), the central player obtains a full half of the total amount of uN (N ) = 1 and the rest of the amount is shared equally among other players. While, in the marginal approach, only terminal players are important, if they exist. Indeed, especially in Lb , Lc , and Le , the entire amount of uN (N ) = 1 is shared only among terminal players, and central players receive none at all.

A difference in the Shapley values on combinatorial structures...

La

Lb

Lc

Ld

Le

Lf

Lg

Lh

Li

11

Fig. 4. Graphs with at most four nodes. Table 1. Comparison of Φ and Ψ .

N(La ) N(Lb ) N(Lc ) N(Ld ) N(Le ) N(Lf ) N(Lg ) N(Lh ) N(Li )

4

Φ

Ψ

( 12 , 12 )

( 12 , 12 ) 1 1 1 (4, 2, 4) ( 18 , 38 , 38 , 18 ) ( 31 , 13 , 13 ) 1 1 1 3 (6, 6, 6, 6) 2 3 3 6 ( 14 , 14 , 14 , 14 ) 1 1 1 1 (4, 4, 4, 4) ( 14 , 14 , 14 , 14 ) 2 3 3 2 ( 10 , 10 , 10 , 10 )

( 24 , 0, 24 ) ( 48 , 0, 0, 48 ) ( 13 , 13 , 13 ) ( 26 , 26 , 26 , 0) 6 4 4 ( 14 , 14 , 14 , 0) 1 1 1 1 (4, 4, 4, 4) ( 14 , 14 , 14 , 14 ) 3 2 2 3 ( 10 , 10 , 10 , 10 )

Concluding Remarks

Let us consider networks like roads, railways, airways, the Internet, others that use nodes or terminals such as airports and railway stations. To maintain such networks, the central hub is most important. The benefits that terminal cities/points receive from direct access to big cities (important nodes or central hubs) are much higher than benefits that big cities receive by connecting to terminal cities/points. Therefore, the Shapley value from the marginal approach can be applied to measure benefits from such networks and that from the cumulative approach to measure the importance for maintaining such networks.

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Katsushige FUJIMOTO

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