Fair distribution of surplus and efficient extensions of the Myerson value

Economics Letters 165 (2018) 1–5

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Fair distribution of surplus and efficient extensions of the Myerson value Xun-Feng Hu a , Deng-Feng Li b, *, Gen-Jiu Xu c a b c

School of Management, Guangzhou University, Guangzhou 510006, China School of Economics and Management, Fuzhou University, No. 2, Xueyuan Road, Daxue New District, Fuzhou District, Fuzhou 350108, China Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China

highlights • We propose two axioms based on the idea of fair distribution of surplus. • They uniquely determine an efficient extension of the Myerson value. • The efficient egalitarian Myerson value is characterized with one of our axioms and the fair distribution of surplus axiom.

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Article history: Received 7 December 2017 Received in revised form 19 January 2018 Accepted 21 January 2018 Available online 31 January 2018

a b s t r a c t In this paper, we propose two axioms based on the idea of fair distribution of surplus. We prove that these two axioms uniquely determine an efficient extension of the Myerson value. Moreover, we also show that another efficient extension of the Myerson value, the efficient egalitarian Myerson value, can be characterized with one of our axioms and the fair distribution of surplus axiom. © 2018 Elsevier B.V. All rights reserved.

JEL classification: C71 D60 MSC: 91A12 Keywords: Transferable utility cooperative game Communication graph Fair distribution of surplus Myerson value Efficient extension

1. Introduction The idea of fair distribution of surplus has a long tradition in economic situations. Recently, Aumann (2010) analyzed the proposed scheme for seven separate allocation problems discussed in the ancient Talmudic and related literature and the modern airport landing problem (Littlechild and Owen, 1973). He proposed a solution for these problems including the eight schemes as special cases. It turns out that the proposed solution is based on the idea of fair distribution of surplus. More precisely, surplus should be divided among related agents equally.

* Corresponding author.

E-mail addresses: [email protected] (X.-F. Hu), [email protected] (D.-F. Li), [email protected] (G.-J. Xu). https://doi.org/10.1016/j.econlet.2018.01.025 0165-1765/© 2018 Elsevier B.V. All rights reserved.

The underlying paper investigates the implication of this idea in the context of transferable utility cooperative games with a communication graph (henceforth graph games) (Myerson, 1977). Specifically, we define two axioms for values of graph games respectively named as fair distribution of surplus between components and fair distribution of surplus within component. These two axioms require that every pair of components and every pair of players in the same component to get the same surplus from participating in the grand coalition, respectively. It turns out that these two axioms uniquely determine an efficient extension of the Myerson value (henceforth My-value) (Myerson, 1977), i.e., an efficient value for graph games coinciding with the My-value when the underlying communication graph is connected. Our extension distributes the surplus of the My-value, i.e., the worth of the grand coalition minus the summation of all players’ My-values, among players with the two-step component-wise egalitarian surplus

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X.-F. Hu et al. / Economics Letters 165 (2018) 1–5

value1 (Béal et al., 2012). Thus, it is named as the efficient twostep egalitarian surplus Myerson value (henceforth ESMy-value). In addition to proposing the ESMy-value, we also prove that the efficient egalitarian Myerson value (henceforth EEMyvalue) (van den Brink et al., 2012) can be characterized as an efficient extension of the My-value satisfying the fair distribution of surplus axiom (van den Brink et al., 2012) and the fair distribution of surplus within component axiom. Thus, the difference between the EEMy-value and the ESMy-value is pinpointed to just one axiom. Moreover, since the EEMy-value is the unique efficient extension of the My-value satisfying fairness (see Béal et al. (2015) for details), our axiomatization of the EEMy-value opens up the possibility of proposing other efficient extensions of the My-value. The rest of the paper is organized as follows. Basic definitions and notations are given in the next section. In Section 3, we define the new axioms and the ESMy-value. Section 4 presents the new axiomatization of the EEMy-value. Finally, Section 5 is devoted to efficient extensions of the My-value for special communication graphs. 2. Definitions and notations A transferable utility cooperative game is an ordered pair (N , v ), where N is a finite player set, and v : 2N → R is a mapping with v (∅) = 0. A subset of N is called a coalition. v (S) is called the worth of coalition S. Given a coalition S ∈ 2N \{∅}, we write the restriction of (N , v ) on S by (S , v|S ), i.e., v|S (T ) = v (T ) for all T ⊆ S. A communication graph is an ordered pair (N , L), where N is a set of nodes representing players, and L ⊆ LN = {{i, j}|i, j ∈ N , i ̸ = j} is a set of edges consisting of pairs of players. When no confusion occurs, we simply write an edge {i, j} as ij. (N , L) is complete if L = LN . The subgraph over S ⊆ N is the restriction of (N , L) on S, denoted by (S , L|S ). A sequence of different players i1 , . . . , ik is a path if il il+1 ∈ L for l = 1, . . . , k − 1. A pair of players i, j ∈ N are connected if i = j or there exists a path i1 , . . . , ik with i1 = i and ik = j. A coalition S ⊆ N is connected if every pair of players belonging to S are connected in (S , L|S ). A coalition K ⊆ N is a component if K is connected, and for every i ∈ N \ K , K ∪ {i} is not connected. Denote the set of all components by C (L). A graph game is an ordered triplet (N , v, L), where (N , v ) is a TU game and (N , L) is a communication graph. Given a coalition S ∈ 2N \{∅}, we write the restriction of (N , v, L) on S by (S , v|S , L|S ). Denote the set of all graph games by GL. A value ϕ over GL is a mapping which associates to every (N , v, L) ∈ GL a vector ϕ (N , v, L) ∈ RN . For every i ∈ N, ϕi (N , v, L) represents the payoff of i according to ϕ . ϕ satisfies

• efficiency (E) if for every (N , v, L) ∈ GL, ∑ ϕi (N , v, L) = v (N); i∈N

• component efficiency (CE) if for every (N , v, L) ∈ GL and T ∈ C (L), ∑ ϕi (N , v, L) = v (T ); i∈T

• fairness (F) if for every (N , v, L) ∈ GL and ij ∈ L, ϕi (N , v, L) − ϕi (N , v, L \ ij) = ϕj (N , v, L) − ϕj (N , v, L \ ij); • component decomposability (CD) if for every (N , v, L) ∈ GL and i ∈ T ∈ C (L), ϕi (N , v, L) = ϕi (T , v|T , L|T ). 1 This value is used to a modified graph game, in which the worths of all proper subsets of the grand coalition are zero, and the worth of the grand coalition is the surplus of the My-value.

Myerson (1977) assumed that only connected players in the communication graph can cooperate with each other and create their joint worth. For every (N , v, L) ∈ GL, he defined a graph restricted game (N , v L ) by

v L (S) =

∑

v (T ) for all S ⊆ N .

T ∈C (L|S )

The My-value of (N , v, L) ∈ GL is the Shapley value (Shapley, 1953) of the graph restricted game (N , v L ). Formally, it is defined for every (N , v, L) ∈ GL and i ∈ N by Myi (N , v, L) =

∑ |S |!(|N | − |S | − 1)! ( ) v L (S ∪ {i}) − v L (S) . |N |!

S ⊆N \{i}

The My-value is the unique value over GL satisfying CE and F (Myerson, 1977). Also, it satisfies CD (Slikker and van den Nouweland, 2001). Considering the fact that the inefficient feature of the Myvalue may incur drawbacks in reality, van den Brink et al. (2012) proposed the EEMy-value as an efficient extension of the My-value. By an efficient extension, we mean a value over GL which satisfies efficiency and coincides with the My-value when the underlying communication graph is connected. Formally, the EEMy-value is defined for every (N , v, L) ∈ GL and i ∈ N by EEMyi (N , v, L) = Myi (N , v, L) +

v (N) −

∑

T ∈C (L)

|N |

v (T )

.

(1)

3. Fair distribution of surplus between components and within component The EEMy-value distributes the surplus of the My-value, i.e., the worth of the grand coalition minus the summation of all players’ My-values, among players equally. Different from this way, in this section, we propose a new efficient extension of the My-value distributing the surplus of the My-value in two steps: First among all components equally and then among all players in the same component equally. We proceed by defining two axioms for values of graph games. Axiom 3.1. Fair distribution of surplus between components (FDSC). For every (N , v, L) ∈ GL and T , T ′ ∈ C (L),

∑

ϕi (N , v, L) −

i∈T

=

∑ i∈T ′

∑

ϕi (T , v|T , L|T )

i∈T

ϕi (N , v, L) −

∑

ϕi (T ′ , v|T ′ , L|T ′ ).

i∈T ′

Axiom 3.2. Fair distribution of surplus within component (FDSI). For every (N , v, L) ∈ GL and i, j ∈ T ∈ C (L),

ϕi (N , v, L) − ϕi (T , v|T , L|T ) = ϕj (N , v, L) − ϕj (T , v|T , L|T ). FDSC and FDSI require a pair of components and a pair of players belonging to the same component to get the same surplus from participating in the grand coalition, respectively. They uniquely determine an efficient extension of the My-value. Theorem 3.1. There exists a unique efficient extension of the Myvalue satisfying FDSC and FDSI. It is the ESMy-value, defined for every (N , v, L) ∈ GL and i ∈ T ∈ C (L) by ESMyi (N , v, L) = Myi (N , v, L) +

v (N) −

∑

T ′ ∈C (L)

|C (L)| · |T |

v (T ′ )

.

(2)

X.-F. Hu et al. / Economics Letters 165 (2018) 1–5

Proof. Existence. First, since the My-value is component efficient, the ESMy-value is an efficient extension of the My-value and satisfies FDSC. And then, since the My-value satisfies CD, it is easy to see that the ESMy-value satisfies FDSI. Uniqueness. Let (N , v, L) ∈ GL and let ϕ be an efficient extension of the My-value satisfying FDSC and FDSI. First, according to FDSC, for every T , T ′ ∈ C (L),

∑

ϕi (N , v, L) −

∑

i∈T

=

ϕi (T , v|T , L|T )

i∈T

∑

ϕi (N , v, L) −

i∈T ′

∑

ϕi (T ′ , v|T ′ , L|T ′ ).

i∈T ′

ϕi (N , v, L) =

i∈T

∑

ϕi (T , v|T , L|T )

+

i∈N

ϕi (N , v, L) −

∑

T ′ ∈C (L)

∑

i∈T ′

ϕi (T ′ , v|T ′ , L|T ′ )

|C (L)|

ϕi (N , v, L) = v (T ) +

v (N) −

∑

T ′ ∈C (L)

v (T ′ )

|C (L)|

i∈T

v (T )

,

In words, QF states that the deletion of an edge in the communication graph should exert the same effect on the quasi-payoffs of the players incident to the deleted edge, where the quasipayoff is measured as the difference in the final payoff minus the component-wise egalitarian surplus value.

≡

.

∑ |S |!(|N | − |S | − 1)! ( ) v¯ L (S ∪ {i}) − v¯ L (S) , |N |!

S ⊆N \{i}

(3)

And then, let i, j ∈ T . According to FDSI,

where (v¯ L )(S) = v L (S) if S ⫋ N and v¯ L (S) = v (N) if S = N. Similarly, the ESMy-value can be defined as the Shapley value of a modified graph game. Formally, for every (N , v, L) ∈ GL and i ∈ T ∈ C (L), ESMyi (N , v, L) = Shi (T , ˜ v |T )

ϕi (N , v, L) − ϕi (T , v|T , L|T ) = ϕj (N , v, L) − ϕj (T , v|T , L|T ).

≡

Let j run over T ,

∑ |S |!(|T | − |S | − 1)! ( ) ˜ v|T (S ∪ {i}) − ˜ v|T (S) , |T |!

S ⊆T \{i}

ϕi (N , v, L)

∑ = ϕi (T , v|T , L|T ) +

T ∈C (L\ij)

|C (L \ ij)| · |Cj (L \ ij)|

where Ci (L) ∈ C (L) represents the component of (N , L) containing i.

.

Since ϕ is an efficient extension of the My-value, we have

∑

v (N) −

)

EEMyi (N , v, L) = Shi (N , v¯ L )

i∈T

∑

− ϕj (N , v, L \ ij) −

∑

Remark 3.2. van den Brink et al. (2012) also defined the EEMyvalue as the Shapley value of the efficient graph restricted game (N , v¯ L ). Formally, for every (N , v, L) ∈ GL and i ∈ N,

Fix T and let T ′ run over C (L),

∑

3

(

j∈T

ϕj (N , v, L) −

∑

j∈ T

ϕj (T , v|T , L|T )

|T |

.

Since ϕ is an efficient extension of the My-value, we get

∑ ϕi (N , v, L) = Myi (T , v|T , L|T ) +

j∈T

ϕj (N , v, L) − v (T ) |T |

.

Using Eq. (3) and the CD of the My-value, we get

ϕi (N , v, L) = Myi (N , v, L) +

v (N) −

∑

T ′ ∈C (L)

v (T ′ )

|C (L)| · |T |

= ESMyi (N , v, L), the desired expression. □ Different from the EEMy-value, which uses the equal division value to distribute the surplus of the My-value, the ESMy-value distributes the surplus with the two-step component-wise egalitarian surplus value. First, the surplus is distributed among all components equally. And then, the payoff of every component in the first step is distributed among players belonging to it equally. Remark 3.1. Although the EEMy-value is the unique efficient extension of the My-value satisfying fairness (see Béal et al. (2015) for details), one can easily verify that the ESMy-value is the unique efficient extension of the My-value satisfying the following variation of fairness.

• Quasi-fairness (QF). For every (N , v, L) ∈ GL and ij ∈ L, ∑ ( ) v (N) − T ∈C(L) v (T ) ϕi (N , v, L) − |C (L)| · |Ci (L)| ( ) ∑ v (N) − T ∈C(L\ij) v (T ) − ϕi (N , v, L \ ij) − |C (L \ ij)| · |Ci (L \ ij)| ∑ ( ) v (N) − T ∈C(L) v (T ) = ϕj (N , v, L) − |C (L)| · |Cj (L)|

where ˜ v (S) = v L (S) if S ∈ 2N \C (L) and ˜ v (S) = v (S)+ otherwise.

∑ v (N)− T ∈C(L) v (T ) |C (L)|

Remark 3.3. The axioms used in Theorem 3.1 are non-redundant. -The EEMy-value is an efficient extension of the My-value satisfying FDSI but not FDSC. -The efficient extension of the My-value ϕ 1 defined for every (N , v, L) ∈ GL and i ∈ T ∈ C (L) by

ϕi1 (N , v, L) ∑ ⎧ v (N) − T ′ ∈C(L) v (T ′ ) ⎪ ⎪ ⎪ ⎨Myi (N , v, L) + 3|C (L)| · |T ∩ N (N , v )| , = ∑ ⎪ 2(v (N) − T ′ ∈C(L) v (T ′ )) ⎪ ⎪ ⎩Myi (N , v, L) + , 3|C (L)| · |T \ N (N , v )|

if i ∈ N (N , v ), if i ̸ ∈ N (N , v ),

satisfies FDSC but not FDSI, where N (N , v ) = i ∈ N |v (S ∪ {i}) = v (S) for all S ⊆ N \ {i}

{

}

represents the null player set of (N , v ). Remark 3.4. Theorem 3.1 can also be viewed as an axiomatization of the ESMy-value in the sense that we split the statement of ‘‘efficient extension of the My-value’’ into two axioms. That is, E and the following:

• Coherence with the My-value for connected graphs (CMC). For every (N , v, L) ∈ GL, if (N , L) is connected, then ϕ (N , v, L) = My(N , v, L).

4. Fair distribution of surplus van den Brink et al. (2012) showed that the EEMy-value is an efficient extension of the My-value satisfying the following variation of FDSC.

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X.-F. Hu et al. / Economics Letters 165 (2018) 1–5

Axiom 4.1. Fair distribution of surplus (FDS). For every (N , v, L) ∈ GL and T , T ′ ∈ C (L),

ϕi (N , v, L) − i∈T ϕi (T , v|T , L|T ) |T | ∑ ∑ ′ ′ ′ ϕ (N , v, L) − ′ i i∈T i∈T ′ ϕi (T , v|T , L|T ) . = |T ′ |

∑

∑

i∈T

FDS requires every component to get the same average surplus from participating in the grand coalition. It is characteristic for the EEMy-value. Theorem 4.1. The EEMy-value is the unique efficient extension of the My-value satisfying FDS and FDSI. Proof. Existence. First, van den Brink et al. (2012) has established that the EEMy-value satisfies FDS. And then, FDSI follows naturally from Eq. (1) and the CD of the My-value. Uniqueness. Let (N , v, L) ∈ GL and let ϕ be an efficient extension of the My-value satisfying FDS and FDSI. First, according to FDS, for every T , T ′ ∈ C (L),

∑ ϕi (N , v, L) − i∈T ϕi (T , v|T , L|T ) |T | ∑ ∑ ′ ′ ′ ϕ (N , v, L) − ′ i i∈T i∈T ′ ϕi (T , v|T , L|T ) = . |T ′ | ∑

ϕi (N , v, L) =

i∈T

∑

ϕi (T , v|T , L|T )

i∈T

i∈N

T ∈C (L) i∈T

Since ϕ is an efficient extension of the My-value, we have

i∈T

ϕi2 (N , v, L) ∑ ⎧ |C (L)| · (v (N) − T ′ ∈C(L) v (T ′ )) ⎪ ⎪ ⎪Myi (N , v, L) + , ⎨ 3|N | · |T ∩ N (N , v )| = ∑ ⎪ 2|C (L)| · (v (N) − T ′ ∈C(L) v (T ′ )) ⎪ ⎪ ⎩Myi (N , v, L) + , 3|N | · |T \ N (N , v )|

if i ∈ N (N , v ), if i ̸ ∈ N (N , v ),

satisfies FDS but not FDSI. The comparison of Theorems 3.1 and 4.1 shows that the difference between the EEMy-value and the ESMY-value lies on the ‘‘surplus part’’ of components. The ESMy-value assigns to every component the same part of the surplus of the My-value, while the EEMy-value allocates the surplus among components proportional to their cardinality. Remark 4.2. Béal et al. (2015) proved that the EEMy-value is the only efficient extension of the My-value satisfying fairness. Since Theorem 4.1 does not impose fairness, it opens up the possibility of proposing new efficient extensions of the My-value.

5. Special communication graphs

⎛ ⎞ ∑ ∑ |T | ⎝∑ ϕi (N , v, L) − + ϕi (T ′ , v|T ′ , L|T ′ )⎠ . |N | ′ ′

∑

by

i∈T

Fix T and let T ′ run over C (L), we get

∑

-The value ϕ 2 defined for every (N , v, L) ∈ GL and i ∈ T ∈ C (L)

⎞ ⎛ ∑ |T | ⎝ v (T ′ )⎠ . ϕi (N , v, L) = v (T ) + v (N) − |N | ′

(4)

T ∈C (L)

And then, take i, j ∈ T . According to FDSI,

ϕi (N , v, L) − ϕi (T , v|T , L|T ) = ϕj (N , v, L) − ϕj (T , v|T , L|T ). Fix i and let j run over T , we get

ϕi (N , v, L)

∑ = ϕi (T , v|T , L|T ) +

j∈T

ϕj (N , v, L) −

∑

j∈ T

ϕj (T , v|T , L|T )

|T |

.

Since ϕ is an efficient extension of the My-value, we have

∑ ϕi (N , v, L) = Myi (T , v|T , L|T ) +

j∈T

ϕj (N , v, L) − v (T ) |T |

.

Using Eq. (4) and the CD of the My-value, we finally get

ϕi (N , v, L) = Myi (T , v|T , L|T ) + = Myi (N , v, L) +

v (N) −

v (N) −

∑

T ′ ∈C (L)

∑ |N |

T ′ ∈C (L)

v (T ′ )

v (T ′ )

|N |

= EEMyi (N , v, L), the desired expression. □ Remark 4.1. The axioms used in Theorem 4.1 are non-redundant. -The ESMy-value satisfies FDSI but not FDS.

Generally, the EEMy-value and the ESMy-value are different from each other, but for some special communication graphs, they are identical. (1) Connected communication graphs. In this case, both the EEMy-value and the ESMy-value coincide with the My-value, and FDS, FDSI, and FDSC hold vacuously. Thus, all the My-value, the EEMy-value, and the ESMy-value can be characterized with E and F (see Myerson (1977) for details). (2) Non-empty unconnected communication graphs consisting of complete subgraphs. In this case, a communication graph can be viewed as a coalition structure (Aumann and Drèze, 1974), and the My-value is equivalent to the Aumann–Drèze value (see Slikker and van den Nouweland (2001) for details). Thus, both the EEMyvalue and the ESMy-value can be viewed as efficient extensions of the Aumanni–Drèze value. Moreover, FDS and FDSC can respectively be viewed as variations of the balanced per capita contributions axiom (Gómez-Rúa and Vidal-Puga, 2011) and the coalitional balanced contributions axiom (Calvo et al., 1996), where the first (second) axiom states that the withdrawal of an element of the coalition structure from the game should exert the same effect on the average (total) payoff of the other elements of the coalition structure. Finally, FDSI can be viewed as a weakening of the population solidarity within unions axiom (Calvo and Gutiérrez, 2010), stating that the withdrawal of a player from the game should exert the same effect on the payoff of players belonging to a different element of the coalition structure. (3) Empty communication graphs. In this case, the My-value degenerates to the dictatorial value (Owen, 1978), assigning to every player his/her own worth. And both the EEMy-value and the ESMyvalue coincide with the equal surplus division value (Driessen and Funaki, 1991), assigning to every player the summation of his/her own worth and the average remaining worth of the grand coalition. On the other hand, FDS and FDSC coincide with each other, while FDSI holds vacuously. Thus, both the EEMy-value and the ESMyvalue can be characterized with E and FDS/FDSC, which also means that the equal surplus division value can be characterized with adaptations of E and FDS/FDSC to transferable utility cooperative game setting. We summarize this paper with a table of axioms which may be satisfied by the My-value, the EEMy-value, and the ESMy-value.

X.-F. Hu et al. / Economics Letters 165 (2018) 1–5 CE My EEMy ESMy

√ × ×

E

× √ √

CD

CMC

F

√

√

√

√

√

× ×

√

×

QF

× × √

FDS

FDSC

FDSI

√

√

√

√ ×

× √

√ √

As mentioned in Remark 3.4, the phrase ‘‘efficient extension of the My-value’’ can be replaced with E and CMC. Thus, according to Theorem 3.1/4.1, the ESMy-value/EEMy value can be characterized with E, CMC, FDSI, and FDSC/FDS. Since this comparable axiomatization contains more axioms than Theorems 3.1 and 4.1, it maybe more illustrative. Acknowledgments This work was supported by the Key Program of National Natural Science Foundation of China (71231003) and the National Natural Science Foundation of China (71271065, 71671053, 71671140). References Aumann, R.J., 2010. Some non-superadditive games, and their Shapley values, in the Talmud. Internat. J. Game Theory 39 (1–2), 3–10. Aumann, R.J., Drèze, J.H., 1974. Cooperative games with coalition structures. Internat. J. Game Theory 3 (4), 217–237.

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