Potential functions for finding stable coalition structures

Operations Research Letters 47 (2019) 478–482

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Potential functions for finding stable coalition structures ∗

Vasily V. Gusev a,b , , Vladimir V. Mazalov a,c a b c

Institute of Applied Mathematical Research, Karelian Research Center, Russian Academy of Sciences, Petrozavodsk, Russia Department of Mathematics, Petrozavodsk State University, Petrozavodsk, Russia Institute of Applied Mathematics, School of Mathematics and Statistics, Qingdao University, Qingdao, People’s Republic of China

article

info

Article history: Received 30 May 2019 Received in revised form 1 August 2019 Accepted 27 August 2019 Available online 2 September 2019 Keywords: Cooperative game with coalition structure Nash stability Aumann–Dreze value Potential functions Multicriteria games c-stability

a b s t r a c t The existence of a Nash-stable coalition structure in cooperative games with the Aumann–Dreze value is investigated. Using the framework of potential functions, it is proved that such a coalition structure exists in any cooperative game. In addition, a similar result is established for some linear values of the game, in particular, the Banzhaf value. For a cooperative game with vector payments, a type of stability based on maximizing the guaranteed payoffs of all players is proposed. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In many contexts, economic agents need to form coalitions to achieve their goals. Each agent seeks to find a group of partners for jointly obtaining a desired result. For example, retailers form alliances to maximize profits. In this case, it is necessary to distribute the joint income between all members of a trading group. Such payoff distribution problems are considered by the theory of cooperative games with coalition structure [3]. The emergence of coalition structures in cooperative games leads to a natural question as follows: what is meant by a stable coalition partition? As a matter of fact, there are many types of stability such as internal stability, external stability, Nash stability, core stability [1,19], etc. Other types of stability can be found in [14]. The classical type of stability is Nash stability. This type of stability arises in economic, social [4,13] and hedonic [5] games. Some results on the stability of coalition structures in graphrestricted games, bank cooperation games and the game with a strong player can be found in [20,21]. In the cited papers, the Nash-stable coalition structures were found in explicit form. In this paper, the existence of a Nash-stable coalition structure in any cooperative game with Aumann–Dreze value will be established by introducing the concept of a potential function for a cooperative game. The definition of a potential function ∗ Correspondence to: IAMR KarRC RAS 11, Pushkinskaya str. Petrozavodsk Karelia 185910, Russia. E-mail addresses: [email protected] (V.V. Gusev), [email protected] (V.V. Mazalov). https://doi.org/10.1016/j.orl.2019.08.006 0167-6377/© 2019 Elsevier B.V. All rights reserved.

in noncooperative game theory was given in [18,24]. It is wellknown that, if there exists a potential function in a strategic game, then this game has a pure strategy Nash equilibrium. In the papers [7,16], the systems of potential equations for finite games were solved. In [8], the relationship between the Boolean and potential games was shown. Potential functions are also used for solving dynamic and differential games [10,11]. The application of potential functions in cooperative game theory can be found in [6]. The goal of this paper is to use the theory of potential functions for cooperative games and also to answer the following question: does a Nash-stable coalition structure exist in any cooperative game with the Aumann–Dreze value? The design procedures of stable coalition structures are also important in multicriteria problems if the members of a coalition obtain vector payoffs in the course of cooperation. For instance, they may be interested in separate incomes and expenses. Another example is project management: while implementing a project, one should consider not only the resulting income but also the period of this project, the quality of work performed, etc. The study of cooperative games with vector payments was started in [25], where the concept of Pareto equilibrium was introduced, and then continued in [22,23]. For each of the criteria, a characteristic function can be constructed, and a Nash-stable coalition structure will correspond to each characteristic function. However, such coalition structures may differ. In this case, players will obtain different payoffs. In what follows, an approach to implement a coalition structure with the maximum guaranteed payoffs of the players will be proposed.

V.V. Gusev and V.V. Mazalov / Operations Research Letters 47 (2019) 478–482

479

2. Nash-stable coalition structure

3. Potential function for Aumann–Dreze value

Consider a cooperative game with coalition structure ⟨N , v, π ⟩, where N denotes the set of players, |N | = n, v : 2N → R, v (∅) = 0 is a characteristic function of this game, and π gives a⋃coalition l partition of the set N, i.e., π = {B1 , B2 , . . . , Bl }, where j=1 Bj = N and ∀i, j ∈ {1, 2, . . . , l}, i ̸ = j : Bi ∩ Bj = ∅. Denote by B(i) the coalition containing player i in the coalition partition π , i.e., i ∈ B(i) ∈ π . Let N be fixed. Denote by G(N) the space of characteristic functions with the set of players N. Also denote by Π (N) the set of all coalition partitions of the set N. Let x : G(N) × Π (N) → ℜn be the value of the cooperative game with coalition structure. In the cooperative game with coalition structure, with each player i associate a value xi (N , v, π ), which will be called the payoff or value of player i. A value x(N , v, π ) = (x1 (N , v, π ), . . . , xn (N , v, π )) will be called a payoff distribution or imputation if it satisfies (1) ∀i ∈ N ∑ : xi (N , v, π ) ≥ v ({i}) (individual rationality) and (2) ∀B ∈ π : i∈B xi (N , v, π ) = v (B) (efficiency).

Consider a cooperative game ⟨N , v, π ⟩. Let the payoff be distributed using the Aumann–Dreze value, which is given by

Definition 1. A value x(N , v, π ) will be said to satisfy component independence if xi (N , v, π ) = xi (B(i), v, π ) and xi (N , v, π ) does not depend on v (K ), K ⊆ N \ B(i), ∀i ∈ N. If x(N , v, π ) satisfies component independence, then the payoff of player i, i ∈ N, is calculated inside the coalition B(i). For example, the Aumann–Dreze value satisfies component independence while the Owen value, the Casajus value and the two-stage Shapley value [15] do not. The next definitions and notations will be needed for the stability analysis of coalition structures. Let B1 , B2 ∈ π , where B1 ̸ = B2 , and denote π−B1 = π \ {B1 } and π−B1 ∪B2 = π \ {B1 , B2 }. Fix a coalition partition π and introduce the following notations: Di (π ), i ∈ N, as the set consisting of the coalition structures, Di (π ) = {π } ∪ {{B(i) \ {i}, A ∪ {i}, π−B(i)∪A }|A = ∅ ∨ A ∈ π−B(i) }. Note that Di (π ) is the set of the coalition structures obtained from π if player i moves to another coalition or forms a coalition himself. For example, let N = {1, 2, 3, 4}, π = {{1, 2}, {3}, {4}}. {

Then D1 (π ) =

{{1, 2}, {3}, {4}}, {{1}, {2}, {3}, {4}}, {{2}, {1, 3}, } {4}}, {{2}, {3}, {1, 4}} . ⋃ Also define D(π ) = i∈N Di .

Definition 2. A coalition structure π is said to be Nash-stable if ∀i ∈ N : xi (N , v, π ) ≥ xi (N , v, ρ ), ∀ρ ∈ Di (π ) for a given value x(N , v, π ) [5,9,12]. In a Nash-stable partition, none of the players wants to leave his coalition. Here is a weaker concept of stability. Definition 3. A coalition structure π is said to be individually stable if there do not exist a player i ∈ N and a coalition structure ρ ∈ Di (π ) such that xi (N , v, ρ ) > xi (N , v, π ) and xk (N , v, ρ ) ≥ xk (N , v, π ), ∀k ∈ B(i) ∈ ρ, k ̸ = i for a given value x(N , v, π ) [5]. In an individually stable coalition structure π , for any player i, i ∈ N, with an incentive to join a coalition B ∈ π there exists at least one player k ∈ B for whom it will be non-beneficial. Note that Nash stability implies individual stability. Definition 4. A function P(N , v, π ) will be called a potential function in a cooperative game with coalition structure ⟨N , v, π ⟩ if ∀i ∈ N , ∀π ∈ Π (N) : xi (N , v, π ) − xi (N , v, ρ ) = P(N , v, π ) − P(N , v, ρ ), ∀ρ ∈ Di (π ) for a given value x(N , v, π ).

φi (N , v, π ) =

(|B(i)| − |K |)! · (|K | − 1)!

∑

|B(i)|! ) · v (K ) − v (K \ {i}) , ∀i ∈ N . K ⊆B(i),i∈K

(

Note that φi (N , v, π ) = φi (B(i), v, π ). Let vS (K ) be an elementary characteristic function, where S is a fixed and non-empty subset of the set N. Using the Möbius transformation, write the characteristic function as

∑

v (K ) =

λS (v ) · vS (K ) =

S ⊆N

λS (v ) =

∑

λS (v ), where

S ⊆K

∑

(−1)|S |−|R| v (R).

R⊆S

Theorem 1. The following statements are true. (1) For the Aumann–Dreze value, the potential function in a cooperative game with coalition structure has the form P(N , v, π ) =

∑ ∑ λ K (v ) |K |

B∈π K ̸=∅ K ⊆B

.

(2) For any cooperative game ⟨N , v, π ⟩, there exists a Nash-stable coalition structure for the Aumann–Dreze value. Proof. (1) Expand the characteristic function v (K ), K ⊆ N, with respect to the basis. Then value can be ∑the Aumann–Dreze λK (v ) calculated as φi (N , v, π ) = K ⊆B(i),i∈K |K | . Therefore, ∀π ∈ Π (N) :

∑

φi (N , v, π ) − φi (N , v, ρ ) =

K ⊆B(i),i∈K

λK (v ) − |K |

∑ K ⊆A∪{i},i∈K

λ K (v ) , |K |

where B(i) ∈ π, A ∪ {i} ∈ ρ and ρ ∈ Di (π ). Now, show that this expression gives the difference of potentials P(N , v, π ) − P(N , v, ρ ). The following chain of equalities is true: P(N , v, π ) − P(N , v, ρ ) =

∑ ∑ λK (v ) ∑ ∑ λK (v ) − |K | |K | K ̸ =∅ K ̸ =∅ B∈π

B∈ρ

K ⊆B

K ⊆B

⎛ ⎜ ∑ λ K (v ) ∑ λ K (v ) + + =⎝ |K | |K | K ̸ =∅ K ̸ =∅ K ⊆A

K ⊆B(i)

∑ B∈π−B(i)∪A

⎞ ∑ λ K (v ) ⎟ ⎠ |K | K ̸ =∅ K ⊆B

⎛ ∑ λ K (v ) ⎜ ∑ λ K (v ) −⎝ + + |K | |K | K ̸ =∅ K ̸ =∅ K ⊆A∪{i}

K ⊆B(i)\{i}

⎛

⎞

∑ λK (v ) ⎟ ⎜ ∑ λ K (v ) =⎝ − ⎠ |K | |K | K ̸ =∅ K ̸ =∅ K ⊆B(i)

K ⊆B(i)\{i}

⎞ ∑ ∑ λK (v ) λ K (v ) ⎟ ⎜ −⎝ − ⎠ | K| |K | K ̸ =∅ K ̸ =∅ ⎛

K ⊆A∪{i}

=

∑ K ⊆B(i),i∈K

λ K (v ) − |K |

K ⊆A

∑ K ⊆A∪{i},i∈K

= φi (N , v, π ) − φi (N , v, ρ ).

λ K (v ) |K |

∑ B∈π−B(i)∪A

⎞ ∑ λK (v ) ⎟ ⎠ |K | K ̸ =∅ K ⊆B

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V.V. Gusev and V.V. Mazalov / Operations Research Letters 47 (2019) 478–482

In final analysis, φi (N , v, π )−φi (N , v, ρ ) = P(N , v, π )−P(N , v, ρ ), ∀i ∈ N , ∀π ∈ Π (N), ∀ρ ∈ Di (π ). Thus, by Definition 4 the function P(N , v, π ) is a potential function for the imputations based on the Aumann–Dreze value. (2) Let π ∗ = argmaxπ ∈Π (N) P(N , v, π ); then ∀i ∈ N , ∀ρ ∈ Di (π ∗ ) : φi (N , v, π ∗ ) − φi (N , v, ρ ) = P(N , v, π ∗ ) − P(N , v, ρ ) ⩾ 0, meaning that π ∗ is a Nash-stable coalition partition. □ 4. Potential function in case of three players Let N = {1, 2, 3}. In this case, |Π (N)| = 5. Expanding the characteristic function v (K ), K ⊆ N, with respect to the basis yields

v (K ) = λ{1} (v ) · v{1} (K ) + λ{2} (v ) · v{2} (K ) + λ{3} (v ) · v{3} (K ) + λ{1,2} (v ) · v{1,2} (K ) + λ{1,3} (v ) · v{1,3} (K ) + λ{2,3} (v ) · v{2,3} (K ) + λ{1,2,3} (v ) · v{1,2,3} (K ),

)

(

)

P N , v, {{2}, {1, 3}} = λ{2} (v ) + λ{1} (v ) + λ{3} (v ) +

(

)

λ{2,3} (v ) 2

λ{1,3} (v ) 2 2

3

2

.

5. Stable coalition partition for identical players Now, consider a cooperative game with coalition structure ⟨N , v, π ⟩ in the case of all identical players. Then the characteristic function depends on the number of players in a given coalition. For the sake of simplicity, we denote it by v (K ) = vk , where k = |K |. Thus, a coalition partition π = {B1 , B2 , . . . , Bl } is completely defined by the number of players belonging to the coalitions Bj , j = 1, . . . , l. Denote by nj = |Bj |, j = 1, . . . , l, the sizes of all coalitions in the partition. Then n1 + · · · + nl = n, and it can be assumed that n1 ≤ n2 ≤ ... ≤ nl . As a result, the potential function takes the form nj ( ) l l ∑ ∑ λ K (v ) ∑ ∑ nj λk (v ) P(N , v, π ) = = . | K| k k K ̸ =∅ j=1

j=1 k=1

K ⊆Bj

In addition,

λK (v ) = λk (v ) =

k−1 ∑

(−1)j

j=0

( ) k j

vk−j ,

r =0

k r

vk−r

for each coalition Bj . Finally, we introduce the change of variables k − r = m and revert the order of summation to obtain nj ( ) k ∑ 1 nj ∑ k=1

k

k

=

nj ∑

(−1)k−m

(

k−m

m=1 nj

vm

∑

(−1)k−m

k=m

)

k

vm

( )( )

1 nj

k

k

m

k

=

nj ∑ vm m=1

m

.

Theorem 2. A stable coalition structure for identical players can be constructed by solving the optimization problem nj l ∑ ∑ vm j=1 m=1

m

→ max

subject to the constraint n1 + · · · + nl = n. In this case, a stable coalition partition can be constructed using a simple algorithm that includes the following steps. Start from the initial partition

For it, the potential is P(N , v, π ) = nv1 . If players 1 and 2 form a coalition, then for the partition

) P N , v, {{1, 2, 3}} = λ{1} (v ) + λ{2} (v ) + λ{3} (v ) λ{1,2} (v ) λ{1,3} (v ) λ{2,3} (v ) + + + +

k

k

( )

,

(

2

k=1

(−1)r

π = {{1}, {2}, . . . , {n}}.

) λ{1,2} (v ) P N , v, {{3}, {1, 2}} = λ{3} (v ) + λ{1} (v ) + λ{2} (v ) + ,

2

nj ( ) k−1 ∑ 1 nj ∑

,

(

λ{1,2,3} (v )

k

k

k=1

=

Thus, the stable coalition partition problem has been reduced to the maximization of a simple-form potential function.

P N , v, {{1}, {2}, {3}} = λ{1} (v ) + λ{2} (v ) + λ{3} (v ), P N , v, {{1}, {2, 3}} = λ{1} (v ) + λ{2} (v ) + λ{3} (v ) +

nj ( ) ∑ nj λ k ( v )

m=1

where λ{1} (v ) = v ({1}), λ{2} (v ) = v ({2}), λ{3} (v ) = v ({3}), λ{1,2} (v ) = v ({1, 2}) − v ({1}) − v ({2}), λ{1,3} (v ) = v ({1, 3}) − v ({1}) − v ({3}), λ{2,3} (v ) = v ({2, 3}) − v ({2}) − v ({3}), λ{1,2,3} (v ) = v ({1, 2, 3}) − v ({1, 2}) − v ({1, 3}) − v ({2, 3}) + v ({1}) + v ({2}) + v ({3}). The values of the potential function for the imputations based on the Aumann–Dreze value are as follows:

(

where k = |K |. Substituting these coefficients into the potential formula gives

π ′ = {{1, 2}, {3}, . . . , {n}} the potential will be P(N , v, π ′ ) = (n − 1)v1 + v2 /2. If v1 ≥ v2 /2, then the initial coalition partition π is stable. If v2 /2 > v1 , then for the coalition π ′ the potential will take a greater value. As a result, pairs will start forming, increasing the potential accordingly. Depending on the parity of n, the partition will be

π ′′ = {{1, 2}, {3, 4}, . . . , {n − 1, n}} or π ′′ = {{1, 2}, {3, 4}, . . . , {n}}. Now, if v2 /2 ≥ v3 /3, then for an even number n the coalition partition π ′′ will be stable. On the other hand, for an odd number n the coalition partition π ′′ will be stable in the case v1 ≥ v3 /3. Otherwise, player n will benefit from joining some pair, e.g., (n − 2, n − 1). If v3 /3 > v2 /2, then triplets of players will start forming, also increasing the potential. This process will continue until reaching a first number k such that

v1 <

v2 2

< ··· <

vk k

,

vk k

≥

vk+1 k+1

.

As a result, the coalition partition consisting of k players and a single coalition of k or less players will be formed. Assume the latter coalition will contain one player only. If v1 ≥ vk+1 /(k + 1), then this will be the stable coalition partition. Otherwise, the potential will increase if this player joins another coalition; hence, the stable coalition partition will be composed of the coalitions of k players and a single coalition of player (k + 1). Similar considerations are applicable to the case in which the smaller coalition contains two or more players.

V.V. Gusev and V.V. Mazalov / Operations Research Letters 47 (2019) 478–482

6. Potential function for linear imputations Denote by Γ (N) the space of all games defined on the set of players N. A payoff distribution satisfies linearity if ∀v, w ∈ Γ (N), ∀α, β ∈ R : xi (N , αv+βw, π ) = α xi (N , v, π )+β xi (N , w, π ), i ∈ N. The next result can be established by analogy with Theorem 1. Theorem 3. Assume a payoff distribution x(N , v, π ) satisfies component independence, linearity and x(B(i), vS , π ) = f (S), S ⊆ B(i), S ̸ = ∅, where ⟨N , vS (K )⟩ is the unanimity game. Then there exists a potential function for the chosen value of the game ⟨N , v, π ⟩ that is given by P(N , v, π ) =

∑∑

f (S)λS (v ).

B∈π S ̸=∅ S ⊆B

Moreover, a Nash-stable coalition structure also exists in this game. Consider the Banzhaf value [2,17] for a cooperative game with coalition structure. Define this value by 1

βi (N , v, π ) =

2|B(i)|−1

∑ (

) v (K ) − v (K \ {i}) .

Table 1 Values of characteristic functions. K

∅

{1}

{2}

{3}

{1, 2}

{1, 3}

{2, 3}

{1, 2, 3}

v1 (K ) v2 (K )

0 0

0 3

0 3

0 3

2 1

2 1

2 1

6 0

w(K , c)

0

3(1 − c)

3(1 − c)

3(1 − c)

1+c

1+c

1+c

6c

Table 2 Players’ payoffs in different coalition partitions.

π

1

2

3

1

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

3(1 − c)

3(1 − c)

3(1 − c)

2

{{1}, {2, 3}}

3(1 − c)

1+c 2

3

{{2}, {1, 3}}

3(1 − c)

1+c 2 1+c 2

4

{{3}, {1, 2}}

1+c 2 1+c 2

1+c 2

3(1 − c)

5

{{1, 2, 3}}

2c

2c

2c

Definition 5. A coalition structure π will be called c-stable in a multicriteria game ⟨N , v, π ⟩ v = (v1 , . . . , vm ) if it satisfies the following requirements.

K ⊆B(i),i∈K

1. The components of the vector c ∗ ∈ C are the solution to the optimization problem

Note that βi (N , v, π ) differs from the classical Banzhaf value because βi (N , v, π ) depends on the coalition structure. The Banzhaf value with coalition structure satisfies component independence and linearity. In addition, for all S ⊆ B(i), S ̸ = ∅,

The value βi (B(i), vS , π ) depends on S only. Then by Theorem 2 the potential function for the Banzhaf value in a cooperative game with coalition structure will be ∑ ∑ λ K (v ) P(N , v, π ) = . 2|K |−1 B∈π K ⊆B K ̸ =∅

Hence, a cooperative game with coalition structure has a Nashstable coalition partition for the Banzhaf value. 7. Stable coalition partition for coalition game with vector payments Now, consider a cooperative game with coalition structure

⟨N , v, π ⟩, where v = (v1 , v2 , . . . , vm ) is a vector function. Co-

operative games with vector payments were pioneered in [25] and then studied in a series of research works [22,23]. The issue of coalition stability seems to be appropriate here, as well. Note that, for each characteristic function vj , there exists a stable coalition partition πj , j = 1, . . . , m; see Theorem 1. However, these coalition structures may differ. Then the players have to choose a preferable coalition structure. This must be a collective decision based on negotiations. Denote C =

(c1 , c2 , . . . , cm )|cj ≥ 0,

⎩

m ∑ j=1

c ∗ = argmax min min φi (N , w, π ). c ∈C

⎫ ⎬

cj = 1

.

⎭

By Theorem 1 a c-stable coalition structure exists for the Aumann–Dreze value. In a c-stable coalition partition, the players agree to choose a vector c for which the least payoff of a player will be maximized. Next, for this vector c a stable coalition structure will be found. Example 1 (Three Players). Let N = {1, 2, 3}. The components of the vector characteristic function v (K ) = (v1 (K ), v2 (K )) are given in Table 1. The characteristic functions v1 (K ) and v2 (K ) are monotonically increasing and monotonically decreasing, respectively. Consider the weighted characteristic function

w(K , c) = c v1 (K ) + (1 − c)v2 (K ), c ∈ [0, 1], whose values are calculated in Table 1. In accordance with Theorem 1, the cooperative game ⟨N , w⟩ has a Nash-stable coalition structure. The components of the Aumann–Dreze value are presented in Table 2. For calculating c ∗ , draw the graphs of the players’ payoffs; see Fig. 1. The solution to the problem mini∈N minπ ∈Π (N) φi (N , w, π ) is the lower envelope curve. This curve achieves maximum at c ∗ = 75 . For c = 57 , there are two c-stable coalition structures—

{{1, 2, 3}} and {{1}, {2}, {3}}.

Example 2 (Identical Players). Assume m characteristic functions {v j (K )}, j = 1, . . . , m, are defined in the cooperative game with identical players. Due to this identity, the payoff of a coalition depends on its size |K | = k only, and hence v j (K ) = v j (k), k = 1, . . . , n. By the definition of c-stability and Theorem 2, the potential function has the form

Consider the weighted characteristic function

w(K , c) =

m ∑

i∈N π∈Π (N)

2. π is Nash stable in the game ⟨N , w (K , c ∗ ), π⟩.

⎧ ⎨ 1 , i ∈ S; βi (B(i), vS , π ) = 2|S |−1 ⎩ 0, i∈ / S.

⎧ ⎨

481

cj vj (K ), c ∈ C .

j=1

Thus, the players have to choose a preferable vector c , c ∈ C .

min k

m 1∑

k

cj v j (k).

j=1

The set of parameters c = (c1 , . . . , cm ) maximizing the potential is fixed, and then a Nash-stable coalition structure will be found.

482

V.V. Gusev and V.V. Mazalov / Operations Research Letters 47 (2019) 478–482

been adopted for the stability analysis of coalition structures. However, noncooperative game theory also includes other concepts such as the ordinal potential and the weight potential. They will be selected for stability analysis in further publications. Potential functions can be employed in cooperative games with vector payments as well. In this case, there exists a Nashstable coalition partition for each characteristic function; see Theorem 1. However, since there are several characteristic functions, these coalition partitions may differ, and the players will obtain different payments. Which coalition partition should be chosen in this circumstance? In this paper, the idea is (1) to find a linear combination of criteria that will guarantee the maximum payoff to the most affected player and then (2) to find a stable coalition partition for this weighted characteristic function. Note that other approaches to introduce the stability of a coalition partition in the multicriteria setup are also possible; e.g., the Nash bargaining solution can be used. This issue will be considered in further research.

Fig. 1. Graphs of players’ payoffs.

Acknowledgment Supported by the Shandong Province ‘‘Double-Hundred Talent Plan’’ (No. WST2017009). References Fig. 2. Players’ payoffs. Table 3 Values of characteristic functions. k

1

2

3

4

5

vk1

5

14

18

12

15

12

vk2

2

6

15

8

20

24

wk

3c + 2

20 − 5c

24 − 12c

8c + 6

3c + 15

4c + 8

6

For example, let N = {1, 2, . . . , 6} and v (K ) = (v 1 (K ), v 2 (K )). As before, suppose all players are identical. In this case, the components of the vector characteristic function can be written as v 1 (K ) = vk1 and v 2 (K ) = vk2 , where k = |K |. The values of the characteristic functions are given in Table 3. In the games v 1 and v 2 , the Nash-stable coalition structures have the form {{1, 2}, {3, 4}, {5, 6}} and {{1, 2, 3}, {4, 5, 6}}, respectively. Other stable coalition structures may exist. The values of the characteristic function w (K , c) = wk = c vk1 + (1 − c)vk2 , c ∈ [0, 1], are presented in Table 3. For player i, the component of the w (B(i)) Aumann–Dreze value is |B(i)| . The graphs of the players’ payoffs in different coalition structures can be seen in Fig. 2. For example, if |B(i)| = 2, then ADi (N , wk , π ) = 4c + 3. The lower envelope curve achieves maximum at c ∗ = 23 . The

coalition structures {{1, 2}, {3, 4}, {5, 6}} and {{1, 2, 3}, {4, 5, 6}} are c −stable, and each of the players will obtain 5 32 as the payoff. 8. Conclusions The framework of potential functions in cooperative game theory with coalition structure allows analyzing the stability of coalition structures. This is especially useful in the problems where a potential function can be found in analytic form. This fact has been demonstrated in the paper for the cooperative game with identical players. Such problems often arise in patrolling and networking games when there is no difference between the players. Future research will be focused on a more general case with several types of players. The results established above can be applied to find stable coalition structures in economic, social, political and other games. In this paper, the classical definition of a potential function has

[1] T. Abe, Stable coalition structures in symmetric majority games: a coincidence between myopia and farsightedness, Theory and Decision (2018) 1–22. [2] R. Amer, F. Carreras, J.M. Gimenez, The modified Banzhaf value for games with coalition structure: an axiomatic characterization, Math. Social Sci. 43 (1) (2002) 45–54. [3] R.J. Aumann, J.H. Dreze, Cooperative games with coalition structures, Internat. J. Game Theory 3 (4) (1974) 217–237. [4] A. Balliu, M. Flammini, G. Melideo, D. Olivetti, Nash stability in social distance games, in: AAAI, 2017, pp. 342–348. [5] A. Bogomolnaia, M.O. Jackson, The stability of hedonic coalition structures, Games Econom. Behav. 38 (2) (2002) 201–230. [6] A. Casajus, F. Huettner, Potential, value, and the multilinear extension, Econom. Lett. 135 (2015) 28–30. [7] D. Cheng, On finite potential games, Automatica 50 (7) (2014) 1793–1801. [8] D. Cheng, T. Liu, From boolean game to potential game, Automatica 96 (2018) 51–60. [9] J. Dreze, J. Greenberg, Hedonic coalitions: optimality and stability, Econometrica 48 (1980) 987–1003. [10] A. Fonseca-Morales, O. Hernandez-Lerma, Potential differential games, Dynam. Games Appl. 8 (2) (2018) 254–279. [11] D. Gonzalez-Sanchez, O. Hernandez-Lerma, A survey of static and dynamic potential games, Sci. China Math. 59 (11) (2016) 2075–2102. [12] J. Greenberg, Pure and local public goods: A game-theoretic approach, in: A. Sandmo (Ed.), Essays in Public Economics, Heath, Lexington, MA, 1978. [13] T. Hiller, On the stability of couples, Games 9 (3) (2018) 48. [14] M. Hoefer, D. Vaz, L. Wagner, Dynamics in matching and coalition formation games with structural constraints, Artif. Intell. 262 (2018) 222–247. [15] Y. Kamijo, A two-step shapley value for cooperative games with coalition structures, Int. Game Theory Rev. 11 (02) (2009) 207–214. [16] X. Liu, J. Zhu, On potential equations of finite games, Automatica 68 (2016) 245–253. [17] V. Mazalov, Mathematical Game Theory and Applications, John Wiley & Sons, 2014. [18] D. Monderer, L.S. Shapley, Potential games, Games Econom. Behav. 14 (1) (1996) 124–143. [19] K. Ohta, N. Barrot, A. Ismaili, Y. Sakurai, M. Yokoo, Core stability in hedonic games among friends and enemies: Impact of neutrals, in: IJCAI, 2017, pp. 359–365. [20] E. Parilina, A. Sedakov, Stable bank cooperation for cost reduction problem, Czech. Econ. Rev. 8 (1) (2014) 7–26. [21] E. Parilina, A. Sedakov, Stable cooperation in a game with a major player, Int. Game Theory Rev. 18 (02) (2016) 1640005–1–1640005–20. [22] F. Patrone, L. Pusillo, S. Tijs, Multicriteria games and potentials, Top 15 (1) (2007) 138–145. [23] G. Pieri, L. Pusillo, Multicriteria partial cooperative games, Appl. Math. 6 (12) (2015) 2125. [24] R.W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, Int. J. Game Theory 2 (1973) 65–67. [25] L.S. Shapley, F.D. Rigby, Equilibrium points in games with vector payoffs, Nav. Res. Logist. Q. 6 (1) (1959) 57–61.