CC BY
6Contributions to Game Theory and Management, XV, 81—95
Two-stage Minimum Cost Spanning Tree Game under Fuzzy
Optimistic Coalition
Zhao Guo1, Dan Wang1, Min Chen1 and Yin Li12
1 St. Petersburg State University, Faculty of Applied Mathematics and Control Processes, 7/9, Universitetskaya nab., St. Petersburg, 198504, Russia 2 School of Mathematics, Harbin Institute of Technology, Harbin, 15000, China E-mail: 13709267195@163.com; wd789564@163.com; cm014033@163.com; liyinrus@outlook.com
Abstract This paper discusses the problem of cost allocation when players have different levels of optimism based on the two-stage minimum spanning tree game, and uses Choquet integral to calculate the characteristic function of fuzzy optimistic coalition and fuzzy pessimistic coalition. It is proved that the subgame of the two-stage clear optimistic coalition minimum cost spanning tree game is also a convex game. Finally, an example is used to prove that the two-stage fuzzy pessimistic coalition minimum cost spanning tree game has a dynamical instability solution.
Keywords: optimistic game, fuzzy game, Choquet integral, spanning tree game.
1. Introduction
This paper studies the minimum cost spanning tree problem under fuzzy coalition. Some marketplaces which located in different geographic locations require the purchase of a commodity that can only be supplied by a common supplier called a source. Agents in each market will pay some cost to make the trade route connect to the source, but they don't care whether they connect directly to the source or indirectly to other markets to complete the transaction. Each of these agents has an optimistic or pessimistic tendency (Cheng, 2021). Optimistic agents believe that other agents will allow themselves to connect to their marketplace, which means that optimistic agents just need to find the closest route to other agents and buy from this agent marketplace at factory price.
(Kruskal,1956) and (Prim,1957) first define the mcstp minimum cost spanning tree (mt) algorithm. (Bird,1976) and (Dutta and Kar, 2004) first introduce two Prim based rules on the problem of allocating mt related costs among agents. Bird associates each mcstp with a cooperative game (TU game) with transferable utility. (Petrosyan, 2014) studies two-stage network games, and defines the Shapley value of cooperative games when the network formed by players in the first stage may influence the actions of players in the second stage. (Cheng, 2021) considers the characteristic function and Shapley value when the players have optimistic and pessimistic properties respectively in the two-stage network game. According to Bird, the worth of a coalition is the cost of connection, assuming that the rest of the agents are not present. Hence, this worth takes the classical stand alone interpretation. The worth of a coalition is simply the best they can do without other players' contributions.
https://doi.org/10.21638/11701/spbu31.2022.07
We consider a more realistic situation based on subjective optimistic game (Cheng, 2021). Every agent has a tendency to be optimistic and pessimistic, but not all agents believe that other agents can convince them. Only those agents whose level of optimism exceeds their threshold will allow them to connect to their network for free. Obviously, compared with the subjective optimistic model of (Cheng, 2021) and (Bergantinos, 2007), that is, as long as one considers myself optimistic, one can convince all other agents. This paper divides optimism into two concepts, and the first is that one can convince others and second is that others can convince me. This indicates my ability to convince other agents, and how much ability other agents need to convince me. Apparently, optimism has changed from a subjective absolute concept to an objective relative optimism. Not all agents think other agents can convince themselves that only agents whose optimism exceeds their own threshold will allow them to connect to their networks for free. We believe that the ability to persuade others and the level that can be persuaded by others are both described as values between the interval [0,1]. The higher the value is, the more likely the agent is to persuade others or the less likely it is to be persuaded by others. The lower the value is, the less likely the agent is to persuade others or the more likely it is to be persuaded by others. This paper only considers a simple optimistic model assuming that each agent's ability to persuade others is equal to the level at which others can persuade. This means how much ability one have to convince other agents, so other agents also need how much ability to convince me.
We consider that the optimism level of each agent is between [0,1]. Obviously, in the cooperative game, because each coalition has a different degree of optimism, there are a variety of agents in the sub-coalition that can be divided into optimistic sets, which means that our coalition status is fuzzy. (Aubin, 1981) first proposed the concept of a fuzzy coalition cooperative game. He pointed out that in some cases, the players do not fully join the coalition, but only participate in the activities of the coalition to a certain extent. Therefore, a number in [0,1] can be used to represent the degree of player participation in a certain coalition. (Butnariu, 1980) proposed a coeur for fuzzy coalition games, and also defined a class of fuzzy games with proportional values Gp(N). (Tsunmi, 2001) pointed out that the characteristic functions of fuzzy games on Gp(N) are neither continuous nor monotonic, and he introduced a class of fuzzy games Gc(N) with Choquet integral expression. The Shapley value formula of fuzzy games on Gc(N) defined by Tsunmi satisfies the four axioms concerning the Shapley function of any fuzzy game that defined by him.
The paper is organized as follows. In Section 2, we consider the fuzzy coalition cooperative game. In Section 3, we introduce the two-stage minimum cost spanning tree game, as well as optimistic and pessimistic spanning tree games. We study the two-stage fuzzy optimistic coalition minimum cost spanning tree game in Section 4. The two-stage fuzzy pessimistic coalition minimum cost spanning tree game is proposed in Section 5.
2. Fuzzy Coalition Cooperative Game Based on Choquet Integral 2.1. Fuzzy coalition
Definition 1. (Yang and Li, 2021) Let (N, v0) e G0(N), N = {1,..., n} and G0(N) is the set of clear coalition cooperative games. For any clear coalition S0 e 2N, the membership degree (S0)j can be used to represent the participation degree of player
i in the clear coalition So. When i G So, then (So)j = 1, and when i G So, then (So)j = 0, so (S0)j G {0,1}. Therefore, the feasible coalition set 2N can be equiva-lently represented by the set {0,1}n
S G 2n ^ So G {0,1}n.
Example 1. N = {1, 2, 3}, clear coalition So = {1, 2}, we know 1 G So, 2 G So, 3 G So, so (So)1 = 1, (So)2 = 1, (So)3 = 0.
According to the idea of fuzzy set definition, the coalition is extended from the set {0,1}n to the set [0,1]n.
Definition 2. (Yang and Li, 2021) Let N = {1,..., n}, if Sc G [0,1]n
Sc : i G N ^ (Sc)i G [0,1],
then Sc is called the fuzzy coalition of N. (Sc)j represents the membership degree of player i to fuzzy coalition Sc, which describes the participation degree of player i in fuzzy coalition Sc.
Definition 3. (Yan and Wang, 2017) The fuzzy coalition Sc is expressed as vector notation
Sc = {(Sc)i, (Sc)2,..., (Sc)i, . . . , (Sc)„},
where i G N .In this paper, the fuzzy coalition Sc uses vector notation, and also defines Zadeh notation and ordered pair notation.
If (Sc)j = 1, it means that player i fully participates in fuzzy coalition Sc. if (Sc)j = 0, it means that player i does not participate in fuzzy coalition Sc. if (Sc)j G (0,1), then it means the player i is partially involved in the fuzzy coalition Sc. In fuzzy coalition, the empty coalition represents the zero vector.
When player i participates in coalition S with (Sc)j which is the degree of optimism, coalition S will become fuzzy coalition Sc. Because the player i G S in S uniquely corresponds to their degree of membership (Sc)j G Sc in Sc, it can also be expressed as i = {i | (Sc)j > 0, (Sc)j G Sc}. This means that when (Sc)j G Sc, player i has already participated in Sc. So (Sc)j G Sc is equivalent to i G Supp(Sc), where Supp(Sc) = {i | (Sc)< > 0, (Sc)< G Sc}.
Example 2. N = {1, 2, 3}, (Sc)1 = 0.5, (Sc)2 = 0.3, (Sc)3 = 0.2, the fuzzy coalition is Sc = {(Sc)i, (Sc)2, (Sc)3} = {0.5,0.3, 0.2}.
Definition 4. (Yang and Li, 2021) The fuzzy coalition cooperative game can be expressed as an ordered pair (Fc(N),vc), where Fc(N) denotes the set [0,1]n of fuzzy coalition on the set N of players in the game, and vc represents the payoff function of the n person fuzzy coalition cooperative game, namely vc : Fc(N) ^ R and vc(0) = 0. The payoff function vc(Sc) represents the expected payoff when the fuzzy coalition Sc cooperates satisfying vc(0) = 0. When (Sc)j only takes 1 or 0, the fuzzy coalition Sc degenerates into a clear coalition, and the corresponding fuzzy coalition cooperative game degenerates into a clear coalition cooperative game.
Therefore, fuzzy coalition cooperative game is an extension of clear coalition cooperative game, and clear coalition cooperative game is a special case of fuzzy coalition cooperative game.
Definition 5. (Yang and Li, 2021) Let (Fc(N),vc) e Gc(N),Gc(N) is the set of fuzzy coalition cooperative games and Sc e [0,1]n, if there is a n-dimensional vector function x = (xi,..., xj,..., xn), it represents the distribution plan of the fuzzy coalition Sc, where xj is the income of the i—th player, when the following conditions are met:
(1) Individual rationality: if i e supp(Sc), we have xj > vc(i), otherwise xj = 0, where Supp(Sc) = {i | (Sc)j > 0, (Sc)j e Sc}.
(2) Collective rationality: Ejes«pp(sc) xj = Vc(Sc).
Then x is said to be a distribution of the game (Fc(N),vc) about the fuzzy coalition Sc.
For the fuzzy coalition game (Fc(N), vc), let Q(Sc) = {(Sc)j | (Sc)j > 0}, (Sc)j e Sc, i e N, where (Sc)j represents the membership degree of the player i to the fuzzy coalition Sc. The elements (membership degree) in Q(Sc) are arranged in the order from small to large 0 < t1 < t2 < • • • < t; < tq(S) < 1. q(Sc) is the number of elements in Q(Sc).
Example 3. N = {1, 2, 3}, the fuzzy coalition Sc = {0.5,0.3, 0.2}. Therefore Q(Sc) = {ti,t2,ts} = {0.2,0.3, 0.5},q(Sc) = 3.
2.2. Characteristic function of fuzzy coalition cooperative games Definition 6. (Yang and Li, 2021) The payoff function vc of the fuzzy coalition game (Fc(N), vc) is a mapping from the fuzzy coalition set Fc(N) to the real number set R, namely vc : Fc(N) ^ R and vc(0) = 0, satisfying
q(Sc)
vc(Sc) = ^ v0([Sc]ti )(t; - tj_i),Sc e [0,1]n
;=i
where [Sc]t; = {i | (Sc)j > t;, (Sc)j e Sc}, t; e Q(Sc). [Sc]t, represents a feasible coalition with membership degree greater than or equal to t;. For any fuzzy coalition Sc, specify t0 = 0, we call vc the fuzzy coalition cooperative game solution of v0 based on Choquet integral. (v0 is the characteristic function of clear coalition cooperative game.)
Example 4. If three logistics service providers N = {1,2,3} fully participate in a cooperative project (clear coalition), the characteristic functions of their clear coalition cooperation game are as follows: v0({1}) = 10,v0({2}) = 10, v0({3}) = 20,v0({1, 2}) = 30,v0({1, 3}) = 50,v0({2, 3}) = 50,v0({1, 2, 3}) = 80. Suppose the participation of {1, 2, 3} is (Sc)i = 0.5, (SJ2 = 0.3, (SJ3 = 0.2. For the fuzzy coalition Sc = {(Sc)i, (Sc)2, (Sc)3} = {0.5,0.3,0.2}. So Q(Sc) = {0.2,0.3,0.5}, q(Sc) = 3.
Table 1. Characteristic function of Fuzzy Games
l 12 3
t 02 0.3 0.5
ti - ti-1 0.2 0.1 0.2
[S]tl {1, 2, 3}{1, 2} {3}
vo ([S]t; ) 80 30 10
vo([S]t;)(ti - ti-1) 16 3 2
The fuzzy characteristic function of the coalition Sc = {(Sc)1, (Sc)2, (Sc)3} is calculated by
3
Vc({(Sc)i, (Sc)2, (Sc)3}) = £ vo([Sc]ti )(t, - t,_i) = 16 + 3 + 2 = 21.
i=i
Similarly, we obtain the following characteristic functions under the fuzzy coalitions
Vc({(Sc)i}) = 10 x 0.5 = 5,
Vc({(Sc)2}) = 10 x 0.3 = 3, Vc({(Sc)3}) = 20 x 0.2 = 4, Vc({(Sc)i, (Sc)2}) = 30 x (0.3 - 0) + 10 x (0.5 - 0.3) = 11, Vc({(Sc)i, (Sc)3}) = 50 x (0.2 - 0) + 10 x (0.5 - 0.2) = 13, Vc({(Sc)2, (Sc)3}) = 50 x (0.2 - 0) + 10 x (0.3 - 0.2) = 11.
2.3. Shapley value of fuzzy optimistic coalition minimun cost spanning tree game
Let vc G Gc(N) and Sc G [0,1]n. For the player i G Supp(Sc) C N, the marginal contribution of the fuzzy coalition Sc is:
P(Sc,i)= Vc(Sc) - Vc(Sc \{i}).
It represents the expected payoff of player i in fuzzy coalition Sc, so the Shapley value of fuzzy coalition cooperative game is:
sh,(Vc)= £ 'T-1^1'' P(Sc,i)
i£ScCN 1 1 '
where Sc is the number of players with non-zero participation in the fuzzy coalition Sc. Let shj(vc) be the ith component of the Shapley value of the fuzzy coalition cooperative game based on Choquet integral.
3. Two-stage Optimistic and Pessimistic Coalition Minimun Cost Spanning Tree Game
3.1. Two-stage pessimistic coalition minimun cost spanning tree game on subgraph
(Li, 2016) Players in each stage of the game will simultaneously choose a strategy. Let us construct a network, and define a cost matrix to obtain the minimum cost spanning tree of the network.
N = {1,2,..., n} is a finite set of players. N' = N U {0}, {0} is the source. A connected graph with respect to N' is represented as G(N', E), where E is the set of all edges. If (i, j) G E, Vi, j G N, then (i, j) is an edge in G(N', E). In the game, the coalition S is satisfied, S C N, S' = S U {0}.
The connection cost matrix between players i and j is C = (cj), cj = j > 0, i = j G N'. where coi = cio, i G N (the cost of connecting i and the source) is a non-negative constant.
In each stage of the game, we use Xj,j to represent the set of strategies of player i e N against player j e N \ i , and Xj,j contains multiple actions xj,j that player i can take againsting player j. The cost of edge (i, j) is defined as: cjj = j = /c(xj,j, xj,j). /c represents the mapping from actions xj,j and xj-,j of players i and j to the cost of edge (i, j). We use Xj to represent the strategy combination of player i e N against any player j e N\ i, Xj = (Xj,i,..., Xj,j-i, Xj,j+i,..., Xj,n) and use X = Xi,..., Xj,..., Xn to represent the strategy combination of all players.
The pessimistic game with spanning tree involves the players in coalition S c N connected with the source {0} without any help from players outside of the coalition S.
Definition 7. (Cheng, 2021)The pessimistic minimum cost spanning tree over the set S c N is defined as:
T(N', Cxs) = arg min Y] cjj,
S (j,j)£G(S',E)
where Cxs is the cost matrix defined by strategy profile xS.
Definition 8. The total cost of edges on the minimum cost spanning tree T(S, Cx) is:
C [T (N' ,Cxs )]= £ cjj,
(j,j)£T(S,CxS)
where Cxs is the cost matrix defined by strategy profile xS. 3.2. Optimistic game with spanning tree on subgraph
In the optimistic game, if the optimistic coalition S is with the help of coalition N \ S, any player in S does not need to connect to the source, but only needs to be indirectly connected to the network of coalition N \ S and {0}. In this case, the optimistic coalition S can complete the trade with the source. Then the total cost of coalition S consists of two parts, one is the total cost of spanning tree S, and the other is the cost of connecting the coalition S to the coalition N\S. It means that using the connection to the source within coalition N\S is cost free for coalition S. In this research, if the coalition N\S supports the coalition S, the costs of the edges that will be provided between them are equal to the costs of the edges in the initial cost matrix.
Definition 9. (Cheng, 2021) The minimum optimistic cost spanning tree for coalition S is defined as follow
T +(S,N'\S,Cxs) = arg min J £ cj + £ c0
G(N>,E) [(j,j)eG(S,E) (o,o')eG(N ',E)
where o e S, o' e N'\S. Cxs is cost matrix defined by strategy profile xS.
Definition 10. The total cost of edges in the minimum cost spanning tree
T +(S,N'\S, Cxs) is
C [T +(S,N '\S,Cxs )]= £ cjj,
(j,j)£T +(S,N'\S,Cxs)
where Cxs is the cost matrix defined by strategy profile xS.
Example 5. N = {1, 2, 3},N' = N U {0},S = {2, 3}, f = x%J x , x%J G Xi,j, G , i = j G N .As shown in Fig.2, player 2 and player 3 choose their action x2,3 = 2, x32 = 5, thus the cost of edge (2, 3) is equal to 10. The total cost of minimum cost spanning tree T(S,CXS ) is:
C [T (S,Cxs )] = 26, and the total cost of minimum cost spanning tree T + (S, N'\S, Cxs ) is:
C[T + (S,N'\S, Cxs )] = 12.
Fig. 1. Pessimistic game and optimistic game on subgraph
3.3. Two-stage minimum cost spanning tree game
We consider a two-stage game, in the first stage player i chooses x1 = (x11,..., x1,n) and constructs network G(N',CX). x\j,i = j G N is the action of player i against player j in the first stage.
In the second stage player m may leave the game, and we discuss each player's action x2 \ {m} when player m leaves the game separately, and each player's action x2 when player m does not leave the game.
Definition 11. (Li, 2016) The probability of player m leaving the game is:
= X) (i,j )eB(m)Cj p = C [T (N ' ,Cxi )] '
where B (m) is the subtree rooted at m in T(N', Cx).
Definition 12. (Li, 2016) In a two-stage minimum cost spanning tree game, the player's total cost is assumed to be the sum of the two-stage player's total cost,
V1 (N ) = C [T (N ' ,Cxi )] + p • C [T (N ' \ {m},Cx2\{m} )] + (1 - p) • C [T (N ' ,Cx2 )],
where N' = N U {0}, as a special player the source {0}, only participates in the construction of the spanning tree, and does not participate in the game. barxi(•), i G S represents the optimal cooperative action combination of player i G N to any player j G N \ i, and x(-) represents the optimal cooperative action combination for all players.
4. Two-stage Fuzzy Optimistic Coalition Game
4.1. Characteristic function of two-stage clear optimistic coalition game [Sc]a
Now let us describe the optimism of player i by membership function (Sc)i; if (Sc)j > (Sc)j, it means that player i is more optimistic than player j. At this point, player j agrees that player i is an optimistic player, and player i is able to convince player j to connect himself to their market. But at this time, player i does not recognize player j as an optimistic player, because player j is less optimistic than player i.
We consider a fuzzy optimistic coalition Sc = j(Sc)i, (Sc)2,..., (Sc)n} composed of all players, and use [Sc]a to represent clear optimistic coalition with the degree of optimism (Sc)j > a, [Sc]a = {i | (Sc)j > a, i G Supp(SC)}. a is called the optimistic confidence level. That is, any player o in [Sc]a can persuade a player o' in N'\[SC]a. Clearly there are multiple possible clear optimistic coalitions when a takes on different values.
Definition 13. When the optimistic confidence level is a, the two-stage fuzzy clear optimistic coalition [Sc]a is V 1+([SC]a).
If m G [Sc]a, [Sc]a C N, and [Sc]a = [Sc]a U {0},
V 1+([Sc]a) = C[T+([Sc]a, N'\[Sc]a, C^a )] + p ■ C[T+([Sc]a\{m}, N'\[Sc]a\{m},
)] + (1 -p)C[T+([Sc]a, N'\[Sc]a, Cf0'" )],
[Sc ] a m } [So]a
where C[Sc]a and C[fc]a\{m} is the cost matrix restricted to [SC]^ If m G [Sc]a, [Sc]a c N, and [Sc]a = [Sc]a U {0},
V 1+([Sc]a) = C[T+([Sc]a,N'\[Sc]a,C|fc]a )] + C[T+([Sc]a,N'\[Sc]a,C;^^c]a )].
[Sc]a [Sc]a
The characteristic function for the coalition [Sc]a in subgame is: V2+ ([Sc]a) = C[T+([Sc]a,N'\[Sc]a,Cjfcla )],
where p =-ij[s — is obviously for fuzzy optimistic coalition Sc,
C[T+([Sc]a ,W'\[Sc]a,C^Sc]a )]
the optimistic confidence level of a will determine the size of the clear optimistic coalition [Sc]a. The larger the value of a, the smaller the number of optimistic players in the optimistic coalition [Sc]a. The characteristic function V 1+([SC]a) of the two-stage game is related to the size of the coalition [Sc]a. This means that we can adjust the size of a to increase or decrease the two-stage optimistic coalition minimum cost spanning tree.
Theorem 1. If cj = j = fc(xj,j, Xj,j) > 0, then the subgame optimistic clear coalition So is a convex game.
V2+(So) - V2+(So\{k}) > 0.
Proof. According to Definition 9 and Definition 10, we find the characteristic function of the minimum cost spanning tree of the clear coalition S0, which satisfies
o e So ,o' e N'\So,
V2+ (So) = g(sA'E) 1 V cij + V W,
{(i,j )GG(S,E) (o,o' )GG(N' ,E) )
we can simply express the above formula as:
V2+ (So) = £ Cij )So + (£ Coo')S0 .
Similarly, according to Definition 9 and Definition 10 we get the characteristic function of the minimum cost spanning tree of the clear coalition S0 \{k}. In this case o e S0\{k},o' e N'\S0\{k},
V2+ (S0\{k})= min { y Cij + V Coo' >,
G(S0\{k},E),G(N' ,E) 1 ^ j f
( °U ^ ^ ( , ) {(i,j)€G(So\{k},E) (o,o')eG(N',E) )
we can simply express the above formula as:
V 2+ (S0\{k }) = (V Cij )S°\{k} + (y coo' )S°\{k}.
Fig. 2. Complete graph G(N',E) and V2+ (So) and V2+ (So\{k})
We consider the first case V2+ (S0) in the optimistic player k and the non-optimistic player o' e N'\S0, and there exist edge (k, o'), k e S0, o' e N'\S0. Figure 2(a) is the complete graph G(N', E). Figure 2(b) is the optimistic coalition S0 = {1, 2, 3, k}. Figure 2(c) is the optimistic coalition S0\{k} = {1, 2, 3}. (X Cij)S° = Cki + Ck3 in Figure 2(b), (ECoo')So = Ck0 + C02. In Figure 2(c) (ECij)S°\{k} = 0, (E Coo')So\{k} = Cki + Ck3 + C02.
When player k is not in optimistic coalition S0, the coalition S0 will be missing all sides (k, j) related to player k ,k e S0, j e S0\{k},
(V Cij)So\{k} = (V Cij)So - y Ckj.
(k,j )GG(S,E)
When player k leaves optimistic coalition S0, it means that we don't care how non-optimistic player k connects to the source,and just consider how the remaining
optimistic coalition S0\{k} connects to the coalition N\k\{0},
(E Coo' )S°\{fc} = E Ckj + ((E Coo' - Cko') .
(fc,j)£G(S,E) Let us combine the above two equations,
(E Cij )S°\{k} + (E Coo' )S°\{k} = (E Cij )S° + (E Coo' )S° - Cko',
V2+(So\{k}) = V2+(So) - Cfco',Cfco' > 0,
so we prove that in the first case V2+(S0) > V2+(S0\{k}).
Similarly, we also consider the second case V2+(S0) in the optimistic player k and the non-optimistic player o' G N'\S0 there is no edge (k, o'), k G S0, o' G N'\S0,
(E Cij )S°\{k} = (E Cij )S° - E Cfcj,
(fc,j)£G(S,E)
(E Coo' )S°\{k} = (E Coo' )S° + E Ckj ,
(fc,j)£G(S,E)
let us combine the two equations above, and we find
V 2+(S0\{k}) = V 2+(S0). Now, we have discussed all possibilities and proved that
V2+(S0) - V2+(S0\{k}) > 0.
4.2. Characteristic function of two-stage fuzzy optimistic coalition Sc
According to Definition 6, when the optimistic confidence level is t;,t; G Q(SC), we define the game characteristic function of the two-stage fuzzy optimistic coalition
Sc.
Definition 14. The payoff function of the fuzzy optimistic coalition Sc, Sc G [0,1]n satisfies
q(Sc)
vc1+(Sc)= E V 1+([Sc]i;)(ti - ti-1), i=1
where [Sc]tl represents the clear optimistic coalition at the confidence level t;. t; G Q(SC), Q(SC) is the order of the elements of the membership degree (Sc)j > 0 in Sc from small to large. q(SC) is the number of elements in Q(SC).
4.3. Shapley value of fuzzy optimistic coalition spanning tree game
Let vc G Gc(N) and Sc G [0,1]n. For the player i G Supp(SC) C N, the marginal contribution of the fuzzy coalition Sc is:
P(Sc,i)= vc+(Sc) - vc+(Sc \{i}).
It represents the expected payoff of player i in fuzzy coalition Sc, so the Shapley value of fuzzy coalition cooperative game is:
«M<i)= £ (|SC|- 1)ir-|SC|)!P(Sc),
i£ScCN 1 |!
where |Sc| is the number of players with non-zero participation in the fuzzy coalition Sc. Let shj(v+) be the ith component of the Shapley value of the fuzzy optimistic coalition spanning tree cooperative game based on Choquet integral.
Example 6. N = {1, 2, 3}, N' = N U {0},coi = cio = 1,co2 = C20 = 80, C03 = c3o = 120. Players can choose strategies such as Tab.2. Assume that fc = Xj,j x Xj, € , Xj,i € Xj,i, i = j € N.
Table 2. The sets of the strategies
C12 X2,1 Cl3 X3,i C23 X3,2
4 5 7 9 4 5
3 12 15 Xi,3 2 14 18 X2,3 4 16 20
4 16 20 5 35 45 6 24 30
At stage 1, player 1 chooses x} 2 = 3, x} 3 = 2, player 2 chooses x 21 = 4, x^ 3 = 4, and player 3 chooses action x3 1 = 7, xg 2 =4. The cost of edge (1, 2) is c} 2 = c21 = 12. The cost of edge (1,3) is c}3 = c31 = 14, and the cost of edge (2,3) is
c23 = c32 = 16
At stage 2, player 1 chooses x21 2 = 3, x12 3 = 2, player 2 chooses x22 1 = 4, x22 3 = 4, and player 3 chooses action x3 2 = 7, x3 , 2 = 4. The cost of edge (1, 2) is cg2 = c221 = 12. The cost of edge (1, 3) is c213 = c321 = 14, and the cost of edge (2, 3) is
c23 = c32 = 16
The fuzzy coalition Sc = {0.5,0.3,0.2}. Therefore Q(Sc) = {t1,t2,t3} = {0.2,0.3, 0.5},q(Sc) = 3.
We consider only the case where player 2 quits in the second stage of the game,
V01+(N) = (1 + 12 + 14) + 0 * (1 + 14) + (1 - 0) * (1 + 12 + 14) = 54,
V01+({1, 2}) = (1 + 12) + 0 * 1 + (1 - 0) * (1 + 12) = 26,
V01+({1, 3}) = (1 + 14) + 0 * (1 + 14) + (1 - 0) * (1 + 14) = 30,
V01+({2, 3}) = (12 + 14) + 0 * (14) + (1 - 0) * (12 + 14) = 52,
V01+({1}) = 1 + 0 * 1 + (1 - 0) * 1 = 2,
V01+({2}) = 12 + 0 * 0+ (1 - 0) * 12 = 24,
V01+({3}) = 14 + 0 * 14 + (1 - 0) * 14 = 28.
According to Definition 14, the characteristic functions of the fuzzy optimistic coalition are:
vc1+({(Sc)1}) = 1 * 0.5 = 0.5, vc1+({(Sc)2}) = 24 * 0.3 = 7.2, vc1+({(Sc)3}) = 28 * 0.2 = 5.6, vc1+({(Sc)1, (Sc)2}) = 26 * 0.3+1 * 0.2 = 8, vc1+({(Sc)1, (Sc)3}) = 30 * 0.2+1 * 0.3 = 6.3, vc1+({(Sc)2, (Sc)3}) = 52 * 0.2 + 24 * 0.1 = 12.4,
vc1+({(Sc)i, (Sc)2, (Sc)3>) = 54 * 0.2 + 26 * 0.1 + 1 * 0.2 = 13.6.
shi(v+) = 4(13.6 - 12.4) + 4(8 - 7.2) + 4(6.3 - 5.6) + 1(0.5 - 0) = 0.8167, 3 6 6 3
sh2(v+) = 1(13.6 - 6.3) + 4(8 - 0.5) + 1 (12.4 - 5.6) + ¿(7.2 - 0) = 7.2167, 3 6 6 3
sh3(v+) = 1(13.6 - 8) + 1(6.3 - 0.5) + 1(12.4 - 7.2) + 1(5.6 - 0) = 5.5667. 3 6 6 3
5. Two-stage Fuzzy Pessimistic Coalition Game
Similarly, we use the membership function (Sc)j to describe the optimism of the player i, then 1 - (Sc)j is the pessimism of the player i. Similarly a is the optimistic confidence level, then 1 - a is the pessimistic confidence level.
Definition 15. Let S- be a fuzzy pessimistic of coalition players
S- = {(S-)i, (S-)2,..., (S-)i,..., (S-)„|, i e N,
where (S-)j = 1 - (Sc)j. (Sc)j is the degree of optimism of player i in the fuzzy optimistic coalition, (S-)j is the pessimism degree of player i in the fuzzy pessimistic coalition.
Definition 16. Let 0 = 1 - a, a be the optimistic confidence level, and 0 be the pessimistic confidence level. [S-]^ be the clear pessimistic coalition under the pessimistic confidence level 0.
[S-k = {i | (S-)i > 0,i e Sc} .
Example 7. Assuming the fuzzy pessimistic coalition S- = {(S-)1, (S-)2, (S-)3} = {0.5,0.7, 0.8}, when a = 0.3, 0 = 1 - a = 0.7, at this time the clear pessimistic coalition [S-]0.7 = {2, 3}.
Definition 17. At a pessimistic confidence level 0, the two-stage minimum cost spanning tree for a clear pessimistic coalition [S-]^ is following two cases. If m e [S-k, [S-k C N, and [S-]^ = [S-^ U {0},
V 1([S-]^) = C[T([S-^,C[f]3 )]+ p • C[T([S-]^Um},^Um} )]
[S- ]* [S- ]*\{m}
+(1 -p)C[T([S-]^)],
[S-
where C[S-^ and C[S-^\{m} is the cost matrix restricted to [S-]^. If m e [S-^, [S-k C N, and [S-^ = [S-]^ U {0},
V1 ([S-k)= C[T([S-]^)]+ C[T([S-]^)]. The characteristic function for the coalition [S-]^ in subgame is: V2([S-]^) = C[T([Sc-]^)].
[S-
5.1. Characteristic function of the two-stage fuzzy pessimistic coalition
game
Definition 18. According to Definition 6 and Definition 16, we get the game characteristic function of the two-stage fuzzy optimistic coalition S—, and it is:
q(Sc)
*1(S—) = E V 1([S-]tL)(tL - tL-l),
L=1
where tL = 1—t;,t; G Q(Sc),tL G Q(S—), Q(Sc) is the optimistic coalition, and Sc is the membership (Sc)j > 0 elements from small to large sort. Q(S—) is the ranking of the elements of membership (S— )j > 0 in the pessimistic coalition S— from small to large. [S— ]tL represents a clear pessimistic coalition under the pessimistic confidence level tL. q(S—) is the number of elements in Q(S—).
5.2. The Shapley value of fuzzy pessimistic coalition minimum cost spanning tree game
The Shapley value of two-stage fuzzy pessimistic coalition spanning tree game
is:
h ( 1) V- CIS—| — 1)!(|N|-|S— |)! . 1 1
shi(v 1)= -[Nn--W^J — V1(S— \{i})),
ies-cw
where |S—| is the number of players in the fuzzy coalition S— with non-zero participation. Let shj(v1) be the i-th component of the Shapley value of the two-stage fuzzy pessimistic coalition minimum cost spanning tree game based on Choquet integral.
Example 8. N = {1, 2, 3}, N' = N U {0}, C01 = C10 = 1, C02 = C20 = 80, C03 = C30 = 120. Players can choose strategies such as Tab 3. Assume that fc = G Xi,j, Xj,i G Xj,i, i = j G N.
Table 3. The sets of the strategies of players
C12 X2,1 C13 X3,i C23 X3,2
4 5 7 9 4 5
3 12 15 Xi,3 2 14 18 X2,3 4 16 20
4 16 20 5 35 45 6 24 30
At stage 1, player 1 chooses x} 2 = 3, x} 3 = 2, player 2 chooses x2, 1 = 4, x^ 3 = 4, and player 3 chooses action x3 = 7, x3 , 2 = 4. The cost of edge (1, 2) is c}2 = c2 } = 12. The cost of edge (1,3) is c }3 = c3 } = 14, and the cost of edge (2,3) is
c23 = c32 = 16.
At stage 2, the player 1 chooses x2 ,2 = 3, x2 ,3 = 2, player 2 chooses x22, = 4, x22,3 = 4, and player 3 chooses action x32, = 7, x23,2 = 4. The cost of edge (1, 2) is c22 = c2 1 = 12. The cost of edge (1, 3) is c23 = c3 2 = 14, and the cost of edge (2,3)
is c23 = c32 = 16.
The fuzizy coalition Sc = {0.5,0.3,0.2}. Therefore Q(Sc) = {t2,t2,t3} = {0.2,0.3, 0.5},q(Sc) = 3.
Now we only consider cases where player 2 exits in the second stage of the game, VQ (N) = (1 + 12+ 14) +0 * (1 + 14) + (1 - 0) * (1 + 12+14) = 54,
V01({1, 2}) = (1 + 12) + 0 * 1 + (1 - 0) * (1 + 12) = 26,
V01({1, 3}) = (1 + 14) + 0 * (1 + 14) + (1 - 0) * (1 + 14) = 30,
V01({2, 3}) = (80 + 16) + 0 * (14) + (1 - 0) * (12 + 14) = 196.8,
Vo({1}) = 1+ 0 * 1 + (1 - 0) * 1=2,
V01({2}) = 80 + 0 * 0 + (1 - 0) * 80 = 160,
V01({3}) = 120 + 0 * 14 + (1 - 0) * 120 = 240.
According to Definition 6 and Definition 17, the characteristic function of the fuzzy pessimistic coalition is shown on Table 4.
Table 4. Two-stage fuzzy pessimistic coalition minimum cost spanning tree game
l 1 2 3
tL 0.5 0.7 0.8
tL-1 — tL 0.5 0.2 0.1
{1, 2, 3} {2, 3} {3}
vo([S-]tL) 54 196.8 240
Vo([S- ]tL )(tL — tL- -1) 27 39.36 24
vl({(Sc-)i, (Sc-)2, (Sc")s}) = 27 + 39.36 + 24 = 90.36, vc1({(S-)i})=0.5 * 2 = 1, vl({(S-)2}) = 160 * 0.7 = 112, vl({(S-)s}) = 240 * 0.8 = 192, vl({(S-)i, (S-)2}) = 26 * 0.5 + 160 * 0.2 = 45, vl({(S-)i, (S-)s}) = 30 * 0.5 + 240 * 0.3 = 87, vl({(S-)2, (S-)3}) = 196.8 * 0.7 + 240 * 0.1 = 161.76.
shi(vl) = 4(90.36 - 161.76) + 4(45 - 112) + 4(87 - 1) + 4(1 - 0) = -20.3, 3 6 6 3
sh2(vl) = 1(90.36 - 87) + 1(45 - 1) + 1(161.76 - 192) + 1(112 - 0) = 40.7467, 3 6 6 3
sh3(vl) = 1(90.36 - 45) + 1(87 - 1) + 1(161.76 - 112) + 1(192 - 0) = 101.7467. 3 6 6 3
In this example of a two-stage fuzzy pessimistic coalition game, there is dynamical instability.
6. Conclusion
This paper analyzes the cost allocation problem about two-stage minimum cost spanning tree game with fuzzy pessimistic coalition and fuzzy optimistic coalition when players have different degrees of optimism, also improves the subjective optimism game model of (Cheng, 2021). We define optimism as behavior that persuades other players in a social activity, and optimism is perceived differently by each player. In this paper, the optimism of each player is represented by a membership function, and the characteristic function of clear optimistic coalition and clear pessimistic coalition are discussed when the ability to convince others is equal to the level of being convinced by others.
Also we define the two-stage optimistic coalition minimum spanning tree game process and two-stage pessimistic coalition minimum spanning tree game process for a given level of optimism, as well as the characteristic function under optimistic and pessimistic fuzzy coalition. It is also proved that the subgame of two-stage clear optimistic coalition minimum cost spaning tree game is a convex game. Finally, the dynamical instability solution of two-stage fuzzy pessimistic coalition minimum cost spaning tree game is proved by an example.
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