CC BY
3Contributions to Game Theory and Management, XIV, 59-71
New Characteristic Function for Two Stage Games with
Spanning Tree
Min Cheng1 and Yin Li1'2
1 St. Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg, 199034, Russia 2 School of Economics and Management, Yanan University, Yan'an, 716000, China E-mail: cm014033@163.com, liyinrus@outlook.com
Abstract Two-stage n-player games with spanning tree are considered. The cooperative behaviour of players is defined. After the first stage, a specified player leaves the game with a probability that depends on the actions of all players in the first stage. A new approach to the construction of the characteristic function is proposed. In the game, all players are connected with the source directly or indirectly. Assume that the players in coalition N \ S have already connected to the source when the players in coalition S C N wish to connect to the source. The players in coalition S could
N\S
new characteristic function is defined in the game, and the Shapley value is constructed. Several results based on the new characteristic function in the two-stage stochastic game are given.
Keywords: dynamic game, minimum cost spanning tree, Shapley value 1. Introduction
In the minimum cost spanning tree game, it is considered that a group of players need to connect the source to get some service or benefit and share the total cost between them. In (Bird, 1976), it is the first time to propose a groundbreaking method to solve this problem, namely the Bird rule, and it is a cost allocation rule in the game with spanning tree. After this, different kinds of solutions in the game with spanning tree have been proposed. Such as the core and nucleolus (Granot and Huberman, 1984), the Folk solution (Feltkamp et al., 1994), the Kar rule (Kar, 2002), and the fair rule (Bergantinos et al, 2007a).
Especially in (Bergantinos et al., 2007b), the author considered that the players in coalition S C N might indirectly connect to the source with the help of the players in the coalition N \ S, while the players in the coalition N \ S already-connected with the source. The game described before is named as the "optimistic" minimum cost spanning tree game.1 In this paper, it is considered to construct the Shapley value in the "optimistic" game.
In (Li, 2016), a two-stage spanning tree game with shock is considered, and assumed that after the first stage in the game, a particular player would leave the game with a probability that depends on the situation in the first stage. In this research the characteristic functions for coalitions are defined by the Bird method(Bird, 1976). The dynamic Shapley value is considered as a solution in the game. In (Yin, 2017), a two-stage spanning tree game with perishable products is studied. In the game, all players need to share the total cost of edges on the minimum cost spanning
S
nected to the source without the help of others. https://doi.org/10.21638/11701/spbu31.2021.05
tree arid the loss caused by the perishable products. The research concluded that the dynamic Shapley value is time inconsistent in the two-stage game with perishable products. Also the two-stage game with forest is studied(Yin. 2017). In this research, there are n players and m > 2 sources in the game. It is proved that using the imputation distribution procedure (IDP)(Petrosyan. 1979). the dynamic Shapley value constructed in the game will be time consistent if there is at least one player who wants to connect to all sources. Furthermore, in (Yin, 2020), the two-stage stochastic game with spanning tree is present. In this research, a transition matrix determined by the players is introduced.
In the paper, an "optimistic" two-stage game with spanning tree is considered. The players in the coalition S c N no longer need to connect with the source directly. If necessary, the coalition S can indirectly connect with the source by the
S
cost matrix is given. In each stage game, arbitrary two players can take actions to redefine the cost of the edge between them. For any two players, if they did not take action together to change the cost of the edge, then the cost of the edge between them will remain fixed as the cost of the edge from the initial cost matrix at the beginning of the game. All players choose their actions simultaneously at the first
m
game with probability p. The probability is defined by actions of all players in the first stage.
Fig. 1. Process of the game
2. The model
2.1. Basic definitions
Let N = {1,2,..., n} be a finite set of players. The source is denoted by {0}, and N' = NU {0}. A graph over N' is denoted by G = (N', E). E is the set of all edges. A pair (i,j) is called an edge in G(N', E), if (i,j) e E, Vi, j e N. A coalition S is the subset of N, where S C ^md S' = S U {0}.
Definition 1. A path in a graph is a finite or infinite sequence of edges that links a
i j i j
be connected. The graph is called a connected graph if any two vertices in a graph are connected.
Definition 2. G(N',E) (or G(S', E)) is defined as the connected graph over N' (or S'). Gn' (or Gs') is a set of connected graphs with vertices in N' (or S'), where S c n.
Example 1. Consider an example. N = {1, 2,3, 4} S = {1,2, 3} C N N' = NU{0} S' = S U {0} {0} is the source. As shown in Fig.2, which represents two different connected graphs Gn' and Gn' over N', and Fig.3 shows two connected graphs Gs> and G S' over S'.
Fig. 2. Connected graphs over N' = {0,1, 2,3,4}
Fig. 3. Connected graphs over S' = {0,1, 2,3}
C = (cij)(n+i)x(n+i) is a c°st matrix, in which all elements represent the costs of edges between vert exes on the graph G(NE). cj = j > 0 is the cost of edge (i,j), Vij G N
The cost matrix associated with G(N',E) (or G(S',E)) is denoted by C (or C S).
Definition 3. In G(NE), an initial cost matrix C0 is defined as
C0 = (c0j)(n+i)x(n+i),c0j = c0i > 0,Vi = j G N'
Consider an example about the initial cost matrix.
Example 2. As shown in Fig.4, N = {1, 2,3, 4, 5,6}, N' = N U {0}.
Fig. 4. Initial cost matrix C
The initial cost matrix associated with the graph G(N', E) is
0 1 2 3 4 5 0 /0 1 2 3 2 4\
C 0
102432 220542 345031 234303 \4 2 2 1 3 0/
2.2. Stage game
In each stage game, all players choose actions simultaneously. Then the cost
G(N', E)
i
xi (xi, 1 , ...? xi,i- 1? xi,i+ 1? ...? xi,n)
where xi,j is a strategy of player i against j, Vi = j e N Xi,j denotes the set of all strategies of player i against j, Vxiij- e Xj
The cost of edge (i,j) is defined as cij = cji = fc(xi,j, xj-ii), where the cost function fc is a mapping from the set of strategies of players i,j to the set of
(i, j) i
j Vi = j e N
{0} i e N
of the edge in the initial cost matrix, i.e
Cio = coi = c°0 = c0i > 0
For strategy profile x = (x1; x2,..., associated with G(N', Cx)
is denoted by Cx = (cij)(„+1)x(n+1}-
Definition 5. (Bird, 1976), the minimum cost spanning tree over N' is defined as follows
E
T(N , Cx) = arg mm ,
( ' } (i,j)eo(N',E)
where Cx = (cij)(n+1)X(n+1) is the cost matrix defined by the strategy profile x (x1,x2, ...,x„) Vi = j e N.
Definition 6. The total cost of edges in the minimum cost spanning tree T(N', Cx) is defined as
C[T(N',Cx)]= ^ cij, Vi = j e N
(i,j)£T (N',Cx)
where Cx = (cj)(n+i)x(n+i) is the cost matrix defined by strategy profile x =
(x1 ? x2? xn)•
Example 5. N = {1,2, 3} N' = N U {0}. The sets of the strategies of the players are shown in Tab.l.
Table 1. The sets of the strategies
C12 X2,1 Cl3 X3,i C23 X3,2
3 5 2 5 3 4
1 3 5 Xi,3 3 6 15 X2,3 2 6 8
2 6 10 4 8 20 4 12 16
Assume that fc = x^- xxj,i; xi,j G X^-, x^ G Xj,i, i = j G N. If player 1 chooses xi,2 = 1,xi,3 = 3 player 2 chooses x2,i = 3,x2,3 = 2, and player 3 chooses strategy x3,i = 2, x3,2 = 3 The cost of edge (1,2) is ci2 = C21 = fc(xi,2, x2,i) = xi,2x2,1 = 3. Similarly, the cost of edge (1,3) is ci3 = c3i = 6, and ^^e cost of edge (2, 3) is C23 = C32 = 6.
x = ( xi , x2 , x3 )
0 12 3 ^ 0 20 15 30 \ C = 1 20 0 3 6 Cx = 2 15 3 0 6 3 \30 6 6 0 J
As shown in Fig.5, on the left side it is an entire graph G(N', Cx) with cost. On the right side it is a minimum cost spanning tree T(N', Cx) on the graph G(N', Gx).
Fig. 5. G(N', Cx) and T(N', Cx)
Thus, the total cost of edges on the minimum cost spanning tree T(N', Cx) is G (T (N', Cx)) = 24 ^^e strategies profiles of player 1, ^d 3 are xi = (1,3), x2 = (3, 2),x3 = (2, 3).
Consider two types of games with spanning trees on subgraph: "optimistic" game with spanning tree (Bergantinos et al., 2007b) and "pessimistic" game with spanning tree.
2.3. "pessimistic" game with spanning tree on subgraph
The "pessimistic" game with spanning tree on subgraph involves the players in coalition S C N connected with the source {0} without any help from players S
Definition 7. (Bird, 1976) The minimum cost spanning tree on coalition S C N in the "pessimistic" game with spanning tree is defined as
T(S, Cxs) = arg min E cij
GeGs'
s (i,j)eo(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(S, Cxs )]= E cij
(i,j)eT(S,CxS)
where Cxs is the cost matrix defined by strat egy profile xs.
2.4. "optimistic" game with spanning tree on subgraph
In contrast to the above, in the optimistic game, if the coalition S C N needs N \ S S
S
cost of connecting the coalition S to the coalition N \ S. It means that using the
N \ S S
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.
S
game is defined as follow
cij
T + (S, N' \ S, Cxs) = arg min {
G(s,E)eGs ,G(N' ,e)e<3n ' (i,j )6G(s,e)
+ E c0O'}
(o,o ') e g(n ' ,E),o e s,o' e N '\s CxS 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 )]= E c
ij
(i,j) e t +(s,N'\s,Cxs)
CxS xs
Example 4. N = {1, 2, 3} N' = N U {0} S = {2, 3} fc = xi,j x xj,i,xi,j G Xj, xj-,i G Xji, i = j G N. As shown in Fig.6, player 2 and player 3 choose their actions x2,3 = 2, x3,2 = 5, thus the cost of edge (2,3) is equal to 10. The total
"Pessimistic" game
Fig. 6. "pessimistic" game and "optimistic" game 011 subgraph
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
2.5. Two-stage game with spanning tree
Let Pm0 denote the path from the source to m on G(N', E). P(m) - an immediate predecessor of m in the minimum cost spanning tree T(N', Cx), if P(m) G Pm0 and (P(m),m) G T(N',Cx).
m
_ 2(i,j)6B(m) Cij
P = C[T (N', Cx)] where B(m) is the set of all edges of the subtree with root m.
The first stage: All players choose their strategies profiles simultaneously:
x (x 1, x2, •••, xn) x1 (x1 , 1, X1 ,2, •••, X2 ,i — 1, x2 ,i+ 1, •••, x2 ,n)
where x1,j is an action of player i against j ,Vi = j G N.
m
of stage game. m
time, just as they behaved in the first stage:
=(
2 2 2 -1, x2, •••, xn
= (x2,
x2 , x2 xi,1 , x
, x , •••, x2 , x2 , •••, x2 ) 1, xi,2, •••, xi,i —1, xi,i+1, •••, xi,n)
where x2,j is an action of player i against ^d x2,j G X2, j, Vi = j G N. m
x2 m = (x2 , x2 , •••, x2 , x2 , •••, x2 ) x \ {m} = (x1, x2, •••, xm—1, xm+1, •••, xn)
x2 = ( x2 , x2 , •••, x2 , x2 , •••, x2 ) xi (xi,1, xi,2, •••, xi,i —1, xi,i+1, •••, xi,n)
where x2 j is an action of player i against ^d x2 „• G Xj Vi = j G N \ {m}.
2
x
2
x
3. Cooperative game
N
Let x(-) denote the strategy of players in the game.
V 1(N') = min{C[T(N', Cxi)] + [pC[T(N' \ {m}, Cx2\W)] + (1 -p)C[T(N', Cx2)]]}
= C[T(N', Cgi)] + [pC[T(N' \ {m}, C,2\{m})] + (1 - p)C[T(N', C,2)]]
where p = )j' x»(^) is the cooperative strategy of player i, and the
strategy profile x(-) is called cooperative strategy profile.
N
m
V2(N' \ {m}) = min C[T(N' \ {m}, Cx2\W)] x2(0
= C[T(N' \ {m}, Cx2)]
where x2(^), i e N\{m} are the cooperative strategies, and the cooperative strategy-profile is x2(-).
m
V 2(N') =^in C [T (N ',Cx2)] = C [T (N ',C,2)]
where x2(-),i e N are cooperative strategies, and the cooperative strategy profile is x2(-).
3.1. The characteristic function in "pessimistic" game with spanning tree
By the Bird(Bird, 1976) method, the characteristic function for the coalition S ^ N is defined. If m e S,
V 1(S') = min{C[T(S, CS )] + [pC[T(S \ {m}, cS^})] + (1 - p)C[T(S, CxSS)]]}
xs (•) s S \ ^ J S
= C[T(S, Cfs)] + [pC[T(S \ {m}, Cg{{mm}})] + (1 - p)C[T(S, C*s)]]
where p = ^p^SBC™? Cf^' C5 and C5\{m} is the cost matrix restricted to S and
S \ {m}. if m e S,
V 1(S') = min [C[T(S, CSS)] + C[T(S, C5S)]] = C[T(S, C*s)] + C[T(S, C*s)]
xs (•) S S S S
where C5 Mid C5\{m} is the cost matrix restricted to S mid S \ {m}. m S
S'
V2(S') = min C[T(S',Cx5S)] = C[T(S',C*s)]
Xs (•) S S
where S c N, S' = S U {0} C5 is the cost matrix restricted to S'.
3.2. The characteristic function in "optimistic" game with spanning tree
The characteristic function for coalition S is defined as follows. If m € S, S C N, and S' = S U {0}
V!+(S') = min{C[T+(S, N' \ S, CS )] + [pC[T+(S \ {m}, N' \ S \ {m}, Cf^™}})]
+ (1 - p)C (T +(N ',CS2))]}
= C[T+(S, N' \ S, CSS)] + [pC[T+ (S \ {m}, N' \ S \ {m}, CS^{g})] +(1 - p)C[T +(S,N' \ S, C;?|)]]
where p = C[T+( j>NAm>cS )]' CS and CS \{m} is the cost matrix restricted to S' S' \ {m}
If m € S S C N, and S' = S U {0}
V 1+(S') = min [C[T +(S, N' \ S, Cg)] + C[T +(S, N' \ S, Cf2)]]
yS
C [T (S, N \ Cxl )] + C [T (S,N \ S,Cx2,
XS (■) 1 1
= C[T + (S, N' \ S, cSi )] + C[T + (S, N' \ S, CS2)]
hs'
where CS is the cost matrix restricted to S, CS is the cost matrix restricted to S'.
S
V2+(S') = min C[T +(S, N' \ S, Cfs)] = C[T +(S, N' \ S, CSs)]
xs (•) s s
where CS is the cost matrix restri cted to S'.
4. The Shapley value in the two stage "optimistic" game with spanning tree.
On the basis of the above discussion, we can define Shapley values in the two stage "optimistic" game with spanning tree.
Sh1+(N', c ) = 1E [v 1+(s; (i) u{i}) - v 1+(s; (i))]
' »en
where n denotes the set of all permutations on N, aid Sn(k) = {i|n(i) < n(k)}. In the similar way, the Shapley value in subgame is defined as follows. m
Sh2+(N' \{m},C) = E [V2+(s;'(i) u{i}) - V2+(S;'(i))]
( )! n'en'
where n' denotes the set of all permutations on N \ {m}, aid Sn'(k) = {i|n'(i) < n'(k)}.
m
Sh2+(N', c ) = 1 E [V 2+(sn (i) u{i}) - V 2+(s; (i))]
' nen
where n denotes the set of all permutations on N, aid Sn(k) = {i|n(i) < n(k)}.
5. Results Theorem 1.
C[T + (S, N' \ S, Cxs)] < C[T(S', N \ S', Cxs)], S C N where CxS is the cost matrix defined by strategy profile
Proof. By definition 9, there are only two possible forms of the minimum cost spanning tree T + (S, N'\S, CxS). The first case is the coalition S connected directly to the source; the second case is connected via one or several Vertexes in the coalition
N\S
Case 1: if the coalition S connected directly to the source {0}, the minimum cost spanning tree T + (S, N' \ S, CxS) is over S' = S U {0}. On the graph G(S', E) with cost matrix CxS, the "optimistic" minimum cost spanning tree is the same to "pessimistic" minimum cost spanning tree, T+(S, N' \ S, CxS) = T(S, N' \ S, CxS). i.e.
C[T +(S, N' \ S, Cxs )] = C[T(S', N \ S', CxS )]
Case 2: if the coalition S connected with the source {0} via one or several Vertexes in the coalition N\S over N', consider using the converse method, such that the total cost of T + (S, N' \ S, Cxs) is larger than the total cost of T(S, N' \ S, Cxs). i.e.
C[T + (S, N' \ S, Cxs)] > C[T(S', N \ S', Cxs)], S C N
Then the coalition S can achieve the case where it is connected to source {0}
{0}
edges associated with the vortexes in N \ S, i.e. such case (C[T + (S, N' \ S, Cxs)] > C[T(S', N \ S', CxS)], S C N) does not exist.
The theorem is proven. □
Theorem 2.
C[T +(S, N' \ S, Cxs)] < C[T +(S, N' \ {m} \ S, Cxs)], S C N where Cxs is the cost matrix defined by stra tegy profile
Proof. Two cases are considered in the minimum cost spanning tree T +(S, N' \ S, Cxs) obtained on the graph G(S', E) with cost matrix Cxs: the first case, in which the help of the player m is needed, i.e. the minimum spanning tree T+(S, N'\S, Cxs) mm Case 1: if the coalition S connected to the source {0} with the help of the player m, the minimum cost spanning tree T +(S, N' \ S, Cxs) can be represented as
T + (S, N' \ S, CxS) = arg min {
( , ( , ) (i,j) e G(S,E)
+ Y" c0 , + c0 ,i
' / y "-oo ' ^ om' J
(o,o ' ) £ G(N '\{m},E),o £ S,o ' £ N '\S\{m}
where Gxs is ^^e cost matrix defined by strategy profile
After the player m leaves the game, there will be three possible changes to the new minimum cost spanning tree T +(S, N' \ {m} \ S, CxS ).
— If the total cost of the minimum cost spanning tree T +(S, N' \ {m} \ S, Cxs)
m
minimum cost spanning tree T +(S, N' \ S, Cxs), i.e.
C[T +(S, N' \ S, Cxs)] = C[T +(S, N' \ {m} \ S, Cxs)], S C N
— If the total cost of the minimum cost spanning tree T + (S, N' \{m}\ S, Cxs) that
m
minimum cost spanning tree T +(S, N' \ S, Cxs), i.e.
C[T +(S, N' \ S, Cxs)] < C[T +(S, N' \ {m} \ S, Cxs)], S C N
— If the total cost of the minimum cost spanning tree T +(S, N' \ {m} \ S, Cxs)
m
minimum cost spanning tree T +(S, N' \ S, Cxs), i.e.
C[T +(S, N' \ S, Cxs)] < C[T +(S, N' \ {m} \ S, Cxs)], S C N
By Definition 9, it means that the minimum spanning tree game T +(S, N' \ S, Cxs) for coalition S is not the minimum total cost, since a much less total
m
the statement of the theorem. Therefore this case does not exist.
S
{0} m
C[T +(S, N' \ S, Cxs)] = C[T +(S, N' \ {m} \ S, Cxs)] The theorem is proven. □
6. Example
The set of players is N = 1,2, 3, the source is {0} N' = N U {0}. Assume that player 3 may leave the game after stage 1. As shown in Fig. 7, there are initial costs of the edges between each player.
Fig. 7. The graph G(N, E) with initial costs of the edges The initial cost matrix C0 is
C0 =
0 1 2 3
0 0 12 10 15 \
1 12 0 1 2
2 10 1 0 2
3 \ 15 2 2 0
Table 2. Strategy set
C12 X2,1 Cl3 X3,i C23 X3 2
2 4 6 2 3 7 1 4 6
1 2 4 6 2 4 6 14 3 3 12 18
3 6 12 18 Xi,3 3 6 9 21 X2,3 6 6 24 36
5 10 20 30 5 10 15 35 7 7 28 42
The strategy set for each player is shown in Tab. 2.
Assume the function fc = xi,j x xj-,i. For example, if player 1 choose action x1,2 = 3 and player 2 choose action x2,1 = 4. The cost of edge (1, 2) will be
C12 = fc(x1,2,x2,1) = x1,2x2,1 = 12.
The cooperative strategies in the game are:
x1 = (x1,x2) = ((1, 2), (1, 2)) x2 = (x2,x2) = ((2, 3), (2, 3))
x3 = (x3,xl) = ((2,1), (2,1))
The probability of player 3 leaving the game is p = 0^467 and V 1(N) = 28.599.
The characteristic function for each coalition and the Shapley value in two-stage "optimistic" game with spanning tree:
V 1+({1}) = 4, V 1+({2}) = 4, V1+ ({3}) = 4^599
V 1+({1, 2}) = 10, V 1+({1, 3}) = 10132, V 1+({2, 3}) = 8^599
Sh1+(N', C) = 9^922, Sh2+(N', C) = 9156, Sh^+(N', C) = 9142
Similarly, the characteristic function for each coalition and Shapley value in subgame:
m
V2(N') = 15, V2+({1}) = 2, V2+({2}) = 2
V2+({3}) = 3, V2+({1, 2}) = 4, V2+({1, 3}) = 5, V2+({2, 3}) = 4 Sh2+(N', C) = 5, Sh2+(N', C) = 4^5, Sfr|+(N', C) = 5^5
m
V2(N') = 12, V2+({1}) = 2, V2+({2}) = 2
V2+({3}) = 0, V2+({1, 2}) = 12, V2+({1, 3}) = 0, V2+({2, 3}) = 0 Sh2+(N', C) = 6, Sh2+(N', C) = 6, Sh3+(N', C) = 0
7. Conclusion
This paper considers the Shapley value in a two-stage "optimistic" game with a
S
S
value in the "optimistic" game with spanning tree is given. Several theorems and examples are provided.
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