CC BY
5Contributions to Game Theory and Management, XIV, 273-289
Dynamic Shapley Value in the Game with Perishable Goods*
Yin Li12, Ovanes Petrosian1 and Jinying Zou1
1 St. Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg 199034, Russia, 2 School of Economies and Management, Yanan University, Yan'an, 716000, China E-mail: liyinrus@outlook.com, petrosian.ovanes@yandex.ru jesco4482@gmail.com
Abstract The paper investigates two-stage stochastic minimum spanning tree games with perishable goods. The cooperative behaviour of the players is defined. At each stage, all players jointly take action to construct a network with a cost matrix. At the second stage, a particular player may leave the game, and the probability of this leaving depends on the cooperative behaviour of all players at the first stage. At each stage game, the total cost of the spanning tree is calculated to include the sum of the costs of the contained edges and the cost of the loss of perishable goods expended on that edge of the spanning tree. The characteristic functions in the game are considered, and the dynamic Shapley values are modified. The time consistency of the dynamic Shapley values is studied.
Keywords: dynamic games, minimum cost spanning tree game, stochastic games.
1. Introduction
This article studies two-stage stochastic minimum cost spanning tree games with perishable goods. Perishable goods are those products whose quality or value decreases over time, such as vegetables, flowers or seafood. Due to differences in the timing of transport and storage methods, perishable goods are more complex in terms of cost loss and risk to players as they are transported through the logistics network. In China, the loss of perishable food products has been calculated to be between 20% and 30% of the production costs of the companies involved (Wan and Cao, 2019). Research on the loss of perishable goods can be traced back to the 1960s and has been systematically reviewed by Nahmias and Janssen et al. in the field of inventory control of perishable goods (Nahmias, 1982; Janssen, 2016). The minimum cost spanning tree game is a classical model of cooperative games derived from the minimum cost spanning tree problem and was formally proposed by Claus and Kleitman (Claus, 1973) in 1973. The most famous solution to the minimum spanning tree game is the Bird allocation proposed by Bird (Bird, 1976). Dutta and Kar (Dutta, 2004) proposed the Dutta-Kar allocation, which satisfies cost monotonicity and is in the core. The literature (Bird, 1976) also introduced the concept of an irreducible core. Another important solution is called the Folk allocation (Bergantinos and Vidal-Puga, 2007) which is the Shapley value of the game corresponding to the irreducible graph.
In dynamic cooperative game theory, the time consistency of the solution is a crucial concern. That is, players should be able to make an agreement at the beginning of the game so that at the end of the game, they receive the gains specified
*This work was supported by the China Scholarship Council (No.201508090030). https://doi.org/10.21638/11701/spbu31.2021.20
in the solution, as previously agreed. In the case of time inconsistency, the players in the game may break the initial agreement. In the literature (Li, 201C), the case in which a minimum spanning tree game was combined with a stochastic game was considered for the first time. It considered the construction of dynamic Shapley values in a two-stage minimum cost spanning tree game. At each stage, when the players take different strategies, the costs of the edges constructed between the players in the network may be different, and thus the associated minimum cost spanning tree may be different, and the cost-sharing solutions obtained will be different. It is assumed in the research that all players in the game adopt cooperative behaviour in order to minimise the expected sum of their costs. The characteristic function is defined. The dynamic Shapley value in the game is constructed, and the problem of time consistency of the solution is considered. Further, in the literature (Li, 2017), a two-stage stochastic minimum cost spanning tree game with perishable goods is discussed.
In the goods transportation model, the valne of a good varies following its utility as the shipping time increases: see Fig. 1. Three cases are possible:
— The bine enrve shows that the good's utility remains the same with an increase in the shipping time (umbrellas, plastic toys for children, etc.).
— The green curve shows that the good's utility grows with an increase in the shipping time (wine, antiques, etc.).
— The red curve shows that that the good's utility drops with an increase in the shipping time. This applies to perishable goods, i.e., the ones becoming cheaper over time (flowers, fruits, etc.).
The perishable goods can be divided into two groups:
1. The goods with a fixed expiration date, without any valne loss on a certain period of time:
2. The continuously perishing goods, with a valne decreasing proportionally to the shipping time.
The second group of perishable goods is studied in this paper.
Product's utility
0
Time
Fig. 1. Three types of goods with different, utility dynamics.
This article follows the research (Li. 2017). At each stage of the game, the players act together to construct a network, and a cost matrix of the network is obtained. The network is built on a set of sources and vertices which represent the players. In the research (Li. 201G), the minimum cost spanning tree is obtained after defining this network with the cost matrix, using algorithms based on classical graph theory (Dijkstra, 1959: Ivruskal, 1956: Prim. 1957). In contrast, in the minimum cost spanning tree games with perishable goods, the total cost of the spanning tree is calculated to include the sum of the costs of the contained edges and the cost of the loss of perishable goods expended on that edge of the spanning tree. It means that the loss of perishable goods is generally less important for the same player on two different spanning trees when the vertex of the player is closer to the source. After the first stage, a particular player leaves the game with a probability that depends on the cooperative behaviour of all players in the first stage. The total cost to the players in this two-stage game is the sum of their costs in the first and second stages. All players select cooperative actions. We consider the strategy profiles of players with the minimum expected total cost of two-stage game as the cooperative strategy. The characteristic functions in the game are considered, and the dynamic Shapley values are modified. The time consistency of the dynamic Shapley values is studied.
2. The Model
First of all. we describe the game model. When defining the two-stage game, we begin with the one-stage game, which is a spanning tree game. Following (Li. 201G), introduce the main definitions and necessary background from graph theory.
Assume that n players are in n different positions. To obtain perishable goods, the players directly or indirectly connect to the source. After the first stage, player m G N can leave the game with a probability p depending on the previous behavior of players. If player m leaves the game, the set of players changes. Otherwise (player m
diagram of this game is presented in Fig. 2.
Fig. 2. The diagram of the game.
Definition 1. Let N = {1,..., n} be a finite set of players, {0} be the source, and N' = N U {0}.
Definition 2. A graph defined on the set N' will be denoted by G(N', E), where E = {(i,j) : (i,j) e L C N'}.
Definition 3. A pair (i,j) is called an edge in a graph G(N',E) if (i,j) e E, Vi, j e N.
Definition 4. Two vertices i mid j e N' are said to be connected in a graph G(N', E) if 3(ii, i2), (i2, is), ..., (in-1, in) : (ifc, ifc+i) e G(N', E), 1 < k < n - 1,
ii = i, and i„ = j.
Definition 5. A graph G(N', E) is said to be connected on the set N' if any two vertices i,j e N' are connected in G(N', E).
Definition 6. The set of all connected graphs on the set N' will be denoted by Gn ' ■
Definition 7. Associate with any md j in a graph G(N', E) cost cj on the edge (i, j), i = j e N', where cj = j e K+ and Vi e N' : cu = The cost cjo = coj ( i, 0) , i e N,
(i, j)
the consumption of some resource (e.g., fuel consumption when shipping a cargo ij ij
The cost on a graph G(N', E) can be represented as a cost matrix
C
{cij }(n+1)x(n+1)-
(1)
Note that a cost matrix is nonnegative and symmetric and has order (n +1) x (n + 1).
Example 1. Consider an example shown in Fig. 3. The red vertex 0 is the source, and the four vertices indicated by green numbers are players. In this example, the graph is complete, and each edge has a specific cost. The corresponding cost matrix is given by
C = {cij }
ij }5x5
0 1 2 3 4
0 ( <x> 6 3 15 11 \
1 6 8 5 10
2 3 8 4 12
3 15 5 4 9
4 11 10 12 9 y
, Vi, j G {0,1, 2, 3, 4}
2.1. Stage Game with Spanning Tree
Definition 8. (Li, 2016) At each stage, player i chooses a strategy vector(Li, 2016)
xi (xi, 1, • • • 7 xi,i— 1 : xi,i+1, • • • 7 ^in')?
where xi,j G Xijj denotes an action of player i against player j, Vi, j G N. Similarly, xji G Xj,i is an action of player j against player i.
Fig. 3. Cost 011 complete graph with four players.
Definition 9. (Li, 2016) The cost on an edge (i, j) of a graph G(N', E) is defined in the following way:
Cij = Cji = fc(xi,j,Xj,i), Ci0 = C0i > 0, Vi, j G N. (2)
where a function fc maps the set of all strategies of players i, j into the set R+ U 1+^} of all admissible costs on the edges (i, j) G E.
In other words, the cost on an edge cij = Cji is completely defined by the strategies xi,j, Xj,i of players i, j, where xi,j G Xi,j,Xj,i G Xj,iy Vi,j G N. The
X=
(xi,..., xn). Note that Cx = {cij}(n+i)X(n+i) is a unique cost matrix constructed by Definition 9. Thus, for a cost matrix Cx = {cij}(n+1)X(n+1), there is a unique graph G(N',E) G Gn' on the set N'. Here Gn' denotes the set of all possible connected graphs on the set N'.
Fig. 4. Graphs with different, costs on edges corresponding to different strategies of players.
Example 2. Consider a three-player game. Let the function /c be /c = x^j x x^j, where x^- e Xj xj,» e Xj,». If players 1-3 choose the strategy profiles
x1 = (x1,2, x1,S) = (3, 10), x2 = (x2,1, x2,S) = (4, 2), xS = (x3,1, xS,2) = (1, 2),
respectively, we easily calcnlate the cost on all edges between the players. In the left-hand part of Fig. 4, the red edges are obtained under the strategy profile (x'1; x2, x3). For example, for the edge (1, 2), we have c12 = c21 = x'12 x x2,1 =3 x 4 = 12. When choosing the strategy profiles
x1 = (x1,2; x1,S) = (5, 1)j x2 = (x2',1,x2',3) = (5, 2)i xS' = (x3',1; xS',2) = (1, 3)
the costs on all edges between the players varies. In the right-hand part of Fig. 4, the blue edges are obtained under strategy profile (x", x2, x3'). For example, for the edge (2,3), we have c2s = cs2 = x2',s x xS',2 = 2 x 3 = 6.
N
T (N',Cx) = argjnin cjj,
where Cx = {cjj}(n+1)X(n+1) denotes a cost matrix on a graph G(N',E), which is uniquely defined by the strategy profile x = (x1;..., xn). Here GN' denotes the set of all possible connected graphs on the set N'.
Definition 11. (Li, 2016) The total cost on the edges of a minimum cost spanning tree T(N', Cx) is given by
C [T (N',CX)]= £ cjj.
(j,j)£T (N ',Cx)
Obviously, a graph with a minimum cost represents a tree. Otherwise, we would eliminate an extra edge to get a connected graph with a smaller cost.
Example 3. As an illustrative example, consider a one-stage three-player game. Let N = {1,2, 3} be the set of players, {0} be the source, and N' = N U {0}. The
N (0, 1) (0, 2),
and (0,3) are fixed, and the cost on these edges is c01 = c10 = 30, c02 = c20 = 51, and c0s = cs0 = 16 respectively. The function cjj- = /c(xj,j-,xj-,j) has the form
fc(xi,j ,xj,i) xi,j x xj,^ Vi, j e N-
(i, j), Vi = j e {1, 2, 3}, Here the red, green, and blue colors indicate different edges. Let X2,s = {2,21} be the set of actions of player 2 against player 3, and Xs,2 = {12,19} be the set of actions of player 3 against player 2, where x2,s e X2,s and xs,2 e Xs,2. If player 1 chooses the action x2,s = 21, md player 2 chooses the action xs,2 = 12, then the cost makes up c2s = /c(x2,s, xs,2) = 21 x 12 = 252.
Different strategies result in different cost matrices, and a minimum spanning tree can be obtained for each cost matrix. Assume that the players choose their strategies, and the strategy profile in the game is
x1 = (x1,2, x1,s) = (6, 3), x2 = (x2,1, x2,s) = (7, 2), xs = (xs,1, xs,2) = (9,12).
Table 1. All possible costs 011 edges under different actions of players
C12 X2,1 C13 X3,i C23 X3,2
7 10 9 11 12 19
6 42 60 Xi,3 3 27 33 X2,3 2 24 39
17 119 170 15 135 165 21 252 399
Fig. 5. Minimum cost spanning tree for graph constructed by players.
The graph with cost corresponding to the strategy profile (9,12) is shown in Fig. 5.
The cost matrix takes the form
0 1 2 3
0 (<x> 30 51 16
1 30 42 27
2 51 42 24
3 16 27 24 )
Cx —
After obtaining the cost matrix, the minimum cost spanning tree T(N',Cx) on the graph consists of edges (0,3), (1,3), mid (2,3); see dotted lines in Fig. 5. Therefore, the total cost in the minimum cost spanning tree T(N',Cx) under the strategy profile x — (x^x2,x3) is
where xi — (xi,2,xi,3)
C[T(N', Cx)] — 67, (6, 3), x2 — (x2,i, x2,3) — (7, 2), x3
(x3,i ,x3,2) — (9,12).
Definition 12. A route in a tree T(N', Cx) is a sequence of vertices ¿i, i2,..., iK and edges (ik, ik+i) for all k G [1, K — K G N+, K > 2. We say that this route passes the edges (ik, ik+i) and the vertices ii, i2,..., iK. If a route ii, i2,..., iK in a tree T(N', Cx) passes none of the edges (ik, ik+i) twice, it is called a path. In this case, vertex ii is called the start vertex of the path, and vertex iK the end vertex of the path. In a tree T(N', Cx), we denote by Pi1 iK a ^^^h fem vertex ii to vertex iK, md by Pm0 a path from any vertex m G N to the source {0}.
Definition 13. Vertex j is called a predecessor of vertex m if Pj0 C Pm0, where Pj0 is a path from s ource {0} to vert ex j in a tree T (N', Cx), j G N.
Definition 14. Consider a tree T(N', Cx). Assume that for any vertex m G N, a path Pm0 from the start vertex m to the end ve rtex {0} passes vert ices mi5 m2,..., mK and edges (mk, mk+i), where k G [1, K — 1], mi — m, mK — {0}, K G N+, and K > 2. Then vertex mk is called a direct predecessor of vertex mk+i.
m P(m)
m m P(m) — m
Definition 15. For any subtree T(S, CS) of a tree T(N',Cx), S C n, consider vertex i G S. If after deleting this vertex i in the subtree T(S, CS), for all j G S\ {i} the path Pj0 does not exist, then the set of all edges of the subtree T(S, CS) is called a branch in the tree T(N', Cx). This set will be denoted by B\
Definition 16. Vertex i G N is said to be terminal in a tree T(N',Cx) if there does not exist a vertex j G N \ {i} such that P0i C P0j mid P0j G T(N ', Cx).
{0}
tree T(N', Cx). The following can be established directly by definition.
— All blue edges (0, j), (j, j2), (j2, j3) make a path Pm0 from source {0} to vertex m
ji , j2 , j3 m j3
is a direct predecessor of vertex m, i.e., P(m) — j3.
— Let S be the set of all red vertices, and m G ^^en Bm is the set of all edges in the yellow domain.
Fig. 6. Path Pmo, direct predecessor of vertex to, predecessor of vertex to, and branch Bm.
m
second stage is calculated as
_ ^2(i,j)eBm Cij
P — C[T(N', Cx)],
where the minimum cost spanning tree T(N ',Cx) is given by Definition 10 on the graph G(N ', E) constructed under the strategy profile x = (x1,..., xn). Due to 15, Bm is a branch in T(N', Cx).
2.2. Minimum Cost Spanning Tree Game with Perishable Goods
i
LV = qj(1 - e Al), Vi e N, (k,1) e Pi0, (3)
qj e (0, +ro),0j e (0, +TO)
where Pj0 denotes the path from source {0} to vertex i, aid qj is the cost of perishable goods of player i. In addition, is the loss parameter of player i due to Vi e N
According to formula (3), the longer the path Pj0 from vert ex i to source {0} is,
i
Different from Definition 10, the definition of a minimum spanning tree with perishable goods is given below.
Definition 19. (Li, 2017) A minimum cost spanning tree with perishable goods on N
T6(N', Cx) = arg .min { V cj + V qj(1 - e ^ )},
GGGn'
N (j,j)GG(N ',E) (k,l)€Pio,(k,l)€G(N ',E),j€N
(4)
where Cx = {cjj-}(n+1)X(n+1) denotes a cost matrix on a graph G(N',E), and the cost matrix Cx itself is determined using a strategy profile x = (x1,..., xn).
Definition 20. (Li, 2017) The total cost on the edges of a minimum cost spanning tree with perishable goods T6(N ', Cx) is given by
C[T6(N',Cx)]= £ cjj + £ qj(1 - e-^),
(j,j)£T e (N ',Cx) (k,l)ePi0,(k,l)eG(N ',E),j£N
where x = (x1,..., xn) denotes the strategy profile of all players.
If player m leaves the game, the set of players will be N \ {m}, and N' \ {m} = N \ {m} U {0}.
Definition 21. (Li, 2017) A minimum cost spanning tree with perishable goods on the set N ' \ {m} is the tree
t6(n' \ {m},Cx\{m}) = arg mm { cjj
G£GN'X{m} (j,j)€G(N'\{m},E)
+ qj(1 - e Ecfci )},
j£N\{m},(k,l)ePi0,(k,l)eG(N '\{m},E)
where x \ {m} = (x1,..., xm-1, xm+1,..., xn) denotes the strategy profile of all
m
Definition 22. (Li, 2017) The total cost on the edges of a minimum cost spanning tree with perishable goods T6(N' \ {m}, Cx\{m}) is given by
C[T6(N ' \{m},CxUm})]— £ Cij
(i,j)£T 0 (N '\{m},C,\{m))
+ 2^ qi(1 —e Ecfci)
ieN\{m},(k,i)ePi0,(k,i)eT 0(n '\{m},cxUm))
where x \ {m} — (xi,..., xm-i, xm+i,..., xn) denotes the strategy profile of all
m
Pm
Definition 17.
3. Description of the Game
Stage 1: At the first stage, the spanning tree game with perishable goods represents a simultaneous n-player game; see Fig. 2. Therefore, at the first stage in vertex zi, the players simultaneously choose their strategies in the form of n-dimensional profiles
x — (xi, . . . , xn),
xi (xi, i, . . . , x i ,i — i, xi,i+i, . . . , xi,n),
Vi — j G N,
where x^- G Xij denotes an action of player i against player j. m
z i, the spanning tree game with perishable goods evolves to the second stage in vertex z2; see Fig. 2. All players again simultaneously choose their strategies in the n
x2 — ( x2 , . . . , x2 ) ,
x (x i, . . . , xn),
x2 — ( x2 , . . . , x2 , x2 , . . . , x2 ) ,
xi — (xi, i, . . . , xi,i— i, xi,i+ i ,...,xi,n), V'/
Vi — j G N,
where x2,j G X?j ^^^^^^s an action of player i against player j.
Then the cost matrix Cx2 — {cij}(n+ :L)x(n+ i)■, Vi — j G N is formed on the graph G(N', E) by the strategy profiles (7); see Definition 9.
If player m leaves the g^e, the set of players will be N\{m}. Then the spanning tree game with perishable goods evolves to the second stage in vertex z3; see Fig. 2.
m,
(n — 1)
x2 m — (x2 , . . . , x2 , x2 , . . . , x2 ) , x \ {m} — (x i, . . . , xm— i, xm+ i, . . . , xn),
x2 — (x2 , . . . , x2 , x2 , . . . , x2 ) , xi — (xi, i, . . . , xi,i— i, xi,i+ i, . . . , xi,n),
Vi — j G N \ {m},
where x2,j G Xi2,j ^^^^^^s an action of player i against player j.
Then the cost matrix Cx2\{m} — {cij}nxn, Vi — j G N \ {m} is formed on the graph G(N' \ {m}, E) by the strategy profiles (3.); see Definition 9.
4. Cooperative Game
Assume that in the two-stage game, the cost on the edges between the players is equal to the sum of corresponding costs at the first and second stages. Consider a cooperative version of the game in which all players jointly choose their strategies to minimize the expected total cost on edges.
N.
z1, z2 is realized during the game with perishable goods. Thus, xj(z1) = x1 and xj(z2) = x2, i e N.
N
V 1(N')
= min |C[T6(N', Cxi)] + [pC[T6(N' \ {m}, Cx2\{m})] + (1 - p)C[T6(N', C^)]] } = C[T6(N', Cxi)] + [pC[T6(N' \ {m}, C,2\{m})] + (1 -p)C[T6(N', C,2)]
where p = ^'¿(N3' )]. The strategies xj(-), i e N, are said to be cooperative, and x(-) = (x1,..., xn) is called a cooperative strategy profile. In addition, C[T6(N', Cx)] and C[T6(N' \ {m}, Cx\{m})] denote the total cost on the edges in the perishable goods trees T6(N', Cx) and T6(N' \ {m}, Cx\{m}), respectively (see Definition 22).
Definition 24. Define the characteristic function for the coalition S C N, S' = S U {0}, in the following way.
If m e S, S' = S U {0}, and xs(■) = {xj(-), i e S}, then
V 1(S') = min {C[T6(S',C< )]
xs (•) ^ S
+
pC[T6(S' \ {m}, eg\{{I}})] + (1 - p)C[T6(S', eg)]] } C[T6(S', eg)] + [pC[T6(S' \ {m}, Cg^™})] + (1 - p)C[T6(S', eg)]
's'
If m e S, S' = S U {0}, and xs(■) = {xj(-), i e S}, then
V 1(S') = min {C[T6(S',CSS)] + C[T6(S',CgS)]} = c [T6 (S', cgi)] + c [T6 (S', eg )],
where p = and CS' or or CS'\{m} denotes the cost matrix restricted
C[T8 (S',CS1 )]
xs
to S' or S' \ {m}. xS(■) are strategy profiles of coalition S.
-,S'\{m} Jxs\{m}
yxs ) 311(1 V" \{""}, ^xs\{m}
In addition, C[T6(S', Cgs )] and C[T6(S' \ {m}, cg\{{m}})] denote the total cost on the edges in the perishable goods trees T6(S', Cgs) mid T6(S' \ {m}, Cg Hm)>
respectively (see Definition 22). If S = 0, then
V1 (0U{0}) = 0.
N
a similar way.
Definition 25. If player m leaves the game after the first stage, then
V2(N' \ {m}) = min C[T6(N' \ {m}, Cx2\W)] = C[T6(N' \ {m}, Cg2\{m})j,
where the strategies x2(^), i G N \ {m}, are said to be cooperative, and x2(-) = (x2,...,xm_ i i,..., xnn) is called a cooperative strategy profile. In addition, C[T6(N' \ {m}, Cx2\{m})] denotes the total cost on the edges in the perishable goods tree T6(N' \ {m},Cx2\{m}) (see Definition 22). m
V2(N') = min C[T6(N', Cx2)] = C[T6(N',0^)], (8)
where the strategies x2(-), i G N, are said to be cooperative, and x2(-) = (x 1 ,..., x^) is called a cooperative strategy profile. In addition, C[T6(N', Cx2)] denotes the total cost on the edges in the perishable goods tree T6(N', Cx2) (see Definition 22).
The characteristic function for the coalition S C N, S' = S U {0}, at the second stage is defined in a similar way:
V2(S') = minC[T6(S', C|2)] = C[T6(S', CS2)], (9)
where S C N, S' = SU {0}^d CS denotes the cost matrix restricted to S'. xS(•)
S
In addition, C[T6(S', CS2)] denotes the total cost on the edges in the perishable
xs
goods tree T6(S',CS2) (
see Definition 22).
Xs
If S = 0, then
V 2(0U{O}) =0.
5. Dynamic Shapley value
Define the dynamic Shapley value, in the game itself and in each subgame. Definition 26. The Shapley value has the form
Shi(N') = ^ E [V 1 (s;w u {i}) - V 1 (S;w)],
where Vi G N, S' = S U {0}, S C N \ {i}, N' = N U {0}, n denotes the set of all permutations on N^d Sn(k) = {i|n(i) < n(k)}.
m
Sh2(N ' \{m}) = ^1T^ E [V2(S4'(i) U{i}) - V2(S;'(i))],
where Vi G N \ {m}, S' = S U {0}, S C N \ {m, i}, N' = N U {0}, ff' denotes the set of all permutations on N \ {m}^d Sn'(k) = {i|n '(i) < n '(k)}. m
sh2(N') = 1E [v 2(s;w U {i}) - V 2(s;w)],
' »en
where Vi G N, S' = S U {0}, S C N \ {i}, N' = N U {0}, n denotes the set of all permutations on N^d Sn(k) = {i|n(i) < n(k)}.
X
Definition 28. The Imputation Distribution Procedure for the Shapley value in the two-stage perishable goods game is a payment scheme P = (P1, P2) given by
P1 = Sh1(N') -pSh2(N' \ {m}) - (1 -p)Sh2(N'), P2 = pSh2(N' \ {m}) + (1 -p)Sh2(N'),
where p = g^efw3 C ^ is the probability that player m will leave the game at the second stage.
The dynamic Shapley values based on the Imputation Distribution Procedure (IDP) is constructed (Petrosyan, 1979).
Definition 29. The Shapley value Sh1 is said to be time-consistent in the perishable goods game if there exists a nonnegative IDP (P1 > 0, P2 > 0, Vi e N) such that
Sh1(N') = P1 + pSh2(N' \ {m}) + (1 -p)Sh2(N'), pSh2(N' \ {m}) + (1 - p)Sh2(N') = P2,
where p = ^^¿(v3 C is the probability that player m will leave the game at the second stage.
P
goods game.
Proposition 1. The IDP P for the dynamic Shapley value Sh1 = (Sh^,..., Sh^) constructed by Definition 28 is time-inconsistent.
For proving this result, we provide an example when the IDP for the dynamic Shapley value is time-inconsistent.
Exam,pie 5. Consider a two-stage perishable goods game of two players as follows. N = {1, 2}, {0} is the source, and N ' = N U {0}. Assume that player 2 may leave the game after the first stage. This game is shown in Fig. 7.
The cost on the edges (0,1), (0, 2), the cost of perishable goods and the loss parameters are shown in Table 2.
Table 2. The cost on the edges (0,1), (0, 2) and other parameters for perishable goods
Player 1 Player 2
C01 C10 C02 C20 qi 0i q2 02
80 80 10 10 4 10 2 15
Suppose that the sets of actions of player 1(2) against player 2(1) are
X1 , 2 = {2, 4},X2 ,1 = {3, 5}. The function fc has the form fc = x1 , 2x2 ,1, where x1 , 2 e X1 , ^d x2 ^ e X2 ,1.
1 clz =fc (X1¡2,x2il) 2 (m)
Fig. 7. Two-stage perishable goods game of two players.
For this two-stage game, we obtain the following cooperative strategy of the player 1: xi = (xi,x i) = (2,2). And cooperative strategy of the player 2: x2 = (x2,x2) = (3,3).
In two-stage games and the subgame, the characteristic functions S ^ N and the Shapley values of the players are shown in Table 3.
Table 3. The characteristic functions S ^ N and the Shapley values of players(p = 0.375).
Stage 1
V 1(N ') V 1({1}U{0}) V 1({2}U{0}) Sh1(N ') Sh2(N ')
61.7219 160.94 23.1074 99.7772 -38.0553
Stage 2(if player 2 leaves the game).
V to N V 2({1}U{0}) V 2({2}U{0}) Sh2(N ') Sh2(N ')
80.47 80.47 0 80.47 0
Stage 2(if player 2 stays in the game).
V 2(N ') V 2({1}U{0}) V 2({2}U{0}) Sh2(N ') Sh2(N ')
19.4127 80.47 11.5537 44.1645 -24.7518
As a result, we obtain the IDP
# « 32.92,^1 « -28.77,^2 - 66.86, « -9.28.
Obviously, the dynamic Shapley value in this example is time-inconsistent.
Theorem 1. In the two-stage game, there exists some player m that may leave the game at the second stage with a probability depending on the choice of all players at the first stage. The dynamic Shapley value constructed by Definition 26 is time-
consistent if, for each stage,
P = 0,
£ c[Te(s'u{i},cSku{i})] > £ c[Te(s',cffc')]
u{i})] ^ v^ C[Te (S ' Cs'
SCN\{i} SCN\{i}
(10)
S' = S U {0},Vi G N,k G {1,2},
where p = CfT%NvBc' ?)] ^ probability that player m will leave the game at the second stage, and (■) are strategy profiles of coalition S.
Proof Using Definitions 27 and 28, we first calculate P2:
p2 = pSh2(N' \ {m}) + (1 -p)Sh2(N') = 0Sh2(N' \ {m}) + (1 - 0)Sh2(N')
= Sh2(N') = nlj £ [V2(s;w U{i}) - V2(s;w)], vi g n, s c n\{i}, s' = su{0}.
' »en
Since p = 0, player m stays in the game at the second stage. For (9) we apply another (equivalent) formula for calculating the Shapley value (Shapley, 1952):
sh2(N') = n £ [cp*(s;w u {i},cSn(i>u{i})] - cp*(s^cJ«)]
' nen
= y |S|!(n -|,S|- 1)! [c[Te(S ' U {i}, CS2u{i})] - c[Te(S ',c|2')]■
^ ^ 71 I
SÇN\{i}
|S|!(n -|S|- 1)!
£ C[te(s;(i) u {¿},cSn<i)U{i})]
SÇN\{i}
- £ C[Te (S',Cf2 )]!, Vi G N, S ' = S U{0}.
-yS' = 2 Ls
SÇN\{i}
From conditions (10) we obtain
£ c[Te(S' u{i},cf2U{i})] > £ c[Te(s ', cSS)]
SÇN\{i} SÇN\{i}
p2 = sh2(N') = 1 £ [v2(s;(i) u {i}) - v2(s;w)] > 0,
where Vi G N, S Ç N \ {i}, S' = S U {0} n denotes the set of all permutations on N, Sn(k) = {i|n(i) < xS(•) = (Xj, i G S) are strategy profiles of coalition
S
Next, we consider p1, Vi G N:
Pi = Sh1(N') - pSh2(N' \ {m}) - (1 - p)Sh2(N') = Sh1(N')-0pSh2(N ' \{m}) - (1 - 0)Sh2(N') = Sh1(N') - Sh2(N').
Using the equivalent formula for the Shapley value together with Definitions 26 and 27, we find:
Sh1(N') - Sh2(N ')
288 Yin Li, Ovanes Petrosian, Jinying Zou
1 . . r. ^ ^rl/nl 1 nrllQl I, T/-2/C'
= n E[vl(sn(i)u{i})-vl(s;(i))j--]t[v2(s;(i)u{i})-v2(s;(i
■wen ' »en
E |S|!(n -n',S|- 1)! [V 1(S' u{i}) - v l(s')]
SCN\{i} '
- E |S|!(n -|s|-1)! [V2(S' u{i}) - V2(S')]
n!
SCN\{i}
= y- |s|!(n -|s|- 1)! [V 1(S' u{i}) - Vl(S') - V2(S' u{i}) + V2(S')], (12) n!
SCN\{i}
where i G N, S' = S u {0}, S C N \ {i}, N' = N u {0}, n denotes the set of all permutations on N, and Sn(k) = {i|n(i) < n(k)}.
We utilize the characteristic function defined as described before. For S C N, if m G S, S' = S u {0}, then
Vl(S') = C[T6(S',C|S)] + [pC[T6(S' \ {m^CX^})] + (1 - p)C[T6(S',C|S )]
= C[T6(S', CSS)] + [0C[T6(S' \ {m}, CX^g)] + (1 - 0)C[T6(S', C|S )]
= C[T6(S', Cg)] + C[T6(S', C|)]. (13)
If m G S, S' = S u {0}, xS(•) = {x^), i G S}, then
Vl(S') = min [C[T6(S',CfS)] + C[T6(S',CfS)]]
= C[T6(S', CSi)] + C[T6(S', C|2)]. (14)
( s' CSS)] + C[T6(S',Cg.
Since p = 0, player m stays in the game at the second stage. Therefore, formulas (8), (9), (12),(13) and (14) imply
Vl(S') - V2(S') = C[T6(S',C|i)] + C[T6(S',CfS)] - C[T6(S',CfS)]
= C[T6(S',C;S^)], S C N,
where xS(•) = (xj, i G S) are strategy profiles of coalition S. Thus, from (12) and (15) we obtain
Shl(N') - Sh2(N') = E |S|!(n -JS|-1)! [(vl(s' u{i}) - v2(s' u{i}))
SCN\{i} !
-(V1(S') - V2(S'))] = E |S|!(n -n|S|-1)! [C[t6(S' u{i},Cju{i})]
SCN\{i} !
-C[T6(S',CxSS)]! = |s|!(n -|,s|-1)! [ V C[T6(S' u {i},CS1u{i})]
xs n! ' ^ xs
(15)
- ]T C[T6 (S',C|S)]
SCN\{i}
SCN\{i} (16)
iN
From (10), (11) and (16) we obtain:
E Cp*(S' U{i},cJu{i})] > E Cp*(S',cg)]
SC«\{i} SC«\{i}
Pi = Sh1(N') - Sh,2(N') > 0,
where Vi g N, S C N \ {i}, S' = S U {0} n denotes the set of all permutations on N, Sn(k) = {i|n(i) < xS(■) = (Xj, i g S) are strategy profiles of coalition
S
The proof of Theorem 1 is complete.
6. Conclusion
In this paper, we study two-stage minimum spanning tree games with perishable goods. A particular player leaves the game at the end of the first stage with a certain probability. This probability depends on the behaviour of all players in the first stage. A dynamic Shapley value is constructed by defining characteristic functions of the cooperative game. Moreover, a theorem is presented in which the dynamic Shapley value is time consistent if specific conditions are satisfied.
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